From 598e24a9c4a81ec9f24f1088d0a7a4a93dc19fd4 Mon Sep 17 00:00:00 2001 From: Vincent Semeria Date: Sat, 9 May 2020 10:09:51 +0200 Subject: Define CRzero and CRone via CR_of_Q Add real numbers up to 10 --- theories/Reals/Abstract/ConstructiveAbs.v | 153 +++++++++++++++--------------- 1 file changed, 75 insertions(+), 78 deletions(-) (limited to 'theories/Reals/Abstract/ConstructiveAbs.v') diff --git a/theories/Reals/Abstract/ConstructiveAbs.v b/theories/Reals/Abstract/ConstructiveAbs.v index d357ad2d54..31397cbddd 100644 --- a/theories/Reals/Abstract/ConstructiveAbs.v +++ b/theories/Reals/Abstract/ConstructiveAbs.v @@ -57,11 +57,11 @@ Proof. - pose proof (CRabs_def R x (CRabs R x)) as [_ H1]. apply H1, CRle_refl. - rewrite <- CRabs_def. split. apply CRle_refl. - apply (CRle_trans _ (CRzero R)). 2: exact H. - apply (CRle_trans _ (CRopp R (CRzero R))). + apply (CRle_trans _ 0). 2: exact H. + apply (CRle_trans _ (CRopp R 0)). intro abs. apply CRopp_lt_cancel in abs. contradiction. - apply (CRplus_le_reg_l (CRzero R)). - apply (CRle_trans _ (CRzero R)). apply CRplus_opp_r. + apply (CRplus_le_reg_l 0). + apply (CRle_trans _ 0). apply CRplus_opp_r. apply CRplus_0_r. Qed. @@ -164,8 +164,7 @@ Lemma CR_of_Q_abs : forall {R : ConstructiveReals} (q : Q), Proof. intros. destruct (Qlt_le_dec 0 q). - apply (CReq_trans _ (CR_of_Q R q)). - apply CRabs_right. apply (CRle_trans _ (CR_of_Q R 0)). - apply CR_of_Q_zero. apply CR_of_Q_le. apply Qlt_le_weak, q0. + apply CRabs_right. apply CR_of_Q_le. apply Qlt_le_weak, q0. apply CR_of_Q_morph. symmetry. apply Qabs_pos, Qlt_le_weak, q0. - apply (CReq_trans _ (CR_of_Q R (-q))). apply (CReq_trans _ (CRabs R (CRopp R (CR_of_Q R q)))). @@ -173,8 +172,7 @@ Proof. 2: apply CR_of_Q_morph; symmetry; apply Qabs_neg, q0. apply (CReq_trans _ (CRopp R (CR_of_Q R q))). 2: apply CReq_sym, CR_of_Q_opp. - apply CRabs_right. apply (CRle_trans _ (CR_of_Q R 0)). - apply CR_of_Q_zero. + apply CRabs_right. apply (CRle_trans _ (CR_of_Q R (-q))). apply CR_of_Q_le. apply (Qplus_le_l _ _ q). ring_simplify. exact q0. apply CR_of_Q_opp. @@ -206,14 +204,14 @@ Proof. destruct (CR_Q_dense R _ _ neg) as [q [H0 H1]]. destruct (Qlt_le_dec 0 q). - destruct (s (CR_of_Q R (-q)) x 0). - rewrite <- CR_of_Q_zero. apply CR_of_Q_lt. + apply CR_of_Q_lt. apply (Qplus_lt_l _ _ q). ring_simplify. exact q0. exfalso. pose proof (CRabs_def R x (CR_of_Q R q)) as [H2 _]. apply H2. clear H2. split. apply CRlt_asym, H0. 2: exact H1. rewrite <- Qopp_involutive, CR_of_Q_opp. apply CRopp_ge_le_contravar, CRlt_asym, c. exact c. - apply (CRlt_le_trans _ _ _ H0). - rewrite <- CR_of_Q_zero. apply CR_of_Q_le. exact q0. + apply CR_of_Q_le. exact q0. Qed. @@ -339,24 +337,23 @@ Proof. left; apply CR_of_Q_pos; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. - rewrite (CR_of_Q_plus R 1 1), CR_of_Q_one, CRmult_plus_distr_l, CRmult_1_r. + rewrite CRmult_1_r. + rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l, CRmult_1_r. rewrite CRabs_right. unfold CRminus. rewrite CRopp_plus_distr, CRplus_assoc, <- (CRplus_assoc y). rewrite CRplus_opp_r, CRplus_0_l, CRopp_involutive. reflexivity. apply (CRmult_lt_compat_r (CR_of_Q R 2)) in H. 2: apply CR_of_Q_pos; reflexivity. - rewrite CRmult_assoc, <- CR_of_Q_mult in H. - setoid_replace ((1 # 2) * 2)%Q with 1%Q in H. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r in H. - rewrite CRmult_comm, (CR_of_Q_plus R 1 1), CR_of_Q_one, CRmult_plus_distr_r, - CRmult_1_l in H. - intro abs. rewrite CRabs_left in H. - unfold CRminus in H. - rewrite CRopp_involutive, CRplus_comm in H. - rewrite CRplus_assoc, <- (CRplus_assoc (-x)), CRplus_opp_l in H. - rewrite CRplus_0_l in H. exact (CRlt_asym _ _ H H). - apply CRlt_asym, abs. + intro abs. contradict H. + apply (CRle_trans _ (x + y - CRabs R (y - x))). + rewrite CRabs_left. 2: apply CRlt_asym, abs. + unfold CRminus. rewrite CRopp_involutive, CRplus_comm. + rewrite CRplus_assoc, <- (CRplus_assoc (-x)), CRplus_opp_l. + rewrite CRplus_0_l, (CR_of_Q_plus R 1 1), CRmult_plus_distr_l. + rewrite CRmult_1_r. apply CRle_refl. + rewrite CRmult_assoc, <- CR_of_Q_mult. + setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. + rewrite CRmult_1_r. apply CRle_refl. Qed. Add Parametric Morphism {R : ConstructiveReals} : CRmin @@ -383,11 +380,11 @@ Lemma CRmin_l : forall {R : ConstructiveReals} (x y : CRcarrier R), Proof. intros. unfold CRmin. apply (CRmult_le_reg_r (CR_of_Q R 2)). - rewrite <- CR_of_Q_zero. apply CR_of_Q_lt; reflexivity. + apply CR_of_Q_lt; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. - rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. + rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r. unfold CRminus. rewrite CRplus_assoc. apply CRplus_le_compat_l. apply (CRplus_le_reg_r (CRabs _ (y + - x)+ -x)). rewrite CRplus_assoc, <- (CRplus_assoc (-CRabs _ (y + - x))). @@ -401,11 +398,11 @@ Lemma CRmin_r : forall {R : ConstructiveReals} (x y : CRcarrier R), Proof. intros. unfold CRmin. apply (CRmult_le_reg_r (CR_of_Q R 2)). - rewrite <- CR_of_Q_zero. apply CR_of_Q_lt; reflexivity. + apply CR_of_Q_lt; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. - rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. + rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r. rewrite (CRplus_comm x). unfold CRminus. rewrite CRplus_assoc. apply CRplus_le_compat_l. apply (CRplus_le_reg_l (-x)). @@ -451,15 +448,15 @@ Proof. intros. unfold CRmin. unfold CRminus. setoid_replace (x + z + - (x + y)) with (z-y). apply (CRmult_eq_reg_r (CR_of_Q _ 2)). - left. rewrite <- CR_of_Q_zero. apply CR_of_Q_lt; reflexivity. + left. apply CR_of_Q_lt; reflexivity. rewrite CRmult_plus_distr_r. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. - rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. + rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r. do 3 rewrite (CRplus_assoc x). apply CRplus_morph. reflexivity. do 2 rewrite <- CRplus_assoc. apply CRplus_morph. 2: reflexivity. rewrite (CRplus_comm x). apply CRplus_assoc. @@ -474,11 +471,11 @@ Lemma CRmin_left : forall {R : ConstructiveReals} (x y : CRcarrier R), Proof. intros. unfold CRmin. apply (CRmult_eq_reg_r (CR_of_Q R 2)). - left. rewrite <- CR_of_Q_zero. apply CR_of_Q_lt; reflexivity. + left. apply CR_of_Q_lt; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. - rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. + rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r. rewrite CRabs_right. unfold CRminus. rewrite CRopp_plus_distr. rewrite CRplus_assoc. apply CRplus_morph. reflexivity. rewrite <- CRplus_assoc, CRplus_opp_r, CRplus_0_l. apply CRopp_involutive. @@ -491,11 +488,11 @@ Lemma CRmin_right : forall {R : ConstructiveReals} (x y : CRcarrier R), Proof. intros. unfold CRmin. apply (CRmult_eq_reg_r (CR_of_Q R 2)). - left. rewrite <- CR_of_Q_zero. apply CR_of_Q_lt; reflexivity. + left. apply CR_of_Q_lt; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. - rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. + rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r. rewrite CRabs_left. unfold CRminus. do 2 rewrite CRopp_plus_distr. rewrite (CRplus_comm x y). rewrite CRplus_assoc. apply CRplus_morph. reflexivity. @@ -510,10 +507,10 @@ Lemma CRmin_lt : forall {R : ConstructiveReals} (x y z : CRcarrier R), Proof. intros. unfold CRmin. apply (CRmult_lt_reg_r (CR_of_Q R 2)). - rewrite <- CR_of_Q_zero. apply CR_of_Q_lt; reflexivity. + apply CR_of_Q_lt; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. apply (CRplus_lt_reg_l _ (CRabs _ (y - x) - (z*CR_of_Q R 2))). unfold CRminus. rewrite CRplus_assoc. rewrite CRplus_opp_l, CRplus_0_r. rewrite (CRplus_comm (CRabs R (y + - x))). @@ -526,7 +523,7 @@ Proof. apply (CRplus_lt_reg_l R (-x)). rewrite CRopp_mult_distr_l. rewrite <- CRplus_assoc, CRplus_opp_l, CRplus_0_l. - rewrite (CR_of_Q_plus R 1 1), CR_of_Q_one, CRmult_plus_distr_l. + rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l. rewrite CRmult_1_r. apply CRplus_le_lt_compat. apply CRlt_asym. apply CRopp_gt_lt_contravar, H. @@ -537,7 +534,7 @@ Proof. apply (CRplus_lt_reg_l R (-y)). rewrite CRopp_mult_distr_l. rewrite <- CRplus_assoc, CRplus_opp_l, CRplus_0_l. - rewrite (CR_of_Q_plus R 1 1), CR_of_Q_one, CRmult_plus_distr_l. + rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l. rewrite CRmult_1_r. apply CRplus_le_lt_compat. apply CRlt_asym. apply CRopp_gt_lt_contravar, H0. @@ -552,12 +549,12 @@ Proof. rewrite (CRabs_morph _ ((x - y + (CRabs _ (a - y) - CRabs _ (a - x))) * CR_of_Q R (1 # 2))). rewrite CRabs_mult, (CRabs_right (CR_of_Q R (1 # 2))). - 2: rewrite <- CR_of_Q_zero; apply CR_of_Q_le; discriminate. + 2: apply CR_of_Q_le; discriminate. apply (CRle_trans _ ((CRabs _ (x - y) * 1 + CRabs _ (x-y) * 1) * CR_of_Q R (1 # 2))). apply CRmult_le_compat_r. - rewrite <- CR_of_Q_zero. apply CR_of_Q_le. discriminate. + apply CR_of_Q_le. discriminate. apply (CRle_trans _ (CRabs _ (x - y) + CRabs _ (CRabs _ (a - y) - CRabs _ (a - x)))). apply CRabs_triang. rewrite CRmult_1_r. apply CRplus_le_compat_l. @@ -568,11 +565,11 @@ Proof. rewrite CRplus_comm, CRopp_plus_distr, <- CRplus_assoc. rewrite CRplus_opp_r, CRopp_involutive, CRplus_0_l. reflexivity. - rewrite <- CRmult_plus_distr_l, <- CR_of_Q_one. + rewrite <- CRmult_plus_distr_l. rewrite <- (CR_of_Q_plus R 1 1). rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 + 1) * (1 # 2))%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. apply CRle_refl. + rewrite CRmult_1_r. apply CRle_refl. unfold CRminus. apply CRmult_morph. 2: reflexivity. do 4 rewrite CRplus_assoc. apply CRplus_morph. reflexivity. rewrite <- CRplus_assoc. rewrite CRplus_comm, CRopp_plus_distr. @@ -587,10 +584,10 @@ Lemma CRmin_glb : forall {R : ConstructiveReals} (x y z:CRcarrier R), Proof. intros. unfold CRmin. apply (CRmult_le_reg_r (CR_of_Q R 2)). - rewrite <- CR_of_Q_zero. apply CR_of_Q_lt; reflexivity. + apply CR_of_Q_lt; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. apply (CRplus_le_reg_l (CRabs _ (y-x) - (z*CR_of_Q R 2))). unfold CRminus. rewrite CRplus_assoc, CRplus_opp_l, CRplus_0_r. rewrite (CRplus_comm (CRabs R (y + - x) + - (z * CR_of_Q R 2))). @@ -601,13 +598,13 @@ Proof. rewrite CRopp_involutive, (CRplus_comm y), CRplus_assoc. apply CRplus_le_compat_l, (CRplus_le_reg_l y). rewrite <- CRplus_assoc, CRplus_opp_r, CRplus_0_l. - rewrite (CR_of_Q_plus R 1 1), CR_of_Q_one, CRmult_plus_distr_l. + rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l. rewrite CRmult_1_r. apply CRplus_le_compat; exact H0. - rewrite (CRplus_comm x), CRplus_assoc. apply CRplus_le_compat_l. apply (CRplus_le_reg_l (-x)). rewrite <- CRplus_assoc, CRplus_opp_l, CRplus_0_l. rewrite CRopp_mult_distr_l. - rewrite (CR_of_Q_plus R 1 1), CR_of_Q_one, CRmult_plus_distr_l. + rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l. rewrite CRmult_1_r. apply CRplus_le_compat; apply CRopp_ge_le_contravar; exact H. Qed. @@ -673,11 +670,11 @@ Lemma CRmax_lub : forall {R : ConstructiveReals} (x y z:CRcarrier R), x <= z -> y <= z -> CRmax x y <= z. Proof. intros. unfold CRmax. - apply (CRmult_le_reg_r (CR_of_Q _ 2)). rewrite <- CR_of_Q_zero. + apply (CRmult_le_reg_r (CR_of_Q _ 2)). apply CR_of_Q_lt; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. apply (CRplus_le_reg_l (-x-y)). rewrite <- CRplus_assoc. unfold CRminus. rewrite <- CRopp_plus_distr, CRplus_opp_l, CRplus_0_l. @@ -687,14 +684,14 @@ Proof. rewrite (CRplus_comm x), CRplus_assoc. apply CRplus_le_compat_l. apply (CRplus_le_reg_l (-x)). rewrite <- CRplus_assoc, CRplus_opp_l, CRplus_0_l. - rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CR_of_Q_one, CRmult_1_r. + rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r. rewrite CRopp_plus_distr. apply CRplus_le_compat; apply CRopp_ge_le_contravar; assumption. - rewrite (CRplus_comm y), CRopp_plus_distr, CRplus_assoc. apply CRplus_le_compat_l. apply (CRplus_le_reg_l y). rewrite <- CRplus_assoc, CRplus_opp_r, CRplus_0_l. - rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CR_of_Q_one, CRmult_1_r. + rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r. apply CRplus_le_compat; assumption. Qed. @@ -702,12 +699,12 @@ Lemma CRmax_l : forall {R : ConstructiveReals} (x y : CRcarrier R), x <= CRmax x y. Proof. intros. unfold CRmax. - apply (CRmult_le_reg_r (CR_of_Q R 2)). rewrite <- CR_of_Q_zero. + apply (CRmult_le_reg_r (CR_of_Q R 2)). apply CR_of_Q_lt; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. - setoid_replace 2%Q with (1+1)%Q. rewrite CR_of_Q_plus, CR_of_Q_one. + rewrite CRmult_1_r. + setoid_replace 2%Q with (1+1)%Q. rewrite CR_of_Q_plus. rewrite CRmult_plus_distr_l, CRmult_1_r, CRplus_assoc. apply CRplus_le_compat_l. apply (CRplus_le_reg_l (-y)). @@ -720,12 +717,12 @@ Lemma CRmax_r : forall {R : ConstructiveReals} (x y : CRcarrier R), y <= CRmax x y. Proof. intros. unfold CRmax. - apply (CRmult_le_reg_r (CR_of_Q _ 2)). rewrite <- CR_of_Q_zero. + apply (CRmult_le_reg_r (CR_of_Q _ 2)). apply CR_of_Q_lt; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. - rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. + rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r. rewrite (CRplus_comm x). rewrite CRplus_assoc. apply CRplus_le_compat_l. apply (CRplus_le_reg_l (-x)). @@ -754,14 +751,14 @@ Proof. intros. unfold CRmax. setoid_replace (x + z - (x + y)) with (z-y). apply (CRmult_eq_reg_r (CR_of_Q _ 2)). - left. rewrite <- CR_of_Q_zero. apply CR_of_Q_lt; reflexivity. + left. apply CR_of_Q_lt; reflexivity. rewrite CRmult_plus_distr_r. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CR_of_Q_one, CRmult_1_r. + rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r. rewrite CRmult_1_r. do 3 rewrite (CRplus_assoc x). apply CRplus_morph. reflexivity. do 2 rewrite <- CRplus_assoc. apply CRplus_morph. 2: reflexivity. @@ -777,11 +774,11 @@ Lemma CRmax_left : forall {R : ConstructiveReals} (x y : CRcarrier R), Proof. intros. unfold CRmax. apply (CRmult_eq_reg_r (CR_of_Q R 2)). - left. rewrite <- CR_of_Q_zero. apply CR_of_Q_lt; reflexivity. + left. apply CR_of_Q_lt; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. - rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. + rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r. rewrite CRplus_assoc. apply CRplus_morph. reflexivity. rewrite CRabs_left. unfold CRminus. rewrite CRopp_plus_distr, CRopp_involutive. rewrite <- CRplus_assoc, CRplus_opp_r, CRplus_0_l. reflexivity. @@ -793,11 +790,11 @@ Lemma CRmax_right : forall {R : ConstructiveReals} (x y : CRcarrier R), Proof. intros. unfold CRmax. apply (CRmult_eq_reg_r (CR_of_Q R 2)). - left. rewrite <- CR_of_Q_zero. apply CR_of_Q_lt; reflexivity. + left. apply CR_of_Q_lt; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. - rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. + rewrite (CR_of_Q_plus _ 1 1), CRmult_plus_distr_l, CRmult_1_r. rewrite (CRplus_comm x y). rewrite CRplus_assoc. apply CRplus_morph. reflexivity. rewrite CRabs_right. unfold CRminus. rewrite CRplus_comm. @@ -812,12 +809,12 @@ Proof. rewrite (CRabs_morph _ ((x - y + (CRabs _ (a - x) - CRabs _ (a - y))) * CR_of_Q R (1 # 2))). rewrite CRabs_mult, (CRabs_right (CR_of_Q R (1 # 2))). - 2: rewrite <- CR_of_Q_zero; apply CR_of_Q_le; discriminate. + 2: apply CR_of_Q_le; discriminate. apply (CRle_trans _ ((CRabs _ (x - y) * 1 + CRabs _ (x-y) * 1) * CR_of_Q R (1 # 2))). apply CRmult_le_compat_r. - rewrite <- CR_of_Q_zero. apply CR_of_Q_le. discriminate. + apply CR_of_Q_le. discriminate. apply (CRle_trans _ (CRabs _ (x - y) + CRabs _ (CRabs _ (a - x) - CRabs _ (a - y)))). apply CRabs_triang. rewrite CRmult_1_r. apply CRplus_le_compat_l. @@ -829,11 +826,11 @@ Proof. rewrite CRplus_comm, CRopp_plus_distr, <- CRplus_assoc. rewrite CRplus_opp_r, CRopp_involutive, CRplus_0_l. reflexivity. - rewrite <- CRmult_plus_distr_l, <- CR_of_Q_one. + rewrite <- CRmult_plus_distr_l. rewrite <- (CR_of_Q_plus R 1 1). rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 + 1) * (1 # 2))%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. apply CRle_refl. + rewrite CRmult_1_r. apply CRle_refl. unfold CRminus. rewrite CRopp_mult_distr_l. rewrite <- CRmult_plus_distr_r. apply CRmult_morph. 2: reflexivity. do 4 rewrite CRplus_assoc. apply CRplus_morph. reflexivity. @@ -849,10 +846,10 @@ Lemma CRmax_lub_lt : forall {R : ConstructiveReals} (x y z : CRcarrier R), Proof. intros. unfold CRmax. apply (CRmult_lt_reg_r (CR_of_Q R 2)). - rewrite <- CR_of_Q_zero. apply CR_of_Q_lt; reflexivity. + apply CR_of_Q_lt; reflexivity. rewrite CRmult_assoc, <- CR_of_Q_mult. setoid_replace ((1 # 2) * 2)%Q with 1%Q. 2: reflexivity. - rewrite CR_of_Q_one, CRmult_1_r. + rewrite CRmult_1_r. apply (CRplus_lt_reg_l _ (-y -x)). unfold CRminus. rewrite CRplus_assoc, <- (CRplus_assoc (-x)), <- (CRplus_assoc (-x)). rewrite CRplus_opp_l, CRplus_0_l, <- CRplus_assoc, CRplus_opp_l, CRplus_0_l. @@ -861,14 +858,14 @@ Proof. apply CRplus_lt_compat_l. apply (CRplus_lt_reg_l _ y). rewrite <- CRplus_assoc, CRplus_opp_r, CRplus_0_l. - rewrite (CR_of_Q_plus R 1 1), CR_of_Q_one, CRmult_plus_distr_l. + rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l. rewrite CRmult_1_r. apply CRplus_le_lt_compat. apply CRlt_asym, H0. exact H0. - rewrite CRopp_plus_distr, CRopp_involutive. rewrite CRplus_assoc. apply CRplus_lt_compat_l. apply (CRplus_lt_reg_l _ x). rewrite <- CRplus_assoc, CRplus_opp_r, CRplus_0_l. - rewrite (CR_of_Q_plus R 1 1), CR_of_Q_one, CRmult_plus_distr_l. + rewrite (CR_of_Q_plus R 1 1), CRmult_plus_distr_l. rewrite CRmult_1_r. apply CRplus_le_lt_compat. apply CRlt_asym, H. exact H. Qed. -- cgit v1.2.3