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author | desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2002-06-11 17:01:24 +0000 |
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committer | desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2002-06-11 17:01:24 +0000 |
commit | 9e5b51066675777240ec2e5b35016686c0c89f41 (patch) | |
tree | f8bcec6a71207d263a295673ffdeb8136d7928e8 /theories/Reals/Ranalysis.v | |
parent | 9b9ed1245225fc95374b070f5f5ed699337448fc (diff) |
Ranalysis.v
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2774 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Reals/Ranalysis.v')
-rw-r--r-- | theories/Reals/Ranalysis.v | 959 |
1 files changed, 5 insertions, 954 deletions
diff --git a/theories/Reals/Ranalysis.v b/theories/Reals/Ranalysis.v index 15b31daea..95046d292 100644 --- a/theories/Reals/Ranalysis.v +++ b/theories/Reals/Ranalysis.v @@ -5,959 +5,10 @@ (* // * This file is distributed under the terms of the *) (* * GNU Lesser General Public License Version 2.1 *) (***********************************************************************) - -(*i $Id$ i*) - -Require Rbase. -Require Rbasic_fun. -Require R_sqr. -Require Rlimit. -Require Rderiv. -Require DiscrR. -Require Rtrigo. - -(****************************************************) -(** Basic operations on functions *) -(****************************************************) -Definition plus_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)+(f2 x)``. -Definition opp_fct [f:R->R] : R->R := [x:R] ``-(f x)``. -Definition mult_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)*(f2 x)``. -Definition mult_real_fct [a:R;f:R->R] : R->R := [x:R] ``a*(f x)``. -Definition minus_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)-(f2 x)``. -Definition div_fct [f1,f2:R->R] : R->R := [x:R] ``(f1 x)/(f2 x)``. -Definition div_real_fct [a:R;f:R->R] : R->R := [x:R] ``a/(f x)``. -Definition comp [f1,f2:R->R] : R->R := [x:R] ``(f1 (f2 x))``. - -(****************************************************) -(** Variations of functions *) -(****************************************************) -Definition increasing [f:R->R] : Prop := (x,y:R) ``x<=y``->``(f x)<=(f y)``. -Definition decreasing [f:R->R] : Prop := (x,y:R) ``x<=y``->``(f y)<=(f x)``. -Definition strict_increasing [f:R->R] : Prop := (x,y:R) ``x<y``->``(f x)<(f y)``. -Definition strict_decreasing [f:R->R] : Prop := (x,y:R) ``x<y``->``(f y)<(f x)``. -Definition constant [f:R->R] : Prop := (x,y:R) ``(f x)==(f y)``. - -(**********) -Axiom fct_eq : (A,B:Type) (f1,f2:A->B) ((x:A)(f1 x)==(f2 x))->f1==f2. - -(**********) -Definition no_cond : R->Prop := [x:R] True. - -(***************************************************) -(** Definition of continuity as a limit *) -(***************************************************) - -(**********) -Definition continuity_pt [f:R->R; x0:R] : Prop := (continue_in f no_cond x0). - -(**********) -Lemma sum_continuous : (f1,f2:R->R; x0:R) (continuity_pt f1 x0) -> (continuity_pt f2 x0) -> (continuity_pt (plus_fct f1 f2) x0). -Unfold continuity_pt plus_fct; Unfold continue_in; Intros; Apply limit_plus; Assumption. -Qed. - -(**********) -Lemma diff_continuous : (f1,f2:R->R; x0:R) (continuity_pt f1 x0) -> (continuity_pt f2 x0) -> (continuity_pt (minus_fct f1 f2) x0). -Unfold continuity_pt minus_fct; Unfold continue_in; Intros; Apply limit_minus; Assumption. -Qed. - -(**********) -Lemma prod_continuous : (f1,f2:R->R; x0:R) (continuity_pt f1 x0) -> (continuity_pt f2 x0) -> (continuity_pt (mult_fct f1 f2) x0). -Unfold continuity_pt mult_fct; Unfold continue_in; Intros; Apply limit_mul; Assumption. -Qed. - -(**********) -Lemma const_continuous : (f:R->R; x0:R) (constant f) -> (continuity_pt f x0). -Unfold constant continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Intros; Exists ``1``; Split; [Apply Rlt_R0_R1 | Intros; Generalize (H x x0); Intro; Rewrite H2; Simpl; Rewrite R_dist_eq; Assumption]. -Qed. - -(**********) -Lemma scal_continuous : (f:R->R;a:R; x0:R) (continuity_pt f x0) -> (continuity_pt (mult_real_fct a f) x0). -Unfold continuity_pt mult_real_fct; Unfold continue_in; Intros; Apply (limit_mul ([x:R] a) f (D_x no_cond x0) a (f x0) x0). -Unfold limit1_in; Unfold limit_in; Intros; Exists ``1``; Split. -Apply Rlt_R0_R1. -Intros; Rewrite R_dist_eq; Assumption. -Assumption. -Qed. - -(**********) -Lemma opp_continuous : (f:R->R; x0:R) (continuity_pt f x0) -> (continuity_pt (opp_fct f) x0). -Unfold continuity_pt opp_fct; Unfold continue_in; Intros; Apply limit_Ropp; Assumption. -Qed. - -(**********) -Lemma inv_continuous : (f:R->R; x0:R) (continuity_pt f x0) -> ~``(f x0)==0`` -> -(continuity_pt ([x:R] ``/(f x)``) x0). -Unfold continuity_pt; Unfold continue_in; Intros; Apply limit_inv; Assumption. -Qed. - -Lemma div_eq_inv : (f1,f2:R->R) (div_fct f1 f2)==(mult_fct f1 ([x:R]``/(f2 x)``)). -Intros; Unfold div_fct; Unfold mult_fct; Unfold Rdiv; Apply fct_eq; Intro x; Reflexivity. -Qed. - -(**********) -Lemma div_continuous : (f1,f2:R->R; x0:R) (continuity_pt f1 x0) -> (continuity_pt f2 x0) -> ~``(f2 x0)==0`` -> (continuity_pt (div_fct f1 f2) x0). -Intros; Rewrite -> (div_eq_inv f1 f2); Apply prod_continuous; [Assumption | Apply inv_continuous; Assumption]. -Qed. - -(**********) -Definition continuity [f:R->R] : Prop := (x:R) (continuity_pt f x). - -Lemma sum_continuity : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (plus_fct f1 f2)). -Unfold continuity; Intros; Apply (sum_continuous f1 f2 x (H x) (H0 x)). -Qed. - -Lemma diff_continuity : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (minus_fct f1 f2)). -Unfold continuity; Intros; Apply (diff_continuous f1 f2 x (H x) (H0 x)). -Qed. - -Lemma prod_continuity : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (mult_fct f1 f2)). -Unfold continuity; Intros; Apply (prod_continuous f1 f2 x (H x) (H0 x)). -Qed. - -Lemma const_continuity : (f:R->R) (constant f) -> (continuity f). -Unfold continuity; Intros; Apply (const_continuous f x H). -Qed. - -Lemma scal_continuity : (f:R->R;a:R) (continuity f) -> (continuity (mult_real_fct a f)). -Unfold continuity; Intros; Apply (scal_continuous f a x (H x)). -Qed. - -Lemma opp_continuity : (f:R->R) (continuity f)->(continuity (opp_fct f)). -Unfold continuity; Intros; Apply (opp_continuous f x (H x)). -Qed. - -Lemma div_continuity : (f1,f2:R->R) (continuity f1)->(continuity f2)->((x:R) ~``(f2 x)==0``)->(continuity (div_fct f1 f2)). -Unfold continuity; Intros; Apply (div_continuous f1 f2 x (H x) (H0 x) (H1 x)). -Qed. - -Lemma inv_continuity : (f:R->R) (continuity f)->((x:R) ~``(f x)==0``)->(continuity ([x:R] ``/(f x)``)). -Unfold continuity; Intros; Apply (inv_continuous f x (H x) (H0 x)). -Qed. - -(*****************************************************) -(** Derivative's definition using Landau's kernel *) -(*****************************************************) -Definition derivable_pt [f:R->R; x:R] : Prop := (EXT l : R | ((eps:R) ``0<eps``->(EXT delta : posreal | ((h:R) ~``h==0``->``(Rabsolu h)<delta`` -> ``(Rabsolu ((((f (x+h))-(f x))/h)-l))<eps``)))). - -Definition derivable [f:R->R] : Prop := (x:R) (derivable_pt f x). - -Parameter derive_pt : (R->R)->R->R. - -Axiom derive_pt_def : (f:R->R;x,l:R) ((eps:R) ``0<eps``->(EXT delta : posreal | ((h:R) ~``h==0``->``(Rabsolu h)<delta`` -> ``(Rabsolu ((((f (x+h))-(f x))/h)-l))<eps``))) <-> (derive_pt f x)==l. - -(**********) -Lemma derive_pt_def_0 : (f:R->R;x,l:R) ((eps:R) ``0<eps``->(EXT delta : posreal | ((h:R) ~``h==0``->``(Rabsolu h)<delta`` -> ``(Rabsolu ((((f (x+h))-(f x))/h)-l))<eps``))) -> (derive_pt f x)==l. -Intros; Elim (derive_pt_def f x l); Intros; Apply (H0 H). -Qed. - -(**********) -Lemma derive_pt_def_1 : (f:R->R;x,l:R) (derive_pt f x)==l -> ((eps:R) ``0<eps``->(EXT delta : posreal | ((h:R) ~``h==0``->``(Rabsolu h)<delta`` -> ``(Rabsolu ((((f (x+h))-(f x))/h)-l))<eps``))). -Intros; Elim (derive_pt_def f x l); Intros; Apply (H2 H eps H0). -Qed. - -(**********) -Definition derive [f:R->R] := [x:R] (derive_pt f x). - -(************************************) -(** Class of differential functions *) -(************************************) -Record Differential : Type := mkDifferential { -d1 :> R->R; -cond_diff : (derivable d1) }. - -Record Differential_D2 : Type := mkDifferential_D2 { -d2 :> R->R; -cond_D1 : (derivable d2); -cond_D2 : (derivable (derive d2)) }. - -(**********) -Lemma derivable_derive : (f:R->R;x:R) (derivable_pt f x) -> (EXT l : R | (derive_pt f x)==l). -Intros f x; Unfold derivable_pt; Intro H; Elim H; Intros l H0; Rewrite (derive_pt_def_0 f x l); [Exists l; Reflexivity | Assumption]. -Qed. - -(**********) -Lemma derive_derivable : (f:R->R;x,l:R) (derive_pt f x)==l -> (derivable_pt f x). -Intros; Unfold derivable_pt; Generalize (derive_pt_def_1 f x l H); Intro H0; Exists l; Assumption. -Qed. - -(********************************************************************) -(** Equivalence of this definition with the one using limit concept *) -(********************************************************************) -Lemma derive_pt_D_in : (f,df:R->R;x:R) (D_in f df no_cond x) <-> (derive_pt -f x)==(df x). -Intros; Split. -Unfold D_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. -Apply derive_pt_def_0. -Intros; Elim (H eps H0); Intros alpha H1; Elim H1; Intros; Exists (mkposreal alpha H2); Intros; Generalize (H3 ``x+h``); Intro; Cut ``x+h-x==h``; [Intro; Cut ``(D_x no_cond x (x+h))``/\``(Rabsolu (x+h-x)) < alpha``; [Intro; Generalize (H6 H8); Rewrite H7; Intro; Assumption | Split; [Unfold D_x; Split; [Unfold no_cond; Trivial | Apply Rminus_not_eq_right; Rewrite H7; Assumption] | Rewrite H7; Assumption]] | Ring]. -Intro; Generalize (derive_pt_def_1 f x (df x) H); Intro; Unfold D_in; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros; Elim (H0 eps H1); Intros alpha H2; Exists (pos alpha); Split. -Apply (cond_pos alpha). -Intros; Elim H3; Intros; Unfold D_x in H4; Elim H4; Intros; Cut ``x0-x<>0``. -Intro; Generalize (H2 ``x0-x`` H8 H5); Replace ``x+(x0-x)`` with x0. -Intro; Assumption. -Ring. -Auto with real. -Qed. - -Definition fct_cte [a:R] : R->R := [x:R]a. - -(***********************************) -(** derivability -> continuity *) -(***********************************) -Theorem derivable_continuous_pt : (f:R->R;x:R) (derivable_pt f x) -> (continuity_pt f x). -Intros. -Generalize (derivable_derive f x H); Intro. -Elim H0; Intros l H1. -Cut l==((fct_cte l) x). -Intro. -Rewrite H2 in H1. -Generalize (derive_pt_D_in f (fct_cte l) x); Intro. -Elim H3; Intros. -Generalize (H5 H1); Intro. -Unfold continuity_pt. -Apply (cont_deriv f (fct_cte l) no_cond x H6). -Unfold fct_cte; Reflexivity. -Qed. -Theorem derivable_continuous : (f:R->R) (derivable f) -> (continuity f). -Unfold derivable continuity; Intros; Apply (derivable_continuous_pt f x (H x)). -Qed. - -(****************************************************************) -(** Main rules *) -(****************************************************************) - -(* Addition *) -Lemma deriv_sum : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> ``(derive_pt (plus_fct f1 f2) x)==(derive_pt f1 x)+(derive_pt f2 x)``. -Intros; Generalize (derivable_derive f1 x H); Intro H1; Generalize (derivable_derive f2 x H0); Intro H2; Elim H1; Clear H1; Intros l1 H1; Elim H2; Clear H2; Intros l2 H2; Unfold plus_fct; Rewrite H1; Rewrite H2; Apply derive_pt_def_0; Intros; Generalize (derive_pt_def_1 f1 x l1 H1); Clear H1; Intro H1; Generalize (derive_pt_def_1 f2 x l2 H2); Clear H2; Intro H2; Cut ~(O=(2)). -Intro Haux; Generalize (lt_INR_0 (2) (neq_O_lt (2) Haux)); Rewrite INR_eq_INR2; Unfold INR2; Intro Haux1; Generalize (Rlt_Rinv ``2`` Haux1); Clear Haux1; Intro Haux1; Generalize (Rmult_lt_pos eps ``/2`` H3 Haux1); Clear Haux1; Intro Haux1; Elim (H1 ``eps/2`` Haux1); Intros delta1 H4; Elim (H2 ``eps/2`` Haux1); Intros delta2 H5; Exists (mkposreal (Rmin delta1 delta2) (Rmin_stable_in_posreal delta1 delta2)); Intros h H6 H7; Unfold plus_fct; Replace ``((f1 (x+h))+(f2 (x+h))-((f1 x)+(f2 x)))/h-(l1+l2)`` with ``(((f1 (x+h))-(f1 x))/h-l1)+(((f2 (x+h))-(f2 x))/h-l2)``. -Apply Rle_lt_trans with ``(Rabsolu ((f1 (x+h))-(f1 x))/h-l1)+(Rabsolu ((f2 (x+h))-(f2 x))/h-l2)``. -Apply Rabsolu_triang. -Generalize (H5 h H6 (Rlt_le_trans (Rabsolu h) (Rmin delta1 delta2) delta2 H7 (Rmin_r delta1 delta2))); Intro H8; Generalize (H4 h H6 (Rlt_le_trans (Rabsolu h) (Rmin delta1 delta2) delta1 H7 (Rmin_l delta1 delta2))); Intro H9. -Generalize (Rplus_lt ``(Rabsolu (((f1 (x+h))-(f1 x))/h-l1))`` ``eps/2`` ``(Rabsolu (((f2 (x+h))-(f2 x))/h-l2))`` ``eps/2`` H9 H8). -Replace ``eps/2+eps/2`` with ``eps``. -Intro H10; Assumption. -Apply double_var. -Unfold Rdiv. -Repeat Rewrite <- (Rmult_sym ``/h``). -Repeat Rewrite Rminus_distr. -Repeat Rewrite Rmult_Rplus_distr. -Unfold Rminus. -Repeat Rewrite Ropp_distr1. -Ring. -Discriminate. -Qed. - -Lemma sum_derivable_pt : (f1,f2:R->R;x:R) (derivable_pt f1 x)->(derivable_pt f2 x)->(derivable_pt (plus_fct f1 f2) x). -Intros; Generalize (derivable_derive f1 x H); Intro; Generalize (derivable_derive f2 x H0); Intro; Elim H1; Clear H1; Intros l1 H1; Elim H2; Clear H2; Intros l2 H2; Apply (derive_derivable (plus_fct f1 f2) x ``l1+l2``); Rewrite <- H1; Rewrite <- H2; Apply deriv_sum; Assumption. -Qed. - -Lemma sum_derivable : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (plus_fct f1 f2)). -Unfold derivable; Intros f1 f2 H1 H2 x; Apply sum_derivable_pt; [Exact (H1 x) | Exact (H2 x)]. -Qed. - -Lemma sum_derivable_pt_var : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> (derivable_pt ([y:R]``(f1 y)+(f2 y)``) x). -Intros; Generalize (sum_derivable_pt f1 f2 x H H0); Unfold plus_fct; Intro; Assumption. -Qed. - -Lemma derive_sum : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> (derive_pt ([y:R]``(f1 y)+(f2 y)``) x)==``(derive_pt f1 x)+(derive_pt f2 x)``. -Intros; Generalize (deriv_sum f1 f2 x H H0); Unfold plus_fct; Intro; Assumption. -Qed. - -(* Opposite *) -Lemma deriv_opposite : (f:R->R;x:R) (derivable_pt f x) -> ``(derive_pt (opp_fct f) x)==-(derive_pt f x)``. -Intros; Generalize (derivable_derive f x H); Intro H0; Elim H0; Intros l H1; Rewrite H1; Unfold opp_fct; Apply derive_pt_def_0; Intros; Generalize (derive_pt_def_1 f x l H1); Intro H3; Elim (H3 eps H2); Intros delta H4; Exists delta; Intros; Replace ``( -(f (x+h))- -(f x))/h- -l`` with ``- (((f (x+h))-(f x))/h-l)``. -Rewrite Rabsolu_Ropp; Apply (H4 h H5 H6). -Unfold Rminus Rdiv; Rewrite Ropp_distr1; Repeat Rewrite Ropp_Ropp; Rewrite <- Ropp_mul1; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Reflexivity. -Qed. - -Lemma opposite_derivable_pt : (f:R->R;x:R) (derivable_pt f x) -> (derivable_pt (opp_fct f) x). -Unfold opp_fct derivable_pt; Intros; Elim H; Intros; Exists ``-x0``; Intros; Elim (H0 eps H1); Intros; Exists x1; Intros; Generalize (H2 h H3 H4); Intro H5; Replace ``( -(f (x+h))- -(f x))/h- -x0`` with ``- (((f (x+h))-(f x))/h-x0)``. -Rewrite Rabsolu_Ropp; Assumption. -Unfold Rminus Rdiv; Rewrite Ropp_distr1; Repeat Rewrite Ropp_Ropp; Rewrite <- Ropp_mul1; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Reflexivity. -Qed. - -Lemma opposite_derivable : (f:R->R) (derivable f) -> (derivable (opp_fct f)). -Unfold derivable; Intros f H1 x; Apply opposite_derivable_pt; Exact (H1 x). -Qed. - -(* Difference *) -Lemma diff_plus_opp : (f1,f2:R->R) (minus_fct f1 f2)==(plus_fct f1 (opp_fct f2)). -Intros; Unfold minus_fct plus_fct opp_fct; Apply fct_eq; Intro x; Ring. -Qed. - -Lemma deriv_diff : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> ``(derive_pt (minus_fct f1 f2) x)==(derive_pt f1 x)-(derive_pt f2 x)``. -Intros; Rewrite diff_plus_opp; Unfold Rminus; Rewrite <- (deriv_opposite f2 x H0); Apply deriv_sum; [Assumption | Apply opposite_derivable_pt; Assumption]. -Qed. - -Lemma diff_derivable_pt : (f1,f2:R->R;x:R) (derivable_pt f1 x)->(derivable_pt f2 x)->(derivable_pt (minus_fct f1 f2) x). -Intros; Rewrite (diff_plus_opp f1 f2); Apply sum_derivable_pt; [Assumption | Apply opposite_derivable_pt; Assumption]. -Qed. - -Lemma diff_derivable : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (minus_fct f1 f2)). -Unfold derivable; Intros f1 f2 H1 H2 x; Apply diff_derivable_pt; [ Exact (H1 x) | Exact (H2 x)]. -Qed. - -Lemma derive_diff : (f1,f2:R->R;x:R) (derivable_pt f1 x) --> (derivable_pt f2 x) -> (derive_pt ([y:R]``(f1 y)-(f2 y)``) x)==``(derive_pt f1 x)-(derive_pt f2 x)``. -Intros; Generalize (deriv_diff f1 f2 x H H0); Unfold minus_fct; Intro; Assumption. -Qed. - -(**********) -Lemma deriv_scal : (f:R->R;a,x:R) (derivable_pt f x) -> ``(derive_pt (mult_real_fct a f) x)==a*(derive_pt f x)``. -Intros f a x Ha; Unfold mult_real_fct; Generalize (derivable_derive f x Ha); Intro Hb; Elim Hb; Intros l Hc; Rewrite Hc; Apply derive_pt_def_0; Generalize (Req_EM a R0); Intro H0; Elim H0; Intro H1. -Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Rewrite H1; Repeat Rewrite Rmult_Ol; Repeat Rewrite minus_R0; Unfold Rdiv; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Assumption. -Intros; Generalize (derive_pt_def_1 f x l Hc); Intro H2; Elim (H2 ``eps/(Rabsolu a)``). -Intros; Exists x0; Intros; Replace ``(a*(f (x+h))-a*(f x))/h-a*l`` with ``a*(((f (x+h))-(f x))/h-l)``. -Rewrite Rabsolu_mult; Replace ``eps`` with ``(Rabsolu a)*(eps/(Rabsolu a))``. -Apply Rlt_monotony. -Apply (Rabsolu_pos_lt a H1). -Apply (H3 h H4 H5). -Rewrite <- Rmult_sym; Unfold Rdiv; Rewrite Rmult_assoc; Rewrite <- (Rinv_l_sym (Rabsolu a)); [Apply Rmult_1r | Apply (Rabsolu_no_R0 a H1)]. -Rewrite Rminus_distr. -Unfold Rdiv. -Rewrite <- Rmult_assoc. -Rewrite Rminus_distr. -Reflexivity. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply (Rabsolu_pos_lt a H1)]. -Qed. - -Lemma scal_derivable_pt : (f:R->R;a:R; x:R) (derivable_pt f x) -> -(derivable_pt (mult_real_fct a f) x). -Unfold mult_real_fct derivable_pt; Intros; Generalize (Req_EM a R0); Intro H0; Elim H0; Intro H1. -Intros; Exists ``0``; Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Rewrite H1; Repeat Rewrite Rmult_Ol; Unfold Rminus; Repeat Rewrite Ropp_O; Repeat Rewrite Rplus_Or; Unfold Rdiv; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Assumption. -Elim H; Intros l H2; Exists ``a*l``; Intros; Elim (H2 ``eps/(Rabsolu a)``); Intros. -Exists x0; Intros; Replace ``(a*(f (x+h))-a*(f x))/h-a*l`` with ``a*(((f (x+h))-(f x))/h-l)``. -Rewrite Rabsolu_mult; Replace ``eps`` with ``(Rabsolu a)*(eps/(Rabsolu a))``. -Apply Rlt_monotony. -Apply (Rabsolu_pos_lt a H1). -Apply (H4 h H5 H6). -Rewrite <- Rmult_sym; Unfold Rdiv; Rewrite Rmult_assoc; Rewrite <- (Rinv_l_sym (Rabsolu a)); [Apply Rmult_1r | Apply (Rabsolu_no_R0 a H1)]. -Rewrite Rminus_distr. -Unfold Rdiv. -Rewrite <- Rmult_assoc. -Rewrite Rminus_distr. -Reflexivity. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply (Rabsolu_pos_lt a H1)]. -Qed. - -Lemma scal_derivable_pt_var : (f:R->R;a:R; x:R) (derivable_pt f x) -> (derivable_pt ([y:R]``a*(f y)``) x). -Intros; Generalize (scal_derivable_pt f a x H); Unfold mult_real_fct; Intro; Assumption. -Qed. - -Lemma scal_derivable : (f:R->R;a:R) (derivable f) -> (derivable (mult_real_fct a f)). -Unfold derivable; Intros f a H1 x; Apply scal_derivable_pt; Exact (H1 x). -Qed. - -Lemma derive_scal : (f:R->R;a,x:R) (derivable_pt f x) -> (derive_pt ([x:R]``a*(f x)``) x)==``a*(derive_pt f x)``. -Intros; Generalize (deriv_scal f a x H); Unfold mult_real_fct; Intro; Assumption. -Qed. - -(* Multiplication *) -Lemma deriv_prod : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> ``(derive_pt (mult_fct f1 f2) x)==(derive_pt f1 x)*(f2 x)+(derive_pt f2 x)*(f1 x)``. -Intros; Generalize (derivable_derive f1 x H); Intro; Generalize (derivable_derive f2 x H0); Intro; Elim H1; Clear H1; Intros l1 H1; Elim H2; Clear H2; Intros l2 H2; Cut l1==((fct_cte l1) x). -Cut l2==((fct_cte l2) x). -Intros; Rewrite H3 in H2; Rewrite H4 in H1; Generalize derive_pt_D_in; Intro; Generalize (H5 f1 (fct_cte l1) x); Intro; Generalize (H5 f2 (fct_cte l2) x); Intro; Elim H6; Elim H7; Intros; Generalize (H11 H1); Intro; Generalize (H9 H2); Intro; Rewrite H1; Rewrite H2; Replace ``(fct_cte l1 x)*(f2 x)+(fct_cte l2 x)*(f1 x)`` with ``((plus_fct (mult_fct (fct_cte l1) f2) (mult_fct f1 (fct_cte l2))) x)``. -Generalize (H5 (mult_fct f1 f2) (plus_fct (mult_fct (fct_cte l1) f2) (mult_fct f1 (fct_cte l2))) x); Intro; Elim H14; Intros; Apply H15; Unfold mult_fct plus_fct; Apply Dmult; Assumption. -Unfold plus_fct mult_fct fct_cte; Ring. -Unfold fct_cte; Reflexivity. -Unfold fct_cte; Reflexivity. -Qed. - -Lemma prod_derivable_pt : (f1,f2:R->R;x:R) (derivable_pt f1 x)->(derivable_pt f2 x)->(derivable_pt (mult_fct f1 f2) x). -Intros; Generalize (deriv_prod f1 f2 x H H0); Intro; Apply (derive_derivable (mult_fct f1 f2) x ``(derive_pt f1 x)*(f2 x)+(derive_pt f2 x)*(f1 x)`` H1). -Qed. - -Lemma prod_derivable : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (mult_fct f1 f2)). -Unfold derivable; Intros f1 f2 H1 H2 x; Apply prod_derivable_pt; [ Exact (H1 x) | Exact (H2 x)]. -Qed. - -Lemma derive_prod : (f1,f2:R->R;x:R) (derivable_pt f1 x) --> (derivable_pt f2 x) -> (derive_pt ([x:R]``(f1 x)*(f2 x)``) x)==``(derive_pt f1 x)*(f2 x)+(derive_pt f2 x)*(f1 x)``. -Intros; Generalize (deriv_prod f1 f2 x H H0); Unfold mult_fct; Intro; Assumption. -Qed. - -(**********) -Lemma deriv_const : (a:R;x:R) (derive_pt ([x:R] a) x)==``0``. -Intros; Apply derive_pt_def_0; Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Replace ``a-a`` with ``0``; [Unfold Rdiv; Rewrite Rmult_Ol; Rewrite minus_R0; Rewrite Rabsolu_R0; Assumption | Ring]. -Qed. - -Lemma const_derivable : (a:R) (derivable ([x:R] a)). -Unfold derivable; Unfold derivable_pt; Intros; Exists ``0``; Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Unfold Rminus; Rewrite Rplus_Ropp_r; Unfold Rdiv; Rewrite Rmult_Ol; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Qed. - -(**********) -Lemma deriv_id : (x:R) (derive_pt ([y:R] y) x)==``1``. -Intro x; Apply derive_pt_def_0; Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Replace ``(x+h-x)/h-1`` with ``0``. -Rewrite Rabsolu_R0; Assumption. -Unfold Rminus; Rewrite Rplus_assoc; Rewrite (Rplus_sym x); Rewrite Rplus_assoc. -Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Unfold Rdiv; Rewrite <- Rinv_r_sym. -Symmetry; Apply Rplus_Ropp_r. -Assumption. -Qed. - -Lemma diff_id : (derivable ([x:R] x)). -Unfold derivable; Intro x; Unfold derivable_pt; Exists ``1``; Intros eps Heps; Exists (mkposreal eps Heps); Intros h H1 H2; Replace ``(x+h-x)/h-1`` with ``0``. -Rewrite Rabsolu_R0; Apply Rle_lt_trans with ``(Rabsolu h)``. -Apply Rabsolu_pos. -Assumption. -Unfold Rminus; Rewrite Rplus_assoc; Rewrite (Rplus_sym x); Rewrite Rplus_assoc. -Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Unfold Rdiv; Rewrite <- Rinv_r_sym. -Symmetry; Apply Rplus_Ropp_r. -Assumption. -Qed. - -(**********) -Lemma sum_fct_cte_derive_pt : (f:R->R;t,a:R) (derivable_pt f t) -> (derive_pt ([x:R]``(f x)+a``) t)==(derive_pt f t). -Intros; Generalize (derivable_derive f t H); Intro; Elim H0; Intros l H1; Rewrite H1; Apply derive_pt_def_0; Intros; Generalize (derive_pt_def_1 f t l H1); Intros; Elim (H3 eps H2); Intros delta H4; Exists delta; Intros; Replace ``(f (t+h))+a-((f t)+a)`` with ``(f (t+h))-(f t)``; [Apply (H4 h H5 H6) | Ring]. -Qed. - -Lemma sum_fct_cte_derivable_pt : (f:R->R;t,a:R) (derivable_pt f t)->(derivable_pt ([t:R]``(f t)+a``) t). -Unfold derivable_pt; Intros; Elim H; Intros; Exists x; Intros; Elim (H0 eps H1); Intros; Exists x0; Intro h; Replace ``(f (t+h))+a-((f t)+a)`` with ``(f (t+h))-(f t)``; [Exact (H2 h) | Ring]. -Qed. - -Lemma sum_fct_cte_derivable : (f:R->R;a:R) (derivable f)->(derivable ([t:R]``(f t)+a``)). -Unfold derivable; Intros; Apply sum_fct_cte_derivable_pt; Apply (H x). -Qed. - -(**********) -Lemma deriv_Rsqr : (x:R) (derive Rsqr x)==``2*x``. -Intro x; Unfold Rsqr; Unfold derive; Apply (derive_pt_def_0 ([x0:R]``x0*x0``) x); Intros eps Heps; Exists (mkposreal eps Heps); Intros h H1 H2; Replace ``((x+h)*(x+h)-x*x)/h-2*x`` with ``h``. -Assumption. -Replace ``(x+h)*(x+h)`` with ``(Rsqr (x+h))``. -Rewrite Rsqr_plus; Unfold Rminus; Repeat Rewrite Rplus_assoc; Rewrite (Rplus_sym (Rsqr x)); Repeat Rewrite Rplus_assoc; Unfold Rsqr; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Unfold Rdiv; Rewrite Rmult_Rplus_distrl. -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Repeat Rewrite Rmult_1r; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r. -Rewrite Rplus_Or; Reflexivity. -Assumption. -Unfold Rsqr; Reflexivity. -Qed. - -Lemma diff_Rsqr : (derivable Rsqr). -Unfold derivable; Intro x; Unfold Rsqr; Unfold derivable_pt; Exists ``2*x``; Intros eps Heps; Exists (mkposreal eps Heps); Intros h H1 H2; Replace ``((x+h)*(x+h)-x*x)/h-2*x`` with ``h``. -Assumption. -Replace ``(x+h)*(x+h)`` with ``(Rsqr (x+h))``. -Rewrite Rsqr_plus; Unfold Rminus; Repeat Rewrite Rplus_assoc; Rewrite (Rplus_sym (Rsqr x)); Repeat Rewrite Rplus_assoc; Unfold Rsqr; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Unfold Rdiv; Rewrite Rmult_Rplus_distrl. -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Repeat Rewrite Rmult_1r; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r. -Rewrite Rplus_Or; Reflexivity. -Assumption. -Unfold Rsqr; Reflexivity. -Qed. - -Lemma Rsqr_derivable_pt : (f:R->R;t:R) (derivable_pt f t) -> (derivable_pt ([x:R](Rsqr (f x))) t). -Unfold Rsqr; Intros; Generalize (prod_derivable_pt f f t H H); Unfold mult_fct; Intro H0; Assumption. -Qed. - -Lemma Rsqr_derivable : (f:R->R) (derivable f)->(derivable ([x:R](Rsqr (f x)))). -Unfold derivable; Intros; Apply (Rsqr_derivable_pt f x (H x)). -Qed. - -(* SQRT *) -Axiom deriv_sqrt : (x:R) ``0<x`` -> (derive sqrt)==[y:R] ``1/(2*(sqrt y))``. - -Lemma eq_fct : (x:R;f1,f2:R->R) f1==f2 -> (f1 x)==(f2 x). -Intros; Rewrite H; Reflexivity. -Qed. - -Lemma diff_sqrt : (x:R) ``0<x`` -> (derivable_pt sqrt x). -Intros; Generalize (deriv_sqrt x H); Unfold derive; Intro; Generalize (eq_fct x ([x:R](derive_pt sqrt x)) ([y:R]``1/(2*(sqrt y))``) H0); Intro; Apply (derive_derivable sqrt x ``1/(2*(sqrt x))`` H1). -Qed. - -(* Composition *) - -Lemma deriv_composition : (f,g:R->R;x:R) (derivable_pt f x) -> (derivable_pt g (f x)) -> ``(derive_pt (comp g f) x)==(derive_pt g (f x))*(derive_pt f x)``. -Intros; Generalize (derivable_derive f x H); Intro; Generalize -(derivable_derive g (f x) H0); Intro; Elim H1; Clear H1; Intros l1 H1; Elim -H2; Clear H2; Intros l2 H2. -Cut l1==((fct_cte l1) x). -Cut l2==((fct_cte l2) x). -Intros; Rewrite H3 in H2; Rewrite H4 in H1; Rewrite H1; Rewrite H2; -Generalize derive_pt_D_in; Intro; Elim (H5 f (fct_cte l1) x); Intros; Elim -(H5 g (fct_cte l2) (f x)); Intros; Generalize (H9 H2); Intro; Generalize (H7 -H1); Intro; Replace ``(fct_cte l2 x)*(fct_cte l1 x)`` with ``((mult_fct -(fct_cte l1) (fct_cte l2)) x)``. -Elim (H5 (comp g f) (mult_fct (fct_cte l1) (fct_cte l2)) x); Intros; Apply -H12. -Generalize (Dcomp no_cond no_cond (fct_cte l1) (fct_cte l2) f g x); Unfold comp mult_fct no_cond D_in; Unfold Dgf; Intros. -Cut (limit1_in [x0:R]``((g (f x0))-(g (f x)))/(x0-x)`` (D_x [_:R]True/\True x) ``(fct_cte l1 x)*(fct_cte l2 (f x))`` x) -> (limit1_in [x0:R]``((g (f x0))-(g (f x)))/(x0-x)`` (D_x [_:R]True x) ``(fct_cte l1 x)*(fct_cte l2 x)`` x). -Intros; Apply H15; Apply H14. -Assumption. -Assumption. -Unfold D_x limit1_in; Unfold limit_in; Intros; Elim (H15 eps H16); Intros; Exists x0; Elim H17; Intros; Split. -Assumption. -Intros; Apply H19; Elim H20; Intros; Elim H21; Intros; Split. -Split. -Split; Trivial. -Assumption. -Assumption. -Unfold mult_fct fct_cte; Rewrite Rmult_sym; Reflexivity. -Unfold fct_cte; Reflexivity. -Unfold fct_cte; Reflexivity. -Qed. - -Lemma composition_derivable : (f,g:R->R;x:R) (derivable_pt f x) -> (derivable_pt g (f x)) -> (derivable_pt (comp g f) x). -Intros; Generalize (deriv_composition f g x H H0); Intro; Apply (derive_derivable (comp g f) x ``(derive_pt g (f x))*(derive_pt f x)`` H1). -Qed. - -Lemma derive_composition : (f,g:R->R;x:R) (derivable_pt f x) -> (derivable_pt g (f x)) -> (derive_pt ([x:R]``(g (f x))``) x)==``(derive_pt g (f x))*(derive_pt f x)``. -Intros; Generalize (deriv_composition f g x H H0); Unfold comp; Intro; Assumption. -Qed. - -Lemma composition_derivable_var : (f,g:R->R;x:R) (derivable_pt f x) -> (derivable_pt g (f x)) -> (derivable_pt ([x:R](g (f x))) x). -Intros; Generalize (composition_derivable f g x H H0); Unfold comp; Intro; Assumption. -Qed. - -Lemma diff_comp : (f,g:R->R) (derivable f)->(derivable g)->(derivable (comp g f)). -Intros f g; Unfold derivable; Intros H1 H2 x; Apply (composition_derivable f g x (H1 x) (H2 (f x))). -Qed. - -Lemma Rsqr_derive : (f:R->R;t:R) (derivable_pt f t)->(derive_pt ([x:R](Rsqr (f x))) t)==(Rmult ``2`` (Rmult (derive_pt f t) (f t))). -Intros; Generalize diff_Rsqr; Unfold derivable; Intro H0; Generalize (deriv_composition f Rsqr t H (H0 (f t))); Unfold comp; Intro H1; Rewrite H1; Generalize (deriv_Rsqr (f t)); Unfold derive; Intro H2; Rewrite H2; Rewrite Rmult_assoc; Rewrite <- (Rmult_sym (derive_pt f t)); Reflexivity. -Qed. - -(* SIN and COS *) -Axiom deriv_sin : (derive sin)==cos. - -Lemma diff_sin : (derivable sin). -Unfold derivable; Intro; Generalize deriv_sin; Unfold derive; Intro; Generalize -(eq_fct x ([x:R](derive_pt sin x)) cos H); Intro; Apply (derive_derivable sin x -(cos x) H0). -Qed. - -Lemma diff_cos : (derivable cos). -Unfold derivable; Intro; Cut ([x:R]``(sin (x+PI/2))``)==cos. -Intro; Rewrite <- H; Apply (composition_derivable_var ([x:R]``x+PI/2``) sin x). -Apply (sum_fct_cte_derivable_pt ([x:R]x) x ``PI/2``); Apply diff_id. -Apply diff_sin. -Apply fct_eq; Intro; Symmetry; Rewrite Rplus_sym; Apply cos_sin. -Qed. - -Lemma derive_pt_sin : (x:R) (derive_pt sin x)==(cos x). -Intro; Generalize deriv_sin; Unfold derive; Intro; Apply (eq_fct x [x:R](derive_pt sin x) cos H). -Qed. - -Lemma deriv_cos : (derive cos)==(opp_fct sin). -Unfold opp_fct derive; Apply fct_eq; Intro; Cut ([x:R]``(sin (x+PI/2))``)==cos. -Intro; Rewrite <- H; Rewrite (derive_composition ([x:R]``x+PI/2``) sin x). -Rewrite (derive_pt_sin ``x+PI/2``); Rewrite (sum_fct_cte_derive_pt ([x:R]``x``) x ``PI/2``). -Generalize (deriv_id x); Intro; Unfold derive in H0; Rewrite H0; Rewrite Rmult_1r; Rewrite Rplus_sym; Rewrite sin_cos; Rewrite Ropp_Ropp; Reflexivity. -Apply diff_id. -Apply (sum_fct_cte_derivable_pt ([x:R]x) x ``PI/2``); Apply diff_id. -Apply diff_sin. -Apply fct_eq; Intro; Symmetry; Rewrite Rplus_sym; Apply cos_sin. -Qed. - -Lemma derive_pt_cos : (x:R) (derive_pt cos x)==``-(sin x)``. -Intro; Generalize deriv_cos; Unfold derive; Intro; Unfold opp_fct in H; Apply (eq_fct x [x:R](derive_pt cos x) [x:R]``-(sin x)`` H). -Qed. - -(************************************************************) -(** Local extremum's condition *) -(************************************************************) -Theorem deriv_maximum : (f:R->R;a,b,c:R) ``a<c``->``c<b``->(derivable_pt f c)->((x:R) ``a<x``->``x<b``->``(f x)<=(f c)``)->``(derive_pt f c)==0``. -Intros; Case (total_order R0 (derive_pt f c)); Intro. -Generalize (derivable_derive f c H1); Intro; Elim H4; Intros l H5; Rewrite H5 in H3; Generalize (derive_pt_def_1 f c l H5); Intro. -Cut ``0<l/2``. -Intro; Elim (H6 ``l/2`` H7); Intros delta H8. -Cut ``0<(b-c)/2``. -Intro; Cut ``(Rmin delta/2 ((b-c)/2))<>0``. -Intro; Cut ``(Rabsolu (Rmin delta/2 ((b-c)/2)))<delta``. -Intro; Generalize (H8 ``(Rmin delta/2 ((b-c)/2))`` H10 H11); Intro; Cut ``0<(Rmin (delta/2) ((b-c)/2))``. -Intro; Cut ``a<c+(Rmin (delta/2) ((b-c)/2))``. -Cut ``c+(Rmin (delta/2) ((b-c)/2))<b``. -Intros; Generalize (H2 ``c+(Rmin (delta/2) ((b-c)/2))`` H15 H14); Intro; Cut ``((f (c+(Rmin (delta/2) ((b-c)/2))))-(f c))/(Rmin (delta/2) ((b-c)/2))<=0``. -Intro; Cut ``-l<0``. -Intro; Unfold Rminus in H12. -Cut ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l<0``. -Intro; Cut ``(Rabsolu (((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l)) < l/2``. -Unfold Rabsolu; Case (case_Rabsolu ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l``); Intro. -Replace `` -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l)`` with ``l+ -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2)))``. -Intro; Generalize (Rlt_compatibility ``-l`` ``l+ -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2)))`` ``l/2`` H20); Repeat Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Replace ``-l+l/2`` with ``-(l/2)``. -Intro; Generalize (Rlt_Ropp ``-(((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2)))`` ``-(l/2)`` H21); Repeat Rewrite Ropp_Ropp; Intro; Generalize (Rlt_trans ``0`` ``l/2`` ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))`` H7 H22); Intro; Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))`` ``0`` H23 H17)). -Pattern 2 l; Rewrite double_var. -Rewrite Ropp_distr1. -Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l. -Symmetry; Apply Rplus_Or. -Ring. -Intro; Generalize (Rle_sym2 ``0`` ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l`` r); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l`` ``0`` H21 H19)). -Assumption. -Rewrite <- Ropp_O; Replace ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l`` with ``-(l+ -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))-(f c))/(Rmin (delta/2) ((b+ -c)/2))))``. -Apply Rgt_Ropp; Change ``0<l+ -(((f (c+(Rmin (delta/2) ((b+ -c)/2))))-(f c))/(Rmin (delta/2) ((b+ -c)/2)))``; Apply gt0_plus_ge0_is_gt0; [Assumption | Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Assumption]. -Ring. -Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. -Replace ``((f (c+(Rmin (delta/2) ((b-c)/2))))-(f c))/(Rmin (delta/2) ((b-c)/2))`` with ``- (((f c)-(f (c+(Rmin (delta/2) ((b-c)/2)))))/(Rmin (delta/2) ((b-c)/2)))``. -Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Unfold Rdiv; Apply Rmult_le_pos; [Generalize (Rle_compatibility_r ``-(f (c+(Rmin (delta*/2) ((b-c)*/2))))`` ``(f (c+(Rmin (delta*/2) ((b-c)*/2))))`` (f c) H16); Rewrite Rplus_Ropp_r; Intro; Assumption | Left; Apply Rlt_Rinv; Assumption]. -Unfold Rdiv. -Rewrite <- Ropp_mul1. -Repeat Rewrite <- (Rmult_sym ``/(Rmin (delta*/2) ((b-c)*/2))``). -Apply r_Rmult_mult with ``(Rmin (delta*/2) ((b-c)*/2))``. -Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_r_sym. -Repeat Rewrite Rmult_1l. -Ring. -Red; Intro. -Unfold Rdiv in H13; Rewrite H17 in H13; Elim (Rlt_antirefl ``0`` H13). -Red; Intro. -Unfold Rdiv in H13; Rewrite H17 in H13; Elim (Rlt_antirefl ``0`` H13). -Generalize (Rmin_r ``(delta/2)`` ``((b-c)/2)``); Intro; Generalize (Rle_compatibility ``c`` ``(Rmin (delta/2) ((b-c)/2))`` ``(b-c)/2`` H14); Intro; Apply Rle_lt_trans with ``c+(b-c)/2``. -Assumption. -Apply Rlt_monotony_contra with ``2``. -Apply Rgt_2_0. -Replace ``2*(c+(b-c)/2)`` with ``c+b``. -Replace ``2*b`` with ``b+b``. -Apply Rlt_compatibility_r; Assumption. -Ring. -Unfold Rdiv; Rewrite Rmult_Rplus_distr. -Repeat Rewrite (Rmult_sym ``2``). -Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Ring. -Apply aze. -Apply Rlt_trans with c. -Assumption. -Pattern 1 c; Rewrite <- (Rplus_Or c); Apply Rlt_compatibility; Assumption. -Cut ``0<delta/2``. -Intro; Apply (Rmin_stable_in_posreal (mkposreal ``delta/2`` H13) (mkposreal ``(b-c)/2`` H9)). -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. -Unfold Rabsolu; Case (case_Rabsolu (Rmin ``delta/2`` ``(b-c)/2``)). -Intro. -Cut ``0<delta/2``. -Intro. -Generalize (Rmin_stable_in_posreal (mkposreal ``delta/2`` H11) (mkposreal ``(b-c)/2`` H9)); Simpl; Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``(Rmin (delta/2) ((b-c)/2))`` ``0`` H12 r)). -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. -Intro; Apply Rle_lt_trans with ``delta/2``. -Apply Rmin_l. -Unfold Rdiv; Apply Rlt_monotony_contra with ``2``. -Apply Rgt_2_0. -Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l. -Replace ``2*delta`` with ``delta+delta``. -Pattern 2 delta; Rewrite <- (Rplus_Or delta); Apply Rlt_compatibility. -Rewrite Rplus_Or; Apply (cond_pos delta). -Symmetry; Apply double. -Apply aze. -Cut ``0<delta/2``. -Intro; Generalize (Rmin_stable_in_posreal (mkposreal ``delta/2`` H10) (mkposreal ``(b-c)/2`` H9)); Simpl; Intro; Red; Intro; Rewrite H12 in H11; Elim (Rlt_antirefl ``0`` H11). -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. -Unfold Rdiv; Apply Rmult_lt_pos. -Generalize (Rlt_compatibility_r ``-c`` c b H0); Rewrite Rplus_Ropp_r; Intro; Assumption. -Apply Rlt_Rinv; Apply Rgt_2_0. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rgt_2_0]. -Elim H3; Intro. -Symmetry; Assumption. -Generalize (derivable_derive f c H1); Intro; Elim H5; Intros l H6; Rewrite H6 in H4; Generalize (derive_pt_def_1 f c l H6); Intro; Cut ``0< -(l/2)``. -Intro; Elim (H7 ``-(l/2)`` H8); Intros delta H9. -Cut ``0<(c-a)/2``. -Intro; Cut ``(Rmax (-(delta/2)) ((a-c)/2))<0``. -Intro; Cut ``(Rmax (-(delta/2)) ((a-c)/2))<>0``. -Intro; Cut ``(Rabsolu (Rmax (-(delta/2)) ((a-c)/2)))<delta``. -Intro; Generalize (H9 ``(Rmax (-(delta/2)) ((a-c)/2))`` H12 H13); Intro; Cut ``a<c+(Rmax (-(delta/2)) ((a-c)/2))``. -Cut ``c+(Rmax (-(delta/2)) ((a-c)/2))<b``. -Intros; Generalize (H2 ``c+(Rmax (-(delta/2)) ((a-c)/2))`` H16 H15); Intro; Cut ``0<=((f (c+(Rmax (-(delta/2)) ((a-c)/2))))-(f c))/(Rmax (-(delta/2)) ((a-c)/2))``. -Intro; Cut ``0< -l``. -Intro; Unfold Rminus in H14; Cut ``0<((f (c+(Rmax (-(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax (-(delta/2)) ((a+ -c)/2))+ -l``. -Intro; Cut ``(Rabsolu (((f (c+(Rmax (-(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax (-(delta/2)) ((a+ -c)/2))+ -l)) < -(l/2)``. -Unfold Rabsolu; Case (case_Rabsolu ``((f (c+(Rmax (-(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax (-(delta/2)) ((a+ -c)/2))+ -l``). -Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``((f (c+(Rmax ( -(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax ( -(delta/2)) ((a+ -c)/2))+ -l`` ``0`` H20 r)). -Intros; Generalize (Rlt_compatibility_r ``l`` ``(((f (c+(Rmax (-(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax (-(delta/2)) ((a+ -c)/2)))+ -l`` ``-(l/2)`` H21); Repeat Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Replace ``-(l/2)+l`` with ``l/2``. -Cut ``l/2<0``. -Intros; Generalize (Rlt_trans ``((f (c+(Rmax ( -(delta/2)) ((a+ -c)/2))))+ -(f c))/(Rmax ( -(delta/2)) ((a+ -c)/2))`` ``l/2`` ``0`` H23 H22); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``((f (c+(Rmax ( -(delta/2)) ((a-c)/2))))-(f c))/(Rmax ( -(delta/2)) ((a-c)/2))`` ``0`` H18 H24)). -Rewrite <- (Ropp_Ropp ``l/2``); Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. -Pattern 3 l; Rewrite double_var. -Ring. -Assumption. -Apply ge0_plus_gt0_is_gt0; Assumption. -Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. -Unfold Rdiv; Replace ``((f (c+(Rmax ( -(delta*/2)) ((a-c)*/2))))-(f c))*/(Rmax ( -(delta*/2)) ((a-c)*/2))`` with ``(-((f (c+(Rmax ( -(delta*/2)) ((a-c)*/2))))-(f c)))*/(-(Rmax ( -(delta*/2)) ((a-c)*/2)))``. -Apply Rmult_le_pos. -Generalize (Rle_compatibility ``-(f (c+(Rmax (-(delta*/2)) ((a-c)*/2))))`` ``(f (c+(Rmax (-(delta*/2)) ((a-c)*/2))))`` (f c) H17); Rewrite Rplus_Ropp_l; Replace ``-((f (c+(Rmax ( -(delta*/2)) ((a-c)*/2))))-(f c))`` with ``-((f (c+(Rmax ( -(delta*/2)) ((a-c)*/2)))))+(f c)``. -Intro; Assumption. -Ring. -Left; Apply Rlt_Rinv; Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. -Unfold Rdiv. -Rewrite <- Ropp_Rinv. -Rewrite Ropp_mul2. -Reflexivity. -Unfold Rdiv in H12; Assumption. -Generalize (Rlt_compatibility c ``(Rmax ( -(delta/2)) ((a-c)/2))`` ``0`` H11); Rewrite Rplus_Or; Intro; Apply Rlt_trans with ``c``; Assumption. -Generalize (RmaxLess2 ``(-(delta/2))`` ``((a-c)/2)``); Intro; Generalize (Rle_compatibility c ``(a-c)/2`` ``(Rmax ( -(delta/2)) ((a-c)/2))`` H15); Intro; Apply Rlt_le_trans with ``c+(a-c)/2``. -Apply Rlt_monotony_contra with ``2``. -Apply Rgt_2_0. -Replace ``2*(c+(a-c)/2)`` with ``a+c``. -Rewrite double. -Apply Rlt_compatibility; Assumption. -Ring. -Rewrite <- Rplus_assoc. -Rewrite <- double_var. -Ring. -Assumption. -Unfold Rabsolu; Case (case_Rabsolu (Rmax ``-(delta/2)`` ``(a-c)/2``)). -Intro; Generalize (RmaxLess1 ``-(delta/2)`` ``(a-c)/2``); Intro; Generalize (Rle_Ropp ``-(delta/2)`` ``(Rmax ( -(delta/2)) ((a-c)/2))`` H13); Rewrite Ropp_Ropp; Intro; Generalize (Rle_sym2 ``-(Rmax ( -(delta/2)) ((a-c)/2))`` ``delta/2`` H14); Intro; Apply Rle_lt_trans with ``delta/2``. -Assumption. -Apply Rlt_monotony_contra with ``2``. -Apply Rgt_2_0. -Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite double. -Pattern 2 delta; Rewrite <- (Rplus_Or delta); Apply Rlt_compatibility; Rewrite Rplus_Or; Apply (cond_pos delta). -Apply aze. -Cut ``-(delta/2) < 0``. -Cut ``(a-c)/2<0``. -Intros; Generalize (Rmax_stable_in_negreal (mknegreal ``-(delta/2)`` H14) (mknegreal ``(a-c)/2`` H13)); Simpl; Intro; Generalize (Rle_sym2 ``0`` ``(Rmax ( -(delta/2)) ((a-c)/2))`` r); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``(Rmax ( -(delta/2)) ((a-c)/2))`` ``0`` H16 H15)). -Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp ``(a-c)/2``); Apply Rlt_Ropp; Replace ``-((a-c)/2)`` with ``(c-a)/2``. -Assumption. -Unfold Rdiv. -Rewrite <- Ropp_mul1. -Rewrite (Ropp_distr2 a c). -Reflexivity. -Rewrite <- Ropp_O; Apply Rlt_Ropp; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply (Rlt_Rinv ``2`` Rgt_2_0)]. -Red; Intro; Rewrite H12 in H11; Elim (Rlt_antirefl ``0`` H11). -Cut ``(a-c)/2<0``. -Intro; Cut ``-(delta/2)<0``. -Intro; Apply (Rmax_stable_in_negreal (mknegreal ``-(delta/2)`` H12) (mknegreal ``(a-c)/2`` H11)). -Rewrite <- Ropp_O; Apply Rlt_Ropp; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply (Rlt_Rinv ``2`` Rgt_2_0)]. -Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp ``(a-c)/2``); Apply Rlt_Ropp; Replace ``-((a-c)/2)`` with ``(c-a)/2``. -Assumption. -Unfold Rdiv. -Rewrite <- Ropp_mul1. -Rewrite (Ropp_distr2 a c). -Reflexivity. -Unfold Rdiv; Apply Rmult_lt_pos; [Generalize (Rlt_compatibility_r ``-a`` a c H); Rewrite Rplus_Ropp_r; Intro; Assumption | Apply (Rlt_Rinv ``2`` Rgt_2_0)]. -Replace ``-(l/2)`` with ``(-l)/2``. -Unfold Rdiv; Apply Rmult_lt_pos. -Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. -Apply (Rlt_Rinv ``2`` Rgt_2_0). -Unfold Rdiv; Apply Ropp_mul1. -Qed. - -Theorem deriv_minimum : (f:R->R;a,b,c:R) ``a<c``->``c<b``->(derivable_pt f c)->((x:R) ``a<x``->``x<b``->``(f c)<=(f x)``)->``(derive_pt f c)==0``. -Intros; Generalize (opposite_derivable_pt f c H1); Intro; Rewrite <- (Ropp_Ropp (derive_pt f c)); Apply eq_RoppO; Rewrite <- (deriv_opposite f c H1); Apply (deriv_maximum (opp_fct f) a b c H H0 H3); Intros; Unfold opp_fct; Apply Rge_Ropp; Apply Rle_sym1; Apply (H2 x H4 H5). -Qed. - -Theorem deriv_constant2 : (f:R->R;a,b,c:R) ``a<c``->``c<b``->(derivable_pt f c)->((x:R) ``a<x``->``x<b``->``(f x)==(f c)``)->``(derive_pt f c)==0``. -Intros; Apply (deriv_maximum f a b c H H0 H1); Intros; Right; Apply (H2 x H3 H4). -Qed. - -(**********) -Lemma nonneg_derivative_0 : (f:R->R) (derivable f)->(increasing f) -> ((x:R) ``0<=(derive_pt f x)``). -Intros; Unfold increasing in H0; Generalize (derivable_derive f x (H x)); Intro; Elim H1; Intros l H2. -Rewrite H2; Case (total_order R0 l); Intro. -Left; Assumption. -Elim H3; Intro. -Right; Assumption. -Generalize (derive_pt_def_1 f x l H2); Intros; Cut ``0< -(l/2)``. -Intro; Elim (H5 ``-(l/2)`` H6); Intros delta H7; Cut ``delta/2<>0``/\``0<delta/2``/\``(Rabsolu delta/2)<delta``. -Intro; Decompose [and] H8; Intros; Generalize (H7 ``delta/2`` H9 H12); Cut ``0<=((f (x+delta/2))-(f x))/(delta/2)``. -Intro; Cut ``0<=((f (x+delta/2))-(f x))/(delta/2)-l``. -Intro; Unfold Rabsolu; Case (case_Rabsolu ``((f (x+delta/2))-(f x))/(delta/2)-l``). -Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``((f (x+delta/2))-(f x))/(delta/2)-l`` ``0`` H13 r)). -Intros; Generalize (Rlt_compatibility_r l ``((f (x+delta/2))-(f x))/(delta/2)-l`` ``-(l/2)`` H14); Unfold Rminus; Replace ``-(l/2)+l`` with ``l/2``. -Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Intro; Generalize (Rle_lt_trans ``0`` ``((f (x+delta/2))-(f x))/(delta/2)`` ``l/2`` H10 H15); Intro; Cut ``l/2<0``. -Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``l/2`` ``0`` H16 H17)). -Rewrite <- Ropp_O in H6; Generalize (Rlt_Ropp ``-0`` ``-(l/2)`` H6); Repeat Rewrite Ropp_Ropp; Intro; Assumption. -Pattern 3 l ; Rewrite double_var. -Ring. -Unfold Rminus; Apply ge0_plus_ge0_is_ge0. -Unfold Rdiv; Apply Rmult_le_pos. -Cut ``x<=(x+(delta*/2))``. -Intro; Generalize (H0 x ``x+(delta*/2)`` H13); Intro; Generalize (Rle_compatibility ``-(f x)`` ``(f x)`` ``(f (x+delta*/2))`` H14); Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Intro; Assumption. -Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Left; Assumption. -Left; Apply Rlt_Rinv; Assumption. -Left; Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. -Unfold Rdiv; Apply Rmult_le_pos. -Cut ``x<=(x+(delta*/2))``. -Intro; Generalize (H0 x ``x+(delta*/2)`` H10); Intro; Generalize (Rle_compatibility ``-(f x)`` ``(f x)`` ``(f (x+delta*/2))`` H13); Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Intro; Assumption. -Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Left; Assumption. -Left; Apply Rlt_Rinv; Assumption. -Split. -Unfold Rdiv; Apply prod_neq_R0. -Generalize (cond_pos delta); Intro; Red; Intro H9; Rewrite H9 in H8; Elim (Rlt_antirefl ``0`` H8). -Apply Rinv_neq_R0; DiscrR. -Split. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. -Unfold Rabsolu; Case (case_Rabsolu ``delta/2``). -Unfold Rdiv; Intro; Generalize (Rlt_monotony_r ``2`` ``delta*/2`` ``0`` Rgt_2_0 r); Rewrite Rmult_Ol; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` delta ``0`` (cond_pos delta) H8)). -DiscrR. -Intro; Unfold Rdiv; Pattern 1 delta; Replace ``(pos delta)`` with ``2*(delta*/2)``. -Replace ``2*(delta*/2)`` with ``delta*/2+delta*/2``. -Pattern 2 delta; Rewrite <- (Rplus_Or ``delta*/2``). -Apply Rlt_compatibility. -Rewrite Rplus_Or. -Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. -Ring. -Rewrite <- Rmult_assoc. -Apply Rinv_r_simpl_m. -Apply aze. -Rewrite <- Ropp_O; Apply Rlt_Ropp; Unfold Rdiv; Generalize (Rlt_monotony_r ``/2`` l ``0`` (Rlt_Rinv ``2`` Rgt_2_0) H4); Rewrite Rmult_Ol; Intro; Assumption. -Qed. - -(**********) -Axiom nonneg_derivative_1 : (f:R->R) (derivable f)->((x:R) ``0<=(derive_pt f x)``) -> (increasing f). - -(**********) -Lemma nonpos_derivative_0 : (f:R->R) (derivable f)->(decreasing f) -> ((x:R) ``(derive_pt f x)<=0``). -Intros; Unfold decreasing in H0; Generalize (derivable_derive f x (H x)); Intro; Elim H1; Intros l H2. -Rewrite H2; Case (total_order l R0); Intro. -Left; Assumption. -Elim H3; Intro. -Right; Assumption. -Generalize (derive_pt_def_1 f x l H2); Intros; Cut ``0< (l/2)``. -Intro; Elim (H5 ``(l/2)`` H6); Intros delta H7; Cut ``delta/2<>0``/\``0<delta/2``/\``(Rabsolu delta/2)<delta``. -Intro; Decompose [and] H8; Intros; Generalize (H7 ``delta/2`` H9 H12); Cut ``((f (x+delta/2))-(f x))/(delta/2)<=0``. -Intro; Cut ``0< -(((f (x+delta/2))-(f x))/(delta/2)-l)``. -Intro; Unfold Rabsolu; Case (case_Rabsolu ``((f (x+delta/2))-(f x))/(delta/2)-l``). -Intros; Generalize (Rlt_compatibility_r ``-l`` ``-(((f (x+delta/2))-(f x))/(delta/2)-l)`` ``(l/2)`` H14); Unfold Rminus. -Replace ``(l/2)+ -l`` with ``-(l/2)``. -Replace `` -(((f (x+delta/2))+ -(f x))/(delta/2)+ -l)+ -l`` with ``-(((f (x+delta/2))+ -(f x))/(delta/2))``. -Intro. -Generalize (Rlt_Ropp ``-(((f (x+delta/2))+ -(f x))/(delta/2))`` ``-(l/2)`` H15). -Repeat Rewrite Ropp_Ropp. -Intro. -Generalize (Rlt_trans ``0`` ``l/2`` ``((f (x+delta/2))-(f x))/(delta/2)`` H6 H16); Intro. -Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``((f (x+delta/2))-(f x))/(delta/2)`` ``0`` H17 H10)). -Ring. -Pattern 3 l; Rewrite double_var. -Ring. -Intros. -Generalize (Rge_Ropp ``((f (x+delta/2))-(f x))/(delta/2)-l`` ``0`` r). -Rewrite Ropp_O. -Intro. -Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``-(((f (x+delta/2))-(f x))/(delta/2)-l)`` ``0`` H13 H15)). -Replace ``-(((f (x+delta/2))-(f x))/(delta/2)-l)`` with ``(((f (x))-(f (x+delta/2)))/(delta/2)) +l``. -Unfold Rminus. -Apply ge0_plus_gt0_is_gt0. -Unfold Rdiv; Apply Rmult_le_pos. -Cut ``x<=(x+(delta*/2))``. -Intro; Generalize (H0 x ``x+(delta*/2)`` H13); Intro; Generalize (Rle_compatibility ``-(f (x+delta/2))`` ``(f (x+delta/2))`` ``(f x)`` H14); Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Intro; Assumption. -Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Left; Assumption. -Left; Apply Rlt_Rinv; Assumption. -Assumption. -Rewrite Ropp_distr2. -Unfold Rminus. -Rewrite (Rplus_sym l). -Unfold Rdiv. -Rewrite <- Ropp_mul1. -Rewrite Ropp_distr1. -Rewrite Ropp_Ropp. -Rewrite (Rplus_sym (f x)). -Reflexivity. -Replace ``((f (x+delta/2))-(f x))/(delta/2)`` with ``-(((f x)-(f (x+delta/2)))/(delta/2))``. -Rewrite <- Ropp_O. -Apply Rge_Ropp. -Apply Rle_sym1. -Unfold Rdiv; Apply Rmult_le_pos. -Cut ``x<=(x+(delta*/2))``. -Intro; Generalize (H0 x ``x+(delta*/2)`` H10); Intro. -Generalize (Rle_compatibility ``-(f (x+delta/2))`` ``(f (x+delta/2))`` ``(f x)`` H13); Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Intro; Assumption. -Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Left; Assumption. -Left; Apply Rlt_Rinv; Assumption. -Unfold Rdiv; Rewrite <- Ropp_mul1. -Rewrite Ropp_distr2. -Reflexivity. -Split. -Unfold Rdiv; Apply prod_neq_R0. -Generalize (cond_pos delta); Intro; Red; Intro H9; Rewrite H9 in H8; Elim (Rlt_antirefl ``0`` H8). -Apply Rinv_neq_R0; DiscrR. -Split. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. -Unfold Rabsolu; Case (case_Rabsolu ``delta/2``). -Unfold Rdiv; Intro; Generalize (Rlt_monotony_r ``2`` ``delta*/2`` ``0`` Rgt_2_0 r); Rewrite Rmult_Ol; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` delta ``0`` (cond_pos delta) H8)). -DiscrR. -Intro; Unfold Rdiv; Pattern 1 delta; Replace ``(pos delta)`` with ``2*(delta*/2)``. -Replace ``2*(delta*/2)`` with ``delta*/2+delta*/2``. -Pattern 2 delta; Rewrite <- (Rplus_Or ``delta*/2``). -Apply Rlt_compatibility. -Rewrite Rplus_Or. -Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Apply Rgt_2_0]. -Ring. -Rewrite <- Rmult_assoc. -Apply Rinv_r_simpl_m. -Apply aze. -Unfold Rdiv; Apply Rmult_lt_pos. -Assumption. -Apply Rlt_Rinv; Apply Rgt_2_0. -Qed. - -(**********) -Lemma increasing_decreasing_opp : (f:R->R) (increasing f) -> (decreasing (opp_fct f)). -Unfold increasing decreasing opp_fct; Intros; Generalize (H x y H0); Intro; Apply Rge_Ropp; Apply Rle_sym1; Assumption. -Qed. - -(**********) -Lemma opp_opp_fct : (f:R->R) (opp_fct (opp_fct f))==f. -Intro; Unfold opp_fct; Apply fct_eq; Intro; Rewrite Ropp_Ropp; Reflexivity. -Qed. - -(**********) -Lemma nonpos_derivative_1 : (f:R->R) (derivable f)->((x:R) ``(derive_pt f x)<=0``) -> (decreasing f). -Intros; Rewrite <- (opp_opp_fct f); Apply increasing_decreasing_opp. -Cut (derivable (opp_fct f)). -Cut (x:R)``0<=(derive_pt (opp_fct f) x)``. -Intros; Apply (nonneg_derivative_1 (opp_fct f) H2 H1). -Intros; Rewrite (deriv_opposite f x (H x)); Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Apply (H0 x). -Apply (opposite_derivable f H). -Qed. - -(**********) -Axiom positive_derivative : (f:R->R) (derivable f)->((x:R) ``0<(derive_pt f x)``)->(strict_increasing f). - -(**********) -Lemma strictincreasing_strictdecreasing_opp : (f:R->R) (strict_increasing f) -> (strict_decreasing (opp_fct f)). -Unfold strict_increasing strict_decreasing opp_fct; Intros; Generalize (H x y H0); Intro; Apply Rlt_Ropp; Assumption. -Qed. - -(**********) -Lemma negative_derivative : (f:R->R) (derivable f)->((x:R) ``(derive_pt f x)<0``)->(strict_decreasing f). -Intros; Rewrite <- (opp_opp_fct f); Apply strictincreasing_strictdecreasing_opp. -Cut (derivable (opp_fct f)). -Cut (x:R)``0<(derive_pt (opp_fct f) x)``. -Intros; Apply (positive_derivative (opp_fct f) H2 H1). -Intros; Rewrite (deriv_opposite f x (H x)); Rewrite <- Ropp_O; Apply Rlt_Ropp; Apply (H0 x). -Apply (opposite_derivable f H). -Qed. - -(**********) -Lemma null_derivative_0 : (f:R->R) (constant f)->((x:R) ``(derive_pt f x)==0``). -Intros; Unfold constant in H; Apply derive_pt_def_0; Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Intros; Rewrite (H x ``x+h``); Unfold Rminus; Unfold Rdiv; Rewrite Rplus_Ropp_r; Rewrite Rmult_Ol; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Qed. - -(**********) -Lemma increasing_decreasing : (f:R->R) (increasing f) -> (decreasing f) -> (constant f). -Unfold increasing decreasing constant; Intros; Case (total_order x y); Intro. -Generalize (Rlt_le x y H1); Intro; Apply (Rle_antisym (f x) (f y) (H x y H2) (H0 x y H2)). -Elim H1; Intro. -Rewrite H2; Reflexivity. -Generalize (Rlt_le y x H2); Intro; Symmetry; Apply (Rle_antisym (f y) (f x) (H y x H3) (H0 y x H3)). -Qed. - -(**********) -Lemma null_derivative_1 : (f:R->R) (derivable f)->((x:R) ``(derive_pt f x)==0``)->(constant f). -Intros. -Cut (x:R)``(derive_pt f x) <= 0``. -Cut (x:R)``0 <= (derive_pt f x)``. -Intros. -Generalize (nonneg_derivative_1 f H H1); Intro. -Generalize (nonpos_derivative_1 f H H2); Intro. -Apply increasing_decreasing; Assumption. -Intro. -Right; Symmetry; Apply (H0 x). -Intro; Right; Apply (H0 x). -Qed. - -(**********) -Axiom derive_increasing_interv_ax : (a,b:R;f:R->R) ``a<b``-> (((t:R) ``a<t<b`` -> ``0<(derive_pt f t)``) -> ((x,y:R) ``a<=x<=b``->``a<=y<=b``->``x<y``->``(f x)<(f y)``)) /\ (((t:R) ``a<t<b`` -> ``0<=(derive_pt f t)``) -> ((x,y:R) ``a<=x<=b``->``a<=y<=b``->``x<y``->``(f x)<=(f y)``)). - -(**********) -Lemma derive_increasing_interv : (a,b:R;f:R->R) ``a<b``-> ((t:R) ``a<t<b`` -> ``0<(derive_pt f t)``) -> ((x,y:R) ``a<=x<=b``->``a<=y<=b``->``x<y``->``(f x)<(f y)``). -Intros; Generalize (derive_increasing_interv_ax a b f H); Intro; Elim H4; Intros H5 _; Apply (H5 H0 x y H1 H2 H3). -Qed. - -(**********) -Lemma derive_increasing_interv_var : (a,b:R;f:R->R) ``a<b``-> ((t:R) ``a<t<b`` -> ``0<=(derive_pt f t)``) -> ((x,y:R) ``a<=x<=b``->``a<=y<=b``->``x<y``->``(f x)<=(f y)``). -Intros; Generalize (derive_increasing_interv_ax a b f H); Intro; Elim H4; Intros _ H5; Apply (H5 H0 x y H1 H2 H3). -Qed. +(*i $Id$ i*) -(**********) -(**********) -Axiom IAF : (f,g:R->R;a,b:R) ``a<=b`` -> (derivable f) -> (derivable g) -> ((c:R) ``a<=c<=b`` -> ``(derive_pt g c)<=(derive_pt f c)``) -> ``(g b)-(g a)<=(f b)-(f a)``. +Require Export Ranalysis1. +Require Export Ranalysis2. +Require Export Ranalysis3. +Require Export Ranalysis4. |