diff options
Diffstat (limited to 'theories7/Reals/Rlimit.v')
| -rw-r--r-- | theories7/Reals/Rlimit.v | 539 |
1 files changed, 0 insertions, 539 deletions
diff --git a/theories7/Reals/Rlimit.v b/theories7/Reals/Rlimit.v deleted file mode 100644 index 3308b2e3..00000000 --- a/theories7/Reals/Rlimit.v +++ /dev/null @@ -1,539 +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: Rlimit.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) - -(*********************************************************) -(* Definition of the limit *) -(* *) -(*********************************************************) - -Require Rbase. -Require Rfunctions. -Require Classical_Prop. -Require Fourier. -V7only [Import R_scope.]. Open Local Scope R_scope. - -(*******************************) -(* Calculus *) -(*******************************) -(*********) -Lemma eps2_Rgt_R0:(eps:R)(Rgt eps R0)-> - (Rgt (Rmult eps (Rinv (Rplus R1 R1))) R0). -Intros;Fourier. -Qed. - -(*********) -Lemma eps2:(eps:R)(Rplus (Rmult eps (Rinv (Rplus R1 R1))) - (Rmult eps (Rinv (Rplus R1 R1))))==eps. -Intro esp. -Assert H := (double_var esp). -Unfold Rdiv in H. -Symmetry; Exact H. -Qed. - -(*********) -Lemma eps4:(eps:R) - (Rplus (Rmult eps (Rinv (Rplus (Rplus R1 R1) (Rplus R1 R1) ))) - (Rmult eps (Rinv (Rplus (Rplus R1 R1) (Rplus R1 R1) ))))== - (Rmult eps (Rinv (Rplus R1 R1))). -Intro eps. -Replace ``2+2`` with ``2*2``. -Pattern 3 eps; Rewrite double_var. -Rewrite (Rmult_Rplus_distrl ``eps/2`` ``eps/2`` ``/2``). -Unfold Rdiv. -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_Rmult. -Reflexivity. -DiscrR. -DiscrR. -Ring. -Qed. - -(*********) -Lemma Rlt_eps2_eps:(eps:R)(Rgt eps R0)-> - (Rlt (Rmult eps (Rinv (Rplus R1 R1))) eps). -Intros. -Pattern 2 eps; Rewrite <- Rmult_1r. -Repeat Rewrite (Rmult_sym eps). -Apply Rlt_monotony_r. -Exact H. -Apply Rlt_monotony_contra with ``2``. -Fourier. -Rewrite Rmult_1r; Rewrite <- Rinv_r_sym. -Fourier. -DiscrR. -Qed. - -(*********) -Lemma Rlt_eps4_eps:(eps:R)(Rgt eps R0)-> - (Rlt (Rmult eps (Rinv (Rplus (Rplus R1 R1) (Rplus R1 R1)))) eps). -Intros. -Replace ``2+2`` with ``4``. -Pattern 2 eps; Rewrite <- Rmult_1r. -Repeat Rewrite (Rmult_sym eps). -Apply Rlt_monotony_r. -Exact H. -Apply Rlt_monotony_contra with ``4``. -Replace ``4`` with ``2*2``. -Apply Rmult_lt_pos; Fourier. -Ring. -Rewrite Rmult_1r; Rewrite <- Rinv_r_sym. -Fourier. -DiscrR. -Ring. -Qed. - -(*********) -Lemma prop_eps:(r:R)((eps:R)(Rgt eps R0)->(Rlt r eps))->(Rle r R0). -Intros;Elim (total_order r R0); Intro. -Apply Rlt_le; Assumption. -Elim H0; Intro. -Apply eq_Rle; Assumption. -Clear H0;Generalize (H r H1); Intro;Generalize (Rlt_antirefl r); - Intro;ElimType False; Auto. -Qed. - -(*********) -Definition mul_factor := [l,l':R](Rinv (Rplus R1 (Rplus (Rabsolu l) - (Rabsolu l')))). - -(*********) -Lemma mul_factor_wd : (l,l':R) - ~(Rplus R1 (Rplus (Rabsolu l) (Rabsolu l')))==R0. -Intros;Rewrite (Rplus_sym R1 (Rplus (Rabsolu l) (Rabsolu l'))); - Apply tech_Rplus. -Cut (Rle (Rabsolu (Rplus l l')) (Rplus (Rabsolu l) (Rabsolu l'))). -Cut (Rle R0 (Rabsolu (Rplus l l'))). -Exact (Rle_trans ? ? ?). -Exact (Rabsolu_pos (Rplus l l')). -Exact (Rabsolu_triang ? ?). -Exact Rlt_R0_R1. -Qed. - -(*********) -Lemma mul_factor_gt:(eps:R)(l,l':R)(Rgt eps R0)-> - (Rgt (Rmult eps (mul_factor l l')) R0). -Intros;Unfold Rgt;Rewrite <- (Rmult_Or eps);Apply Rlt_monotony. -Assumption. -Unfold mul_factor;Apply Rlt_Rinv; - Cut (Rle R1 (Rplus R1 (Rplus (Rabsolu l) (Rabsolu l')))). -Cut (Rlt R0 R1). -Exact (Rlt_le_trans ? ? ?). -Exact Rlt_R0_R1. -Replace (Rle R1 (Rplus R1 (Rplus (Rabsolu l) (Rabsolu l')))) - with (Rle (Rplus R1 R0) (Rplus R1 (Rplus (Rabsolu l) (Rabsolu l')))). -Apply Rle_compatibility. -Cut (Rle (Rabsolu (Rplus l l')) (Rplus (Rabsolu l) (Rabsolu l'))). -Cut (Rle R0 (Rabsolu (Rplus l l'))). -Exact (Rle_trans ? ? ?). -Exact (Rabsolu_pos ?). -Exact (Rabsolu_triang ? ?). -Rewrite (proj1 ? ? (Rplus_ne R1));Trivial. -Qed. - -(*********) -Lemma mul_factor_gt_f:(eps:R)(l,l':R)(Rgt eps R0)-> - (Rgt (Rmin R1 (Rmult eps (mul_factor l l'))) R0). -Intros;Apply Rmin_Rgt_r;Split. -Exact Rlt_R0_R1. -Exact (mul_factor_gt eps l l' H). -Qed. - - -(*******************************) -(* Metric space *) -(*******************************) - -(*********) -Record Metric_Space:Type:= { - Base:Type; - dist:Base->Base->R; - dist_pos:(x,y:Base)(Rge (dist x y) R0); - dist_sym:(x,y:Base)(dist x y)==(dist y x); - dist_refl:(x,y:Base)((dist x y)==R0<->x==y); - dist_tri:(x,y,z:Base)(Rle (dist x y) - (Rplus (dist x z) (dist z y))) }. - -(*******************************) -(* Limit in Metric space *) -(*******************************) - -(*********) -Definition limit_in:= - [X:Metric_Space; X':Metric_Space; f:(Base X)->(Base X'); - D:(Base X)->Prop; x0:(Base X); l:(Base X')] - (eps:R)(Rgt eps R0)-> - (EXT alp:R | (Rgt alp R0)/\(x:(Base X))(D x)/\ - (Rlt (dist X x x0) alp)-> - (Rlt (dist X' (f x) l) eps)). - -(*******************************) -(* R is a metric space *) -(*******************************) - -(*********) -Definition R_met:Metric_Space:=(Build_Metric_Space R R_dist - R_dist_pos R_dist_sym R_dist_refl R_dist_tri). - -(*******************************) -(* Limit 1 arg *) -(*******************************) -(*********) -Definition Dgf:=[Df,Dg:R->Prop][f:R->R][x:R](Df x)/\(Dg (f x)). - -(*********) -Definition limit1_in:(R->R)->(R->Prop)->R->R->Prop:= - [f:R->R; D:R->Prop; l:R; x0:R](limit_in R_met R_met f D x0 l). - -(*********) -Lemma tech_limit:(f:R->R)(D:R->Prop)(l:R)(x0:R)(D x0)-> - (limit1_in f D l x0)->l==(f x0). -Intros f D l x0 H H0. -Case (Rabsolu_pos (Rminus (f x0) l)); Intros H1. -Absurd (Rlt (dist R_met (f x0) l) (dist R_met (f x0) l)). -Apply Rlt_antirefl. -Case (H0 (dist R_met (f x0) l)); Auto. -Intros alpha1 (H2, H3); Apply H3; Auto; Split; Auto. -Case (dist_refl R_met x0 x0); Intros Hr1 Hr2; Rewrite Hr2; Auto. -Case (dist_refl R_met (f x0) l); Intros Hr1 Hr2; Apply sym_eqT; Auto. -Qed. - -(*********) -Lemma tech_limit_contr:(f:R->R)(D:R->Prop)(l:R)(x0:R)(D x0)->~l==(f x0) - ->~(limit1_in f D l x0). -Intros;Generalize (tech_limit f D l x0);Tauto. -Qed. - -(*********) -Lemma lim_x:(D:R->Prop)(x0:R)(limit1_in [x:R]x D x0 x0). -Unfold limit1_in; Unfold limit_in; Simpl; Intros;Split with eps; - Split; Auto;Intros;Elim H0; Intros; Auto. -Qed. - -(*********) -Lemma limit_plus:(f,g:R->R)(D:R->Prop)(l,l':R)(x0:R) - (limit1_in f D l x0)->(limit1_in g D l' x0)-> - (limit1_in [x:R](Rplus (f x) (g x)) D (Rplus l l') x0). -Intros;Unfold limit1_in; Unfold limit_in; Simpl; Intros; - Elim (H (Rmult eps (Rinv (Rplus R1 R1))) (eps2_Rgt_R0 eps H1)); - Elim (H0 (Rmult eps (Rinv (Rplus R1 R1))) (eps2_Rgt_R0 eps H1)); - Simpl;Clear H H0; Intros; Elim H; Elim H0; Clear H H0; Intros; - Split with (Rmin x1 x); Split. -Exact (Rmin_Rgt_r x1 x R0 (conj ? ? H H2)). -Intros;Elim H4; Clear H4; Intros; - Cut (Rlt (Rplus (R_dist (f x2) l) (R_dist (g x2) l')) eps). - Cut (Rle (R_dist (Rplus (f x2) (g x2)) (Rplus l l')) - (Rplus (R_dist (f x2) l) (R_dist (g x2) l'))). -Exact (Rle_lt_trans ? ? ?). -Exact (R_dist_plus ? ? ? ?). -Elim (Rmin_Rgt_l x1 x (R_dist x2 x0) H5); Clear H5; Intros. -Generalize (H3 x2 (conj (D x2) (Rlt (R_dist x2 x0) x) H4 H6)); - Generalize (H0 x2 (conj (D x2) (Rlt (R_dist x2 x0) x1) H4 H5)); - Intros; - Replace eps - with (Rplus (Rmult eps (Rinv (Rplus R1 R1))) - (Rmult eps (Rinv (Rplus R1 R1)))). -Exact (Rplus_lt ? ? ? ? H7 H8). -Exact (eps2 eps). -Qed. - -(*********) -Lemma limit_Ropp:(f:R->R)(D:R->Prop)(l:R)(x0:R) - (limit1_in f D l x0)->(limit1_in [x:R](Ropp (f x)) D (Ropp l) x0). -Unfold limit1_in;Unfold limit_in;Simpl;Intros;Elim (H eps H0);Clear H; - Intros;Elim H;Clear H;Intros;Split with x;Split;Auto;Intros; - Generalize (H1 x1 H2);Clear H1;Intro;Unfold R_dist;Unfold Rminus; - Rewrite (Ropp_Ropp l);Rewrite (Rplus_sym (Ropp (f x1)) l); - Fold (Rminus l (f x1));Fold (R_dist l (f x1));Rewrite R_dist_sym; - Assumption. -Qed. - -(*********) -Lemma limit_minus:(f,g:R->R)(D:R->Prop)(l,l':R)(x0:R) - (limit1_in f D l x0)->(limit1_in g D l' x0)-> - (limit1_in [x:R](Rminus (f x) (g x)) D (Rminus l l') x0). -Intros;Unfold Rminus;Generalize (limit_Ropp g D l' x0 H0);Intro; - Exact (limit_plus f [x:R](Ropp (g x)) D l (Ropp l') x0 H H1). -Qed. - -(*********) -Lemma limit_free:(f:R->R)(D:R->Prop)(x:R)(x0:R) - (limit1_in [h:R](f x) D (f x) x0). -Unfold limit1_in;Unfold limit_in;Simpl;Intros;Split with eps;Split; - Auto;Intros;Elim (R_dist_refl (f x) (f x));Intros a b; - Rewrite (b (refl_eqT R (f x)));Unfold Rgt in H;Assumption. -Qed. - -(*********) -Lemma limit_mul:(f,g:R->R)(D:R->Prop)(l,l':R)(x0:R) - (limit1_in f D l x0)->(limit1_in g D l' x0)-> - (limit1_in [x:R](Rmult (f x) (g x)) D (Rmult l l') x0). -Intros;Unfold limit1_in; Unfold limit_in; Simpl; Intros; - Elim (H (Rmin R1 (Rmult eps (mul_factor l l'))) - (mul_factor_gt_f eps l l' H1)); - Elim (H0 (Rmult eps (mul_factor l l')) (mul_factor_gt eps l l' H1)); - Clear H H0; Simpl; Intros; Elim H; Elim H0; Clear H H0; Intros; - Split with (Rmin x1 x); Split. -Exact (Rmin_Rgt_r x1 x R0 (conj ? ? H H2)). -Intros; Elim H4; Clear H4; Intros;Unfold R_dist; - Replace (Rminus (Rmult (f x2) (g x2)) (Rmult l l')) with - (Rplus (Rmult (f x2) (Rminus (g x2) l')) (Rmult l' (Rminus (f x2) l))). -Cut (Rlt (Rplus (Rabsolu (Rmult (f x2) (Rminus (g x2) l'))) (Rabsolu (Rmult l' - (Rminus (f x2) l)))) eps). -Cut (Rle (Rabsolu (Rplus (Rmult (f x2) (Rminus (g x2) l')) (Rmult l' (Rminus - (f x2) l)))) (Rplus (Rabsolu (Rmult (f x2) (Rminus (g x2) l'))) (Rabsolu - (Rmult l' (Rminus (f x2) l))))). -Exact (Rle_lt_trans ? ? ?). -Exact (Rabsolu_triang ? ?). -Rewrite (Rabsolu_mult (f x2) (Rminus (g x2) l')); - Rewrite (Rabsolu_mult l' (Rminus (f x2) l)); - Cut (Rle (Rplus (Rmult (Rplus R1 (Rabsolu l)) (Rmult eps (mul_factor l l'))) - (Rmult (Rabsolu l') (Rmult eps (mul_factor l l')))) eps). -Cut (Rlt (Rplus (Rmult (Rabsolu (f x2)) (Rabsolu (Rminus (g x2) l'))) (Rmult - (Rabsolu l') (Rabsolu (Rminus (f x2) l)))) (Rplus (Rmult (Rplus R1 (Rabsolu - l)) (Rmult eps (mul_factor l l'))) (Rmult (Rabsolu l') (Rmult eps - (mul_factor l l'))))). -Exact (Rlt_le_trans ? ? ?). -Elim (Rmin_Rgt_l x1 x (R_dist x2 x0) H5); Clear H5; Intros; - Generalize (H0 x2 (conj (D x2) (Rlt (R_dist x2 x0) x1) H4 H5));Intro; - Generalize (Rmin_Rgt_l ? ? ? H7);Intro;Elim H8;Intros;Clear H0 H8; - Apply Rplus_lt_le_lt. -Apply Rmult_lt_0. -Apply Rle_sym1. -Exact (Rabsolu_pos (Rminus (g x2) l')). -Rewrite (Rplus_sym R1 (Rabsolu l));Unfold Rgt;Apply Rlt_r_plus_R1; - Exact (Rabsolu_pos l). -Unfold R_dist in H9; - Apply (Rlt_anti_compatibility (Ropp (Rabsolu l)) (Rabsolu (f x2)) - (Rplus R1 (Rabsolu l))). -Rewrite <- (Rplus_assoc (Ropp (Rabsolu l)) R1 (Rabsolu l)); - Rewrite (Rplus_sym (Ropp (Rabsolu l)) R1); - Rewrite (Rplus_assoc R1 (Ropp (Rabsolu l)) (Rabsolu l)); - Rewrite (Rplus_Ropp_l (Rabsolu l)); - Rewrite (proj1 ? ? (Rplus_ne R1)); - Rewrite (Rplus_sym (Ropp (Rabsolu l)) (Rabsolu (f x2))); - Generalize H9; -Cut (Rle (Rminus (Rabsolu (f x2)) (Rabsolu l)) (Rabsolu (Rminus (f x2) l))). -Exact (Rle_lt_trans ? ? ?). -Exact (Rabsolu_triang_inv ? ?). -Generalize (H3 x2 (conj (D x2) (Rlt (R_dist x2 x0) x) H4 H6));Trivial. -Apply Rle_monotony. -Exact (Rabsolu_pos l'). -Unfold Rle;Left;Assumption. -Rewrite (Rmult_sym (Rplus R1 (Rabsolu l)) (Rmult eps (mul_factor l l'))); - Rewrite (Rmult_sym (Rabsolu l') (Rmult eps (mul_factor l l'))); - Rewrite <- (Rmult_Rplus_distr - (Rmult eps (mul_factor l l')) - (Rplus R1 (Rabsolu l)) - (Rabsolu l')); - Rewrite (Rmult_assoc eps (mul_factor l l') (Rplus (Rplus R1 (Rabsolu l)) - (Rabsolu l'))); - Rewrite (Rplus_assoc R1 (Rabsolu l) (Rabsolu l'));Unfold mul_factor; - Rewrite (Rinv_l (Rplus R1 (Rplus (Rabsolu l) (Rabsolu l'))) - (mul_factor_wd l l')); - Rewrite (proj1 ? ? (Rmult_ne eps));Apply eq_Rle;Trivial. -Ring. -Qed. - -(*********) -Definition adhDa:(R->Prop)->R->Prop:=[D:R->Prop][a:R] - (alp:R)(Rgt alp R0)->(EXT x:R | (D x)/\(Rlt (R_dist x a) alp)). - -(*********) -Lemma single_limit:(f:R->R)(D:R->Prop)(l:R)(l':R)(x0:R) - (adhDa D x0)->(limit1_in f D l x0)->(limit1_in f D l' x0)->l==l'. -Unfold limit1_in; Unfold limit_in; Intros. -Cut (eps:R)(Rgt eps R0)->(Rlt (dist R_met l l') - (Rmult (Rplus R1 R1) eps)). -Clear H0 H1;Unfold dist; Unfold R_met; Unfold R_dist; - Unfold Rabsolu;Case (case_Rabsolu (Rminus l l')); Intros. -Cut (eps:R)(Rgt eps R0)->(Rlt (Ropp (Rminus l l')) eps). -Intro;Generalize (prop_eps (Ropp (Rminus l l')) H1);Intro; - Generalize (Rlt_RoppO (Rminus l l') r); Intro;Unfold Rgt in H3; - Generalize (Rle_not (Ropp (Rminus l l')) R0 H3); Intro; - ElimType False; Auto. -Intros;Cut (Rgt (Rmult eps (Rinv (Rplus R1 R1))) R0). -Intro;Generalize (H0 (Rmult eps (Rinv (Rplus R1 R1))) H2); - Rewrite (Rmult_sym eps (Rinv (Rplus R1 R1))); - Rewrite <- (Rmult_assoc (Rplus R1 R1) (Rinv (Rplus R1 R1)) eps); - Rewrite (Rinv_r (Rplus R1 R1)). -Elim (Rmult_ne eps);Intros a b;Rewrite b;Clear a b;Trivial. -Apply (imp_not_Req (Rplus R1 R1) R0);Right;Generalize Rlt_R0_R1;Intro; - Unfold Rgt;Generalize (Rlt_compatibility R1 R0 R1 H3);Intro; - Elim (Rplus_ne R1);Intros a b;Rewrite a in H4;Clear a b; - Apply (Rlt_trans R0 R1 (Rplus R1 R1) H3 H4). -Unfold Rgt;Unfold Rgt in H1; - Rewrite (Rmult_sym eps(Rinv (Rplus R1 R1))); - Rewrite <-(Rmult_Or (Rinv (Rplus R1 R1))); - Apply (Rlt_monotony (Rinv (Rplus R1 R1)) R0 eps);Auto. -Apply (Rlt_Rinv (Rplus R1 R1));Cut (Rlt R1 (Rplus R1 R1)). -Intro;Apply (Rlt_trans R0 R1 (Rplus R1 R1) Rlt_R0_R1 H2). -Generalize (Rlt_compatibility R1 R0 R1 Rlt_R0_R1);Elim (Rplus_ne R1); - Intros a b;Rewrite a;Clear a b;Trivial. -(**) -Cut (eps:R)(Rgt eps R0)->(Rlt (Rminus l l') eps). -Intro;Generalize (prop_eps (Rminus l l') H1);Intro; - Elim (Rle_le_eq (Rminus l l') R0);Intros a b;Clear b; - Apply (Rminus_eq l l');Apply a;Split. -Assumption. -Apply (Rle_sym2 R0 (Rminus l l') r). -Intros;Cut (Rgt (Rmult eps (Rinv (Rplus R1 R1))) R0). -Intro;Generalize (H0 (Rmult eps (Rinv (Rplus R1 R1))) H2); - Rewrite (Rmult_sym eps (Rinv (Rplus R1 R1))); - Rewrite <- (Rmult_assoc (Rplus R1 R1) (Rinv (Rplus R1 R1)) eps); - Rewrite (Rinv_r (Rplus R1 R1)). -Elim (Rmult_ne eps);Intros a b;Rewrite b;Clear a b;Trivial. -Apply (imp_not_Req (Rplus R1 R1) R0);Right;Generalize Rlt_R0_R1;Intro; - Unfold Rgt;Generalize (Rlt_compatibility R1 R0 R1 H3);Intro; - Elim (Rplus_ne R1);Intros a b;Rewrite a in H4;Clear a b; - Apply (Rlt_trans R0 R1 (Rplus R1 R1) H3 H4). -Unfold Rgt;Unfold Rgt in H1; - Rewrite (Rmult_sym eps(Rinv (Rplus R1 R1))); - Rewrite <-(Rmult_Or (Rinv (Rplus R1 R1))); - Apply (Rlt_monotony (Rinv (Rplus R1 R1)) R0 eps);Auto. -Apply (Rlt_Rinv (Rplus R1 R1));Cut (Rlt R1 (Rplus R1 R1)). -Intro;Apply (Rlt_trans R0 R1 (Rplus R1 R1) Rlt_R0_R1 H2). -Generalize (Rlt_compatibility R1 R0 R1 Rlt_R0_R1);Elim (Rplus_ne R1); - Intros a b;Rewrite a;Clear a b;Trivial. -(**) -Intros;Unfold adhDa in H;Elim (H0 eps H2);Intros;Elim (H1 eps H2); - Intros;Clear H0 H1;Elim H3;Elim H4;Clear H3 H4;Intros; - Simpl;Simpl in H1 H4;Generalize (Rmin_Rgt x x1 R0);Intro;Elim H5; - Intros;Clear H5; - Elim (H (Rmin x x1) (H7 (conj (Rgt x R0) (Rgt x1 R0) H3 H0))); - Intros; Elim H5;Intros;Clear H5 H H6 H7; - Generalize (Rmin_Rgt x x1 (R_dist x2 x0));Intro;Elim H; - Intros;Clear H H6;Unfold Rgt in H5;Elim (H5 H9);Intros;Clear H5 H9; - Generalize (H1 x2 (conj (D x2) (Rlt (R_dist x2 x0) x1) H8 H6)); - Generalize (H4 x2 (conj (D x2) (Rlt (R_dist x2 x0) x) H8 H)); - Clear H8 H H6 H1 H4 H0 H3;Intros; - Generalize (Rplus_lt (R_dist (f x2) l) eps (R_dist (f x2) l') eps - H H0); Unfold R_dist;Intros; - Rewrite (Rabsolu_minus_sym (f x2) l) in H1; - Rewrite (Rmult_sym (Rplus R1 R1) eps);Rewrite (Rmult_Rplus_distr eps R1 R1); - Elim (Rmult_ne eps);Intros a b;Rewrite a;Clear a b; - Generalize (R_dist_tri l l' (f x2));Unfold R_dist;Intros; - Apply (Rle_lt_trans (Rabsolu (Rminus l l')) - (Rplus (Rabsolu (Rminus l (f x2))) (Rabsolu (Rminus (f x2) l'))) - (Rplus eps eps) H3 H1). -Qed. - -(*********) -Lemma limit_comp:(f,g:R->R)(Df,Dg:R->Prop)(l,l':R)(x0:R) - (limit1_in f Df l x0)->(limit1_in g Dg l' l)-> - (limit1_in [x:R](g (f x)) (Dgf Df Dg f) l' x0). -Unfold limit1_in limit_in Dgf;Simpl. -Intros f g Df Dg l l' x0 Hf Hg eps eps_pos. -Elim (Hg eps eps_pos). -Intros alpg lg. -Elim (Hf alpg). -2: Tauto. -Intros alpf lf. -Exists alpf. -Intuition. -Qed. - -(*********) - -Lemma limit_inv : (f:R->R)(D:R->Prop)(l:R)(x0:R) (limit1_in f D l x0)->~(l==R0)->(limit1_in [x:R](Rinv (f x)) D (Rinv l) x0). -Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Elim (H ``(Rabsolu l)/2``). -Intros delta1 H2; Elim (H ``eps*((Rsqr l)/2)``). -Intros delta2 H3; Elim H2; Elim H3; Intros; Exists (Rmin delta1 delta2); Split. -Unfold Rmin; Case (total_order_Rle delta1 delta2); Intro; Assumption. -Intro; Generalize (H5 x); Clear H5; Intro H5; Generalize (H7 x); Clear H7; Intro H7; Intro H10; Elim H10; Intros; Cut (D x)/\``(Rabsolu (x-x0))<delta1``. -Cut (D x)/\``(Rabsolu (x-x0))<delta2``. -Intros; Generalize (H5 H11); Clear H5; Intro H5; Generalize (H7 H12); Clear H7; Intro H7; Generalize (Rabsolu_triang_inv l (f x)); Intro; Rewrite Rabsolu_minus_sym in H7; Generalize (Rle_lt_trans ``(Rabsolu l)-(Rabsolu (f x))`` ``(Rabsolu (l-(f x)))`` ``(Rabsolu l)/2`` H13 H7); Intro; Generalize (Rlt_compatibility ``(Rabsolu (f x))-(Rabsolu l)/2`` ``(Rabsolu l)-(Rabsolu (f x))`` ``(Rabsolu l)/2`` H14); Replace ``(Rabsolu (f x))-(Rabsolu l)/2+((Rabsolu l)-(Rabsolu (f x)))`` with ``(Rabsolu l)/2``. -Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Intro; Cut ~``(f x)==0``. -Intro; Replace ``/(f x)+ -/l`` with ``(l-(f x))*/(l*(f x))``. -Rewrite Rabsolu_mult; Rewrite Rabsolu_Rinv. -Cut ``/(Rabsolu (l*(f x)))<2/(Rsqr l)``. -Intro; Rewrite Rabsolu_minus_sym in H5; Cut ``0<=/(Rabsolu (l*(f x)))``. -Intro; Generalize (Rmult_lt2 ``(Rabsolu (l-(f x)))`` ``eps*(Rsqr l)/2`` ``/(Rabsolu (l*(f x)))`` ``2/(Rsqr l)`` (Rabsolu_pos ``l-(f x)``) H18 H5 H17); Replace ``eps*(Rsqr l)/2*2/(Rsqr l)`` with ``eps``. -Intro; Assumption. -Unfold Rdiv; Unfold Rsqr; Rewrite Rinv_Rmult. -Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym l). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Rewrite (Rmult_sym l). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Reflexivity. -DiscrR. -Exact H0. -Exact H0. -Exact H0. -Exact H0. -Left; Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Apply prod_neq_R0; Assumption. -Rewrite Rmult_sym; Rewrite Rabsolu_mult; Rewrite Rinv_Rmult. -Rewrite (Rsqr_abs l); Unfold Rsqr; Unfold Rdiv; Rewrite Rinv_Rmult. -Repeat Rewrite <- Rmult_assoc; Apply Rlt_monotony_r. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. -Apply Rlt_monotony_contra with ``(Rabsolu (f x))*(Rabsolu l)*/2``. -Repeat Apply Rmult_lt_pos. -Apply Rabsolu_pos_lt; Assumption. -Apply Rabsolu_pos_lt; Assumption. -Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H17; Generalize (lt_INR_0 (2) (neq_O_lt (2) H17)); Unfold INR; Intro H18; Assumption | Discriminate]. -Replace ``(Rabsolu (f x))*(Rabsolu l)*/2*/(Rabsolu (f x))`` with ``(Rabsolu l)/2``. -Replace ``(Rabsolu (f x))*(Rabsolu l)*/2*(2*/(Rabsolu l))`` with ``(Rabsolu (f x))``. -Assumption. -Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym (Rabsolu l)). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Reflexivity. -DiscrR. -Apply Rabsolu_no_R0. -Assumption. -Unfold Rdiv. -Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym (Rabsolu (f x))). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Reflexivity. -Apply Rabsolu_no_R0; Assumption. -Apply Rabsolu_no_R0; Assumption. -Apply Rabsolu_no_R0; Assumption. -Apply Rabsolu_no_R0; Assumption. -Apply Rabsolu_no_R0; Assumption. -Apply prod_neq_R0; Assumption. -Rewrite (Rinv_Rmult ? ? H0 H16). -Unfold Rminus; Rewrite Rmult_Rplus_distrl. -Rewrite <- Rmult_assoc. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l. -Rewrite Ropp_mul1. -Rewrite (Rmult_sym (f x)). -Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Reflexivity. -Assumption. -Assumption. -Red; Intro; Rewrite H16 in H15; Rewrite Rabsolu_R0 in H15; Cut ``0<(Rabsolu l)/2``. -Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``(Rabsolu l)/2`` ``0`` H17 H15)). -Unfold Rdiv; Apply Rmult_lt_pos. -Apply Rabsolu_pos_lt; Assumption. -Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H17; Generalize (lt_INR_0 (2) (neq_O_lt (2) H17)); Unfold INR; Intro; Assumption | Discriminate]. -Pattern 3 (Rabsolu l); Rewrite double_var. -Ring. -Split; [Assumption | Apply Rlt_le_trans with (Rmin delta1 delta2); [Assumption | Apply Rmin_r]]. -Split; [Assumption | Apply Rlt_le_trans with (Rmin delta1 delta2); [Assumption | Apply Rmin_l]]. -Change ``0<eps*(Rsqr l)/2``; Unfold Rdiv; Repeat Rewrite Rmult_assoc; Repeat Apply Rmult_lt_pos. -Assumption. -Apply Rsqr_pos_lt; Assumption. -Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H3; Generalize (lt_INR_0 (2) (neq_O_lt (2) H3)); Unfold INR; Intro; Assumption | Discriminate]. -Change ``0<(Rabsolu l)/2``; Unfold Rdiv; Apply Rmult_lt_pos; [Apply Rabsolu_pos_lt; Assumption | Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H3; Generalize (lt_INR_0 (2) (neq_O_lt (2) H3)); Unfold INR; Intro; Assumption | Discriminate]]. -Qed. |