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Diffstat (limited to 'theories/Reals/R_Ifp.v')
| -rw-r--r-- | theories/Reals/R_Ifp.v | 545 |
1 files changed, 545 insertions, 0 deletions
diff --git a/theories/Reals/R_Ifp.v b/theories/Reals/R_Ifp.v new file mode 100644 index 00000000..289b1921 --- /dev/null +++ b/theories/Reals/R_Ifp.v @@ -0,0 +1,545 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: R_Ifp.v,v 1.14.2.1 2004/07/16 19:31:12 herbelin Exp $ i*) + +(**********************************************************) +(** Complements for the reals.Integer and fractional part *) +(* *) +(**********************************************************) + +Require Import Rbase. +Require Import Omega. +Open Local Scope R_scope. + +(*********************************************************) +(** 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. +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. +intros; generalize (Rplus_le_compat_l 1 (IZR z) r H); intro; clear H; + rewrite (Rplus_comm 1 (IZR z)) in H1; rewrite (Rplus_comm 1 r) in H1; + cut (1 = IZR 1); auto with zarith real. +intro; generalize H1; pattern 1 at 1 in |- *; rewrite H; intro; clear H H1; + rewrite <- (plus_IZR z 1) in H2; apply (tech_up r (z + 1)); + auto with zarith real. +Qed. + +(**********) +Lemma fp_R0 : frac_part 0 = 0. +unfold frac_part in |- *; unfold Int_part in |- *; elim (archimed 0); intros; + unfold Rminus in |- *; elim (Rplus_ne (- IZR (up 0 - 1))); + intros a b; rewrite b; clear a b; rewrite <- Z_R_minus; + cut (up 0 = 1%Z). +intro; rewrite H1; + rewrite (Rminus_diag_eq (IZR 1) (IZR 1) (refl_equal (IZR 1))); + apply Ropp_0. +elim (archimed 0); intros; clear H2; unfold Rgt in H1; + rewrite (Rminus_0_r (IZR (up 0))) in H0; generalize (lt_O_IZR (up 0) H1); + intro; clear H1; generalize (le_IZR_R1 (up 0) H0); + intro; clear H H0; omega. +Qed. + +(**********) +Lemma for_base_fp : forall r:R, IZR (up r) - r > 0 /\ IZR (up r) - r <= 1. +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. +intro; unfold frac_part in |- *; unfold Int_part in |- *; split. + (*sup a O*) +cut (r - IZR (up r) >= -1). +rewrite <- Z_R_minus; simpl in |- *; intro; unfold Rminus in |- *; + rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; + fold (r - IZR (up r)) in |- *; fold (r - IZR (up r) - -1) in |- *; + 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. + (*inf a 1*) +cut (r - IZR (up r) < 0). +rewrite <- Z_R_minus; simpl in |- *; intro; unfold Rminus in |- *; + rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; + fold (r - IZR (up r)) in |- *; rewrite Ropp_involutive; + elim (Rplus_ne 1); intros a b; pattern 1 at 2 in |- *; + 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. + +(*********************************************************) +(** Properties *) +(*********************************************************) + +(**********) +Lemma base_Int_part : + forall r:R, IZR (Int_part r) <= r /\ IZR (Int_part r) - r > -1. +intro; unfold Int_part in |- *; elim (archimed r); intros. +split; rewrite <- (Z_R_minus (up r) 1); simpl in |- *. +generalize (Rle_minus (IZR (up r) - r) 1 H0); intro; unfold Rminus in H1; + rewrite (Rplus_assoc (IZR (up r)) (- r) (-1)) in H1; + rewrite (Rplus_comm (- r) (-1)) in H1; + rewrite <- (Rplus_assoc (IZR (up r)) (-1) (- r)) in H1; + fold (IZR (up r) - 1) in H1; fold (IZR (up r) - 1 - r) in H1; + apply Rminus_le; auto with zarith real. +generalize (Rplus_gt_compat_l (-1) (IZR (up r)) r H); intro; + rewrite (Rplus_comm (-1) (IZR (up r))) in H1; + generalize (Rplus_gt_compat_l (- r) (IZR (up r) + -1) (-1 + r) H1); + intro; clear H H0 H1; rewrite (Rplus_comm (- r) (IZR (up r) + -1)) in H2; + fold (IZR (up r) - 1) in H2; fold (IZR (up r) - 1 - r) in H2; + rewrite (Rplus_comm (- r) (-1 + r)) in H2; + rewrite (Rplus_assoc (-1) r (- r)) in H2; rewrite (Rplus_opp_r r) in H2; + elim (Rplus_ne (-1)); intros a b; rewrite a in H2; + clear a b; auto with zarith real. +Qed. + +(**********) +Lemma Int_part_INR : forall n:nat, Int_part (INR n) = Z_of_nat n. +intros n; unfold Int_part in |- *. +cut (up (INR n) = (Z_of_nat n + Z_of_nat 1)%Z). +intros H'; rewrite H'; simpl in |- *; ring. +apply sym_equal; 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 Zplus_comm; rewrite <- Znat.inj_plus; simpl in |- *; auto. +rewrite plus_IZR; simpl in |- *; 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. +unfold frac_part in |- *; 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. +red in |- *; 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. +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; + cut (1 = IZR 1); auto with zarith real. +intro; rewrite H1 in H; clear H1; + 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 in |- *; + 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. +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; + cut (1 = IZR 1); auto with zarith real. +intro; rewrite H1 in H; rewrite H1 in H0; clear H1; + 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 in |- *; + 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. +intros; unfold frac_part in |- *; generalize (Rminus_Int_part1 r1 r2 H); + intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1) (Int_part r2)); + unfold Rminus in |- *; + 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. +intros; unfold frac_part in |- *; 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 in |- *; + 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 in |- *; + 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. +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; + rewrite <- (Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2)) 1 1) in H0; + cut (1 = IZR 1); auto with zarith real. +intro; rewrite H1 in H0; rewrite H1 in H; clear H1; + 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 in |- *; 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. +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; cut (1 = IZR 1); + auto with zarith real. +intro; rewrite H in H1; clear H; + 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 in |- *; + 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. +intros; unfold frac_part in |- *; 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 in |- *; + unfold Rminus at 3 4 in |- *; + 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 in |- *; + 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. +intros; unfold frac_part in |- *; generalize (plus_Int_part2 r1 r2 H); intro; + rewrite H0; rewrite (plus_IZR (Int_part r1) (Int_part r2)); + unfold Rminus at 2 3 in |- *; + 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 in |- *; trivial with zarith real. +Qed.
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