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diff --git a/theories7/Reals/Rbasic_fun.v b/theories7/Reals/Rbasic_fun.v deleted file mode 100644 index 3d143e34..00000000 --- a/theories7/Reals/Rbasic_fun.v +++ /dev/null @@ -1,476 +0,0 @@ -(************************************************************************) -(* 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: Rbasic_fun.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) - -(*********************************************************) -(** Complements for the real numbers *) -(* *) -(*********************************************************) - -Require Rbase. -Require R_Ifp. -Require Fourier. -V7only [Import R_scope.]. Open Local Scope R_scope. - -Implicit Variable Type r:R. - -(*******************************) -(** Rmin *) -(*******************************) - -(*********) -Definition Rmin :R->R->R:=[x,y:R] - Cases (total_order_Rle x y) of - (leftT _) => x - | (rightT _) => y - end. - -(*********) -Lemma Rmin_Rgt_l:(r1,r2,r:R)(Rgt (Rmin r1 r2) r) -> - ((Rgt r1 r)/\(Rgt r2 r)). -Intros r1 r2 r;Unfold Rmin;Case (total_order_Rle r1 r2);Intros. -Split. -Assumption. -Unfold Rgt;Unfold Rgt in H;Exact (Rlt_le_trans r r1 r2 H r0). -Split. -Generalize (not_Rle r1 r2 n);Intro;Exact (Rgt_trans r1 r2 r H0 H). -Assumption. -Qed. - -(*********) -Lemma Rmin_Rgt_r:(r1,r2,r:R)(((Rgt r1 r)/\(Rgt r2 r)) -> - (Rgt (Rmin r1 r2) r)). -Intros;Unfold Rmin;Case (total_order_Rle r1 r2);Elim H;Clear H;Intros; - Assumption. -Qed. - -(*********) -Lemma Rmin_Rgt:(r1,r2,r:R)(Rgt (Rmin r1 r2) r)<-> - ((Rgt r1 r)/\(Rgt r2 r)). -Intros; Split. -Exact (Rmin_Rgt_l r1 r2 r). -Exact (Rmin_Rgt_r r1 r2 r). -Qed. - -(*********) -Lemma Rmin_l : (x,y:R) ``(Rmin x y)<=x``. -Intros; Unfold Rmin; Case (total_order_Rle x y); Intro H1; [Right; Reflexivity | Auto with real]. -Qed. - -(*********) -Lemma Rmin_r : (x,y:R) ``(Rmin x y)<=y``. -Intros; Unfold Rmin; Case (total_order_Rle x y); Intro H1; [Assumption | Auto with real]. -Qed. - -(*********) -Lemma Rmin_sym : (a,b:R) (Rmin a b)==(Rmin b a). -Intros; Unfold Rmin; Case (total_order_Rle a b); Case (total_order_Rle b a); Intros; Try Reflexivity Orelse (Apply Rle_antisym; Assumption Orelse Auto with real). -Qed. - -(*********) -Lemma Rmin_stable_in_posreal : (x,y:posreal) ``0<(Rmin x y)``. -Intros; Apply Rmin_Rgt_r; Split; [Apply (cond_pos x) | Apply (cond_pos y)]. -Qed. - -(*******************************) -(** Rmax *) -(*******************************) - -(*********) -Definition Rmax :R->R->R:=[x,y:R] - Cases (total_order_Rle x y) of - (leftT _) => y - | (rightT _) => x - end. - -(*********) -Lemma Rmax_Rle:(r1,r2,r:R)(Rle r (Rmax r1 r2))<-> - ((Rle r r1)\/(Rle r r2)). -Intros;Split. -Unfold Rmax;Case (total_order_Rle r1 r2);Intros;Auto. -Intro;Unfold Rmax;Case (total_order_Rle r1 r2);Elim H;Clear H;Intros;Auto. -Apply (Rle_trans r r1 r2);Auto. -Generalize (not_Rle r1 r2 n);Clear n;Intro;Unfold Rgt in H0; - Apply (Rlt_le r r1 (Rle_lt_trans r r2 r1 H H0)). -Qed. - -Lemma RmaxLess1: (r1, r2 : R) (Rle r1 (Rmax r1 r2)). -Intros r1 r2; Unfold Rmax; Case (total_order_Rle r1 r2); Auto with real. -Qed. - -Lemma RmaxLess2: (r1, r2 : R) (Rle r2 (Rmax r1 r2)). -Intros r1 r2; Unfold Rmax; Case (total_order_Rle r1 r2); Auto with real. -Qed. - -Lemma RmaxSym: (p, q : R) (Rmax p q) == (Rmax q p). -Intros p q; Unfold Rmax; - Case (total_order_Rle p q); Case (total_order_Rle q p); Auto; Intros H1 H2; - Apply Rle_antisym; Auto with real. -Qed. - -Lemma RmaxRmult: - (p, q, r : R) - (Rle R0 r) -> (Rmax (Rmult r p) (Rmult r q)) == (Rmult r (Rmax p q)). -Intros p q r H; Unfold Rmax. -Case (total_order_Rle p q); Case (total_order_Rle (Rmult r p) (Rmult r q)); - Auto; Intros H1 H2; Auto. -Case H; Intros E1. -Case H1; Auto with real. -Rewrite <- E1; Repeat Rewrite Rmult_Ol; Auto. -Case H; Intros E1. -Case H2; Auto with real. -Apply Rle_monotony_contra with z := r; Auto. -Rewrite <- E1; Repeat Rewrite Rmult_Ol; Auto. -Qed. - -Lemma Rmax_stable_in_negreal : (x,y:negreal) ``(Rmax x y)<0``. -Intros; Unfold Rmax; Case (total_order_Rle x y); Intro; [Apply (cond_neg y) | Apply (cond_neg x)]. -Qed. - -(*******************************) -(** Rabsolu *) -(*******************************) - -(*********) -Lemma case_Rabsolu:(r:R)(sumboolT (Rlt r R0) (Rge r R0)). -Intro;Generalize (total_order_Rle R0 r);Intro X;Elim X;Intro;Clear X. -Right;Apply (Rle_sym1 R0 r a). -Left;Fold (Rgt R0 r);Apply (not_Rle R0 r b). -Qed. - -(*********) -Definition Rabsolu:R->R:= - [r:R](Cases (case_Rabsolu r) of - (leftT _) => (Ropp r) - |(rightT _) => r - end). - -(*********) -Lemma Rabsolu_R0:(Rabsolu R0)==R0. -Unfold Rabsolu;Case (case_Rabsolu R0);Auto;Intro. -Generalize (Rlt_antirefl R0);Intro;ElimType False;Auto. -Qed. - -Lemma Rabsolu_R1: (Rabsolu R1)==R1. -Unfold Rabsolu; Case (case_Rabsolu R1); Auto with real. -Intros H; Absurd ``1 < 0``;Auto with real. -Qed. - -(*********) -Lemma Rabsolu_no_R0:(r:R)~r==R0->~(Rabsolu r)==R0. -Intros;Unfold Rabsolu;Case (case_Rabsolu r);Intro;Auto. -Apply Ropp_neq;Auto. -Qed. - -(*********) -Lemma Rabsolu_left: (r:R)(Rlt r R0)->((Rabsolu r) == (Ropp r)). -Intros;Unfold Rabsolu;Case (case_Rabsolu r);Trivial;Intro;Absurd (Rge r R0). -Exact (Rlt_ge_not r R0 H). -Assumption. -Qed. - -(*********) -Lemma Rabsolu_right: (r:R)(Rge r R0)->((Rabsolu r) == r). -Intros;Unfold Rabsolu;Case (case_Rabsolu r);Intro. -Absurd (Rge r R0). -Exact (Rlt_ge_not r R0 r0). -Assumption. -Trivial. -Qed. - -Lemma Rabsolu_left1: (a : R) (Rle a R0) -> (Rabsolu a) == (Ropp a). -Intros a H; Case H; Intros H1. -Apply Rabsolu_left; Auto. -Rewrite H1; Simpl; Rewrite Rabsolu_right; Auto with real. -Qed. - -(*********) -Lemma Rabsolu_pos:(x:R)(Rle R0 (Rabsolu x)). -Intros;Unfold Rabsolu;Case (case_Rabsolu x);Intro. -Generalize (Rlt_Ropp x R0 r);Intro;Unfold Rgt in H; - Rewrite Ropp_O in H;Unfold Rle;Left;Assumption. -Apply Rle_sym2;Assumption. -Qed. - -Lemma Rle_Rabsolu: - (x:R) (Rle x (Rabsolu x)). -Intro; Unfold Rabsolu;Case (case_Rabsolu x);Intros;Fourier. -Qed. - -(*********) -Lemma Rabsolu_pos_eq:(x:R)(Rle R0 x)->(Rabsolu x)==x. -Intros;Unfold Rabsolu;Case (case_Rabsolu x);Intro; - [Generalize (Rle_not R0 x r);Intro;ElimType False;Auto|Trivial]. -Qed. - -(*********) -Lemma Rabsolu_Rabsolu:(x:R)(Rabsolu (Rabsolu x))==(Rabsolu x). -Intro;Apply (Rabsolu_pos_eq (Rabsolu x) (Rabsolu_pos x)). -Qed. - -(*********) -Lemma Rabsolu_pos_lt:(x:R)(~x==R0)->(Rlt R0 (Rabsolu x)). -Intros;Generalize (Rabsolu_pos x);Intro;Unfold Rle in H0; - Elim H0;Intro;Auto. -ElimType False;Clear H0;Elim H;Clear H;Generalize H1; - Unfold Rabsolu;Case (case_Rabsolu x);Intros;Auto. -Clear r H1; Generalize (Rplus_plus_r x R0 (Ropp x) H0); - Rewrite (let (H1,H2)=(Rplus_ne x) in H1);Rewrite (Rplus_Ropp_r x);Trivial. -Qed. - -(*********) -Lemma Rabsolu_minus_sym:(x,y:R) - (Rabsolu (Rminus x y))==(Rabsolu (Rminus y x)). -Intros;Unfold Rabsolu;Case (case_Rabsolu (Rminus x y)); - Case (case_Rabsolu (Rminus y x));Intros. - Generalize (Rminus_lt y x r);Generalize (Rminus_lt x y r0);Intros; - Generalize (Rlt_antisym x y H);Intro;ElimType False;Auto. -Rewrite (Ropp_distr2 x y);Trivial. -Rewrite (Ropp_distr2 y x);Trivial. -Unfold Rge in r r0;Elim r;Elim r0;Intros;Clear r r0. -Generalize (Rgt_RoppO (Rminus x y) H);Rewrite (Ropp_distr2 x y); - Intro;Unfold Rgt in H0;Generalize (Rlt_antisym R0 (Rminus y x) H0); - Intro;ElimType False;Auto. -Rewrite (Rminus_eq x y H);Trivial. -Rewrite (Rminus_eq y x H0);Trivial. -Rewrite (Rminus_eq y x H0);Trivial. -Qed. - -(*********) -Lemma Rabsolu_mult:(x,y:R) - (Rabsolu (Rmult x y))==(Rmult (Rabsolu x) (Rabsolu y)). -Intros;Unfold Rabsolu;Case (case_Rabsolu (Rmult x y)); - Case (case_Rabsolu x);Case (case_Rabsolu y);Intros;Auto. -Generalize (Rlt_anti_monotony y x R0 r r0);Intro; - Rewrite (Rmult_Or y) in H;Generalize (Rlt_antisym (Rmult x y) R0 r1); - Intro;Unfold Rgt in H;ElimType False;Rewrite (Rmult_sym y x) in H; - Auto. -Rewrite (Ropp_mul1 x y);Trivial. -Rewrite (Rmult_sym x (Ropp y));Rewrite (Ropp_mul1 y x); - Rewrite (Rmult_sym x y);Trivial. -Unfold Rge in r r0;Elim r;Elim r0;Clear r r0;Intros;Unfold Rgt in H H0. -Generalize (Rlt_monotony x R0 y H H0);Intro;Rewrite (Rmult_Or x) in H1; - Generalize (Rlt_antisym (Rmult x y) R0 r1);Intro;ElimType False;Auto. -Rewrite H in r1;Rewrite (Rmult_Ol y) in r1;Generalize (Rlt_antirefl R0); - Intro;ElimType False;Auto. -Rewrite H0 in r1;Rewrite (Rmult_Or x) in r1;Generalize (Rlt_antirefl R0); - Intro;ElimType False;Auto. -Rewrite H0 in r1;Rewrite (Rmult_Or x) in r1;Generalize (Rlt_antirefl R0); - Intro;ElimType False;Auto. -Rewrite (Ropp_mul2 x y);Trivial. -Unfold Rge in r r1;Elim r;Elim r1;Clear r r1;Intros;Unfold Rgt in H0 H. -Generalize (Rlt_monotony y x R0 H0 r0);Intro;Rewrite (Rmult_Or y) in H1; - Rewrite (Rmult_sym y x) in H1; - Generalize (Rlt_antisym (Rmult x y) R0 H1);Intro;ElimType False;Auto. -Generalize (imp_not_Req x R0 (or_introl (Rlt x R0) (Rgt x R0) r0)); - Generalize (imp_not_Req y R0 (or_intror (Rlt y R0) (Rgt y R0) H0));Intros; - Generalize (without_div_Od x y H);Intro;Elim H3;Intro;ElimType False; - Auto. -Rewrite H0 in H;Rewrite (Rmult_Or x) in H;Unfold Rgt in H; - Generalize (Rlt_antirefl R0);Intro;ElimType False;Auto. -Rewrite H0;Rewrite (Rmult_Or x);Rewrite (Rmult_Or (Ropp x));Trivial. -Unfold Rge in r0 r1;Elim r0;Elim r1;Clear r0 r1;Intros;Unfold Rgt in H0 H. -Generalize (Rlt_monotony x y R0 H0 r);Intro;Rewrite (Rmult_Or x) in H1; - Generalize (Rlt_antisym (Rmult x y) R0 H1);Intro;ElimType False;Auto. -Generalize (imp_not_Req y R0 (or_introl (Rlt y R0) (Rgt y R0) r)); - Generalize (imp_not_Req R0 x (or_introl (Rlt R0 x) (Rgt R0 x) H0));Intros; - Generalize (without_div_Od x y H);Intro;Elim H3;Intro;ElimType False; - Auto. -Rewrite H0 in H;Rewrite (Rmult_Ol y) in H;Unfold Rgt in H; - Generalize (Rlt_antirefl R0);Intro;ElimType False;Auto. -Rewrite H0;Rewrite (Rmult_Ol y);Rewrite (Rmult_Ol (Ropp y));Trivial. -Qed. - -(*********) -Lemma Rabsolu_Rinv:(r:R)(~r==R0)->(Rabsolu (Rinv r))== - (Rinv (Rabsolu r)). -Intro;Unfold Rabsolu;Case (case_Rabsolu r); - Case (case_Rabsolu (Rinv r));Auto;Intros. -Apply Ropp_Rinv;Auto. -Generalize (Rlt_Rinv2 r r1);Intro;Unfold Rge in r0;Elim r0;Intros. -Unfold Rgt in H1;Generalize (Rlt_antisym R0 (Rinv r) H1);Intro; - ElimType False;Auto. -Generalize - (imp_not_Req (Rinv r) R0 - (or_introl (Rlt (Rinv r) R0) (Rgt (Rinv r) R0) H0));Intro; - ElimType False;Auto. -Unfold Rge in r1;Elim r1;Clear r1;Intro. -Unfold Rgt in H0;Generalize (Rlt_antisym R0 (Rinv r) - (Rlt_Rinv r H0));Intro;ElimType False;Auto. -ElimType False;Auto. -Qed. - -Lemma Rabsolu_Ropp: - (x:R) (Rabsolu (Ropp x))==(Rabsolu x). -Intro;Cut (Ropp x)==(Rmult (Ropp R1) x). -Intros; Rewrite H. -Rewrite Rabsolu_mult. -Cut (Rabsolu (Ropp R1))==R1. -Intros; Rewrite H0. -Ring. -Unfold Rabsolu; Case (case_Rabsolu (Ropp R1)). -Intro; Ring. -Intro H0;Generalize (Rle_sym2 R0 (Ropp R1) H0);Intros. -Generalize (Rle_Ropp R0 (Ropp R1) H1). -Rewrite Ropp_Ropp; Rewrite Ropp_O. -Intro;Generalize (Rle_not R1 R0 Rlt_R0_R1);Intro; - Generalize (Rle_sym2 R1 R0 H2);Intro; - ElimType False;Auto. -Ring. -Qed. - -(*********) -Lemma Rabsolu_triang:(a,b:R)(Rle (Rabsolu (Rplus a b)) - (Rplus (Rabsolu a) (Rabsolu b))). -Intros a b;Unfold Rabsolu;Case (case_Rabsolu (Rplus a b)); - Case (case_Rabsolu a);Case (case_Rabsolu b);Intros. -Apply (eq_Rle (Ropp (Rplus a b)) (Rplus (Ropp a) (Ropp b))); - Rewrite (Ropp_distr1 a b);Reflexivity. -(**) -Rewrite (Ropp_distr1 a b); - Apply (Rle_compatibility (Ropp a) (Ropp b) b); - Unfold Rle;Unfold Rge in r;Elim r;Intro. -Left;Unfold Rgt in H;Generalize (Rlt_compatibility (Ropp b) R0 b H); - Intro;Elim (Rplus_ne (Ropp b));Intros v w;Rewrite v in H0;Clear v w; - Rewrite (Rplus_Ropp_l b) in H0;Apply (Rlt_trans (Ropp b) R0 b H0 H). -Right;Rewrite H;Apply Ropp_O. -(**) -Rewrite (Ropp_distr1 a b); - Rewrite (Rplus_sym (Ropp a) (Ropp b)); - Rewrite (Rplus_sym a (Ropp b)); - Apply (Rle_compatibility (Ropp b) (Ropp a) a); - Unfold Rle;Unfold Rge in r0;Elim r0;Intro. -Left;Unfold Rgt in H;Generalize (Rlt_compatibility (Ropp a) R0 a H); - Intro;Elim (Rplus_ne (Ropp a));Intros v w;Rewrite v in H0;Clear v w; - Rewrite (Rplus_Ropp_l a) in H0;Apply (Rlt_trans (Ropp a) R0 a H0 H). -Right;Rewrite H;Apply Ropp_O. -(**) -ElimType False;Generalize (Rge_plus_plus_r a b R0 r);Intro; - Elim (Rplus_ne a);Intros v w;Rewrite v in H;Clear v w; - Generalize (Rge_trans (Rplus a b) a R0 H r0);Intro;Clear H; - Unfold Rge in H0;Elim H0;Intro;Clear H0. -Unfold Rgt in H;Generalize (Rlt_antisym (Rplus a b) R0 r1);Intro;Auto. -Absurd (Rplus a b)==R0;Auto. -Apply (imp_not_Req (Rplus a b) R0);Left;Assumption. -(**) -ElimType False;Generalize (Rlt_compatibility a b R0 r);Intro; - Elim (Rplus_ne a);Intros v w;Rewrite v in H;Clear v w; - Generalize (Rlt_trans (Rplus a b) a R0 H r0);Intro;Clear H; - Unfold Rge in r1;Elim r1;Clear r1;Intro. -Unfold Rgt in H; - Generalize (Rlt_trans (Rplus a b) R0 (Rplus a b) H0 H);Intro; - Apply (Rlt_antirefl (Rplus a b));Assumption. -Rewrite H in H0;Apply (Rlt_antirefl R0);Assumption. -(**) -Rewrite (Rplus_sym a b);Rewrite (Rplus_sym (Ropp a) b); - Apply (Rle_compatibility b a (Ropp a)); - Apply (Rminus_le a (Ropp a));Unfold Rminus;Rewrite (Ropp_Ropp a); - Generalize (Rlt_compatibility a a R0 r0);Clear r r1;Intro; - Elim (Rplus_ne a);Intros v w;Rewrite v in H;Clear v w; - Generalize (Rlt_trans (Rplus a a) a R0 H r0);Intro; - Apply (Rlt_le (Rplus a a) R0 H0). -(**) -Apply (Rle_compatibility a b (Ropp b)); - Apply (Rminus_le b (Ropp b));Unfold Rminus;Rewrite (Ropp_Ropp b); - Generalize (Rlt_compatibility b b R0 r);Clear r0 r1;Intro; - Elim (Rplus_ne b);Intros v w;Rewrite v in H;Clear v w; - Generalize (Rlt_trans (Rplus b b) b R0 H r);Intro; - Apply (Rlt_le (Rplus b b) R0 H0). -(**) -Unfold Rle;Right;Reflexivity. -Qed. - -(*********) -Lemma Rabsolu_triang_inv:(a,b:R)(Rle (Rminus (Rabsolu a) (Rabsolu b)) - (Rabsolu (Rminus a b))). -Intros; - Apply (Rle_anti_compatibility (Rabsolu b) - (Rminus (Rabsolu a) (Rabsolu b)) (Rabsolu (Rminus a b))); - Unfold Rminus; - Rewrite <- (Rplus_assoc (Rabsolu b) (Rabsolu a) (Ropp (Rabsolu b))); - Rewrite (Rplus_sym (Rabsolu b) (Rabsolu a)); - Rewrite (Rplus_assoc (Rabsolu a) (Rabsolu b) (Ropp (Rabsolu b))); - Rewrite (Rplus_Ropp_r (Rabsolu b)); - Rewrite (proj1 ? ? (Rplus_ne (Rabsolu a))); - Replace (Rabsolu a) with (Rabsolu (Rplus a R0)). - Rewrite <- (Rplus_Ropp_r b); - Rewrite <- (Rplus_assoc a b (Ropp b)); - Rewrite (Rplus_sym a b); - Rewrite (Rplus_assoc b a (Ropp b)). - Exact (Rabsolu_triang b (Rplus a (Ropp b))). - Rewrite (proj1 ? ? (Rplus_ne a));Trivial. -Qed. - -(* ||a|-|b||<=|a-b| *) -Lemma Rabsolu_triang_inv2 : (a,b:R) ``(Rabsolu ((Rabsolu a)-(Rabsolu b)))<=(Rabsolu (a-b))``. -Cut (a,b:R) ``(Rabsolu b)<=(Rabsolu a)``->``(Rabsolu ((Rabsolu a)-(Rabsolu b))) <= (Rabsolu (a-b))``. -Intros; NewDestruct (total_order (Rabsolu a) (Rabsolu b)) as [Hlt|[Heq|Hgt]]. -Rewrite <- (Rabsolu_Ropp ``(Rabsolu a)-(Rabsolu b)``); Rewrite <- (Rabsolu_Ropp ``a-b``); Do 2 Rewrite Ropp_distr2. -Apply H; Left; Assumption. -Rewrite Heq; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rabsolu_pos. -Apply H; Left; Assumption. -Intros; Replace ``(Rabsolu ((Rabsolu a)-(Rabsolu b)))`` with ``(Rabsolu a)-(Rabsolu b)``. -Apply Rabsolu_triang_inv. -Rewrite (Rabsolu_right ``(Rabsolu a)-(Rabsolu b)``); [Reflexivity | Apply Rle_sym1; Apply Rle_anti_compatibility with (Rabsolu b); Rewrite Rplus_Or; Replace ``(Rabsolu b)+((Rabsolu a)-(Rabsolu b))`` with (Rabsolu a); [Assumption | Ring]]. -Qed. - -(*********) -Lemma Rabsolu_def1:(x,a:R)(Rlt x a)->(Rlt (Ropp a) x)->(Rlt (Rabsolu x) a). -Unfold Rabsolu;Intros;Case (case_Rabsolu x);Intro. -Generalize (Rlt_Ropp (Ropp a) x H0);Unfold Rgt;Rewrite Ropp_Ropp;Intro; - Assumption. -Assumption. -Qed. - -(*********) -Lemma Rabsolu_def2:(x,a:R)(Rlt (Rabsolu x) a)->(Rlt x a)/\(Rlt (Ropp a) x). -Unfold Rabsolu;Intro x;Case (case_Rabsolu x);Intros. -Generalize (Rlt_RoppO x r);Unfold Rgt;Intro; - Generalize (Rlt_trans R0 (Ropp x) a H0 H);Intro;Split. -Apply (Rlt_trans x R0 a r H1). -Generalize (Rlt_Ropp (Ropp x) a H);Rewrite (Ropp_Ropp x);Unfold Rgt;Trivial. -Fold (Rgt a x) in H;Generalize (Rgt_ge_trans a x R0 H r);Intro; - Generalize (Rgt_RoppO a H0);Intro;Fold (Rgt R0 (Ropp a)); - Generalize (Rge_gt_trans x R0 (Ropp a) r H1);Unfold Rgt;Intro;Split; - Assumption. -Qed. - -Lemma RmaxAbs: - (p, q, r : R) - (Rle p q) -> (Rle q r) -> (Rle (Rabsolu q) (Rmax (Rabsolu p) (Rabsolu r))). -Intros p q r H' H'0; Case (Rle_or_lt R0 p); Intros H'1. -Repeat Rewrite Rabsolu_right; Auto with real. -Apply Rle_trans with r; Auto with real. -Apply RmaxLess2; Auto. -Apply Rge_trans with p; Auto with real; Apply Rge_trans with q; Auto with real. -Apply Rge_trans with p; Auto with real. -Rewrite (Rabsolu_left p); Auto. -Case (Rle_or_lt R0 q); Intros H'2. -Repeat Rewrite Rabsolu_right; Auto with real. -Apply Rle_trans with r; Auto. -Apply RmaxLess2; Auto. -Apply Rge_trans with q; Auto with real. -Rewrite (Rabsolu_left q); Auto. -Case (Rle_or_lt R0 r); Intros H'3. -Repeat Rewrite Rabsolu_right; Auto with real. -Apply Rle_trans with (Ropp p); Auto with real. -Apply RmaxLess1; Auto. -Rewrite (Rabsolu_left r); Auto. -Apply Rle_trans with (Ropp p); Auto with real. -Apply RmaxLess1; Auto. -Qed. - -Lemma Rabsolu_Zabs: (z : Z) (Rabsolu (IZR z)) == (IZR (Zabs z)). -Intros z; Case z; Simpl; Auto with real. -Apply Rabsolu_right; Auto with real. -Intros p0; Apply Rabsolu_right; Auto with real zarith. -Intros p0; Rewrite Rabsolu_Ropp. -Apply Rabsolu_right; Auto with real zarith. -Qed. - |