Library Coq.Reals.R_Ifp
Complements for the reals.Integer and fractional part
Definition Int_part (r:R) : Z := (up r - 1)%Z.
Definition frac_part (r:R) : R := r - IZR (Int_part r).
Lemma tech_up : forall (r:R) (z:Z), r < IZR z -> IZR z <= r + 1 -> z = up r.
Proof.
intros; generalize (archimed r); intro; elim H1; intros; clear H1;
unfold Rgt in H2; unfold Rminus in H3;
generalize (Rplus_le_compat_l r (IZR (up r) + - r) 1 H3);
intro; clear H3; rewrite (Rplus_comm (IZR (up r)) (- r)) in H1;
rewrite <- (Rplus_assoc r (- r) (IZR (up r))) in H1;
rewrite (Rplus_opp_r r) in H1; elim (Rplus_ne (IZR (up r)));
intros a b; rewrite b in H1; clear a b; apply (single_z_r_R1 r z (up r));
auto with zarith real.
Qed.
Lemma up_tech :
forall (r:R) (z:Z), IZR z <= r -> r < IZR (z + 1) -> (z + 1)%Z = up r.
Proof.
intros.
apply tech_up with (1 := H0).
rewrite plus_IZR.
now apply Rplus_le_compat_r.
Qed.
Lemma fp_R0 : frac_part 0 = 0.
Proof.
unfold frac_part, Int_part.
replace (up 0) with 1%Z.
now rewrite <- minus_IZR.
destruct (archimed 0) as [H1 H2].
apply lt_IZR in H1.
rewrite <- minus_IZR in H2.
apply le_IZR in H2.
omega.
Qed.
Lemma for_base_fp : forall r:R, IZR (up r) - r > 0 /\ IZR (up r) - r <= 1.
Proof.
intro; split; cut (IZR (up r) > r /\ IZR (up r) - r <= 1).
intro; elim H; intros.
apply (Rgt_minus (IZR (up r)) r H0).
apply archimed.
intro; elim H; intros.
exact H1.
apply archimed.
Qed.
Lemma base_fp : forall r:R, frac_part r >= 0 /\ frac_part r < 1.
Proof.
intro; unfold frac_part; unfold Int_part; split.
cut (r - IZR (up r) >= -1).
rewrite <- Z_R_minus; simpl; intro; unfold Rminus;
rewrite Ropp_plus_distr; rewrite <- Rplus_assoc;
fold (r - IZR (up r)); fold (r - IZR (up r) - -1);
apply Rge_minus; auto with zarith real.
rewrite <- Ropp_minus_distr; apply Ropp_le_ge_contravar; elim (for_base_fp r);
auto with zarith real.
cut (r - IZR (up r) < 0).
rewrite <- Z_R_minus; simpl; intro; unfold Rminus;
rewrite Ropp_plus_distr; rewrite <- Rplus_assoc;
fold (r - IZR (up r)); rewrite Ropp_involutive;
elim (Rplus_ne 1); intros a b; pattern 1 at 2;
rewrite <- a; clear a b; rewrite (Rplus_comm (r - IZR (up r)) 1);
apply Rplus_lt_compat_l; auto with zarith real.
elim (for_base_fp r); intros; rewrite <- Ropp_0; rewrite <- Ropp_minus_distr;
apply Ropp_gt_lt_contravar; auto with zarith real.
Qed.
Definition frac_part (r:R) : R := r - IZR (Int_part r).
Lemma tech_up : forall (r:R) (z:Z), r < IZR z -> IZR z <= r + 1 -> z = up r.
Proof.
intros; generalize (archimed r); intro; elim H1; intros; clear H1;
unfold Rgt in H2; unfold Rminus in H3;
generalize (Rplus_le_compat_l r (IZR (up r) + - r) 1 H3);
intro; clear H3; rewrite (Rplus_comm (IZR (up r)) (- r)) in H1;
rewrite <- (Rplus_assoc r (- r) (IZR (up r))) in H1;
rewrite (Rplus_opp_r r) in H1; elim (Rplus_ne (IZR (up r)));
intros a b; rewrite b in H1; clear a b; apply (single_z_r_R1 r z (up r));
auto with zarith real.
Qed.
Lemma up_tech :
forall (r:R) (z:Z), IZR z <= r -> r < IZR (z + 1) -> (z + 1)%Z = up r.
Proof.
intros.
apply tech_up with (1 := H0).
rewrite plus_IZR.
now apply Rplus_le_compat_r.
Qed.
Lemma fp_R0 : frac_part 0 = 0.
Proof.
unfold frac_part, Int_part.
replace (up 0) with 1%Z.
now rewrite <- minus_IZR.
destruct (archimed 0) as [H1 H2].
apply lt_IZR in H1.
rewrite <- minus_IZR in H2.
apply le_IZR in H2.
omega.
Qed.
Lemma for_base_fp : forall r:R, IZR (up r) - r > 0 /\ IZR (up r) - r <= 1.
Proof.
intro; split; cut (IZR (up r) > r /\ IZR (up r) - r <= 1).
intro; elim H; intros.
apply (Rgt_minus (IZR (up r)) r H0).
apply archimed.
intro; elim H; intros.
exact H1.
apply archimed.
Qed.
Lemma base_fp : forall r:R, frac_part r >= 0 /\ frac_part r < 1.
Proof.
intro; unfold frac_part; unfold Int_part; split.
cut (r - IZR (up r) >= -1).
rewrite <- Z_R_minus; simpl; intro; unfold Rminus;
rewrite Ropp_plus_distr; rewrite <- Rplus_assoc;
fold (r - IZR (up r)); fold (r - IZR (up r) - -1);
apply Rge_minus; auto with zarith real.
rewrite <- Ropp_minus_distr; apply Ropp_le_ge_contravar; elim (for_base_fp r);
auto with zarith real.
cut (r - IZR (up r) < 0).
rewrite <- Z_R_minus; simpl; intro; unfold Rminus;
rewrite Ropp_plus_distr; rewrite <- Rplus_assoc;
fold (r - IZR (up r)); rewrite Ropp_involutive;
elim (Rplus_ne 1); intros a b; pattern 1 at 2;
rewrite <- a; clear a b; rewrite (Rplus_comm (r - IZR (up r)) 1);
apply Rplus_lt_compat_l; auto with zarith real.
elim (for_base_fp r); intros; rewrite <- Ropp_0; rewrite <- Ropp_minus_distr;
apply Ropp_gt_lt_contravar; auto with zarith real.
Qed.
Lemma base_Int_part :
forall r:R, IZR (Int_part r) <= r /\ IZR (Int_part r) - r > -1.
Proof.
intro; unfold Int_part; elim (archimed r); intros.
split; rewrite <- (Z_R_minus (up r) 1); simpl.
apply Rminus_le.
replace (IZR (up r) - 1 - r) with (IZR (up r) - r - 1) by ring.
now apply Rle_minus.
apply Rminus_gt.
replace (IZR (up r) - 1 - r - -1) with (IZR (up r) - r) by ring.
now apply Rgt_minus.
Qed.
Lemma Int_part_INR : forall n:nat, Int_part (INR n) = Z.of_nat n.
Proof.
intros n; unfold Int_part.
cut (up (INR n) = (Z.of_nat n + Z.of_nat 1)%Z).
intros H'; rewrite H'; simpl; ring.
symmetry; apply tech_up; auto.
replace (Z.of_nat n + Z.of_nat 1)%Z with (Z.of_nat (S n)).
repeat rewrite <- INR_IZR_INZ.
apply lt_INR; auto.
rewrite Z.add_comm; rewrite <- Znat.Nat2Z.inj_add; simpl; auto.
rewrite plus_IZR; simpl; auto with real.
repeat rewrite <- INR_IZR_INZ; auto with real.
Qed.
Lemma fp_nat : forall r:R, frac_part r = 0 -> exists c : Z, r = IZR c.
Proof.
unfold frac_part; intros; split with (Int_part r);
apply Rminus_diag_uniq; auto with zarith real.
Qed.
Lemma R0_fp_O : forall r:R, 0 <> frac_part r -> 0 <> r.
Proof.
red; intros; rewrite <- H0 in H; generalize fp_R0; intro;
auto with zarith real.
Qed.
Lemma Rminus_Int_part1 :
forall r1 r2:R,
frac_part r1 >= frac_part r2 ->
Int_part (r1 - r2) = (Int_part r1 - Int_part r2)%Z.
Proof.
intros; elim (base_fp r1); elim (base_fp r2); intros;
generalize (Rge_le (frac_part r2) 0 H0); intro; clear H0;
generalize (Ropp_le_ge_contravar 0 (frac_part r2) H4);
intro; clear H4; rewrite Ropp_0 in H0;
generalize (Rge_le 0 (- frac_part r2) H0); intro;
clear H0; generalize (Rge_le (frac_part r1) 0 H2);
intro; clear H2; generalize (Ropp_lt_gt_contravar (frac_part r2) 1 H1);
intro; clear H1; unfold Rgt in H2;
generalize
(sum_inequa_Rle_lt 0 (frac_part r1) 1 (-1) (- frac_part r2) 0 H0 H3 H2 H4);
intro; elim H1; intros; clear H1; elim (Rplus_ne 1);
intros a b; rewrite a in H6; clear a b H5;
generalize (Rge_minus (frac_part r1) (frac_part r2) H);
intro; clear H; fold (frac_part r1 - frac_part r2) in H6;
generalize (Rge_le (frac_part r1 - frac_part r2) 0 H1);
intro; clear H1 H3 H4 H0 H2; unfold frac_part in H6, H;
unfold Rminus in H6, H;
rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))) in H;
rewrite (Ropp_involutive (IZR (Int_part r2))) in H;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2)))
in H;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2)))
in H; rewrite (Rplus_comm (- IZR (Int_part r1)) (- r2)) in H;
rewrite (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))) in H;
rewrite <- (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)))
in H; rewrite (Rplus_comm (- IZR (Int_part r1)) (IZR (Int_part r2))) in H;
fold (r1 - r2) in H; fold (IZR (Int_part r2) - IZR (Int_part r1)) in H;
generalize
(Rplus_le_compat_l (IZR (Int_part r1) - IZR (Int_part r2)) 0
(r1 - r2 + (IZR (Int_part r2) - IZR (Int_part r1))) H);
intro; clear H;
rewrite (Rplus_comm (r1 - r2) (IZR (Int_part r2) - IZR (Int_part r1))) in H0;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2))
(IZR (Int_part r2) - IZR (Int_part r1)) (r1 - r2))
in H0; unfold Rminus in H0; fold (r1 - r2) in H0;
rewrite
(Rplus_assoc (IZR (Int_part r1)) (- IZR (Int_part r2))
(IZR (Int_part r2) + - IZR (Int_part r1))) in H0;
rewrite <-
(Rplus_assoc (- IZR (Int_part r2)) (IZR (Int_part r2))
(- IZR (Int_part r1))) in H0;
rewrite (Rplus_opp_l (IZR (Int_part r2))) in H0;
elim (Rplus_ne (- IZR (Int_part r1))); intros a b;
rewrite b in H0; clear a b;
elim (Rplus_ne (IZR (Int_part r1) + - IZR (Int_part r2)));
intros a b; rewrite a in H0; clear a b;
rewrite (Rplus_opp_r (IZR (Int_part r1))) in H0; elim (Rplus_ne (r1 - r2));
intros a b; rewrite b in H0; clear a b;
fold (IZR (Int_part r1) - IZR (Int_part r2)) in H0;
rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))) in H6;
rewrite (Ropp_involutive (IZR (Int_part r2))) in H6;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2)))
in H6;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2)))
in H6; rewrite (Rplus_comm (- IZR (Int_part r1)) (- r2)) in H6;
rewrite (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))) in H6;
rewrite <- (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)))
in H6;
rewrite (Rplus_comm (- IZR (Int_part r1)) (IZR (Int_part r2))) in H6;
fold (r1 - r2) in H6; fold (IZR (Int_part r2) - IZR (Int_part r1)) in H6;
generalize
(Rplus_lt_compat_l (IZR (Int_part r1) - IZR (Int_part r2))
(r1 - r2 + (IZR (Int_part r2) - IZR (Int_part r1))) 1 H6);
intro; clear H6;
rewrite (Rplus_comm (r1 - r2) (IZR (Int_part r2) - IZR (Int_part r1))) in H;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2))
(IZR (Int_part r2) - IZR (Int_part r1)) (r1 - r2))
in H;
rewrite <- (Ropp_minus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H;
rewrite (Rplus_opp_r (IZR (Int_part r1) - IZR (Int_part r2))) in H;
elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H;
clear a b; rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0;
rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H.
rewrite <- (plus_IZR (Int_part r1 - Int_part r2) 1) in H;
generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2) H0 H);
intros; clear H H0; unfold Int_part at 1;
omega.
Qed.
Lemma Rminus_Int_part2 :
forall r1 r2:R,
frac_part r1 < frac_part r2 ->
Int_part (r1 - r2) = (Int_part r1 - Int_part r2 - 1)%Z.
Proof.
intros; elim (base_fp r1); elim (base_fp r2); intros;
generalize (Rge_le (frac_part r2) 0 H0); intro; clear H0;
generalize (Ropp_le_ge_contravar 0 (frac_part r2) H4);
intro; clear H4; rewrite Ropp_0 in H0;
generalize (Rge_le 0 (- frac_part r2) H0); intro;
clear H0; generalize (Rge_le (frac_part r1) 0 H2);
intro; clear H2; generalize (Ropp_lt_gt_contravar (frac_part r2) 1 H1);
intro; clear H1; unfold Rgt in H2;
generalize
(sum_inequa_Rle_lt 0 (frac_part r1) 1 (-1) (- frac_part r2) 0 H0 H3 H2 H4);
intro; elim H1; intros; clear H1; elim (Rplus_ne (-1));
intros a b; rewrite b in H5; clear a b H6;
generalize (Rlt_minus (frac_part r1) (frac_part r2) H);
intro; clear H; fold (frac_part r1 - frac_part r2) in H5;
clear H3 H4 H0 H2; unfold frac_part in H5, H1; unfold Rminus in H5, H1;
rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))) in H5;
rewrite (Ropp_involutive (IZR (Int_part r2))) in H5;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2)))
in H5;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2)))
in H5; rewrite (Rplus_comm (- IZR (Int_part r1)) (- r2)) in H5;
rewrite (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))) in H5;
rewrite <- (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)))
in H5;
rewrite (Rplus_comm (- IZR (Int_part r1)) (IZR (Int_part r2))) in H5;
fold (r1 - r2) in H5; fold (IZR (Int_part r2) - IZR (Int_part r1)) in H5;
generalize
(Rplus_lt_compat_l (IZR (Int_part r1) - IZR (Int_part r2)) (-1)
(r1 - r2 + (IZR (Int_part r2) - IZR (Int_part r1))) H5);
intro; clear H5;
rewrite (Rplus_comm (r1 - r2) (IZR (Int_part r2) - IZR (Int_part r1))) in H;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2))
(IZR (Int_part r2) - IZR (Int_part r1)) (r1 - r2))
in H; unfold Rminus in H; fold (r1 - r2) in H;
rewrite
(Rplus_assoc (IZR (Int_part r1)) (- IZR (Int_part r2))
(IZR (Int_part r2) + - IZR (Int_part r1))) in H;
rewrite <-
(Rplus_assoc (- IZR (Int_part r2)) (IZR (Int_part r2))
(- IZR (Int_part r1))) in H;
rewrite (Rplus_opp_l (IZR (Int_part r2))) in H;
elim (Rplus_ne (- IZR (Int_part r1))); intros a b;
rewrite b in H; clear a b; rewrite (Rplus_opp_r (IZR (Int_part r1))) in H;
elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H;
clear a b; fold (IZR (Int_part r1) - IZR (Int_part r2)) in H;
fold (IZR (Int_part r1) - IZR (Int_part r2) - 1) in H;
rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))) in H1;
rewrite (Ropp_involutive (IZR (Int_part r2))) in H1;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2)))
in H1;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2)))
in H1; rewrite (Rplus_comm (- IZR (Int_part r1)) (- r2)) in H1;
rewrite (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))) in H1;
rewrite <- (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)))
in H1;
rewrite (Rplus_comm (- IZR (Int_part r1)) (IZR (Int_part r2))) in H1;
fold (r1 - r2) in H1; fold (IZR (Int_part r2) - IZR (Int_part r1)) in H1;
generalize
(Rplus_lt_compat_l (IZR (Int_part r1) - IZR (Int_part r2))
(r1 - r2 + (IZR (Int_part r2) - IZR (Int_part r1))) 0 H1);
intro; clear H1;
rewrite (Rplus_comm (r1 - r2) (IZR (Int_part r2) - IZR (Int_part r1))) in H0;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2))
(IZR (Int_part r2) - IZR (Int_part r1)) (r1 - r2))
in H0;
rewrite <- (Ropp_minus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H0;
rewrite (Rplus_opp_r (IZR (Int_part r1) - IZR (Int_part r2))) in H0;
elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H0;
clear a b; rewrite <- (Rplus_opp_l 1) in H0;
rewrite <- (Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2)) (-(1)) 1)
in H0; fold (IZR (Int_part r1) - IZR (Int_part r2) - 1) in H0;
rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0;
rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H;
auto with zarith real.
change (_ + -1) with (IZR (Int_part r1 - Int_part r2) - 1) in H;
rewrite (Z_R_minus (Int_part r1 - Int_part r2) 1) in H;
rewrite (Z_R_minus (Int_part r1 - Int_part r2) 1) in H0;
rewrite <- (plus_IZR (Int_part r1 - Int_part r2 - 1) 1) in H0;
generalize (Rlt_le (IZR (Int_part r1 - Int_part r2 - 1)) (r1 - r2) H);
intro; clear H;
generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2 - 1) H1 H0);
intros; clear H0 H1; unfold Int_part at 1;
omega.
Qed.
Lemma Rminus_fp1 :
forall r1 r2:R,
frac_part r1 >= frac_part r2 ->
frac_part (r1 - r2) = frac_part r1 - frac_part r2.
Proof.
intros; unfold frac_part; generalize (Rminus_Int_part1 r1 r2 H);
intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));
unfold Rminus;
rewrite (Ropp_plus_distr (IZR (Int_part r1)) (- IZR (Int_part r2)));
rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2)));
rewrite (Ropp_involutive (IZR (Int_part r2)));
rewrite (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)));
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2)));
rewrite (Rplus_comm (- r2) (- IZR (Int_part r1)));
auto with zarith real.
Qed.
Lemma Rminus_fp2 :
forall r1 r2:R,
frac_part r1 < frac_part r2 ->
frac_part (r1 - r2) = frac_part r1 - frac_part r2 + 1.
Proof.
intros; unfold frac_part; generalize (Rminus_Int_part2 r1 r2 H);
intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1 - Int_part r2) 1);
rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));
unfold Rminus;
rewrite
(Ropp_plus_distr (IZR (Int_part r1) + - IZR (Int_part r2)) (- IZR 1))
; rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2)));
rewrite (Ropp_involutive (IZR 1));
rewrite (Ropp_involutive (IZR (Int_part r2)));
rewrite (Ropp_plus_distr (IZR (Int_part r1)));
rewrite (Ropp_involutive (IZR (Int_part r2))); simpl;
rewrite <-
(Rplus_assoc (r1 + - r2) (- IZR (Int_part r1) + IZR (Int_part r2)) 1)
; rewrite (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)));
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2)));
rewrite (Rplus_comm (- r2) (- IZR (Int_part r1)));
auto with zarith real.
Qed.
Lemma plus_Int_part1 :
forall r1 r2:R,
frac_part r1 + frac_part r2 >= 1 ->
Int_part (r1 + r2) = (Int_part r1 + Int_part r2 + 1)%Z.
Proof.
intros; generalize (Rge_le (frac_part r1 + frac_part r2) 1 H); intro; clear H;
elim (base_fp r1); elim (base_fp r2); intros; clear H H2;
generalize (Rplus_lt_compat_l (frac_part r2) (frac_part r1) 1 H3);
intro; clear H3; generalize (Rplus_lt_compat_l 1 (frac_part r2) 1 H1);
intro; clear H1; rewrite (Rplus_comm 1 (frac_part r2)) in H2;
generalize
(Rlt_trans (frac_part r2 + frac_part r1) (frac_part r2 + 1) 2 H H2);
intro; clear H H2; rewrite (Rplus_comm (frac_part r2) (frac_part r1)) in H1;
unfold frac_part in H0, H1; unfold Rminus in H0, H1;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)))
in H1; rewrite (Rplus_comm r2 (- IZR (Int_part r2))) in H1;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2)
in H1;
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2) in H1;
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)))
in H1;
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H1;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)))
in H0; rewrite (Rplus_comm r2 (- IZR (Int_part r2))) in H0;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2)
in H0;
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2) in H0;
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)))
in H0;
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H0;
generalize
(Rplus_le_compat_l (IZR (Int_part r1) + IZR (Int_part r2)) 1
(r1 + r2 + - (IZR (Int_part r1) + IZR (Int_part r2))) H0);
intro; clear H0;
generalize
(Rplus_lt_compat_l (IZR (Int_part r1) + IZR (Int_part r2))
(r1 + r2 + - (IZR (Int_part r1) + IZR (Int_part r2))) 2 H1);
intro; clear H1;
rewrite (Rplus_comm (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))))
in H;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2))
(- (IZR (Int_part r1) + IZR (Int_part r2))) (r1 + r2))
in H; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H;
elim (Rplus_ne (r1 + r2)); intros a b; rewrite b in H;
clear a b;
rewrite (Rplus_comm (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))))
in H0;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2))
(- (IZR (Int_part r1) + IZR (Int_part r2))) (r1 + r2))
in H0; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H0;
elim (Rplus_ne (r1 + r2)); intros a b; rewrite b in H0;
clear a b;
change 2 with (1 + 1) in H0;
rewrite <- (Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2)) 1 1) in H0;
auto with zarith real.
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H;
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H0;
rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H;
rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H0;
rewrite <- (plus_IZR (Int_part r1 + Int_part r2 + 1) 1) in H0;
generalize (up_tech (r1 + r2) (Int_part r1 + Int_part r2 + 1) H H0);
intro; clear H H0; unfold Int_part at 1; omega.
Qed.
Lemma plus_Int_part2 :
forall r1 r2:R,
frac_part r1 + frac_part r2 < 1 ->
Int_part (r1 + r2) = (Int_part r1 + Int_part r2)%Z.
Proof.
intros; elim (base_fp r1); elim (base_fp r2); intros; clear H1 H3;
generalize (Rge_le (frac_part r2) 0 H0); intro; clear H0;
generalize (Rge_le (frac_part r1) 0 H2); intro; clear H2;
generalize (Rplus_le_compat_l (frac_part r1) 0 (frac_part r2) H1);
intro; clear H1; elim (Rplus_ne (frac_part r1)); intros a b;
rewrite a in H2; clear a b;
generalize (Rle_trans 0 (frac_part r1) (frac_part r1 + frac_part r2) H0 H2);
intro; clear H0 H2; unfold frac_part in H, H1; unfold Rminus in H, H1;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)))
in H1; rewrite (Rplus_comm r2 (- IZR (Int_part r2))) in H1;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2)
in H1;
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2) in H1;
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)))
in H1;
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H1;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)))
in H; rewrite (Rplus_comm r2 (- IZR (Int_part r2))) in H;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2) in H;
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2) in H;
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)))
in H;
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H;
generalize
(Rplus_le_compat_l (IZR (Int_part r1) + IZR (Int_part r2)) 0
(r1 + r2 + - (IZR (Int_part r1) + IZR (Int_part r2))) H1);
intro; clear H1;
generalize
(Rplus_lt_compat_l (IZR (Int_part r1) + IZR (Int_part r2))
(r1 + r2 + - (IZR (Int_part r1) + IZR (Int_part r2))) 1 H);
intro; clear H;
rewrite (Rplus_comm (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))))
in H1;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2))
(- (IZR (Int_part r1) + IZR (Int_part r2))) (r1 + r2))
in H1; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H1;
elim (Rplus_ne (r1 + r2)); intros a b; rewrite b in H1;
clear a b;
rewrite (Rplus_comm (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))))
in H0;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2))
(- (IZR (Int_part r1) + IZR (Int_part r2))) (r1 + r2))
in H0; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H0;
elim (Rplus_ne (IZR (Int_part r1) + IZR (Int_part r2)));
intros a b; rewrite a in H0; clear a b; elim (Rplus_ne (r1 + r2));
intros a b; rewrite b in H0; clear a b.
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H0;
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H1;
rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H1;
generalize (up_tech (r1 + r2) (Int_part r1 + Int_part r2) H0 H1);
intro; clear H0 H1; unfold Int_part at 1;
omega.
Qed.
Lemma plus_frac_part1 :
forall r1 r2:R,
frac_part r1 + frac_part r2 >= 1 ->
frac_part (r1 + r2) = frac_part r1 + frac_part r2 - 1.
Proof.
intros; unfold frac_part; generalize (plus_Int_part1 r1 r2 H); intro;
rewrite H0; rewrite (plus_IZR (Int_part r1 + Int_part r2) 1);
rewrite (plus_IZR (Int_part r1) (Int_part r2)); simpl;
unfold Rminus at 3 4;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)));
rewrite (Rplus_comm r2 (- IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2);
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2);
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)));
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2)));
unfold Rminus;
rewrite
(Rplus_assoc (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))) (-(1)))
; rewrite <- (Ropp_plus_distr (IZR (Int_part r1) + IZR (Int_part r2)) 1);
trivial with zarith real.
Qed.
Lemma plus_frac_part2 :
forall r1 r2:R,
frac_part r1 + frac_part r2 < 1 ->
frac_part (r1 + r2) = frac_part r1 + frac_part r2.
Proof.
intros; unfold frac_part; generalize (plus_Int_part2 r1 r2 H); intro;
rewrite H0; rewrite (plus_IZR (Int_part r1) (Int_part r2));
unfold Rminus at 2 3;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)));
rewrite (Rplus_comm r2 (- IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2);
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2);
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)));
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2)));
unfold Rminus; trivial with zarith real.
Qed.
forall r:R, IZR (Int_part r) <= r /\ IZR (Int_part r) - r > -1.
Proof.
intro; unfold Int_part; elim (archimed r); intros.
split; rewrite <- (Z_R_minus (up r) 1); simpl.
apply Rminus_le.
replace (IZR (up r) - 1 - r) with (IZR (up r) - r - 1) by ring.
now apply Rle_minus.
apply Rminus_gt.
replace (IZR (up r) - 1 - r - -1) with (IZR (up r) - r) by ring.
now apply Rgt_minus.
Qed.
Lemma Int_part_INR : forall n:nat, Int_part (INR n) = Z.of_nat n.
Proof.
intros n; unfold Int_part.
cut (up (INR n) = (Z.of_nat n + Z.of_nat 1)%Z).
intros H'; rewrite H'; simpl; ring.
symmetry; apply tech_up; auto.
replace (Z.of_nat n + Z.of_nat 1)%Z with (Z.of_nat (S n)).
repeat rewrite <- INR_IZR_INZ.
apply lt_INR; auto.
rewrite Z.add_comm; rewrite <- Znat.Nat2Z.inj_add; simpl; auto.
rewrite plus_IZR; simpl; auto with real.
repeat rewrite <- INR_IZR_INZ; auto with real.
Qed.
Lemma fp_nat : forall r:R, frac_part r = 0 -> exists c : Z, r = IZR c.
Proof.
unfold frac_part; intros; split with (Int_part r);
apply Rminus_diag_uniq; auto with zarith real.
Qed.
Lemma R0_fp_O : forall r:R, 0 <> frac_part r -> 0 <> r.
Proof.
red; intros; rewrite <- H0 in H; generalize fp_R0; intro;
auto with zarith real.
Qed.
Lemma Rminus_Int_part1 :
forall r1 r2:R,
frac_part r1 >= frac_part r2 ->
Int_part (r1 - r2) = (Int_part r1 - Int_part r2)%Z.
Proof.
intros; elim (base_fp r1); elim (base_fp r2); intros;
generalize (Rge_le (frac_part r2) 0 H0); intro; clear H0;
generalize (Ropp_le_ge_contravar 0 (frac_part r2) H4);
intro; clear H4; rewrite Ropp_0 in H0;
generalize (Rge_le 0 (- frac_part r2) H0); intro;
clear H0; generalize (Rge_le (frac_part r1) 0 H2);
intro; clear H2; generalize (Ropp_lt_gt_contravar (frac_part r2) 1 H1);
intro; clear H1; unfold Rgt in H2;
generalize
(sum_inequa_Rle_lt 0 (frac_part r1) 1 (-1) (- frac_part r2) 0 H0 H3 H2 H4);
intro; elim H1; intros; clear H1; elim (Rplus_ne 1);
intros a b; rewrite a in H6; clear a b H5;
generalize (Rge_minus (frac_part r1) (frac_part r2) H);
intro; clear H; fold (frac_part r1 - frac_part r2) in H6;
generalize (Rge_le (frac_part r1 - frac_part r2) 0 H1);
intro; clear H1 H3 H4 H0 H2; unfold frac_part in H6, H;
unfold Rminus in H6, H;
rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))) in H;
rewrite (Ropp_involutive (IZR (Int_part r2))) in H;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2)))
in H;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2)))
in H; rewrite (Rplus_comm (- IZR (Int_part r1)) (- r2)) in H;
rewrite (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))) in H;
rewrite <- (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)))
in H; rewrite (Rplus_comm (- IZR (Int_part r1)) (IZR (Int_part r2))) in H;
fold (r1 - r2) in H; fold (IZR (Int_part r2) - IZR (Int_part r1)) in H;
generalize
(Rplus_le_compat_l (IZR (Int_part r1) - IZR (Int_part r2)) 0
(r1 - r2 + (IZR (Int_part r2) - IZR (Int_part r1))) H);
intro; clear H;
rewrite (Rplus_comm (r1 - r2) (IZR (Int_part r2) - IZR (Int_part r1))) in H0;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2))
(IZR (Int_part r2) - IZR (Int_part r1)) (r1 - r2))
in H0; unfold Rminus in H0; fold (r1 - r2) in H0;
rewrite
(Rplus_assoc (IZR (Int_part r1)) (- IZR (Int_part r2))
(IZR (Int_part r2) + - IZR (Int_part r1))) in H0;
rewrite <-
(Rplus_assoc (- IZR (Int_part r2)) (IZR (Int_part r2))
(- IZR (Int_part r1))) in H0;
rewrite (Rplus_opp_l (IZR (Int_part r2))) in H0;
elim (Rplus_ne (- IZR (Int_part r1))); intros a b;
rewrite b in H0; clear a b;
elim (Rplus_ne (IZR (Int_part r1) + - IZR (Int_part r2)));
intros a b; rewrite a in H0; clear a b;
rewrite (Rplus_opp_r (IZR (Int_part r1))) in H0; elim (Rplus_ne (r1 - r2));
intros a b; rewrite b in H0; clear a b;
fold (IZR (Int_part r1) - IZR (Int_part r2)) in H0;
rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))) in H6;
rewrite (Ropp_involutive (IZR (Int_part r2))) in H6;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2)))
in H6;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2)))
in H6; rewrite (Rplus_comm (- IZR (Int_part r1)) (- r2)) in H6;
rewrite (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))) in H6;
rewrite <- (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)))
in H6;
rewrite (Rplus_comm (- IZR (Int_part r1)) (IZR (Int_part r2))) in H6;
fold (r1 - r2) in H6; fold (IZR (Int_part r2) - IZR (Int_part r1)) in H6;
generalize
(Rplus_lt_compat_l (IZR (Int_part r1) - IZR (Int_part r2))
(r1 - r2 + (IZR (Int_part r2) - IZR (Int_part r1))) 1 H6);
intro; clear H6;
rewrite (Rplus_comm (r1 - r2) (IZR (Int_part r2) - IZR (Int_part r1))) in H;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2))
(IZR (Int_part r2) - IZR (Int_part r1)) (r1 - r2))
in H;
rewrite <- (Ropp_minus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H;
rewrite (Rplus_opp_r (IZR (Int_part r1) - IZR (Int_part r2))) in H;
elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H;
clear a b; rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0;
rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H.
rewrite <- (plus_IZR (Int_part r1 - Int_part r2) 1) in H;
generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2) H0 H);
intros; clear H H0; unfold Int_part at 1;
omega.
Qed.
Lemma Rminus_Int_part2 :
forall r1 r2:R,
frac_part r1 < frac_part r2 ->
Int_part (r1 - r2) = (Int_part r1 - Int_part r2 - 1)%Z.
Proof.
intros; elim (base_fp r1); elim (base_fp r2); intros;
generalize (Rge_le (frac_part r2) 0 H0); intro; clear H0;
generalize (Ropp_le_ge_contravar 0 (frac_part r2) H4);
intro; clear H4; rewrite Ropp_0 in H0;
generalize (Rge_le 0 (- frac_part r2) H0); intro;
clear H0; generalize (Rge_le (frac_part r1) 0 H2);
intro; clear H2; generalize (Ropp_lt_gt_contravar (frac_part r2) 1 H1);
intro; clear H1; unfold Rgt in H2;
generalize
(sum_inequa_Rle_lt 0 (frac_part r1) 1 (-1) (- frac_part r2) 0 H0 H3 H2 H4);
intro; elim H1; intros; clear H1; elim (Rplus_ne (-1));
intros a b; rewrite b in H5; clear a b H6;
generalize (Rlt_minus (frac_part r1) (frac_part r2) H);
intro; clear H; fold (frac_part r1 - frac_part r2) in H5;
clear H3 H4 H0 H2; unfold frac_part in H5, H1; unfold Rminus in H5, H1;
rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))) in H5;
rewrite (Ropp_involutive (IZR (Int_part r2))) in H5;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2)))
in H5;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2)))
in H5; rewrite (Rplus_comm (- IZR (Int_part r1)) (- r2)) in H5;
rewrite (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))) in H5;
rewrite <- (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)))
in H5;
rewrite (Rplus_comm (- IZR (Int_part r1)) (IZR (Int_part r2))) in H5;
fold (r1 - r2) in H5; fold (IZR (Int_part r2) - IZR (Int_part r1)) in H5;
generalize
(Rplus_lt_compat_l (IZR (Int_part r1) - IZR (Int_part r2)) (-1)
(r1 - r2 + (IZR (Int_part r2) - IZR (Int_part r1))) H5);
intro; clear H5;
rewrite (Rplus_comm (r1 - r2) (IZR (Int_part r2) - IZR (Int_part r1))) in H;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2))
(IZR (Int_part r2) - IZR (Int_part r1)) (r1 - r2))
in H; unfold Rminus in H; fold (r1 - r2) in H;
rewrite
(Rplus_assoc (IZR (Int_part r1)) (- IZR (Int_part r2))
(IZR (Int_part r2) + - IZR (Int_part r1))) in H;
rewrite <-
(Rplus_assoc (- IZR (Int_part r2)) (IZR (Int_part r2))
(- IZR (Int_part r1))) in H;
rewrite (Rplus_opp_l (IZR (Int_part r2))) in H;
elim (Rplus_ne (- IZR (Int_part r1))); intros a b;
rewrite b in H; clear a b; rewrite (Rplus_opp_r (IZR (Int_part r1))) in H;
elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H;
clear a b; fold (IZR (Int_part r1) - IZR (Int_part r2)) in H;
fold (IZR (Int_part r1) - IZR (Int_part r2) - 1) in H;
rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))) in H1;
rewrite (Ropp_involutive (IZR (Int_part r2))) in H1;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2)))
in H1;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2)))
in H1; rewrite (Rplus_comm (- IZR (Int_part r1)) (- r2)) in H1;
rewrite (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))) in H1;
rewrite <- (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)))
in H1;
rewrite (Rplus_comm (- IZR (Int_part r1)) (IZR (Int_part r2))) in H1;
fold (r1 - r2) in H1; fold (IZR (Int_part r2) - IZR (Int_part r1)) in H1;
generalize
(Rplus_lt_compat_l (IZR (Int_part r1) - IZR (Int_part r2))
(r1 - r2 + (IZR (Int_part r2) - IZR (Int_part r1))) 0 H1);
intro; clear H1;
rewrite (Rplus_comm (r1 - r2) (IZR (Int_part r2) - IZR (Int_part r1))) in H0;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2))
(IZR (Int_part r2) - IZR (Int_part r1)) (r1 - r2))
in H0;
rewrite <- (Ropp_minus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H0;
rewrite (Rplus_opp_r (IZR (Int_part r1) - IZR (Int_part r2))) in H0;
elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H0;
clear a b; rewrite <- (Rplus_opp_l 1) in H0;
rewrite <- (Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2)) (-(1)) 1)
in H0; fold (IZR (Int_part r1) - IZR (Int_part r2) - 1) in H0;
rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0;
rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H;
auto with zarith real.
change (_ + -1) with (IZR (Int_part r1 - Int_part r2) - 1) in H;
rewrite (Z_R_minus (Int_part r1 - Int_part r2) 1) in H;
rewrite (Z_R_minus (Int_part r1 - Int_part r2) 1) in H0;
rewrite <- (plus_IZR (Int_part r1 - Int_part r2 - 1) 1) in H0;
generalize (Rlt_le (IZR (Int_part r1 - Int_part r2 - 1)) (r1 - r2) H);
intro; clear H;
generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2 - 1) H1 H0);
intros; clear H0 H1; unfold Int_part at 1;
omega.
Qed.
Lemma Rminus_fp1 :
forall r1 r2:R,
frac_part r1 >= frac_part r2 ->
frac_part (r1 - r2) = frac_part r1 - frac_part r2.
Proof.
intros; unfold frac_part; generalize (Rminus_Int_part1 r1 r2 H);
intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));
unfold Rminus;
rewrite (Ropp_plus_distr (IZR (Int_part r1)) (- IZR (Int_part r2)));
rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2)));
rewrite (Ropp_involutive (IZR (Int_part r2)));
rewrite (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)));
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2)));
rewrite (Rplus_comm (- r2) (- IZR (Int_part r1)));
auto with zarith real.
Qed.
Lemma Rminus_fp2 :
forall r1 r2:R,
frac_part r1 < frac_part r2 ->
frac_part (r1 - r2) = frac_part r1 - frac_part r2 + 1.
Proof.
intros; unfold frac_part; generalize (Rminus_Int_part2 r1 r2 H);
intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1 - Int_part r2) 1);
rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));
unfold Rminus;
rewrite
(Ropp_plus_distr (IZR (Int_part r1) + - IZR (Int_part r2)) (- IZR 1))
; rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2)));
rewrite (Ropp_involutive (IZR 1));
rewrite (Ropp_involutive (IZR (Int_part r2)));
rewrite (Ropp_plus_distr (IZR (Int_part r1)));
rewrite (Ropp_involutive (IZR (Int_part r2))); simpl;
rewrite <-
(Rplus_assoc (r1 + - r2) (- IZR (Int_part r1) + IZR (Int_part r2)) 1)
; rewrite (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2)));
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2)));
rewrite (Rplus_comm (- r2) (- IZR (Int_part r1)));
auto with zarith real.
Qed.
Lemma plus_Int_part1 :
forall r1 r2:R,
frac_part r1 + frac_part r2 >= 1 ->
Int_part (r1 + r2) = (Int_part r1 + Int_part r2 + 1)%Z.
Proof.
intros; generalize (Rge_le (frac_part r1 + frac_part r2) 1 H); intro; clear H;
elim (base_fp r1); elim (base_fp r2); intros; clear H H2;
generalize (Rplus_lt_compat_l (frac_part r2) (frac_part r1) 1 H3);
intro; clear H3; generalize (Rplus_lt_compat_l 1 (frac_part r2) 1 H1);
intro; clear H1; rewrite (Rplus_comm 1 (frac_part r2)) in H2;
generalize
(Rlt_trans (frac_part r2 + frac_part r1) (frac_part r2 + 1) 2 H H2);
intro; clear H H2; rewrite (Rplus_comm (frac_part r2) (frac_part r1)) in H1;
unfold frac_part in H0, H1; unfold Rminus in H0, H1;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)))
in H1; rewrite (Rplus_comm r2 (- IZR (Int_part r2))) in H1;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2)
in H1;
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2) in H1;
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)))
in H1;
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H1;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)))
in H0; rewrite (Rplus_comm r2 (- IZR (Int_part r2))) in H0;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2)
in H0;
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2) in H0;
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)))
in H0;
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H0;
generalize
(Rplus_le_compat_l (IZR (Int_part r1) + IZR (Int_part r2)) 1
(r1 + r2 + - (IZR (Int_part r1) + IZR (Int_part r2))) H0);
intro; clear H0;
generalize
(Rplus_lt_compat_l (IZR (Int_part r1) + IZR (Int_part r2))
(r1 + r2 + - (IZR (Int_part r1) + IZR (Int_part r2))) 2 H1);
intro; clear H1;
rewrite (Rplus_comm (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))))
in H;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2))
(- (IZR (Int_part r1) + IZR (Int_part r2))) (r1 + r2))
in H; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H;
elim (Rplus_ne (r1 + r2)); intros a b; rewrite b in H;
clear a b;
rewrite (Rplus_comm (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))))
in H0;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2))
(- (IZR (Int_part r1) + IZR (Int_part r2))) (r1 + r2))
in H0; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H0;
elim (Rplus_ne (r1 + r2)); intros a b; rewrite b in H0;
clear a b;
change 2 with (1 + 1) in H0;
rewrite <- (Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2)) 1 1) in H0;
auto with zarith real.
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H;
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H0;
rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H;
rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H0;
rewrite <- (plus_IZR (Int_part r1 + Int_part r2 + 1) 1) in H0;
generalize (up_tech (r1 + r2) (Int_part r1 + Int_part r2 + 1) H H0);
intro; clear H H0; unfold Int_part at 1; omega.
Qed.
Lemma plus_Int_part2 :
forall r1 r2:R,
frac_part r1 + frac_part r2 < 1 ->
Int_part (r1 + r2) = (Int_part r1 + Int_part r2)%Z.
Proof.
intros; elim (base_fp r1); elim (base_fp r2); intros; clear H1 H3;
generalize (Rge_le (frac_part r2) 0 H0); intro; clear H0;
generalize (Rge_le (frac_part r1) 0 H2); intro; clear H2;
generalize (Rplus_le_compat_l (frac_part r1) 0 (frac_part r2) H1);
intro; clear H1; elim (Rplus_ne (frac_part r1)); intros a b;
rewrite a in H2; clear a b;
generalize (Rle_trans 0 (frac_part r1) (frac_part r1 + frac_part r2) H0 H2);
intro; clear H0 H2; unfold frac_part in H, H1; unfold Rminus in H, H1;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)))
in H1; rewrite (Rplus_comm r2 (- IZR (Int_part r2))) in H1;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2)
in H1;
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2) in H1;
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)))
in H1;
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H1;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)))
in H; rewrite (Rplus_comm r2 (- IZR (Int_part r2))) in H;
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2) in H;
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2) in H;
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)))
in H;
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H;
generalize
(Rplus_le_compat_l (IZR (Int_part r1) + IZR (Int_part r2)) 0
(r1 + r2 + - (IZR (Int_part r1) + IZR (Int_part r2))) H1);
intro; clear H1;
generalize
(Rplus_lt_compat_l (IZR (Int_part r1) + IZR (Int_part r2))
(r1 + r2 + - (IZR (Int_part r1) + IZR (Int_part r2))) 1 H);
intro; clear H;
rewrite (Rplus_comm (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))))
in H1;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2))
(- (IZR (Int_part r1) + IZR (Int_part r2))) (r1 + r2))
in H1; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H1;
elim (Rplus_ne (r1 + r2)); intros a b; rewrite b in H1;
clear a b;
rewrite (Rplus_comm (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))))
in H0;
rewrite <-
(Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2))
(- (IZR (Int_part r1) + IZR (Int_part r2))) (r1 + r2))
in H0; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H0;
elim (Rplus_ne (IZR (Int_part r1) + IZR (Int_part r2)));
intros a b; rewrite a in H0; clear a b; elim (Rplus_ne (r1 + r2));
intros a b; rewrite b in H0; clear a b.
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H0;
rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H1;
rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H1;
generalize (up_tech (r1 + r2) (Int_part r1 + Int_part r2) H0 H1);
intro; clear H0 H1; unfold Int_part at 1;
omega.
Qed.
Lemma plus_frac_part1 :
forall r1 r2:R,
frac_part r1 + frac_part r2 >= 1 ->
frac_part (r1 + r2) = frac_part r1 + frac_part r2 - 1.
Proof.
intros; unfold frac_part; generalize (plus_Int_part1 r1 r2 H); intro;
rewrite H0; rewrite (plus_IZR (Int_part r1 + Int_part r2) 1);
rewrite (plus_IZR (Int_part r1) (Int_part r2)); simpl;
unfold Rminus at 3 4;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)));
rewrite (Rplus_comm r2 (- IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2);
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2);
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)));
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2)));
unfold Rminus;
rewrite
(Rplus_assoc (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))) (-(1)))
; rewrite <- (Ropp_plus_distr (IZR (Int_part r1) + IZR (Int_part r2)) 1);
trivial with zarith real.
Qed.
Lemma plus_frac_part2 :
forall r1 r2:R,
frac_part r1 + frac_part r2 < 1 ->
frac_part (r1 + r2) = frac_part r1 + frac_part r2.
Proof.
intros; unfold frac_part; generalize (plus_Int_part2 r1 r2 H); intro;
rewrite H0; rewrite (plus_IZR (Int_part r1) (Int_part r2));
unfold Rminus at 2 3;
rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2)));
rewrite (Rplus_comm r2 (- IZR (Int_part r2)));
rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2);
rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2);
rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2)));
rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2)));
unfold Rminus; trivial with zarith real.
Qed.