diff options
| author |  herbelin <herbelin@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2003-11-29 17:28:49 +0000 |
|---|---|---|
| committer | herbelin <herbelin@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2003-11-29 17:28:49 +0000 |
| commit | 9a6e3fe764dc2543dfa94de20fe5eec42d6be705 (patch) | |
| tree | 77c0021911e3696a8c98e35a51840800db4be2a9 /theories/Reals | |
| parent | 9058fb97426307536f56c3e7447be2f70798e081 (diff) | |
Remplacement des fichiers .v ancienne syntaxe de theories, contrib et states par les fichiers nouvelle syntaxe
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@5027 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Reals')
53 files changed, 29225 insertions, 21382 deletions
diff --git a/theories/Reals/Alembert.v b/theories/Reals/Alembert.v index 455803aa1..7d8a93914 100644 --- a/theories/Reals/Alembert.v +++ b/theories/Reals/Alembert.v @@ -8,12 +8,12 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Rseries. -Require SeqProp. -Require PartSum. -Require Max. +Require Import Rbase. +Require Import Rfunctions. +Require Import Rseries. +Require Import SeqProp. +Require Import PartSum. +Require Import Max. Open Local Scope R_scope. @@ -21,529 +21,706 @@ Open Local Scope R_scope. (* Various versions of the criterion of D'Alembert *) (***************************************************) -Lemma Alembert_C1 : (An:nat->R) ((n:nat)``0<(An n)``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) R0) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). -Intros An H H0. -Cut (sigTT R [l:R](is_lub (EUn [N:nat](sum_f_R0 An N)) l)) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). -Intro; Apply X. -Apply complet. -Unfold Un_cv in H0; Unfold bound; Cut ``0</2``; [Intro | Apply Rlt_Rinv; Sup0]. -Elim (H0 ``/2`` H1); Intros. -Exists ``(sum_f_R0 An x)+2*(An (S x))``. -Unfold is_upper_bound; Intros; Unfold EUn in H3; Elim H3; Intros. -Rewrite H4; Assert H5 := (lt_eq_lt_dec x1 x). -Elim H5; Intros. -Elim a; Intro. -Replace (sum_f_R0 An x) with (Rplus (sum_f_R0 An x1) (sum_f_R0 [i:nat](An (plus (S x1) i)) (minus x (S x1)))). -Pattern 1 (sum_f_R0 An x1); Rewrite <- Rplus_Or; Rewrite Rplus_assoc; Apply Rle_compatibility. -Left; Apply gt0_plus_gt0_is_gt0. -Apply tech1; Intros; Apply H. -Apply Rmult_lt_pos; [Sup0 | Apply H]. -Symmetry; Apply tech2; Assumption. -Rewrite b; Pattern 1 (sum_f_R0 An x); Rewrite <- Rplus_Or; Apply Rle_compatibility. -Left; Apply Rmult_lt_pos; [Sup0 | Apply H]. -Replace (sum_f_R0 An x1) with (Rplus (sum_f_R0 An x) (sum_f_R0 [i:nat](An (plus (S x) i)) (minus x1 (S x)))). -Apply Rle_compatibility. -Cut (Rle (sum_f_R0 [i:nat](An (plus (S x) i)) (minus x1 (S x))) (Rmult (An (S x)) (sum_f_R0 [i:nat](pow ``/2`` i) (minus x1 (S x))))). -Intro; Apply Rle_trans with (Rmult (An (S x)) (sum_f_R0 [i:nat](pow ``/2`` i) (minus x1 (S x)))). -Assumption. -Rewrite <- (Rmult_sym (An (S x))); Apply Rle_monotony. -Left; Apply H. -Rewrite tech3. -Replace ``1-/2`` with ``/2``. -Unfold Rdiv; Rewrite Rinv_Rinv. -Pattern 3 ``2``; Rewrite <- Rmult_1r; Rewrite <- (Rmult_sym ``2``); Apply Rle_monotony. -Left; Sup0. -Left; Apply Rlt_anti_compatibility with ``(pow (/2) (S (minus x1 (S x))))``. -Replace ``(pow (/2) (S (minus x1 (S x))))+(1-(pow (/2) (S (minus x1 (S x)))))`` with R1; [Idtac | Ring]. -Rewrite <- (Rplus_sym ``1``); Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility. -Apply pow_lt; Apply Rlt_Rinv; Sup0. -DiscrR. -Apply r_Rmult_mult with ``2``. -Rewrite Rminus_distr; Rewrite <- Rinv_r_sym. -Ring. -DiscrR. -DiscrR. -Pattern 3 R1; Replace R1 with ``/1``; [Apply tech7; DiscrR | Apply Rinv_R1]. -Replace (An (S x)) with (An (plus (S x) O)). -Apply (tech6 [i:nat](An (plus (S x) i)) ``/2``). -Left; Apply Rlt_Rinv; Sup0. -Intro; Cut (n:nat)(ge n x)->``(An (S n))</2*(An n)``. -Intro; Replace (plus (S x) (S i)) with (S (plus (S x) i)). -Apply H6; Unfold ge; Apply tech8. -Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Do 2 Rewrite S_INR; Ring. -Intros; Unfold R_dist in H2; Apply Rlt_monotony_contra with ``/(An n)``. -Apply Rlt_Rinv; Apply H. -Do 2 Rewrite (Rmult_sym ``/(An n)``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Replace ``(An (S n))*/(An n)`` with ``(Rabsolu ((Rabsolu ((An (S n))/(An n)))-0))``. -Apply H2; Assumption. -Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Rewrite Rabsolu_right. -Unfold Rdiv; Reflexivity. -Left; Unfold Rdiv; Change ``0<(An (S n))*/(An n)``; Apply Rmult_lt_pos; [Apply H | Apply Rlt_Rinv; Apply H]. -Red; Intro; Assert H8 := (H n); Rewrite H7 in H8; Elim (Rlt_antirefl ? H8). -Replace (plus (S x) O) with (S x); [Reflexivity | Ring]. -Symmetry; Apply tech2; Assumption. -Exists (sum_f_R0 An O); Unfold EUn; Exists O; Reflexivity. -Intro; Elim X; Intros. -Apply Specif.existT with x; Apply tech10; [Unfold Un_growing; Intro; Rewrite tech5; Pattern 1 (sum_f_R0 An n); Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Apply H | Apply p]. +Lemma Alembert_C1 : + forall An:nat -> R, + (forall n:nat, 0 < An n) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l). +intros An H H0. +cut + (sigT (fun l:R => is_lub (EUn (fun N:nat => sum_f_R0 An N)) l) -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l)). +intro; apply X. +apply completeness. +unfold Un_cv in H0; unfold bound in |- *; cut (0 < / 2); + [ intro | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (H0 (/ 2) H1); intros. +exists (sum_f_R0 An x + 2 * An (S x)). +unfold is_upper_bound in |- *; intros; unfold EUn in H3; elim H3; intros. +rewrite H4; assert (H5 := lt_eq_lt_dec x1 x). +elim H5; intros. +elim a; intro. +replace (sum_f_R0 An x) with + (sum_f_R0 An x1 + sum_f_R0 (fun i:nat => An (S x1 + i)%nat) (x - S x1)). +pattern (sum_f_R0 An x1) at 1 in |- *; rewrite <- Rplus_0_r; + rewrite Rplus_assoc; apply Rplus_le_compat_l. +left; apply Rplus_lt_0_compat. +apply tech1; intros; apply H. +apply Rmult_lt_0_compat; [ prove_sup0 | apply H ]. +symmetry in |- *; apply tech2; assumption. +rewrite b; pattern (sum_f_R0 An x) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l. +left; apply Rmult_lt_0_compat; [ prove_sup0 | apply H ]. +replace (sum_f_R0 An x1) with + (sum_f_R0 An x + sum_f_R0 (fun i:nat => An (S x + i)%nat) (x1 - S x)). +apply Rplus_le_compat_l. +cut + (sum_f_R0 (fun i:nat => An (S x + i)%nat) (x1 - S x) <= + An (S x) * sum_f_R0 (fun i:nat => (/ 2) ^ i) (x1 - S x)). +intro; + apply Rle_trans with + (An (S x) * sum_f_R0 (fun i:nat => (/ 2) ^ i) (x1 - S x)). +assumption. +rewrite <- (Rmult_comm (An (S x))); apply Rmult_le_compat_l. +left; apply H. +rewrite tech3. +replace (1 - / 2) with (/ 2). +unfold Rdiv in |- *; rewrite Rinv_involutive. +pattern 2 at 3 in |- *; rewrite <- Rmult_1_r; rewrite <- (Rmult_comm 2); + apply Rmult_le_compat_l. +left; prove_sup0. +left; apply Rplus_lt_reg_r with ((/ 2) ^ S (x1 - S x)). +replace ((/ 2) ^ S (x1 - S x) + (1 - (/ 2) ^ S (x1 - S x))) with 1; + [ idtac | ring ]. +rewrite <- (Rplus_comm 1); pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_lt_compat_l. +apply pow_lt; apply Rinv_0_lt_compat; prove_sup0. +discrR. +apply Rmult_eq_reg_l with 2. +rewrite Rmult_minus_distr_l; rewrite <- Rinv_r_sym. +ring. +discrR. +discrR. +pattern 1 at 3 in |- *; replace 1 with (/ 1); + [ apply tech7; discrR | apply Rinv_1 ]. +replace (An (S x)) with (An (S x + 0)%nat). +apply (tech6 (fun i:nat => An (S x + i)%nat) (/ 2)). +left; apply Rinv_0_lt_compat; prove_sup0. +intro; cut (forall n:nat, (n >= x)%nat -> An (S n) < / 2 * An n). +intro; replace (S x + S i)%nat with (S (S x + i)). +apply H6; unfold ge in |- *; apply tech8. +apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; do 2 rewrite S_INR; ring. +intros; unfold R_dist in H2; apply Rmult_lt_reg_l with (/ An n). +apply Rinv_0_lt_compat; apply H. +do 2 rewrite (Rmult_comm (/ An n)); rewrite Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; + replace (An (S n) * / An n) with (Rabs (Rabs (An (S n) / An n) - 0)). +apply H2; assumption. +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + rewrite Rabs_Rabsolu; rewrite Rabs_right. +unfold Rdiv in |- *; reflexivity. +left; unfold Rdiv in |- *; change (0 < An (S n) * / An n) in |- *; + apply Rmult_lt_0_compat; [ apply H | apply Rinv_0_lt_compat; apply H ]. +red in |- *; intro; assert (H8 := H n); rewrite H7 in H8; + elim (Rlt_irrefl _ H8). +replace (S x + 0)%nat with (S x); [ reflexivity | ring ]. +symmetry in |- *; apply tech2; assumption. +exists (sum_f_R0 An 0); unfold EUn in |- *; exists 0%nat; reflexivity. +intro; elim X; intros. +apply existT with x; apply tech10; + [ unfold Un_growing in |- *; intro; rewrite tech5; + pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l; left; apply H + | apply p ]. Qed. -Lemma Alembert_C2 : (An:nat->R) ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) R0) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). -Intros. -Pose Vn := [i:nat]``(2*(Rabsolu (An i))+(An i))/2``. -Pose Wn := [i:nat]``(2*(Rabsolu (An i))-(An i))/2``. -Cut (n:nat)``0<(Vn n)``. -Intro; Cut (n:nat)``0<(Wn n)``. -Intro; Cut (Un_cv [n:nat](Rabsolu ``(Vn (S n))/(Vn n)``) ``0``). -Intro; Cut (Un_cv [n:nat](Rabsolu ``(Wn (S n))/(Wn n)``) ``0``). -Intro; Assert H5 := (Alembert_C1 Vn H1 H3). -Assert H6 := (Alembert_C1 Wn H2 H4). -Elim H5; Intros. -Elim H6; Intros. -Apply Specif.existT with ``x-x0``; Unfold Un_cv; Unfold Un_cv in p; Unfold Un_cv in p0; Intros; Cut ``0<eps/2``. -Intro; Elim (p ``eps/2`` H8); Clear p; Intros. -Elim (p0 ``eps/2`` H8); Clear p0; Intros. -Pose N := (max x1 x2). -Exists N; Intros; Replace (sum_f_R0 An n) with (Rminus (sum_f_R0 Vn n) (sum_f_R0 Wn n)). -Unfold R_dist; Replace (Rminus (Rminus (sum_f_R0 Vn n) (sum_f_R0 Wn n)) (Rminus x x0)) with (Rplus (Rminus (sum_f_R0 Vn n) x) (Ropp (Rminus (sum_f_R0 Wn n) x0))); [Idtac | Ring]; Apply Rle_lt_trans with (Rplus (Rabsolu (Rminus (sum_f_R0 Vn n) x)) (Rabsolu (Ropp (Rminus (sum_f_R0 Wn n) x0)))). -Apply Rabsolu_triang. -Rewrite Rabsolu_Ropp; Apply Rlt_le_trans with ``eps/2+eps/2``. -Apply Rplus_lt. -Unfold R_dist in H9; Apply H9; Unfold ge; Apply le_trans with N; [Unfold N; Apply le_max_l | Assumption]. -Unfold R_dist in H10; Apply H10; Unfold ge; Apply le_trans with N; [Unfold N; Apply le_max_r | Assumption]. -Right; Symmetry; Apply double_var. -Symmetry; Apply tech11; Intro; Unfold Vn Wn; Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/2``); Apply r_Rmult_mult with ``2``. -Rewrite Rminus_distr; Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Ring. -DiscrR. -DiscrR. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Cut (n:nat)``/2*(Rabsolu (An n))<=(Wn n)<=(3*/2)*(Rabsolu (An n))``. -Intro; Cut (n:nat)``/(Wn n)<=2*/(Rabsolu (An n))``. -Intro; Cut (n:nat)``(Wn (S n))/(Wn n)<=3*(Rabsolu (An (S n))/(An n))``. -Intro; Unfold Un_cv; Intros; Unfold Un_cv in H0; Cut ``0<eps/3``. -Intro; Elim (H0 ``eps/3`` H8); Intros. -Exists x; Intros. -Assert H11 := (H9 n H10). -Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Unfold R_dist in H11; Unfold Rminus in H11; Rewrite Ropp_O in H11; Rewrite Rplus_Or in H11; Rewrite Rabsolu_Rabsolu in H11; Rewrite Rabsolu_right. -Apply Rle_lt_trans with ``3*(Rabsolu ((An (S n))/(An n)))``. -Apply H6. -Apply Rlt_monotony_contra with ``/3``. -Apply Rlt_Rinv; Sup0. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]; Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Unfold Rdiv in H11; Exact H11. -Left; Change ``0<(Wn (S n))/(Wn n)``; Unfold Rdiv; Apply Rmult_lt_pos. -Apply H2. -Apply Rlt_Rinv; Apply H2. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Intro; Unfold Rdiv; Rewrite Rabsolu_mult; Rewrite <- Rmult_assoc; Replace ``3`` with ``2*(3*/2)``; [Idtac | Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m; DiscrR]; Apply Rle_trans with ``(Wn (S n))*2*/(Rabsolu (An n))``. -Rewrite Rmult_assoc; Apply Rle_monotony. -Left; Apply H2. -Apply H5. -Rewrite Rabsolu_Rinv. -Replace ``(Wn (S n))*2*/(Rabsolu (An n))`` with ``(2*/(Rabsolu (An n)))*(Wn (S n))``; [Idtac | Ring]; Replace ``2*(3*/2)*(Rabsolu (An (S n)))*/(Rabsolu (An n))`` with ``(2*/(Rabsolu (An n)))*((3*/2)*(Rabsolu (An (S n))))``; [Idtac | Ring]; Apply Rle_monotony. -Left; Apply Rmult_lt_pos. -Sup0. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Apply H. -Elim (H4 (S n)); Intros; Assumption. -Apply H. -Intro; Apply Rle_monotony_contra with (Wn n). -Apply H2. -Rewrite <- Rinv_r_sym. -Apply Rle_monotony_contra with (Rabsolu (An n)). -Apply Rabsolu_pos_lt; Apply H. -Rewrite Rmult_1r; Replace ``(Rabsolu (An n))*((Wn n)*(2*/(Rabsolu (An n))))`` with ``2*(Wn n)*((Rabsolu (An n))*/(Rabsolu (An n)))``; [Idtac | Ring]; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Apply Rle_monotony_contra with ``/2``. -Apply Rlt_Rinv; Sup0. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Elim (H4 n); Intros; Assumption. -DiscrR. -Apply Rabsolu_no_R0; Apply H. -Red; Intro; Assert H6 := (H2 n); Rewrite H5 in H6; Elim (Rlt_antirefl ? H6). -Intro; Split. -Unfold Wn; Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Apply Rle_monotony. -Left; Apply Rlt_Rinv; Sup0. -Pattern 1 (Rabsolu (An n)); Rewrite <- Rplus_Or; Rewrite double; Unfold Rminus; Rewrite Rplus_assoc; Apply Rle_compatibility. -Apply Rle_anti_compatibility with (An n). -Rewrite Rplus_Or; Rewrite (Rplus_sym (An n)); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply Rle_Rabsolu. -Unfold Wn; Unfold Rdiv; Repeat Rewrite <- (Rmult_sym ``/2``); Repeat Rewrite Rmult_assoc; Apply Rle_monotony. -Left; Apply Rlt_Rinv; Sup0. -Unfold Rminus; Rewrite double; Replace ``3*(Rabsolu (An n))`` with ``(Rabsolu (An n))+(Rabsolu (An n))+(Rabsolu (An n))``; [Idtac | Ring]; Repeat Rewrite Rplus_assoc; Repeat Apply Rle_compatibility. -Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. -Cut (n:nat)``/2*(Rabsolu (An n))<=(Vn n)<=(3*/2)*(Rabsolu (An n))``. -Intro; Cut (n:nat)``/(Vn n)<=2*/(Rabsolu (An n))``. -Intro; Cut (n:nat)``(Vn (S n))/(Vn n)<=3*(Rabsolu (An (S n))/(An n))``. -Intro; Unfold Un_cv; Intros; Unfold Un_cv in H1; Cut ``0<eps/3``. -Intro; Elim (H0 ``eps/3`` H7); Intros. -Exists x; Intros. -Assert H10 := (H8 n H9). -Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Unfold R_dist in H10; Unfold Rminus in H10; Rewrite Ropp_O in H10; Rewrite Rplus_Or in H10; Rewrite Rabsolu_Rabsolu in H10; Rewrite Rabsolu_right. -Apply Rle_lt_trans with ``3*(Rabsolu ((An (S n))/(An n)))``. -Apply H5. -Apply Rlt_monotony_contra with ``/3``. -Apply Rlt_Rinv; Sup0. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]; Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Unfold Rdiv in H10; Exact H10. -Left; Change ``0<(Vn (S n))/(Vn n)``; Unfold Rdiv; Apply Rmult_lt_pos. -Apply H1. -Apply Rlt_Rinv; Apply H1. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Intro; Unfold Rdiv; Rewrite Rabsolu_mult; Rewrite <- Rmult_assoc; Replace ``3`` with ``2*(3*/2)``; [Idtac | Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m; DiscrR]; Apply Rle_trans with ``(Vn (S n))*2*/(Rabsolu (An n))``. -Rewrite Rmult_assoc; Apply Rle_monotony. -Left; Apply H1. -Apply H4. -Rewrite Rabsolu_Rinv. -Replace ``(Vn (S n))*2*/(Rabsolu (An n))`` with ``(2*/(Rabsolu (An n)))*(Vn (S n))``; [Idtac | Ring]; Replace ``2*(3*/2)*(Rabsolu (An (S n)))*/(Rabsolu (An n))`` with ``(2*/(Rabsolu (An n)))*((3*/2)*(Rabsolu (An (S n))))``; [Idtac | Ring]; Apply Rle_monotony. -Left; Apply Rmult_lt_pos. -Sup0. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Apply H. -Elim (H3 (S n)); Intros; Assumption. -Apply H. -Intro; Apply Rle_monotony_contra with (Vn n). -Apply H1. -Rewrite <- Rinv_r_sym. -Apply Rle_monotony_contra with (Rabsolu (An n)). -Apply Rabsolu_pos_lt; Apply H. -Rewrite Rmult_1r; Replace ``(Rabsolu (An n))*((Vn n)*(2*/(Rabsolu (An n))))`` with ``2*(Vn n)*((Rabsolu (An n))*/(Rabsolu (An n)))``; [Idtac | Ring]; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Apply Rle_monotony_contra with ``/2``. -Apply Rlt_Rinv; Sup0. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Elim (H3 n); Intros; Assumption. -DiscrR. -Apply Rabsolu_no_R0; Apply H. -Red; Intro; Assert H5 := (H1 n); Rewrite H4 in H5; Elim (Rlt_antirefl ? H5). -Intro; Split. -Unfold Vn; Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Apply Rle_monotony. -Left; Apply Rlt_Rinv; Sup0. -Pattern 1 (Rabsolu (An n)); Rewrite <- Rplus_Or; Rewrite double; Rewrite Rplus_assoc; Apply Rle_compatibility. -Apply Rle_anti_compatibility with ``-(An n)``; Rewrite Rplus_Or; Rewrite <- (Rplus_sym (An n)); Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. -Unfold Vn; Unfold Rdiv; Repeat Rewrite <- (Rmult_sym ``/2``); Repeat Rewrite Rmult_assoc; Apply Rle_monotony. -Left; Apply Rlt_Rinv; Sup0. -Unfold Rminus; Rewrite double; Replace ``3*(Rabsolu (An n))`` with ``(Rabsolu (An n))+(Rabsolu (An n))+(Rabsolu (An n))``; [Idtac | Ring]; Repeat Rewrite Rplus_assoc; Repeat Apply Rle_compatibility; Apply Rle_Rabsolu. -Intro; Unfold Wn; Unfold Rdiv; Rewrite <- (Rmult_Or ``/2``); Rewrite <- (Rmult_sym ``/2``); Apply Rlt_monotony. -Apply Rlt_Rinv; Sup0. -Apply Rlt_anti_compatibility with (An n); Rewrite Rplus_Or; Unfold Rminus; Rewrite (Rplus_sym (An n)); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply Rle_lt_trans with (Rabsolu (An n)). -Apply Rle_Rabsolu. -Rewrite double; Pattern 1 (Rabsolu (An n)); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rabsolu_pos_lt; Apply H. -Intro; Unfold Vn; Unfold Rdiv; Rewrite <- (Rmult_Or ``/2``); Rewrite <- (Rmult_sym ``/2``); Apply Rlt_monotony. -Apply Rlt_Rinv; Sup0. -Apply Rlt_anti_compatibility with ``-(An n)``; Rewrite Rplus_Or; Unfold Rminus; Rewrite (Rplus_sym ``-(An n)``); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Apply Rle_lt_trans with (Rabsolu (An n)). -Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. -Rewrite double; Pattern 1 (Rabsolu (An n)); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rabsolu_pos_lt; Apply H. +Lemma Alembert_C2 : + forall An:nat -> R, + (forall n:nat, An n <> 0) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l). +intros. +pose (Vn := fun i:nat => (2 * Rabs (An i) + An i) / 2). +pose (Wn := fun i:nat => (2 * Rabs (An i) - An i) / 2). +cut (forall n:nat, 0 < Vn n). +intro; cut (forall n:nat, 0 < Wn n). +intro; cut (Un_cv (fun n:nat => Rabs (Vn (S n) / Vn n)) 0). +intro; cut (Un_cv (fun n:nat => Rabs (Wn (S n) / Wn n)) 0). +intro; assert (H5 := Alembert_C1 Vn H1 H3). +assert (H6 := Alembert_C1 Wn H2 H4). +elim H5; intros. +elim H6; intros. +apply existT with (x - x0); unfold Un_cv in |- *; unfold Un_cv in p; + unfold Un_cv in p0; intros; cut (0 < eps / 2). +intro; elim (p (eps / 2) H8); clear p; intros. +elim (p0 (eps / 2) H8); clear p0; intros. +pose (N := max x1 x2). +exists N; intros; + replace (sum_f_R0 An n) with (sum_f_R0 Vn n - sum_f_R0 Wn n). +unfold R_dist in |- *; + replace (sum_f_R0 Vn n - sum_f_R0 Wn n - (x - x0)) with + (sum_f_R0 Vn n - x + - (sum_f_R0 Wn n - x0)); [ idtac | ring ]; + apply Rle_lt_trans with + (Rabs (sum_f_R0 Vn n - x) + Rabs (- (sum_f_R0 Wn n - x0))). +apply Rabs_triang. +rewrite Rabs_Ropp; apply Rlt_le_trans with (eps / 2 + eps / 2). +apply Rplus_lt_compat. +unfold R_dist in H9; apply H9; unfold ge in |- *; apply le_trans with N; + [ unfold N in |- *; apply le_max_l | assumption ]. +unfold R_dist in H10; apply H10; unfold ge in |- *; apply le_trans with N; + [ unfold N in |- *; apply le_max_r | assumption ]. +right; symmetry in |- *; apply double_var. +symmetry in |- *; apply tech11; intro; unfold Vn, Wn in |- *; + unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ 2)); + apply Rmult_eq_reg_l with 2. +rewrite Rmult_minus_distr_l; repeat rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +ring. +discrR. +discrR. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +cut (forall n:nat, / 2 * Rabs (An n) <= Wn n <= 3 * / 2 * Rabs (An n)). +intro; cut (forall n:nat, / Wn n <= 2 * / Rabs (An n)). +intro; cut (forall n:nat, Wn (S n) / Wn n <= 3 * Rabs (An (S n) / An n)). +intro; unfold Un_cv in |- *; intros; unfold Un_cv in H0; cut (0 < eps / 3). +intro; elim (H0 (eps / 3) H8); intros. +exists x; intros. +assert (H11 := H9 n H10). +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold R_dist in H11; + unfold Rminus in H11; rewrite Ropp_0 in H11; rewrite Rplus_0_r in H11; + rewrite Rabs_Rabsolu in H11; rewrite Rabs_right. +apply Rle_lt_trans with (3 * Rabs (An (S n) / An n)). +apply H6. +apply Rmult_lt_reg_l with (/ 3). +apply Rinv_0_lt_compat; prove_sup0. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]; + rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H11; + exact H11. +left; change (0 < Wn (S n) / Wn n) in |- *; unfold Rdiv in |- *; + apply Rmult_lt_0_compat. +apply H2. +apply Rinv_0_lt_compat; apply H2. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +intro; unfold Rdiv in |- *; rewrite Rabs_mult; rewrite <- Rmult_assoc; + replace 3 with (2 * (3 * / 2)); + [ idtac | rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR ]; + apply Rle_trans with (Wn (S n) * 2 * / Rabs (An n)). +rewrite Rmult_assoc; apply Rmult_le_compat_l. +left; apply H2. +apply H5. +rewrite Rabs_Rinv. +replace (Wn (S n) * 2 * / Rabs (An n)) with (2 * / Rabs (An n) * Wn (S n)); + [ idtac | ring ]; + replace (2 * (3 * / 2) * Rabs (An (S n)) * / Rabs (An n)) with + (2 * / Rabs (An n) * (3 * / 2 * Rabs (An (S n)))); + [ idtac | ring ]; apply Rmult_le_compat_l. +left; apply Rmult_lt_0_compat. +prove_sup0. +apply Rinv_0_lt_compat; apply Rabs_pos_lt; apply H. +elim (H4 (S n)); intros; assumption. +apply H. +intro; apply Rmult_le_reg_l with (Wn n). +apply H2. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with (Rabs (An n)). +apply Rabs_pos_lt; apply H. +rewrite Rmult_1_r; + replace (Rabs (An n) * (Wn n * (2 * / Rabs (An n)))) with + (2 * Wn n * (Rabs (An n) * / Rabs (An n))); [ idtac | ring ]; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; apply Rmult_le_reg_l with (/ 2). +apply Rinv_0_lt_compat; prove_sup0. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; elim (H4 n); intros; assumption. +discrR. +apply Rabs_no_R0; apply H. +red in |- *; intro; assert (H6 := H2 n); rewrite H5 in H6; + elim (Rlt_irrefl _ H6). +intro; split. +unfold Wn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; prove_sup0. +pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; rewrite double; + unfold Rminus in |- *; rewrite Rplus_assoc; apply Rplus_le_compat_l. +apply Rplus_le_reg_l with (An n). +rewrite Rplus_0_r; rewrite (Rplus_comm (An n)); rewrite Rplus_assoc; + rewrite Rplus_opp_l; rewrite Rplus_0_r; apply RRle_abs. +unfold Wn in |- *; unfold Rdiv in |- *; repeat rewrite <- (Rmult_comm (/ 2)); + repeat rewrite Rmult_assoc; apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; prove_sup0. +unfold Rminus in |- *; rewrite double; + replace (3 * Rabs (An n)) with (Rabs (An n) + Rabs (An n) + Rabs (An n)); + [ idtac | ring ]; repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l. +rewrite <- Rabs_Ropp; apply RRle_abs. +cut (forall n:nat, / 2 * Rabs (An n) <= Vn n <= 3 * / 2 * Rabs (An n)). +intro; cut (forall n:nat, / Vn n <= 2 * / Rabs (An n)). +intro; cut (forall n:nat, Vn (S n) / Vn n <= 3 * Rabs (An (S n) / An n)). +intro; unfold Un_cv in |- *; intros; unfold Un_cv in H1; cut (0 < eps / 3). +intro; elim (H0 (eps / 3) H7); intros. +exists x; intros. +assert (H10 := H8 n H9). +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold R_dist in H10; + unfold Rminus in H10; rewrite Ropp_0 in H10; rewrite Rplus_0_r in H10; + rewrite Rabs_Rabsolu in H10; rewrite Rabs_right. +apply Rle_lt_trans with (3 * Rabs (An (S n) / An n)). +apply H5. +apply Rmult_lt_reg_l with (/ 3). +apply Rinv_0_lt_compat; prove_sup0. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]; + rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H10; + exact H10. +left; change (0 < Vn (S n) / Vn n) in |- *; unfold Rdiv in |- *; + apply Rmult_lt_0_compat. +apply H1. +apply Rinv_0_lt_compat; apply H1. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +intro; unfold Rdiv in |- *; rewrite Rabs_mult; rewrite <- Rmult_assoc; + replace 3 with (2 * (3 * / 2)); + [ idtac | rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR ]; + apply Rle_trans with (Vn (S n) * 2 * / Rabs (An n)). +rewrite Rmult_assoc; apply Rmult_le_compat_l. +left; apply H1. +apply H4. +rewrite Rabs_Rinv. +replace (Vn (S n) * 2 * / Rabs (An n)) with (2 * / Rabs (An n) * Vn (S n)); + [ idtac | ring ]; + replace (2 * (3 * / 2) * Rabs (An (S n)) * / Rabs (An n)) with + (2 * / Rabs (An n) * (3 * / 2 * Rabs (An (S n)))); + [ idtac | ring ]; apply Rmult_le_compat_l. +left; apply Rmult_lt_0_compat. +prove_sup0. +apply Rinv_0_lt_compat; apply Rabs_pos_lt; apply H. +elim (H3 (S n)); intros; assumption. +apply H. +intro; apply Rmult_le_reg_l with (Vn n). +apply H1. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with (Rabs (An n)). +apply Rabs_pos_lt; apply H. +rewrite Rmult_1_r; + replace (Rabs (An n) * (Vn n * (2 * / Rabs (An n)))) with + (2 * Vn n * (Rabs (An n) * / Rabs (An n))); [ idtac | ring ]; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; apply Rmult_le_reg_l with (/ 2). +apply Rinv_0_lt_compat; prove_sup0. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; elim (H3 n); intros; assumption. +discrR. +apply Rabs_no_R0; apply H. +red in |- *; intro; assert (H5 := H1 n); rewrite H4 in H5; + elim (Rlt_irrefl _ H5). +intro; split. +unfold Vn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; prove_sup0. +pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; rewrite double; + rewrite Rplus_assoc; apply Rplus_le_compat_l. +apply Rplus_le_reg_l with (- An n); rewrite Rplus_0_r; + rewrite <- (Rplus_comm (An n)); rewrite <- Rplus_assoc; + rewrite Rplus_opp_l; rewrite Rplus_0_l; rewrite <- Rabs_Ropp; + apply RRle_abs. +unfold Vn in |- *; unfold Rdiv in |- *; repeat rewrite <- (Rmult_comm (/ 2)); + repeat rewrite Rmult_assoc; apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; prove_sup0. +unfold Rminus in |- *; rewrite double; + replace (3 * Rabs (An n)) with (Rabs (An n) + Rabs (An n) + Rabs (An n)); + [ idtac | ring ]; repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l; + apply RRle_abs. +intro; unfold Wn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_0_r (/ 2)); + rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l. +apply Rinv_0_lt_compat; prove_sup0. +apply Rplus_lt_reg_r with (An n); rewrite Rplus_0_r; unfold Rminus in |- *; + rewrite (Rplus_comm (An n)); rewrite Rplus_assoc; + rewrite Rplus_opp_l; rewrite Rplus_0_r; + apply Rle_lt_trans with (Rabs (An n)). +apply RRle_abs. +rewrite double; pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_lt_compat_l; apply Rabs_pos_lt; apply H. +intro; unfold Vn in |- *; unfold Rdiv in |- *; rewrite <- (Rmult_0_r (/ 2)); + rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l. +apply Rinv_0_lt_compat; prove_sup0. +apply Rplus_lt_reg_r with (- An n); rewrite Rplus_0_r; unfold Rminus in |- *; + rewrite (Rplus_comm (- An n)); rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; + apply Rle_lt_trans with (Rabs (An n)). +rewrite <- Rabs_Ropp; apply RRle_abs. +rewrite double; pattern (Rabs (An n)) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_lt_compat_l; apply Rabs_pos_lt; apply H. Qed. -Lemma AlembertC3_step1 : (An:nat->R;x:R) ``x<>0`` -> ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) ``0``) -> (SigT R [l:R](Pser An x l)). -Intros; Pose Bn := [i:nat]``(An i)*(pow x i)``. -Cut (n:nat)``(Bn n)<>0``. -Intro; Cut (Un_cv [n:nat](Rabsolu ``(Bn (S n))/(Bn n)``) ``0``). -Intro; Assert H4 := (Alembert_C2 Bn H2 H3). -Elim H4; Intros. -Apply Specif.existT with x0; Unfold Bn in p; Apply tech12; Assumption. -Unfold Un_cv; Intros; Unfold Un_cv in H1; Cut ``0<eps/(Rabsolu x)``. -Intro; Elim (H1 ``eps/(Rabsolu x)`` H4); Intros. -Exists x0; Intros; Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Unfold Bn; Replace ``((An (S n))*(pow x (S n)))/((An n)*(pow x n))`` with ``(An (S n))/(An n)*x``. -Rewrite Rabsolu_mult; Apply Rlt_monotony_contra with ``/(Rabsolu x)``. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. -Rewrite <- (Rmult_sym (Rabsolu x)); Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Unfold Rdiv in H5; Replace ``(Rabsolu ((An (S n))/(An n)))`` with ``(R_dist (Rabsolu ((An (S n))*/(An n))) 0)``. -Apply H5; Assumption. -Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Unfold Rdiv; Reflexivity. -Apply Rabsolu_no_R0; Assumption. -Replace (S n) with (plus n (1)); [Idtac | Ring]; Rewrite pow_add; Unfold Rdiv; Rewrite Rinv_Rmult. -Replace ``(An (plus n (S O)))*((pow x n)*(pow x (S O)))*(/(An n)*/(pow x n))`` with ``(An (plus n (S O)))*(pow x (S O))*/(An n)*((pow x n)*/(pow x n))``; [Idtac | Ring]; Rewrite <- Rinv_r_sym. -Simpl; Ring. -Apply pow_nonzero; Assumption. -Apply H0. -Apply pow_nonzero; Assumption. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption]. -Intro; Unfold Bn; Apply prod_neq_R0; [Apply H0 | Apply pow_nonzero; Assumption]. +Lemma AlembertC3_step1 : + forall (An:nat -> R) (x:R), + x <> 0 -> + (forall n:nat, An n <> 0) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 -> + sigT (fun l:R => Pser An x l). +intros; pose (Bn := fun i:nat => An i * x ^ i). +cut (forall n:nat, Bn n <> 0). +intro; cut (Un_cv (fun n:nat => Rabs (Bn (S n) / Bn n)) 0). +intro; assert (H4 := Alembert_C2 Bn H2 H3). +elim H4; intros. +apply existT with x0; unfold Bn in p; apply tech12; assumption. +unfold Un_cv in |- *; intros; unfold Un_cv in H1; cut (0 < eps / Rabs x). +intro; elim (H1 (eps / Rabs x) H4); intros. +exists x0; intros; unfold R_dist in |- *; unfold Rminus in |- *; + rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; + unfold Bn in |- *; + replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x). +rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs x). +apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. +rewrite <- (Rmult_comm (Rabs x)); rewrite <- Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H5; + replace (Rabs (An (S n) / An n)) with (R_dist (Rabs (An (S n) * / An n)) 0). +apply H5; assumption. +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; rewrite Rabs_Rabsolu; unfold Rdiv in |- *; + reflexivity. +apply Rabs_no_R0; assumption. +replace (S n) with (n + 1)%nat; [ idtac | ring ]; rewrite pow_add; + unfold Rdiv in |- *; rewrite Rinv_mult_distr. +replace (An (n + 1)%nat * (x ^ n * x ^ 1) * (/ An n * / x ^ n)) with + (An (n + 1)%nat * x ^ 1 * / An n * (x ^ n * / x ^ n)); + [ idtac | ring ]; rewrite <- Rinv_r_sym. +simpl in |- *; ring. +apply pow_nonzero; assumption. +apply H0. +apply pow_nonzero; assumption. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ]. +intro; unfold Bn in |- *; apply prod_neq_R0; + [ apply H0 | apply pow_nonzero; assumption ]. Qed. -Lemma AlembertC3_step2 : (An:nat->R;x:R) ``x==0`` -> (SigT R [l:R](Pser An x l)). -Intros; Apply Specif.existT with (An O). -Unfold Pser; Unfold infinit_sum; Intros; Exists O; Intros; Replace (sum_f_R0 [n0:nat]``(An n0)*(pow x n0)`` n) with (An O). -Unfold R_dist; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Induction n. -Simpl; Ring. -Rewrite tech5; Rewrite Hrecn; [Rewrite H; Simpl; Ring | Unfold ge; Apply le_O_n]. +Lemma AlembertC3_step2 : + forall (An:nat -> R) (x:R), x = 0 -> sigT (fun l:R => Pser An x l). +intros; apply existT with (An 0%nat). +unfold Pser in |- *; unfold infinit_sum in |- *; intros; exists 0%nat; intros; + replace (sum_f_R0 (fun n0:nat => An n0 * x ^ n0) n) with (An 0%nat). +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; assumption. +induction n as [| n Hrecn]. +simpl in |- *; ring. +rewrite tech5; rewrite Hrecn; + [ rewrite H; simpl in |- *; ring | unfold ge in |- *; apply le_O_n ]. Qed. (* An useful criterion of convergence for power series *) -Theorem Alembert_C3 : (An:nat->R;x:R) ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) ``0``) -> (SigT R [l:R](Pser An x l)). -Intros; Case (total_order_T x R0); Intro. -Elim s; Intro. -Cut ``x<>0``. -Intro; Apply AlembertC3_step1; Assumption. -Red; Intro; Rewrite H1 in a; Elim (Rlt_antirefl ? a). -Apply AlembertC3_step2; Assumption. -Cut ``x<>0``. -Intro; Apply AlembertC3_step1; Assumption. -Red; Intro; Rewrite H1 in r; Elim (Rlt_antirefl ? r). +Theorem Alembert_C3 : + forall (An:nat -> R) (x:R), + (forall n:nat, An n <> 0) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) 0 -> + sigT (fun l:R => Pser An x l). +intros; case (total_order_T x 0); intro. +elim s; intro. +cut (x <> 0). +intro; apply AlembertC3_step1; assumption. +red in |- *; intro; rewrite H1 in a; elim (Rlt_irrefl _ a). +apply AlembertC3_step2; assumption. +cut (x <> 0). +intro; apply AlembertC3_step1; assumption. +red in |- *; intro; rewrite H1 in r; elim (Rlt_irrefl _ r). Qed. -Lemma Alembert_C4 : (An:nat->R;k:R) ``0<=k<1`` -> ((n:nat)``0<(An n)``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). -Intros An k Hyp H H0. -Cut (sigTT R [l:R](is_lub (EUn [N:nat](sum_f_R0 An N)) l)) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). -Intro; Apply X. -Apply complet. -Assert H1 := (tech13 ? ? Hyp H0). -Elim H1; Intros. -Elim H2; Intros. -Elim H4; Intros. -Unfold bound; Exists ``(sum_f_R0 An x0)+/(1-x)*(An (S x0))``. -Unfold is_upper_bound; Intros; Unfold EUn in H6. -Elim H6; Intros. -Rewrite H7. -Assert H8 := (lt_eq_lt_dec x2 x0). -Elim H8; Intros. -Elim a; Intro. -Replace (sum_f_R0 An x0) with (Rplus (sum_f_R0 An x2) (sum_f_R0 [i:nat](An (plus (S x2) i)) (minus x0 (S x2)))). -Pattern 1 (sum_f_R0 An x2); Rewrite <- Rplus_Or. -Rewrite Rplus_assoc; Apply Rle_compatibility. -Left; Apply gt0_plus_gt0_is_gt0. -Apply tech1. -Intros; Apply H. -Apply Rmult_lt_pos. -Apply Rlt_Rinv; Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or; Replace ``x+(1-x)`` with R1; [Elim H3; Intros; Assumption | Ring]. -Apply H. -Symmetry; Apply tech2; Assumption. -Rewrite b; Pattern 1 (sum_f_R0 An x0); Rewrite <- Rplus_Or; Apply Rle_compatibility. -Left; Apply Rmult_lt_pos. -Apply Rlt_Rinv; Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or; Replace ``x+(1-x)`` with R1; [Elim H3; Intros; Assumption | Ring]. -Apply H. -Replace (sum_f_R0 An x2) with (Rplus (sum_f_R0 An x0) (sum_f_R0 [i:nat](An (plus (S x0) i)) (minus x2 (S x0)))). -Apply Rle_compatibility. -Cut (Rle (sum_f_R0 [i:nat](An (plus (S x0) i)) (minus x2 (S x0))) (Rmult (An (S x0)) (sum_f_R0 [i:nat](pow x i) (minus x2 (S x0))))). -Intro; Apply Rle_trans with (Rmult (An (S x0)) (sum_f_R0 [i:nat](pow x i) (minus x2 (S x0)))). -Assumption. -Rewrite <- (Rmult_sym (An (S x0))); Apply Rle_monotony. -Left; Apply H. -Rewrite tech3. -Unfold Rdiv; Apply Rle_monotony_contra with ``1-x``. -Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or. -Replace ``x+(1-x)`` with R1; [Elim H3; Intros; Assumption | Ring]. -Do 2 Rewrite (Rmult_sym ``1-x``). -Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Apply Rle_anti_compatibility with ``(pow x (S (minus x2 (S x0))))``. -Replace ``(pow x (S (minus x2 (S x0))))+(1-(pow x (S (minus x2 (S x0)))))`` with R1; [Idtac | Ring]. -Rewrite <- (Rplus_sym R1); Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rle_compatibility. -Left; Apply pow_lt. -Apply Rle_lt_trans with k. -Elim Hyp; Intros; Assumption. -Elim H3; Intros; Assumption. -Apply Rminus_eq_contra. -Red; Intro. -Elim H3; Intros. -Rewrite H10 in H12; Elim (Rlt_antirefl ? H12). -Red; Intro. -Elim H3; Intros. -Rewrite H10 in H12; Elim (Rlt_antirefl ? H12). -Replace (An (S x0)) with (An (plus (S x0) O)). -Apply (tech6 [i:nat](An (plus (S x0) i)) x). -Left; Apply Rle_lt_trans with k. -Elim Hyp; Intros; Assumption. -Elim H3; Intros; Assumption. -Intro. -Cut (n:nat)(ge n x0)->``(An (S n))<x*(An n)``. -Intro. -Replace (plus (S x0) (S i)) with (S (plus (S x0) i)). -Apply H9. -Unfold ge. -Apply tech8. - Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Do 2 Rewrite S_INR; Ring. -Intros. -Apply Rlt_monotony_contra with ``/(An n)``. -Apply Rlt_Rinv; Apply H. -Do 2 Rewrite (Rmult_sym ``/(An n)``). -Rewrite Rmult_assoc. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r. -Replace ``(An (S n))*/(An n)`` with ``(Rabsolu ((An (S n))/(An n)))``. -Apply H5; Assumption. -Rewrite Rabsolu_right. -Unfold Rdiv; Reflexivity. -Left; Unfold Rdiv; Change ``0<(An (S n))*/(An n)``; Apply Rmult_lt_pos. -Apply H. -Apply Rlt_Rinv; Apply H. -Red; Intro. -Assert H11 := (H n). -Rewrite H10 in H11; Elim (Rlt_antirefl ? H11). -Replace (plus (S x0) O) with (S x0); [Reflexivity | Ring]. -Symmetry; Apply tech2; Assumption. -Exists (sum_f_R0 An O); Unfold EUn; Exists O; Reflexivity. -Intro; Elim X; Intros. -Apply Specif.existT with x; Apply tech10; [Unfold Un_growing; Intro; Rewrite tech5; Pattern 1 (sum_f_R0 An n); Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Apply H | Apply p]. +Lemma Alembert_C4 : + forall (An:nat -> R) (k:R), + 0 <= k < 1 -> + (forall n:nat, 0 < An n) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) k -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l). +intros An k Hyp H H0. +cut + (sigT (fun l:R => is_lub (EUn (fun N:nat => sum_f_R0 An N)) l) -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l)). +intro; apply X. +apply completeness. +assert (H1 := tech13 _ _ Hyp H0). +elim H1; intros. +elim H2; intros. +elim H4; intros. +unfold bound in |- *; exists (sum_f_R0 An x0 + / (1 - x) * An (S x0)). +unfold is_upper_bound in |- *; intros; unfold EUn in H6. +elim H6; intros. +rewrite H7. +assert (H8 := lt_eq_lt_dec x2 x0). +elim H8; intros. +elim a; intro. +replace (sum_f_R0 An x0) with + (sum_f_R0 An x2 + sum_f_R0 (fun i:nat => An (S x2 + i)%nat) (x0 - S x2)). +pattern (sum_f_R0 An x2) at 1 in |- *; rewrite <- Rplus_0_r. +rewrite Rplus_assoc; apply Rplus_le_compat_l. +left; apply Rplus_lt_0_compat. +apply tech1. +intros; apply H. +apply Rmult_lt_0_compat. +apply Rinv_0_lt_compat; apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; + replace (x + (1 - x)) with 1; [ elim H3; intros; assumption | ring ]. +apply H. +symmetry in |- *; apply tech2; assumption. +rewrite b; pattern (sum_f_R0 An x0) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l. +left; apply Rmult_lt_0_compat. +apply Rinv_0_lt_compat; apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; + replace (x + (1 - x)) with 1; [ elim H3; intros; assumption | ring ]. +apply H. +replace (sum_f_R0 An x2) with + (sum_f_R0 An x0 + sum_f_R0 (fun i:nat => An (S x0 + i)%nat) (x2 - S x0)). +apply Rplus_le_compat_l. +cut + (sum_f_R0 (fun i:nat => An (S x0 + i)%nat) (x2 - S x0) <= + An (S x0) * sum_f_R0 (fun i:nat => x ^ i) (x2 - S x0)). +intro; + apply Rle_trans with (An (S x0) * sum_f_R0 (fun i:nat => x ^ i) (x2 - S x0)). +assumption. +rewrite <- (Rmult_comm (An (S x0))); apply Rmult_le_compat_l. +left; apply H. +rewrite tech3. +unfold Rdiv in |- *; apply Rmult_le_reg_l with (1 - x). +apply Rplus_lt_reg_r with x; rewrite Rplus_0_r. +replace (x + (1 - x)) with 1; [ elim H3; intros; assumption | ring ]. +do 2 rewrite (Rmult_comm (1 - x)). +rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; apply Rplus_le_reg_l with (x ^ S (x2 - S x0)). +replace (x ^ S (x2 - S x0) + (1 - x ^ S (x2 - S x0))) with 1; + [ idtac | ring ]. +rewrite <- (Rplus_comm 1); pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l. +left; apply pow_lt. +apply Rle_lt_trans with k. +elim Hyp; intros; assumption. +elim H3; intros; assumption. +apply Rminus_eq_contra. +red in |- *; intro. +elim H3; intros. +rewrite H10 in H12; elim (Rlt_irrefl _ H12). +red in |- *; intro. +elim H3; intros. +rewrite H10 in H12; elim (Rlt_irrefl _ H12). +replace (An (S x0)) with (An (S x0 + 0)%nat). +apply (tech6 (fun i:nat => An (S x0 + i)%nat) x). +left; apply Rle_lt_trans with k. +elim Hyp; intros; assumption. +elim H3; intros; assumption. +intro. +cut (forall n:nat, (n >= x0)%nat -> An (S n) < x * An n). +intro. +replace (S x0 + S i)%nat with (S (S x0 + i)). +apply H9. +unfold ge in |- *. +apply tech8. + apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; do 2 rewrite S_INR; + ring. +intros. +apply Rmult_lt_reg_l with (/ An n). +apply Rinv_0_lt_compat; apply H. +do 2 rewrite (Rmult_comm (/ An n)). +rewrite Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r. +replace (An (S n) * / An n) with (Rabs (An (S n) / An n)). +apply H5; assumption. +rewrite Rabs_right. +unfold Rdiv in |- *; reflexivity. +left; unfold Rdiv in |- *; change (0 < An (S n) * / An n) in |- *; + apply Rmult_lt_0_compat. +apply H. +apply Rinv_0_lt_compat; apply H. +red in |- *; intro. +assert (H11 := H n). +rewrite H10 in H11; elim (Rlt_irrefl _ H11). +replace (S x0 + 0)%nat with (S x0); [ reflexivity | ring ]. +symmetry in |- *; apply tech2; assumption. +exists (sum_f_R0 An 0); unfold EUn in |- *; exists 0%nat; reflexivity. +intro; elim X; intros. +apply existT with x; apply tech10; + [ unfold Un_growing in |- *; intro; rewrite tech5; + pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l; left; apply H + | apply p ]. Qed. -Lemma Alembert_C5 : (An:nat->R;k:R) ``0<=k<1`` -> ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). -Intros. -Cut (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)) -> (SigT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). -Intro Hyp0; Apply Hyp0. -Apply cv_cauchy_2. -Apply cauchy_abs. -Apply cv_cauchy_1. -Cut (SigT R [l:R](Un_cv [N:nat](sum_f_R0 [i:nat](Rabsolu (An i)) N) l)) -> (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 [i:nat](Rabsolu (An i)) N) l)). -Intro Hyp; Apply Hyp. -Apply (Alembert_C4 [i:nat](Rabsolu (An i)) k). -Assumption. -Intro; Apply Rabsolu_pos_lt; Apply H0. -Unfold Un_cv. -Unfold Un_cv in H1. -Unfold Rdiv. -Intros. -Elim (H1 eps H2); Intros. -Exists x; Intros. -Rewrite <- Rabsolu_Rinv. -Rewrite <- Rabsolu_mult. -Rewrite Rabsolu_Rabsolu. -Unfold Rdiv in H3; Apply H3; Assumption. -Apply H0. -Intro. -Elim X; Intros. -Apply existTT with x. -Assumption. -Intro. -Elim X; Intros. -Apply Specif.existT with x. -Assumption. +Lemma Alembert_C5 : + forall (An:nat -> R) (k:R), + 0 <= k < 1 -> + (forall n:nat, An n <> 0) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) k -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l). +intros. +cut + (sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l) -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l)). +intro Hyp0; apply Hyp0. +apply cv_cauchy_2. +apply cauchy_abs. +apply cv_cauchy_1. +cut + (sigT + (fun l:R => Un_cv (fun N:nat => sum_f_R0 (fun i:nat => Rabs (An i)) N) l) -> + sigT + (fun l:R => Un_cv (fun N:nat => sum_f_R0 (fun i:nat => Rabs (An i)) N) l)). +intro Hyp; apply Hyp. +apply (Alembert_C4 (fun i:nat => Rabs (An i)) k). +assumption. +intro; apply Rabs_pos_lt; apply H0. +unfold Un_cv in |- *. +unfold Un_cv in H1. +unfold Rdiv in |- *. +intros. +elim (H1 eps H2); intros. +exists x; intros. +rewrite <- Rabs_Rinv. +rewrite <- Rabs_mult. +rewrite Rabs_Rabsolu. +unfold Rdiv in H3; apply H3; assumption. +apply H0. +intro. +elim X; intros. +apply existT with x. +assumption. +intro. +elim X; intros. +apply existT with x. +assumption. Qed. (* Convergence of power series in D(O,1/k) *) (* k=0 is described in Alembert_C3 *) -Lemma Alembert_C6 : (An:nat->R;x,k:R) ``0<k`` -> ((n:nat)``(An n)<>0``) -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> ``(Rabsolu x)</k`` -> (SigT R [l:R](Pser An x l)). -Intros. -Cut (SigT R [l:R](Un_cv [N:nat](sum_f_R0 [i:nat]``(An i)*(pow x i)`` N) l)). -Intro. -Elim X; Intros. -Apply Specif.existT with x0. -Apply tech12; Assumption. -Case (total_order_T x R0); Intro. -Elim s; Intro. -EApply Alembert_C5 with ``k*(Rabsolu x)``. -Split. -Unfold Rdiv; Apply Rmult_le_pos. -Left; Assumption. -Left; Apply Rabsolu_pos_lt. -Red; Intro; Rewrite H3 in a; Elim (Rlt_antirefl ? a). -Apply Rlt_monotony_contra with ``/k``. -Apply Rlt_Rinv; Assumption. -Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Rewrite Rmult_1r; Assumption. -Red; Intro; Rewrite H3 in H; Elim (Rlt_antirefl ? H). -Intro; Apply prod_neq_R0. -Apply H0. -Apply pow_nonzero. -Red; Intro; Rewrite H3 in a; Elim (Rlt_antirefl ? a). -Unfold Un_cv; Unfold Un_cv in H1. -Intros. -Cut ``0<eps/(Rabsolu x)``. -Intro. -Elim (H1 ``eps/(Rabsolu x)`` H4); Intros. -Exists x0. -Intros. -Replace ``((An (S n))*(pow x (S n)))/((An n)*(pow x n))`` with ``(An (S n))/(An n)*x``. -Unfold R_dist. -Rewrite Rabsolu_mult. -Replace ``(Rabsolu ((An (S n))/(An n)))*(Rabsolu x)-k*(Rabsolu x)`` with ``(Rabsolu x)*((Rabsolu ((An (S n))/(An n)))-k)``; [Idtac | Ring]. -Rewrite Rabsolu_mult. -Rewrite Rabsolu_Rabsolu. -Apply Rlt_monotony_contra with ``/(Rabsolu x)``. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt. -Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a). -Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Rewrite <- (Rmult_sym eps). -Unfold R_dist in H5. -Unfold Rdiv; Unfold Rdiv in H5; Apply H5; Assumption. -Apply Rabsolu_no_R0. -Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a). -Unfold Rdiv; Replace (S n) with (plus n (1)); [Idtac | Ring]. -Rewrite pow_add. -Simpl. -Rewrite Rmult_1r. -Rewrite Rinv_Rmult. -Replace ``(An (plus n (S O)))*((pow x n)*x)*(/(An n)*/(pow x n))`` with ``(An (plus n (S O)))*/(An n)*x*((pow x n)*/(pow x n))``; [Idtac | Ring]. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Reflexivity. -Apply pow_nonzero. -Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a). -Apply H0. -Apply pow_nonzero. -Red; Intro; Rewrite H7 in a; Elim (Rlt_antirefl ? a). -Unfold Rdiv; Apply Rmult_lt_pos. -Assumption. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt. -Red; Intro H7; Rewrite H7 in a; Elim (Rlt_antirefl ? a). -Apply Specif.existT with (An O). -Unfold Un_cv. -Intros. -Exists O. -Intros. -Unfold R_dist. -Replace (sum_f_R0 [i:nat]``(An i)*(pow x i)`` n) with (An O). -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Induction n. -Simpl; Ring. -Rewrite tech5. -Rewrite <- Hrecn. -Rewrite b; Simpl; Ring. -Unfold ge; Apply le_O_n. -EApply Alembert_C5 with ``k*(Rabsolu x)``. -Split. -Unfold Rdiv; Apply Rmult_le_pos. -Left; Assumption. -Left; Apply Rabsolu_pos_lt. -Red; Intro; Rewrite H3 in r; Elim (Rlt_antirefl ? r). -Apply Rlt_monotony_contra with ``/k``. -Apply Rlt_Rinv; Assumption. -Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Rewrite Rmult_1r; Assumption. -Red; Intro; Rewrite H3 in H; Elim (Rlt_antirefl ? H). -Intro; Apply prod_neq_R0. -Apply H0. -Apply pow_nonzero. -Red; Intro; Rewrite H3 in r; Elim (Rlt_antirefl ? r). -Unfold Un_cv; Unfold Un_cv in H1. -Intros. -Cut ``0<eps/(Rabsolu x)``. -Intro. -Elim (H1 ``eps/(Rabsolu x)`` H4); Intros. -Exists x0. -Intros. -Replace ``((An (S n))*(pow x (S n)))/((An n)*(pow x n))`` with ``(An (S n))/(An n)*x``. -Unfold R_dist. -Rewrite Rabsolu_mult. -Replace ``(Rabsolu ((An (S n))/(An n)))*(Rabsolu x)-k*(Rabsolu x)`` with ``(Rabsolu x)*((Rabsolu ((An (S n))/(An n)))-k)``; [Idtac | Ring]. -Rewrite Rabsolu_mult. -Rewrite Rabsolu_Rabsolu. -Apply Rlt_monotony_contra with ``/(Rabsolu x)``. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt. -Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r). -Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Rewrite <- (Rmult_sym eps). -Unfold R_dist in H5. -Unfold Rdiv; Unfold Rdiv in H5; Apply H5; Assumption. -Apply Rabsolu_no_R0. -Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r). -Unfold Rdiv; Replace (S n) with (plus n (1)); [Idtac | Ring]. -Rewrite pow_add. -Simpl. -Rewrite Rmult_1r. -Rewrite Rinv_Rmult. -Replace ``(An (plus n (S O)))*((pow x n)*x)*(/(An n)*/(pow x n))`` with ``(An (plus n (S O)))*/(An n)*x*((pow x n)*/(pow x n))``; [Idtac | Ring]. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Reflexivity. -Apply pow_nonzero. -Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r). -Apply H0. -Apply pow_nonzero. -Red; Intro; Rewrite H7 in r; Elim (Rlt_antirefl ? r). -Unfold Rdiv; Apply Rmult_lt_pos. -Assumption. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt. -Red; Intro H7; Rewrite H7 in r; Elim (Rlt_antirefl ? r). -Qed. +Lemma Alembert_C6 : + forall (An:nat -> R) (x k:R), + 0 < k -> + (forall n:nat, An n <> 0) -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) k -> + Rabs x < / k -> sigT (fun l:R => Pser An x l). +intros. +cut + (sigT + (fun l:R => Un_cv (fun N:nat => sum_f_R0 (fun i:nat => An i * x ^ i) N) l)). +intro. +elim X; intros. +apply existT with x0. +apply tech12; assumption. +case (total_order_T x 0); intro. +elim s; intro. +eapply Alembert_C5 with (k * Rabs x). +split. +unfold Rdiv in |- *; apply Rmult_le_pos. +left; assumption. +left; apply Rabs_pos_lt. +red in |- *; intro; rewrite H3 in a; elim (Rlt_irrefl _ a). +apply Rmult_lt_reg_l with (/ k). +apply Rinv_0_lt_compat; assumption. +rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +rewrite Rmult_1_r; assumption. +red in |- *; intro; rewrite H3 in H; elim (Rlt_irrefl _ H). +intro; apply prod_neq_R0. +apply H0. +apply pow_nonzero. +red in |- *; intro; rewrite H3 in a; elim (Rlt_irrefl _ a). +unfold Un_cv in |- *; unfold Un_cv in H1. +intros. +cut (0 < eps / Rabs x). +intro. +elim (H1 (eps / Rabs x) H4); intros. +exists x0. +intros. +replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x). +unfold R_dist in |- *. +rewrite Rabs_mult. +replace (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) with + (Rabs x * (Rabs (An (S n) / An n) - k)); [ idtac | ring ]. +rewrite Rabs_mult. +rewrite Rabs_Rabsolu. +apply Rmult_lt_reg_l with (/ Rabs x). +apply Rinv_0_lt_compat; apply Rabs_pos_lt. +red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). +rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +rewrite <- (Rmult_comm eps). +unfold R_dist in H5. +unfold Rdiv in |- *; unfold Rdiv in H5; apply H5; assumption. +apply Rabs_no_R0. +red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). +unfold Rdiv in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. +rewrite pow_add. +simpl in |- *. +rewrite Rmult_1_r. +rewrite Rinv_mult_distr. +replace (An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)) with + (An (n + 1)%nat * / An n * x * (x ^ n * / x ^ n)); + [ idtac | ring ]. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; reflexivity. +apply pow_nonzero. +red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). +apply H0. +apply pow_nonzero. +red in |- *; intro; rewrite H7 in a; elim (Rlt_irrefl _ a). +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +assumption. +apply Rinv_0_lt_compat; apply Rabs_pos_lt. +red in |- *; intro H7; rewrite H7 in a; elim (Rlt_irrefl _ a). +apply existT with (An 0%nat). +unfold Un_cv in |- *. +intros. +exists 0%nat. +intros. +unfold R_dist in |- *. +replace (sum_f_R0 (fun i:nat => An i * x ^ i) n) with (An 0%nat). +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. +induction n as [| n Hrecn]. +simpl in |- *; ring. +rewrite tech5. +rewrite <- Hrecn. +rewrite b; simpl in |- *; ring. +unfold ge in |- *; apply le_O_n. +eapply Alembert_C5 with (k * Rabs x). +split. +unfold Rdiv in |- *; apply Rmult_le_pos. +left; assumption. +left; apply Rabs_pos_lt. +red in |- *; intro; rewrite H3 in r; elim (Rlt_irrefl _ r). +apply Rmult_lt_reg_l with (/ k). +apply Rinv_0_lt_compat; assumption. +rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +rewrite Rmult_1_r; assumption. +red in |- *; intro; rewrite H3 in H; elim (Rlt_irrefl _ H). +intro; apply prod_neq_R0. +apply H0. +apply pow_nonzero. +red in |- *; intro; rewrite H3 in r; elim (Rlt_irrefl _ r). +unfold Un_cv in |- *; unfold Un_cv in H1. +intros. +cut (0 < eps / Rabs x). +intro. +elim (H1 (eps / Rabs x) H4); intros. +exists x0. +intros. +replace (An (S n) * x ^ S n / (An n * x ^ n)) with (An (S n) / An n * x). +unfold R_dist in |- *. +rewrite Rabs_mult. +replace (Rabs (An (S n) / An n) * Rabs x - k * Rabs x) with + (Rabs x * (Rabs (An (S n) / An n) - k)); [ idtac | ring ]. +rewrite Rabs_mult. +rewrite Rabs_Rabsolu. +apply Rmult_lt_reg_l with (/ Rabs x). +apply Rinv_0_lt_compat; apply Rabs_pos_lt. +red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). +rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +rewrite <- (Rmult_comm eps). +unfold R_dist in H5. +unfold Rdiv in |- *; unfold Rdiv in H5; apply H5; assumption. +apply Rabs_no_R0. +red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). +unfold Rdiv in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. +rewrite pow_add. +simpl in |- *. +rewrite Rmult_1_r. +rewrite Rinv_mult_distr. +replace (An (n + 1)%nat * (x ^ n * x) * (/ An n * / x ^ n)) with + (An (n + 1)%nat * / An n * x * (x ^ n * / x ^ n)); + [ idtac | ring ]. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; reflexivity. +apply pow_nonzero. +red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). +apply H0. +apply pow_nonzero. +red in |- *; intro; rewrite H7 in r; elim (Rlt_irrefl _ r). +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +assumption. +apply Rinv_0_lt_compat; apply Rabs_pos_lt. +red in |- *; intro H7; rewrite H7 in r; elim (Rlt_irrefl _ r). +Qed.
\ No newline at end of file diff --git a/theories/Reals/AltSeries.v b/theories/Reals/AltSeries.v index c35f18a73..e9be3fc02 100644 --- a/theories/Reals/AltSeries.v +++ b/theories/Reals/AltSeries.v @@ -8,156 +8,204 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Rseries. -Require SeqProp. -Require PartSum. -Require Max. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import Rseries. +Require Import SeqProp. +Require Import PartSum. +Require Import Max. Open Local Scope R_scope. (**********) -Definition tg_alt [Un:nat->R] : nat->R := [i:nat]``(pow (-1) i)*(Un i)``. -Definition positivity_seq [Un:nat->R] : Prop := (n:nat)``0<=(Un n)``. +Definition tg_alt (Un:nat -> R) (i:nat) : R := (-1) ^ i * Un i. +Definition positivity_seq (Un:nat -> R) : Prop := forall n:nat, 0 <= Un n. -Lemma CV_ALT_step0 : (Un:nat->R) (Un_decreasing Un) -> (Un_growing [N:nat](sum_f_R0 (tg_alt Un) (S (mult (2) N)))). -Intros; Unfold Un_growing; Intro. -Cut (mult (S (S O)) (S n)) = (S (S (mult (2) n))). -Intro; Rewrite H0. -Do 4 Rewrite tech5; Repeat Rewrite Rplus_assoc; Apply Rle_compatibility. -Pattern 1 (tg_alt Un (S (mult (S (S O)) n))); Rewrite <- Rplus_Or. -Apply Rle_compatibility. -Unfold tg_alt; Rewrite <- H0; Rewrite pow_1_odd; Rewrite pow_1_even; Rewrite Rmult_1l. -Apply Rle_anti_compatibility with ``(Un (S (mult (S (S O)) (S n))))``. -Rewrite Rplus_Or; Replace ``(Un (S (mult (S (S O)) (S n))))+((Un (mult (S (S O)) (S n)))+ -1*(Un (S (mult (S (S O)) (S n)))))`` with ``(Un (mult (S (S O)) (S n)))``; [Idtac | Ring]. -Apply H. -Cut (n:nat) (S n)=(plus n (1)); [Intro | Intro; Ring]. -Rewrite (H0 n); Rewrite (H0 (S (mult (2) n))); Rewrite (H0 (mult (2) n)); Ring. +Lemma CV_ALT_step0 : + forall Un:nat -> R, + Un_decreasing Un -> + Un_growing (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))). +intros; unfold Un_growing in |- *; intro. +cut ((2 * S n)%nat = S (S (2 * n))). +intro; rewrite H0. +do 4 rewrite tech5; repeat rewrite Rplus_assoc; apply Rplus_le_compat_l. +pattern (tg_alt Un (S (2 * n))) at 1 in |- *; rewrite <- Rplus_0_r. +apply Rplus_le_compat_l. +unfold tg_alt in |- *; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even; + rewrite Rmult_1_l. +apply Rplus_le_reg_l with (Un (S (2 * S n))). +rewrite Rplus_0_r; + replace (Un (S (2 * S n)) + (Un (2 * S n)%nat + -1 * Un (S (2 * S n)))) with + (Un (2 * S n)%nat); [ idtac | ring ]. +apply H. +cut (forall n:nat, S n = (n + 1)%nat); [ intro | intro; ring ]. +rewrite (H0 n); rewrite (H0 (S (2 * n))); rewrite (H0 (2 * n)%nat); ring. Qed. -Lemma CV_ALT_step1 : (Un:nat->R) (Un_decreasing Un) -> (Un_decreasing [N:nat](sum_f_R0 (tg_alt Un) (mult (2) N))). -Intros; Unfold Un_decreasing; Intro. -Cut (mult (S (S O)) (S n)) = (S (S (mult (2) n))). -Intro; Rewrite H0; Do 2 Rewrite tech5; Repeat Rewrite Rplus_assoc. -Pattern 2 (sum_f_R0 (tg_alt Un) (mult (S (S O)) n)); Rewrite <- Rplus_Or. -Apply Rle_compatibility. -Unfold tg_alt; Rewrite <- H0; Rewrite pow_1_odd; Rewrite pow_1_even; Rewrite Rmult_1l. -Apply Rle_anti_compatibility with ``(Un (S (mult (S (S O)) n)))``. -Rewrite Rplus_Or; Replace ``(Un (S (mult (S (S O)) n)))+( -1*(Un (S (mult (S (S O)) n)))+(Un (mult (S (S O)) (S n))))`` with ``(Un (mult (S (S O)) (S n)))``; [Idtac | Ring]. -Rewrite H0; Apply H. -Cut (n:nat) (S n)=(plus n (1)); [Intro | Intro; Ring]. -Rewrite (H0 n); Rewrite (H0 (S (mult (2) n))); Rewrite (H0 (mult (2) n)); Ring. +Lemma CV_ALT_step1 : + forall Un:nat -> R, + Un_decreasing Un -> + Un_decreasing (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N)). +intros; unfold Un_decreasing in |- *; intro. +cut ((2 * S n)%nat = S (S (2 * n))). +intro; rewrite H0; do 2 rewrite tech5; repeat rewrite Rplus_assoc. +pattern (sum_f_R0 (tg_alt Un) (2 * n)) at 2 in |- *; rewrite <- Rplus_0_r. +apply Rplus_le_compat_l. +unfold tg_alt in |- *; rewrite <- H0; rewrite pow_1_odd; rewrite pow_1_even; + rewrite Rmult_1_l. +apply Rplus_le_reg_l with (Un (S (2 * n))). +rewrite Rplus_0_r; + replace (Un (S (2 * n)) + (-1 * Un (S (2 * n)) + Un (2 * S n)%nat)) with + (Un (2 * S n)%nat); [ idtac | ring ]. +rewrite H0; apply H. +cut (forall n:nat, S n = (n + 1)%nat); [ intro | intro; ring ]. +rewrite (H0 n); rewrite (H0 (S (2 * n))); rewrite (H0 (2 * n)%nat); ring. Qed. (**********) -Lemma CV_ALT_step2 : (Un:nat->R;N:nat) (Un_decreasing Un) -> (positivity_seq Un) -> (Rle (sum_f_R0 [i:nat](tg_alt Un (S i)) (S (mult (2) N))) R0). -Intros; Induction N. -Simpl; Unfold tg_alt; Simpl; Rewrite Rmult_1r. -Replace ``-1* -1*(Un (S (S O)))`` with (Un (S (S O))); [Idtac | Ring]. -Apply Rle_anti_compatibility with ``(Un (S O))``; Rewrite Rplus_Or. -Replace ``(Un (S O))+ (-1*(Un (S O))+(Un (S (S O))))`` with (Un (S (S O))); [Apply H | Ring]. -Cut (S (mult (2) (S N))) = (S (S (S (mult (2) N)))). -Intro; Rewrite H1; Do 2 Rewrite tech5. -Apply Rle_trans with (sum_f_R0 [i:nat](tg_alt Un (S i)) (S (mult (S (S O)) N))). -Pattern 2 (sum_f_R0 [i:nat](tg_alt Un (S i)) (S (mult (S (S O)) N))); Rewrite <- Rplus_Or. -Rewrite Rplus_assoc; Apply Rle_compatibility. -Unfold tg_alt; Rewrite <- H1. -Rewrite pow_1_odd. -Cut (S (S (mult (2) (S N)))) = (mult (2) (S (S N))). -Intro; Rewrite H2; Rewrite pow_1_even; Rewrite Rmult_1l; Rewrite <- H2. -Apply Rle_anti_compatibility with ``(Un (S (mult (S (S O)) (S N))))``. -Rewrite Rplus_Or; Replace ``(Un (S (mult (S (S O)) (S N))))+( -1*(Un (S (mult (S (S O)) (S N))))+(Un (S (S (mult (S (S O)) (S N))))))`` with ``(Un (S (S (mult (S (S O)) (S N)))))``; [Idtac | Ring]. -Apply H. -Apply INR_eq; Rewrite mult_INR; Repeat Rewrite S_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Apply HrecN. -Apply INR_eq; Repeat Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Lemma CV_ALT_step2 : + forall (Un:nat -> R) (N:nat), + Un_decreasing Un -> + positivity_seq Un -> + sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N)) <= 0. +intros; induction N as [| N HrecN]. +simpl in |- *; unfold tg_alt in |- *; simpl in |- *; rewrite Rmult_1_r. +replace (-1 * -1 * Un 2%nat) with (Un 2%nat); [ idtac | ring ]. +apply Rplus_le_reg_l with (Un 1%nat); rewrite Rplus_0_r. +replace (Un 1%nat + (-1 * Un 1%nat + Un 2%nat)) with (Un 2%nat); + [ apply H | ring ]. +cut (S (2 * S N) = S (S (S (2 * N)))). +intro; rewrite H1; do 2 rewrite tech5. +apply Rle_trans with (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N))). +pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * N))) at 2 in |- *; + rewrite <- Rplus_0_r. +rewrite Rplus_assoc; apply Rplus_le_compat_l. +unfold tg_alt in |- *; rewrite <- H1. +rewrite pow_1_odd. +cut (S (S (2 * S N)) = (2 * S (S N))%nat). +intro; rewrite H2; rewrite pow_1_even; rewrite Rmult_1_l; rewrite <- H2. +apply Rplus_le_reg_l with (Un (S (2 * S N))). +rewrite Rplus_0_r; + replace (Un (S (2 * S N)) + (-1 * Un (S (2 * S N)) + Un (S (S (2 * S N))))) + with (Un (S (S (2 * S N)))); [ idtac | ring ]. +apply H. +apply INR_eq; rewrite mult_INR; repeat rewrite S_INR; rewrite mult_INR; + repeat rewrite S_INR; ring. +apply HrecN. +apply INR_eq; repeat rewrite S_INR; do 2 rewrite mult_INR; + repeat rewrite S_INR; ring. Qed. (* A more general inequality *) -Lemma CV_ALT_step3 : (Un:nat->R;N:nat) (Un_decreasing Un) -> (positivity_seq Un) -> (Rle (sum_f_R0 [i:nat](tg_alt Un (S i)) N) R0). -Intros; Induction N. -Simpl; Unfold tg_alt; Simpl; Rewrite Rmult_1r. -Apply Rle_anti_compatibility with (Un (S O)). -Rewrite Rplus_Or; Replace ``(Un (S O))+ -1*(Un (S O))`` with R0; [Apply H0 | Ring]. -Assert H1 := (even_odd_cor N). -Elim H1; Intros. -Elim H2; Intro. -Rewrite H3; Apply CV_ALT_step2; Assumption. -Rewrite H3; Rewrite tech5. -Apply Rle_trans with (sum_f_R0 [i:nat](tg_alt Un (S i)) (S (mult (S (S O)) x))). -Pattern 2 (sum_f_R0 [i:nat](tg_alt Un (S i)) (S (mult (S (S O)) x))); Rewrite <- Rplus_Or. -Apply Rle_compatibility. -Unfold tg_alt; Simpl. -Replace (plus x (plus x O)) with (mult (2) x); [Idtac | Ring]. -Rewrite pow_1_even. -Replace `` -1*( -1*( -1*1))*(Un (S (S (S (mult (S (S O)) x)))))`` with ``-(Un (S (S (S (mult (S (S O)) x)))))``; [Idtac | Ring]. -Apply Rle_anti_compatibility with (Un (S (S (S (mult (S (S O)) x))))). -Rewrite Rplus_Or; Rewrite Rplus_Ropp_r. -Apply H0. -Apply CV_ALT_step2; Assumption. +Lemma CV_ALT_step3 : + forall (Un:nat -> R) (N:nat), + Un_decreasing Un -> + positivity_seq Un -> sum_f_R0 (fun i:nat => tg_alt Un (S i)) N <= 0. +intros; induction N as [| N HrecN]. +simpl in |- *; unfold tg_alt in |- *; simpl in |- *; rewrite Rmult_1_r. +apply Rplus_le_reg_l with (Un 1%nat). +rewrite Rplus_0_r; replace (Un 1%nat + -1 * Un 1%nat) with 0; + [ apply H0 | ring ]. +assert (H1 := even_odd_cor N). +elim H1; intros. +elim H2; intro. +rewrite H3; apply CV_ALT_step2; assumption. +rewrite H3; rewrite tech5. +apply Rle_trans with (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * x))). +pattern (sum_f_R0 (fun i:nat => tg_alt Un (S i)) (S (2 * x))) at 2 in |- *; + rewrite <- Rplus_0_r. +apply Rplus_le_compat_l. +unfold tg_alt in |- *; simpl in |- *. +replace (x + (x + 0))%nat with (2 * x)%nat; [ idtac | ring ]. +rewrite pow_1_even. +replace (-1 * (-1 * (-1 * 1)) * Un (S (S (S (2 * x))))) with + (- Un (S (S (S (2 * x))))); [ idtac | ring ]. +apply Rplus_le_reg_l with (Un (S (S (S (2 * x))))). +rewrite Rplus_0_r; rewrite Rplus_opp_r. +apply H0. +apply CV_ALT_step2; assumption. Qed. (**********) -Lemma CV_ALT_step4 : (Un:nat->R) (Un_decreasing Un) -> (positivity_seq Un) -> (has_ub [N:nat](sum_f_R0 (tg_alt Un) (S (mult (2) N)))). -Intros; Unfold has_ub; Unfold bound. -Exists ``(Un O)``. -Unfold is_upper_bound; Intros; Elim H1; Intros. -Rewrite H2; Rewrite decomp_sum. -Replace (tg_alt Un O) with ``(Un O)``. -Pattern 2 ``(Un O)``; Rewrite <- Rplus_Or. -Apply Rle_compatibility. -Apply CV_ALT_step3; Assumption. -Unfold tg_alt; Simpl; Ring. -Apply lt_O_Sn. +Lemma CV_ALT_step4 : + forall Un:nat -> R, + Un_decreasing Un -> + positivity_seq Un -> + has_ub (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))). +intros; unfold has_ub in |- *; unfold bound in |- *. +exists (Un 0%nat). +unfold is_upper_bound in |- *; intros; elim H1; intros. +rewrite H2; rewrite decomp_sum. +replace (tg_alt Un 0) with (Un 0%nat). +pattern (Un 0%nat) at 2 in |- *; rewrite <- Rplus_0_r. +apply Rplus_le_compat_l. +apply CV_ALT_step3; assumption. +unfold tg_alt in |- *; simpl in |- *; ring. +apply lt_O_Sn. Qed. (* This lemma gives an interesting result about alternated series *) -Lemma CV_ALT : (Un:nat->R) (Un_decreasing Un) -> (positivity_seq Un) -> (Un_cv Un R0) -> (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 (tg_alt Un) N) l)). -Intros. -Assert H2 := (CV_ALT_step0 ? H). -Assert H3 := (CV_ALT_step4 ? H H0). -Assert X := (growing_cv ? H2 H3). -Elim X; Intros. -Apply existTT with x. -Unfold Un_cv; Unfold R_dist; Unfold Un_cv in H1; Unfold R_dist in H1; Unfold Un_cv in p; Unfold R_dist in p. -Intros; Cut ``0<eps/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. -Elim (H1 ``eps/2`` H5); Intros N2 H6. -Elim (p ``eps/2`` H5); Intros N1 H7. -Pose N := (max (S (mult (2) N1)) N2). -Exists N; Intros. -Assert H9 := (even_odd_cor n). -Elim H9; Intros P H10. -Cut (le N1 P). -Intro; Elim H10; Intro. -Replace ``(sum_f_R0 (tg_alt Un) n)-x`` with ``((sum_f_R0 (tg_alt Un) (S n))-x)+(-(tg_alt Un (S n)))``. -Apply Rle_lt_trans with ``(Rabsolu ((sum_f_R0 (tg_alt Un) (S n))-x))+(Rabsolu (-(tg_alt Un (S n))))``. -Apply Rabsolu_triang. -Rewrite (double_var eps); Apply Rplus_lt. -Rewrite H12; Apply H7; Assumption. -Rewrite Rabsolu_Ropp; Unfold tg_alt; Rewrite Rabsolu_mult; Rewrite pow_1_abs; Rewrite Rmult_1l; Unfold Rminus in H6; Rewrite Ropp_O in H6; Rewrite <- (Rplus_Or (Un (S n))); Apply H6. -Unfold ge; Apply le_trans with n. -Apply le_trans with N; [Unfold N; Apply le_max_r | Assumption]. -Apply le_n_Sn. -Rewrite tech5; Ring. -Rewrite H12; Apply Rlt_trans with ``eps/2``. -Apply H7; Assumption. -Unfold Rdiv; Apply Rlt_monotony_contra with ``2``. -Sup0. -Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Rewrite Rmult_1r | DiscrR]. -Rewrite RIneq.double. -Pattern 1 eps; Rewrite <- (Rplus_Or eps); Apply Rlt_compatibility; Assumption. -Elim H10; Intro; Apply le_double. -Rewrite <- H11; Apply le_trans with N. -Unfold N; Apply le_trans with (S (mult (2) N1)); [Apply le_n_Sn | Apply le_max_l]. -Assumption. -Apply lt_n_Sm_le. -Rewrite <- H11. -Apply lt_le_trans with N. -Unfold N; Apply lt_le_trans with (S (mult (2) N1)). -Apply lt_n_Sn. -Apply le_max_l. -Assumption. +Lemma CV_ALT : + forall Un:nat -> R, + Un_decreasing Un -> + positivity_seq Un -> + Un_cv Un 0 -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l). +intros. +assert (H2 := CV_ALT_step0 _ H). +assert (H3 := CV_ALT_step4 _ H H0). +assert (X := growing_cv _ H2 H3). +elim X; intros. +apply existT with x. +unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1; + unfold R_dist in H1; unfold Un_cv in p; unfold R_dist in p. +intros; cut (0 < eps / 2); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H1 (eps / 2) H5); intros N2 H6. +elim (p (eps / 2) H5); intros N1 H7. +pose (N := max (S (2 * N1)) N2). +exists N; intros. +assert (H9 := even_odd_cor n). +elim H9; intros P H10. +cut (N1 <= P)%nat. +intro; elim H10; intro. +replace (sum_f_R0 (tg_alt Un) n - x) with + (sum_f_R0 (tg_alt Un) (S n) - x + - tg_alt Un (S n)). +apply Rle_lt_trans with + (Rabs (sum_f_R0 (tg_alt Un) (S n) - x) + Rabs (- tg_alt Un (S n))). +apply Rabs_triang. +rewrite (double_var eps); apply Rplus_lt_compat. +rewrite H12; apply H7; assumption. +rewrite Rabs_Ropp; unfold tg_alt in |- *; rewrite Rabs_mult; + rewrite pow_1_abs; rewrite Rmult_1_l; unfold Rminus in H6; + rewrite Ropp_0 in H6; rewrite <- (Rplus_0_r (Un (S n))); + apply H6. +unfold ge in |- *; apply le_trans with n. +apply le_trans with N; [ unfold N in |- *; apply le_max_r | assumption ]. +apply le_n_Sn. +rewrite tech5; ring. +rewrite H12; apply Rlt_trans with (eps / 2). +apply H7; assumption. +unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym; + [ rewrite Rmult_1_r | discrR ]. +rewrite double. +pattern eps at 1 in |- *; rewrite <- (Rplus_0_r eps); apply Rplus_lt_compat_l; + assumption. +elim H10; intro; apply le_double. +rewrite <- H11; apply le_trans with N. +unfold N in |- *; apply le_trans with (S (2 * N1)); + [ apply le_n_Sn | apply le_max_l ]. +assumption. +apply lt_n_Sm_le. +rewrite <- H11. +apply lt_le_trans with N. +unfold N in |- *; apply lt_le_trans with (S (2 * N1)). +apply lt_n_Sn. +apply le_max_l. +assumption. Qed. (************************************************) @@ -165,198 +213,236 @@ Qed. (* *) (* Applications: PI, cos, sin *) (************************************************) -Theorem alternated_series : (Un:nat->R) (Un_decreasing Un) -> (Un_cv Un R0) -> (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 (tg_alt Un) N) l)). -Intros; Apply CV_ALT. -Assumption. -Unfold positivity_seq; Apply decreasing_ineq; Assumption. -Assumption. +Theorem alternated_series : + forall Un:nat -> R, + Un_decreasing Un -> + Un_cv Un 0 -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l). +intros; apply CV_ALT. +assumption. +unfold positivity_seq in |- *; apply decreasing_ineq; assumption. +assumption. Qed. -Theorem alternated_series_ineq : (Un:nat->R;l:R;N:nat) (Un_decreasing Un) -> (Un_cv Un R0) -> (Un_cv [N:nat](sum_f_R0 (tg_alt Un) N) l) -> ``(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) N)))<=l<=(sum_f_R0 (tg_alt Un) (mult (S (S O)) N))``. -Intros. -Cut (Un_cv [N:nat](sum_f_R0 (tg_alt Un) (mult (2) N)) l). -Cut (Un_cv [N:nat](sum_f_R0 (tg_alt Un) (S (mult (2) N))) l). -Intros; Split. -Apply (growing_ineq [N:nat](sum_f_R0 (tg_alt Un) (S (mult (2) N)))). -Apply CV_ALT_step0; Assumption. -Assumption. -Apply (decreasing_ineq [N:nat](sum_f_R0 (tg_alt Un) (mult (2) N))). -Apply CV_ALT_step1; Assumption. -Assumption. -Unfold Un_cv; Unfold R_dist; Unfold Un_cv in H1; Unfold R_dist in H1; Intros. -Elim (H1 eps H2); Intros. -Exists x; Intros. -Apply H3. -Unfold ge; Apply le_trans with (mult (2) n). -Apply le_trans with n. -Assumption. -Assert H5 := (mult_O_le n (2)). -Elim H5; Intro. -Cut ~(O)=(2); [Intro; Elim H7; Symmetry; Assumption | Discriminate]. -Assumption. -Apply le_n_Sn. -Unfold Un_cv; Unfold R_dist; Unfold Un_cv in H1; Unfold R_dist in H1; Intros. -Elim (H1 eps H2); Intros. -Exists x; Intros. -Apply H3. -Unfold ge; Apply le_trans with n. -Assumption. -Assert H5 := (mult_O_le n (2)). -Elim H5; Intro. -Cut ~(O)=(2); [Intro; Elim H7; Symmetry; Assumption | Discriminate]. -Assumption. +Theorem alternated_series_ineq : + forall (Un:nat -> R) (l:R) (N:nat), + Un_decreasing Un -> + Un_cv Un 0 -> + Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) N) l -> + sum_f_R0 (tg_alt Un) (S (2 * N)) <= l <= sum_f_R0 (tg_alt Un) (2 * N). +intros. +cut (Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N)) l). +cut (Un_cv (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N))) l). +intros; split. +apply (growing_ineq (fun N:nat => sum_f_R0 (tg_alt Un) (S (2 * N)))). +apply CV_ALT_step0; assumption. +assumption. +apply (decreasing_ineq (fun N:nat => sum_f_R0 (tg_alt Un) (2 * N))). +apply CV_ALT_step1; assumption. +assumption. +unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1; + unfold R_dist in H1; intros. +elim (H1 eps H2); intros. +exists x; intros. +apply H3. +unfold ge in |- *; apply le_trans with (2 * n)%nat. +apply le_trans with n. +assumption. +assert (H5 := mult_O_le n 2). +elim H5; intro. +cut (0%nat <> 2%nat); + [ intro; elim H7; symmetry in |- *; assumption | discriminate ]. +assumption. +apply le_n_Sn. +unfold Un_cv in |- *; unfold R_dist in |- *; unfold Un_cv in H1; + unfold R_dist in H1; intros. +elim (H1 eps H2); intros. +exists x; intros. +apply H3. +unfold ge in |- *; apply le_trans with n. +assumption. +assert (H5 := mult_O_le n 2). +elim H5; intro. +cut (0%nat <> 2%nat); + [ intro; elim H7; symmetry in |- *; assumption | discriminate ]. +assumption. Qed. (************************************) (* Application : construction of PI *) (************************************) -Definition PI_tg := [n:nat]``/(INR (plus (mult (S (S O)) n) (S O)))``. +Definition PI_tg (n:nat) := / INR (2 * n + 1). -Lemma PI_tg_pos : (n:nat)``0<=(PI_tg n)``. -Intro; Unfold PI_tg; Left; Apply Rlt_Rinv; Apply lt_INR_0; Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_O_Sn | Ring]. +Lemma PI_tg_pos : forall n:nat, 0 <= PI_tg n. +intro; unfold PI_tg in |- *; left; apply Rinv_0_lt_compat; apply lt_INR_0; + replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ]. Qed. -Lemma PI_tg_decreasing : (Un_decreasing PI_tg). -Unfold PI_tg Un_decreasing; Intro. -Apply Rle_monotony_contra with ``(INR (plus (mult (S (S O)) n) (S O)))``. -Apply lt_INR_0. -Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_O_Sn | Ring]. -Rewrite <- Rinv_r_sym. -Apply Rle_monotony_contra with ``(INR (plus (mult (S (S O)) (S n)) (S O)))``. -Apply lt_INR_0. -Replace (plus (mult (2) (S n)) (1)) with (S (mult (2) (S n))); [Apply lt_O_Sn | Ring]. -Rewrite (Rmult_sym ``(INR (plus (mult (S (S O)) (S n)) (S O)))``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Do 2 Rewrite Rmult_1r; Apply le_INR. -Replace (plus (mult (2) (S n)) (1)) with (S (S (plus (mult (2) n) (1)))). -Apply le_trans with (S (plus (mult (2) n) (1))); Apply le_n_Sn. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Apply not_O_INR; Discriminate. -Apply not_O_INR; Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Discriminate | Ring]. +Lemma PI_tg_decreasing : Un_decreasing PI_tg. +unfold PI_tg, Un_decreasing in |- *; intro. +apply Rmult_le_reg_l with (INR (2 * n + 1)). +apply lt_INR_0. +replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ]. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with (INR (2 * S n + 1)). +apply lt_INR_0. +replace (2 * S n + 1)%nat with (S (2 * S n)); [ apply lt_O_Sn | ring ]. +rewrite (Rmult_comm (INR (2 * S n + 1))); rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +do 2 rewrite Rmult_1_r; apply le_INR. +replace (2 * S n + 1)%nat with (S (S (2 * n + 1))). +apply le_trans with (S (2 * n + 1)); apply le_n_Sn. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite plus_INR; + do 2 rewrite mult_INR; repeat rewrite S_INR; ring. +apply not_O_INR; discriminate. +apply not_O_INR; replace (2 * n + 1)%nat with (S (2 * n)); + [ discriminate | ring ]. Qed. -Lemma PI_tg_cv : (Un_cv PI_tg R0). -Unfold Un_cv; Unfold R_dist; Intros. -Cut ``0<2*eps``; [Intro | Apply Rmult_lt_pos; [Sup0 | Assumption]]. -Assert H1 := (archimed ``/(2*eps)``). -Cut (Zle `0` ``(up (/(2*eps)))``). -Intro; Assert H3 := (IZN ``(up (/(2*eps)))`` H2). -Elim H3; Intros N H4. -Cut (lt O N). -Intro; Exists N; Intros. -Cut (lt O n). -Intro; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_right. -Unfold PI_tg; Apply Rlt_trans with ``/(INR (mult (S (S O)) n))``. -Apply Rlt_monotony_contra with ``(INR (mult (S (S O)) n))``. -Apply lt_INR_0. -Replace (mult (2) n) with (plus n n); [Idtac | Ring]. -Apply lt_le_trans with n. -Assumption. -Apply le_plus_l. -Rewrite <- Rinv_r_sym. -Apply Rlt_monotony_contra with ``(INR (plus (mult (S (S O)) n) (S O)))``. -Apply lt_INR_0. -Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_O_Sn | Ring]. -Rewrite (Rmult_sym ``(INR (plus (mult (S (S O)) n) (S O)))``). -Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Do 2 Rewrite Rmult_1r; Apply lt_INR. -Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_n_Sn | Ring]. -Apply not_O_INR; Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Discriminate | Ring]. -Replace n with (S (pred n)). -Apply not_O_INR; Discriminate. -Symmetry; Apply S_pred with O. -Assumption. -Apply Rle_lt_trans with ``/(INR (mult (S (S O)) N))``. -Apply Rle_monotony_contra with ``(INR (mult (S (S O)) N))``. -Rewrite mult_INR; Apply Rmult_lt_pos; [Simpl; Sup0 | Apply lt_INR_0; Assumption]. -Rewrite <- Rinv_r_sym. -Apply Rle_monotony_contra with ``(INR (mult (S (S O)) n))``. -Rewrite mult_INR; Apply Rmult_lt_pos; [Simpl; Sup0 | Apply lt_INR_0; Assumption]. -Rewrite (Rmult_sym (INR (mult (S (S O)) n))); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Do 2 Rewrite Rmult_1r; Apply le_INR. -Apply mult_le; Assumption. -Replace n with (S (pred n)). -Apply not_O_INR; Discriminate. -Symmetry; Apply S_pred with O. -Assumption. -Replace N with (S (pred N)). -Apply not_O_INR; Discriminate. -Symmetry; Apply S_pred with O. -Assumption. -Rewrite mult_INR. -Rewrite Rinv_Rmult. -Replace (INR (S (S O))) with ``2``; [Idtac | Reflexivity]. -Apply Rlt_monotony_contra with ``2``. -Sup0. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Idtac | DiscrR]. -Rewrite Rmult_1l; Apply Rlt_monotony_contra with (INR N). -Apply lt_INR_0; Assumption. -Rewrite <- Rinv_r_sym. -Apply Rlt_monotony_contra with ``/(2*eps)``. -Apply Rlt_Rinv; Assumption. -Rewrite Rmult_1r; Replace ``/(2*eps)*((INR N)*(2*eps))`` with ``(INR N)*((2*eps)*/(2*eps))``; [Idtac | Ring]. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Replace (INR N) with (IZR (INZ N)). -Rewrite <- H4. -Elim H1; Intros; Assumption. -Symmetry; Apply INR_IZR_INZ. -Apply prod_neq_R0; [DiscrR | Red; Intro; Rewrite H8 in H; Elim (Rlt_antirefl ? H)]. -Apply not_O_INR. -Red; Intro; Rewrite H8 in H5; Elim (lt_n_n ? H5). -Replace (INR (S (S O))) with ``2``; [DiscrR | Reflexivity]. -Apply not_O_INR. -Red; Intro; Rewrite H8 in H5; Elim (lt_n_n ? H5). -Apply Rle_sym1; Apply PI_tg_pos. -Apply lt_le_trans with N; Assumption. -Elim H1; Intros H5 _. -Assert H6 := (lt_eq_lt_dec O N). -Elim H6; Intro. -Elim a; Intro. -Assumption. -Rewrite <- b in H4. -Rewrite H4 in H5. -Simpl in H5. -Cut ``0</(2*eps)``; [Intro | Apply Rlt_Rinv; Assumption]. -Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H7 H5)). -Elim (lt_n_O ? b). -Apply le_IZR. -Simpl. -Left; Apply Rlt_trans with ``/(2*eps)``. -Apply Rlt_Rinv; Assumption. -Elim H1; Intros; Assumption. +Lemma PI_tg_cv : Un_cv PI_tg 0. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +cut (0 < 2 * eps); + [ intro | apply Rmult_lt_0_compat; [ prove_sup0 | assumption ] ]. +assert (H1 := archimed (/ (2 * eps))). +cut (0 <= up (/ (2 * eps)))%Z. +intro; assert (H3 := IZN (up (/ (2 * eps))) H2). +elim H3; intros N H4. +cut (0 < N)%nat. +intro; exists N; intros. +cut (0 < n)%nat. +intro; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + rewrite Rabs_right. +unfold PI_tg in |- *; apply Rlt_trans with (/ INR (2 * n)). +apply Rmult_lt_reg_l with (INR (2 * n)). +apply lt_INR_0. +replace (2 * n)%nat with (n + n)%nat; [ idtac | ring ]. +apply lt_le_trans with n. +assumption. +apply le_plus_l. +rewrite <- Rinv_r_sym. +apply Rmult_lt_reg_l with (INR (2 * n + 1)). +apply lt_INR_0. +replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ]. +rewrite (Rmult_comm (INR (2 * n + 1))). +rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +do 2 rewrite Rmult_1_r; apply lt_INR. +replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_n_Sn | ring ]. +apply not_O_INR; replace (2 * n + 1)%nat with (S (2 * n)); + [ discriminate | ring ]. +replace n with (S (pred n)). +apply not_O_INR; discriminate. +symmetry in |- *; apply S_pred with 0%nat. +assumption. +apply Rle_lt_trans with (/ INR (2 * N)). +apply Rmult_le_reg_l with (INR (2 * N)). +rewrite mult_INR; apply Rmult_lt_0_compat; + [ simpl in |- *; prove_sup0 | apply lt_INR_0; assumption ]. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with (INR (2 * n)). +rewrite mult_INR; apply Rmult_lt_0_compat; + [ simpl in |- *; prove_sup0 | apply lt_INR_0; assumption ]. +rewrite (Rmult_comm (INR (2 * n))); rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +do 2 rewrite Rmult_1_r; apply le_INR. +apply (fun m n p:nat => mult_le_compat_l p n m); assumption. +replace n with (S (pred n)). +apply not_O_INR; discriminate. +symmetry in |- *; apply S_pred with 0%nat. +assumption. +replace N with (S (pred N)). +apply not_O_INR; discriminate. +symmetry in |- *; apply S_pred with 0%nat. +assumption. +rewrite mult_INR. +rewrite Rinv_mult_distr. +replace (INR 2) with 2; [ idtac | reflexivity ]. +apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; [ idtac | discrR ]. +rewrite Rmult_1_l; apply Rmult_lt_reg_l with (INR N). +apply lt_INR_0; assumption. +rewrite <- Rinv_r_sym. +apply Rmult_lt_reg_l with (/ (2 * eps)). +apply Rinv_0_lt_compat; assumption. +rewrite Rmult_1_r; + replace (/ (2 * eps) * (INR N * (2 * eps))) with + (INR N * (2 * eps * / (2 * eps))); [ idtac | ring ]. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; replace (INR N) with (IZR (Z_of_nat N)). +rewrite <- H4. +elim H1; intros; assumption. +symmetry in |- *; apply INR_IZR_INZ. +apply prod_neq_R0; + [ discrR | red in |- *; intro; rewrite H8 in H; elim (Rlt_irrefl _ H) ]. +apply not_O_INR. +red in |- *; intro; rewrite H8 in H5; elim (lt_irrefl _ H5). +replace (INR 2) with 2; [ discrR | reflexivity ]. +apply not_O_INR. +red in |- *; intro; rewrite H8 in H5; elim (lt_irrefl _ H5). +apply Rle_ge; apply PI_tg_pos. +apply lt_le_trans with N; assumption. +elim H1; intros H5 _. +assert (H6 := lt_eq_lt_dec 0 N). +elim H6; intro. +elim a; intro. +assumption. +rewrite <- b in H4. +rewrite H4 in H5. +simpl in H5. +cut (0 < / (2 * eps)); [ intro | apply Rinv_0_lt_compat; assumption ]. +elim (Rlt_irrefl _ (Rlt_trans _ _ _ H7 H5)). +elim (lt_n_O _ b). +apply le_IZR. +simpl in |- *. +left; apply Rlt_trans with (/ (2 * eps)). +apply Rinv_0_lt_compat; assumption. +elim H1; intros; assumption. Qed. -Lemma exist_PI : (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 (tg_alt PI_tg) N) l)). -Apply alternated_series. -Apply PI_tg_decreasing. -Apply PI_tg_cv. +Lemma exist_PI : + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 (tg_alt PI_tg) N) l). +apply alternated_series. +apply PI_tg_decreasing. +apply PI_tg_cv. Qed. (* Now, PI is defined *) -Definition PI : R := (Rmult ``4`` (Cases exist_PI of (existTT a b) => a end)). +Definition PI : R := 4 * match exist_PI with + | existT a b => a + end. (* We can get an approximation of PI with the following inequality *) -Lemma PI_ineq : (N:nat) ``(sum_f_R0 (tg_alt PI_tg) (S (mult (S (S O)) N)))<=PI/4<=(sum_f_R0 (tg_alt PI_tg) (mult (S (S O)) N))``. -Intro; Apply alternated_series_ineq. -Apply PI_tg_decreasing. -Apply PI_tg_cv. -Unfold PI; Case exist_PI; Intro. -Replace ``(4*x)/4`` with x. -Trivial. -Unfold Rdiv; Rewrite (Rmult_sym ``4``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r; Reflexivity | DiscrR]. +Lemma PI_ineq : + forall N:nat, + sum_f_R0 (tg_alt PI_tg) (S (2 * N)) <= PI / 4 <= + sum_f_R0 (tg_alt PI_tg) (2 * N). +intro; apply alternated_series_ineq. +apply PI_tg_decreasing. +apply PI_tg_cv. +unfold PI in |- *; case exist_PI; intro. +replace (4 * x / 4) with x. +trivial. +unfold Rdiv in |- *; rewrite (Rmult_comm 4); rewrite Rmult_assoc; + rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r; reflexivity | discrR ]. Qed. -Lemma PI_RGT_0 : ``0<PI``. -Assert H := (PI_ineq O). -Apply Rlt_monotony_contra with ``/4``. -Apply Rlt_Rinv; Sup0. -Rewrite Rmult_Or; Rewrite Rmult_sym. -Elim H; Clear H; Intros H _. -Unfold Rdiv in H; Apply Rlt_le_trans with ``(sum_f_R0 (tg_alt PI_tg) (S (mult (S (S O)) O)))``. -Simpl; Unfold tg_alt; Simpl; Rewrite Rmult_1l; Rewrite Rmult_1r; Apply Rlt_anti_compatibility with ``(PI_tg (S O))``. -Rewrite Rplus_Or; Replace ``(PI_tg (S O))+((PI_tg O)+ -1*(PI_tg (S O)))`` with ``(PI_tg O)``; [Unfold PI_tg | Ring]. -Simpl; Apply Rinv_lt. -Rewrite Rmult_1l; Replace ``2+1`` with ``3``; [Sup0 | Ring]. -Rewrite Rplus_sym; Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Sup0. -Assumption. -Qed. +Lemma PI_RGT_0 : 0 < PI. +assert (H := PI_ineq 0). +apply Rmult_lt_reg_l with (/ 4). +apply Rinv_0_lt_compat; prove_sup0. +rewrite Rmult_0_r; rewrite Rmult_comm. +elim H; clear H; intros H _. +unfold Rdiv in H; + apply Rlt_le_trans with (sum_f_R0 (tg_alt PI_tg) (S (2 * 0))). +simpl in |- *; unfold tg_alt in |- *; simpl in |- *; rewrite Rmult_1_l; + rewrite Rmult_1_r; apply Rplus_lt_reg_r with (PI_tg 1). +rewrite Rplus_0_r; + replace (PI_tg 1 + (PI_tg 0 + -1 * PI_tg 1)) with (PI_tg 0); + [ unfold PI_tg in |- * | ring ]. +simpl in |- *; apply Rinv_lt_contravar. +rewrite Rmult_1_l; replace (2 + 1) with 3; [ prove_sup0 | ring ]. +rewrite Rplus_comm; pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_lt_compat_l; prove_sup0. +assumption. +Qed.
\ No newline at end of file diff --git a/theories/Reals/ArithProp.v b/theories/Reals/ArithProp.v index 7ec8ad1ed..72c99fc10 100644 --- a/theories/Reals/ArithProp.v +++ b/theories/Reals/ArithProp.v @@ -8,127 +8,171 @@ (*i $Id$ i*) -Require Rbase. -Require Rbasic_fun. -Require Even. -Require Div2. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rbasic_fun. +Require Import Even. +Require Import Div2. Open Local Scope Z_scope. Open Local Scope R_scope. -Lemma minus_neq_O : (n,i:nat) (lt i n) -> ~(minus n i)=O. -Intros; Red; Intro. -Cut (n,m:nat) (le m n) -> (minus n m)=O -> n=m. -Intro; Assert H2 := (H1 ? ? (lt_le_weak ? ? H) H0); Rewrite H2 in H; Elim (lt_n_n ? H). -Pose R := [n,m:nat](le m n)->(minus n m)=(0)->n=m. -Cut ((n,m:nat)(R n m)) -> ((n0,m:nat)(le m n0)->(minus n0 m)=(0)->n0=m). -Intro; Apply H1. -Apply nat_double_ind. -Unfold R; Intros; Inversion H2; Reflexivity. -Unfold R; Intros; Simpl in H3; Assumption. -Unfold R; Intros; Simpl in H4; Assert H5 := (le_S_n ? ? H3); Assert H6 := (H2 H5 H4); Rewrite H6; Reflexivity. -Unfold R; Intros; Apply H1; Assumption. +Lemma minus_neq_O : forall n i:nat, (i < n)%nat -> (n - i)%nat <> 0%nat. +intros; red in |- *; intro. +cut (forall n m:nat, (m <= n)%nat -> (n - m)%nat = 0%nat -> n = m). +intro; assert (H2 := H1 _ _ (lt_le_weak _ _ H) H0); rewrite H2 in H; + elim (lt_irrefl _ H). +pose (R := fun n m:nat => (m <= n)%nat -> (n - m)%nat = 0%nat -> n = m). +cut + ((forall n m:nat, R n m) -> + forall n0 m:nat, (m <= n0)%nat -> (n0 - m)%nat = 0%nat -> n0 = m). +intro; apply H1. +apply nat_double_ind. +unfold R in |- *; intros; inversion H2; reflexivity. +unfold R in |- *; intros; simpl in H3; assumption. +unfold R in |- *; intros; simpl in H4; assert (H5 := le_S_n _ _ H3); + assert (H6 := H2 H5 H4); rewrite H6; reflexivity. +unfold R in |- *; intros; apply H1; assumption. Qed. -Lemma le_minusni_n : (n,i:nat) (le i n)->(le (minus n i) n). -Pose R := [m,n:nat] (le n m) -> (le (minus m n) m). -Cut ((m,n:nat)(R m n)) -> ((n,i:nat)(le i n)->(le (minus n i) n)). -Intro; Apply H. -Apply nat_double_ind. -Unfold R; Intros; Simpl; Apply le_n. -Unfold R; Intros; Simpl; Apply le_n. -Unfold R; Intros; Simpl; Apply le_trans with n. -Apply H0; Apply le_S_n; Assumption. -Apply le_n_Sn. -Unfold R; Intros; Apply H; Assumption. +Lemma le_minusni_n : forall n i:nat, (i <= n)%nat -> (n - i <= n)%nat. +pose (R := fun m n:nat => (n <= m)%nat -> (m - n <= m)%nat). +cut + ((forall m n:nat, R m n) -> forall n i:nat, (i <= n)%nat -> (n - i <= n)%nat). +intro; apply H. +apply nat_double_ind. +unfold R in |- *; intros; simpl in |- *; apply le_n. +unfold R in |- *; intros; simpl in |- *; apply le_n. +unfold R in |- *; intros; simpl in |- *; apply le_trans with n. +apply H0; apply le_S_n; assumption. +apply le_n_Sn. +unfold R in |- *; intros; apply H; assumption. Qed. -Lemma lt_minus_O_lt : (m,n:nat) (lt m n) -> (lt O (minus n m)). -Intros n m; Pattern n m; Apply nat_double_ind; [ - Intros; Rewrite <- minus_n_O; Assumption -| Intros; Elim (lt_n_O ? H) -| Intros; Simpl; Apply H; Apply lt_S_n; Assumption]. +Lemma lt_minus_O_lt : forall m n:nat, (m < n)%nat -> (0 < n - m)%nat. +intros n m; pattern n, m in |- *; apply nat_double_ind; + [ intros; rewrite <- minus_n_O; assumption + | intros; elim (lt_n_O _ H) + | intros; simpl in |- *; apply H; apply lt_S_n; assumption ]. Qed. -Lemma even_odd_cor : (n:nat) (EX p : nat | n=(mult (2) p)\/n=(S (mult (2) p))). -Intro. -Assert H := (even_or_odd n). -Exists (div2 n). -Assert H0 := (even_odd_double n). -Elim H0; Intros. -Elim H1; Intros H3 _. -Elim H2; Intros H4 _. -Replace (mult (2) (div2 n)) with (Div2.double (div2 n)). -Elim H; Intro. -Left. -Apply H3; Assumption. -Right. -Apply H4; Assumption. -Unfold Div2.double; Ring. +Lemma even_odd_cor : + forall n:nat, exists p : nat | n = (2 * p)%nat \/ n = S (2 * p). +intro. +assert (H := even_or_odd n). +exists (div2 n). +assert (H0 := even_odd_double n). +elim H0; intros. +elim H1; intros H3 _. +elim H2; intros H4 _. +replace (2 * div2 n)%nat with (double (div2 n)). +elim H; intro. +left. +apply H3; assumption. +right. +apply H4; assumption. +unfold double in |- *; ring. Qed. (* 2m <= 2n => m<=n *) -Lemma le_double : (m,n:nat) (le (mult (2) m) (mult (2) n)) -> (le m n). -Intros; Apply INR_le. -Assert H1 := (le_INR ? ? H). -Do 2 Rewrite mult_INR in H1. -Apply Rle_monotony_contra with ``(INR (S (S O)))``. -Replace (INR (S (S O))) with ``2``; [Sup0 | Reflexivity]. -Assumption. +Lemma le_double : forall m n:nat, (2 * m <= 2 * n)%nat -> (m <= n)%nat. +intros; apply INR_le. +assert (H1 := le_INR _ _ H). +do 2 rewrite mult_INR in H1. +apply Rmult_le_reg_l with (INR 2). +replace (INR 2) with 2; [ prove_sup0 | reflexivity ]. +assumption. Qed. (* Here, we have the euclidian division *) (* This lemma is used in the proof of sin_eq_0 : (sin x)=0<->x=kPI *) -Lemma euclidian_division : (x,y:R) ``y<>0`` -> (EXT k:Z | (EXT r : R | ``x==(IZR k)*y+r``/\``0<=r<(Rabsolu y)``)). -Intros. -Pose k0 := Cases (case_Rabsolu y) of - (leftT _) => (Zminus `1` (up ``x/-y``)) - | (rightT _) => (Zminus (up ``x/y``) `1`) end. -Exists k0. -Exists ``x-(IZR k0)*y``. -Split. -Ring. -Unfold k0; Case (case_Rabsolu y); Intro. -Assert H0 := (archimed ``x/-y``); Rewrite <- Z_R_minus; Simpl; Unfold Rminus. -Replace ``-((1+ -(IZR (up (x/( -y)))))*y)`` with ``((IZR (up (x/-y)))-1)*y``; [Idtac | Ring]. -Split. -Apply Rle_monotony_contra with ``/-y``. -Apply Rlt_Rinv; Apply Rgt_RO_Ropp; Exact r. -Rewrite Rmult_Or; Rewrite (Rmult_sym ``/-y``); Rewrite Rmult_Rplus_distrl; Rewrite <- Ropp_Rinv; [Idtac | Assumption]. -Rewrite Rmult_assoc; Repeat Rewrite Ropp_mul3; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r | Assumption]. -Apply Rle_anti_compatibility with ``(IZR (up (x/( -y))))-x/( -y)``. -Rewrite Rplus_Or; Unfold Rdiv; Pattern 4 ``/-y``; Rewrite <- Ropp_Rinv; [Idtac | Assumption]. -Replace ``(IZR (up (x*/ -y)))-x* -/y+( -(x*/y)+ -((IZR (up (x*/ -y)))-1))`` with R1; [Idtac | Ring]. -Elim H0; Intros _ H1; Unfold Rdiv in H1; Exact H1. -Rewrite (Rabsolu_left ? r); Apply Rlt_monotony_contra with ``/-y``. -Apply Rlt_Rinv; Apply Rgt_RO_Ropp; Exact r. -Rewrite <- Rinv_l_sym. -Rewrite (Rmult_sym ``/-y``); Rewrite Rmult_Rplus_distrl; Rewrite <- Ropp_Rinv; [Idtac | Assumption]. -Rewrite Rmult_assoc; Repeat Rewrite Ropp_mul3; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r | Assumption]; Apply Rlt_anti_compatibility with ``((IZR (up (x/( -y))))-1)``. -Replace ``(IZR (up (x/( -y))))-1+1`` with ``(IZR (up (x/( -y))))``; [Idtac | Ring]. -Replace ``(IZR (up (x/( -y))))-1+( -(x*/y)+ -((IZR (up (x/( -y))))-1))`` with ``-(x*/y)``; [Idtac | Ring]. -Rewrite <- Ropp_mul3; Rewrite (Ropp_Rinv ? H); Elim H0; Unfold Rdiv; Intros H1 _; Exact H1. -Apply Ropp_neq; Assumption. -Assert H0 := (archimed ``x/y``); Rewrite <- Z_R_minus; Simpl; Cut ``0<y``. -Intro; Unfold Rminus; Replace ``-(((IZR (up (x/y)))+ -1)*y)`` with ``(1-(IZR (up (x/y))))*y``; [Idtac | Ring]. -Split. -Apply Rle_monotony_contra with ``/y``. -Apply Rlt_Rinv; Assumption. -Rewrite Rmult_Or; Rewrite (Rmult_sym ``/y``); Rewrite Rmult_Rplus_distrl; Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r | Assumption]; Apply Rle_anti_compatibility with ``(IZR (up (x/y)))-x/y``; Rewrite Rplus_Or; Unfold Rdiv; Replace ``(IZR (up (x*/y)))-x*/y+(x*/y+(1-(IZR (up (x*/y)))))`` with R1; [Idtac | Ring]; Elim H0; Intros _ H2; Unfold Rdiv in H2; Exact H2. -Rewrite (Rabsolu_right ? r); Apply Rlt_monotony_contra with ``/y``. -Apply Rlt_Rinv; Assumption. -Rewrite <- (Rinv_l_sym ? H); Rewrite (Rmult_sym ``/y``); Rewrite Rmult_Rplus_distrl; Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r | Assumption]; Apply Rlt_anti_compatibility with ``((IZR (up (x/y)))-1)``; Replace ``(IZR (up (x/y)))-1+1`` with ``(IZR (up (x/y)))``; [Idtac | Ring]; Replace ``(IZR (up (x/y)))-1+(x*/y+(1-(IZR (up (x/y)))))`` with ``x*/y``; [Idtac | Ring]; Elim H0; Unfold Rdiv; Intros H2 _; Exact H2. -Case (total_order_T R0 y); Intro. -Elim s; Intro. -Assumption. -Elim H; Symmetry; Exact b. -Assert H1 := (Rle_sym2 ? ? r); Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 r0)). +Lemma euclidian_division : + forall x y:R, + y <> 0 -> + exists k : Z | ( exists r : R | x = IZR k * y + r /\ 0 <= r < Rabs y). +intros. +pose + (k0 := + match Rcase_abs y with + | left _ => (1 - up (x / - y))%Z + | right _ => (up (x / y) - 1)%Z + end). +exists k0. +exists (x - IZR k0 * y). +split. +ring. +unfold k0 in |- *; case (Rcase_abs y); intro. +assert (H0 := archimed (x / - y)); rewrite <- Z_R_minus; simpl in |- *; + unfold Rminus in |- *. +replace (- ((1 + - IZR (up (x / - y))) * y)) with + ((IZR (up (x / - y)) - 1) * y); [ idtac | ring ]. +split. +apply Rmult_le_reg_l with (/ - y). +apply Rinv_0_lt_compat; apply Ropp_0_gt_lt_contravar; exact r. +rewrite Rmult_0_r; rewrite (Rmult_comm (/ - y)); rewrite Rmult_plus_distr_r; + rewrite <- Ropp_inv_permute; [ idtac | assumption ]. +rewrite Rmult_assoc; repeat rewrite Ropp_mult_distr_r_reverse; + rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r | assumption ]. +apply Rplus_le_reg_l with (IZR (up (x / - y)) - x / - y). +rewrite Rplus_0_r; unfold Rdiv in |- *; pattern (/ - y) at 4 in |- *; + rewrite <- Ropp_inv_permute; [ idtac | assumption ]. +replace + (IZR (up (x * / - y)) - x * - / y + + (- (x * / y) + - (IZR (up (x * / - y)) - 1))) with 1; + [ idtac | ring ]. +elim H0; intros _ H1; unfold Rdiv in H1; exact H1. +rewrite (Rabs_left _ r); apply Rmult_lt_reg_l with (/ - y). +apply Rinv_0_lt_compat; apply Ropp_0_gt_lt_contravar; exact r. +rewrite <- Rinv_l_sym. +rewrite (Rmult_comm (/ - y)); rewrite Rmult_plus_distr_r; + rewrite <- Ropp_inv_permute; [ idtac | assumption ]. +rewrite Rmult_assoc; repeat rewrite Ropp_mult_distr_r_reverse; + rewrite <- Rinv_r_sym; [ rewrite Rmult_1_r | assumption ]; + apply Rplus_lt_reg_r with (IZR (up (x / - y)) - 1). +replace (IZR (up (x / - y)) - 1 + 1) with (IZR (up (x / - y))); + [ idtac | ring ]. +replace (IZR (up (x / - y)) - 1 + (- (x * / y) + - (IZR (up (x / - y)) - 1))) + with (- (x * / y)); [ idtac | ring ]. +rewrite <- Ropp_mult_distr_r_reverse; rewrite (Ropp_inv_permute _ H); elim H0; + unfold Rdiv in |- *; intros H1 _; exact H1. +apply Ropp_neq_0_compat; assumption. +assert (H0 := archimed (x / y)); rewrite <- Z_R_minus; simpl in |- *; + cut (0 < y). +intro; unfold Rminus in |- *; + replace (- ((IZR (up (x / y)) + -1) * y)) with ((1 - IZR (up (x / y))) * y); + [ idtac | ring ]. +split. +apply Rmult_le_reg_l with (/ y). +apply Rinv_0_lt_compat; assumption. +rewrite Rmult_0_r; rewrite (Rmult_comm (/ y)); rewrite Rmult_plus_distr_r; + rewrite Rmult_assoc; rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_r | assumption ]; + apply Rplus_le_reg_l with (IZR (up (x / y)) - x / y); + rewrite Rplus_0_r; unfold Rdiv in |- *; + replace + (IZR (up (x * / y)) - x * / y + (x * / y + (1 - IZR (up (x * / y))))) with + 1; [ idtac | ring ]; elim H0; intros _ H2; unfold Rdiv in H2; + exact H2. +rewrite (Rabs_right _ r); apply Rmult_lt_reg_l with (/ y). +apply Rinv_0_lt_compat; assumption. +rewrite <- (Rinv_l_sym _ H); rewrite (Rmult_comm (/ y)); + rewrite Rmult_plus_distr_r; rewrite Rmult_assoc; rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_r | assumption ]; + apply Rplus_lt_reg_r with (IZR (up (x / y)) - 1); + replace (IZR (up (x / y)) - 1 + 1) with (IZR (up (x / y))); + [ idtac | ring ]; + replace (IZR (up (x / y)) - 1 + (x * / y + (1 - IZR (up (x / y))))) with + (x * / y); [ idtac | ring ]; elim H0; unfold Rdiv in |- *; + intros H2 _; exact H2. +case (total_order_T 0 y); intro. +elim s; intro. +assumption. +elim H; symmetry in |- *; exact b. +assert (H1 := Rge_le _ _ r); elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 r0)). Qed. -Lemma tech8 : (n,i:nat) (le n (plus (S n) i)). -Intros; Induction i. -Replace (plus (S n) O) with (S n); [Apply le_n_Sn | Ring]. -Replace (plus (S n) (S i)) with (S (plus (S n) i)). -Apply le_S; Assumption. -Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Do 2 Rewrite S_INR; Ring. -Qed. +Lemma tech8 : forall n i:nat, (n <= S n + i)%nat. +intros; induction i as [| i Hreci]. +replace (S n + 0)%nat with (S n); [ apply le_n_Sn | ring ]. +replace (S n + S i)%nat with (S (S n + i)). +apply le_S; assumption. +apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; do 2 rewrite S_INR; ring. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Binomial.v b/theories/Reals/Binomial.v index 5bbf8c7dd..e8173b82e 100644 --- a/theories/Reals/Binomial.v +++ b/theories/Reals/Binomial.v @@ -8,174 +8,197 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require PartSum. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import PartSum. Open Local Scope R_scope. -Definition C [n,p:nat] : R := ``(INR (fact n))/((INR (fact p))*(INR (fact (minus n p))))``. +Definition C (n p:nat) : R := + INR (fact n) / (INR (fact p) * INR (fact (n - p))). -Lemma pascal_step1 : (n,i:nat) (le i n) -> (C n i) == (C n (minus n i)). -Intros; Unfold C; Replace (minus n (minus n i)) with i. -Rewrite Rmult_sym. -Reflexivity. -Apply plus_minus; Rewrite plus_sym; Apply le_plus_minus; Assumption. +Lemma pascal_step1 : forall n i:nat, (i <= n)%nat -> C n i = C n (n - i). +intros; unfold C in |- *; replace (n - (n - i))%nat with i. +rewrite Rmult_comm. +reflexivity. +apply plus_minus; rewrite plus_comm; apply le_plus_minus; assumption. Qed. -Lemma pascal_step2 : (n,i:nat) (le i n) -> (C (S n) i) == ``(INR (S n))/(INR (minus (S n) i))*(C n i)``. -Intros; Unfold C; Replace (minus (S n) i) with (S (minus n i)). -Cut (n:nat) (fact (S n))=(mult (S n) (fact n)). -Intro; Repeat Rewrite H0. -Unfold Rdiv; Repeat Rewrite mult_INR; Repeat Rewrite Rinv_Rmult. -Ring. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply not_O_INR; Discriminate. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply prod_neq_R0. -Apply not_O_INR; Discriminate. -Apply INR_fact_neq_0. -Intro; Reflexivity. -Apply minus_Sn_m; Assumption. +Lemma pascal_step2 : + forall n i:nat, + (i <= n)%nat -> C (S n) i = INR (S n) / INR (S n - i) * C n i. +intros; unfold C in |- *; replace (S n - i)%nat with (S (n - i)). +cut (forall n:nat, fact (S n) = (S n * fact n)%nat). +intro; repeat rewrite H0. +unfold Rdiv in |- *; repeat rewrite mult_INR; repeat rewrite Rinv_mult_distr. +ring. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply not_O_INR; discriminate. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply prod_neq_R0. +apply not_O_INR; discriminate. +apply INR_fact_neq_0. +intro; reflexivity. +apply minus_Sn_m; assumption. Qed. -Lemma pascal_step3 : (n,i:nat) (lt i n) -> (C n (S i)) == ``(INR (minus n i))/(INR (S i))*(C n i)``. -Intros; Unfold C. -Cut (n:nat) (fact (S n))=(mult (S n) (fact n)). -Intro. -Cut (minus n i) = (S (minus n (S i))). -Intro. -Pattern 2 (minus n i); Rewrite H1. -Repeat Rewrite H0; Unfold Rdiv; Repeat Rewrite mult_INR; Repeat Rewrite Rinv_Rmult. -Rewrite <- H1; Rewrite (Rmult_sym ``/(INR (minus n i))``); Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym (INR (minus n i))); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Ring. -Apply not_O_INR; Apply minus_neq_O; Assumption. -Apply not_O_INR; Discriminate. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply prod_neq_R0; [Apply not_O_INR; Discriminate | Apply INR_fact_neq_0]. -Apply not_O_INR; Discriminate. -Apply INR_fact_neq_0. -Apply prod_neq_R0; [Apply not_O_INR; Discriminate | Apply INR_fact_neq_0]. -Apply INR_fact_neq_0. -Rewrite minus_Sn_m. -Simpl; Reflexivity. -Apply lt_le_S; Assumption. -Intro; Reflexivity. +Lemma pascal_step3 : + forall n i:nat, (i < n)%nat -> C n (S i) = INR (n - i) / INR (S i) * C n i. +intros; unfold C in |- *. +cut (forall n:nat, fact (S n) = (S n * fact n)%nat). +intro. +cut ((n - i)%nat = S (n - S i)). +intro. +pattern (n - i)%nat at 2 in |- *; rewrite H1. +repeat rewrite H0; unfold Rdiv in |- *; repeat rewrite mult_INR; + repeat rewrite Rinv_mult_distr. +rewrite <- H1; rewrite (Rmult_comm (/ INR (n - i))); + repeat rewrite Rmult_assoc; rewrite (Rmult_comm (INR (n - i))); + repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +ring. +apply not_O_INR; apply minus_neq_O; assumption. +apply not_O_INR; discriminate. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ]. +apply not_O_INR; discriminate. +apply INR_fact_neq_0. +apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ]. +apply INR_fact_neq_0. +rewrite minus_Sn_m. +simpl in |- *; reflexivity. +apply lt_le_S; assumption. +intro; reflexivity. Qed. (**********) -Lemma pascal : (n,i:nat) (lt i n) -> ``(C n i)+(C n (S i))==(C (S n) (S i))``. -Intros. -Rewrite pascal_step3; [Idtac | Assumption]. -Replace ``(C n i)+(INR (minus n i))/(INR (S i))*(C n i)`` with ``(C n i)*(1+(INR (minus n i))/(INR (S i)))``; [Idtac | Ring]. -Replace ``1+(INR (minus n i))/(INR (S i))`` with ``(INR (S n))/(INR (S i))``. -Rewrite pascal_step1. -Rewrite Rmult_sym; Replace (S i) with (minus (S n) (minus n i)). -Rewrite <- pascal_step2. -Apply pascal_step1. -Apply le_trans with n. -Apply le_minusni_n. -Apply lt_le_weak; Assumption. -Apply le_n_Sn. -Apply le_minusni_n. -Apply lt_le_weak; Assumption. -Rewrite <- minus_Sn_m. -Cut (minus n (minus n i))=i. -Intro; Rewrite H0; Reflexivity. -Symmetry; Apply plus_minus. -Rewrite plus_sym; Rewrite le_plus_minus_r. -Reflexivity. -Apply lt_le_weak; Assumption. -Apply le_minusni_n; Apply lt_le_weak; Assumption. -Apply lt_le_weak; Assumption. -Unfold Rdiv. -Repeat Rewrite S_INR. -Rewrite minus_INR. -Cut ``((INR i)+1)<>0``. -Intro. -Apply r_Rmult_mult with ``(INR i)+1``; [Idtac | Assumption]. -Rewrite Rmult_Rplus_distr. -Rewrite Rmult_1r. -Do 2 Rewrite (Rmult_sym ``(INR i)+1``). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym; [Idtac | Assumption]. -Ring. -Rewrite <- S_INR. -Apply not_O_INR; Discriminate. -Apply lt_le_weak; Assumption. +Lemma pascal : + forall n i:nat, (i < n)%nat -> C n i + C n (S i) = C (S n) (S i). +intros. +rewrite pascal_step3; [ idtac | assumption ]. +replace (C n i + INR (n - i) / INR (S i) * C n i) with + (C n i * (1 + INR (n - i) / INR (S i))); [ idtac | ring ]. +replace (1 + INR (n - i) / INR (S i)) with (INR (S n) / INR (S i)). +rewrite pascal_step1. +rewrite Rmult_comm; replace (S i) with (S n - (n - i))%nat. +rewrite <- pascal_step2. +apply pascal_step1. +apply le_trans with n. +apply le_minusni_n. +apply lt_le_weak; assumption. +apply le_n_Sn. +apply le_minusni_n. +apply lt_le_weak; assumption. +rewrite <- minus_Sn_m. +cut ((n - (n - i))%nat = i). +intro; rewrite H0; reflexivity. +symmetry in |- *; apply plus_minus. +rewrite plus_comm; rewrite le_plus_minus_r. +reflexivity. +apply lt_le_weak; assumption. +apply le_minusni_n; apply lt_le_weak; assumption. +apply lt_le_weak; assumption. +unfold Rdiv in |- *. +repeat rewrite S_INR. +rewrite minus_INR. +cut (INR i + 1 <> 0). +intro. +apply Rmult_eq_reg_l with (INR i + 1); [ idtac | assumption ]. +rewrite Rmult_plus_distr_l. +rewrite Rmult_1_r. +do 2 rewrite (Rmult_comm (INR i + 1)). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym; [ idtac | assumption ]. +ring. +rewrite <- S_INR. +apply not_O_INR; discriminate. +apply lt_le_weak; assumption. Qed. (*********************) (*********************) -Lemma binomial : (x,y:R;n:nat) ``(pow (x+y) n)``==(sum_f_R0 [i:nat]``(C n i)*(pow x i)*(pow y (minus n i))`` n). -Intros; Induction n. -Unfold C; Simpl; Unfold Rdiv; Repeat Rewrite Rmult_1r; Rewrite Rinv_R1; Ring. -Pattern 1 (S n); Replace (S n) with (plus n (1)); [Idtac | Ring]. -Rewrite pow_add; Rewrite Hrecn. -Replace ``(pow (x+y) (S O))`` with ``x+y``; [Idtac | Simpl; Ring]. -Rewrite tech5. -Cut (p:nat)(C p p)==R1. -Cut (p:nat)(C p O)==R1. -Intros; Rewrite H0; Rewrite <- minus_n_n; Rewrite Rmult_1l. -Replace (pow y O) with R1; [Rewrite Rmult_1r | Simpl; Reflexivity]. -Induction n. -Simpl; Do 2 Rewrite H; Ring. +Lemma binomial : + forall (x y:R) (n:nat), + (x + y) ^ n = sum_f_R0 (fun i:nat => C n i * x ^ i * y ^ (n - i)) n. +intros; induction n as [| n Hrecn]. +unfold C in |- *; simpl in |- *; unfold Rdiv in |- *; + repeat rewrite Rmult_1_r; rewrite Rinv_1; ring. +pattern (S n) at 1 in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. +rewrite pow_add; rewrite Hrecn. +replace ((x + y) ^ 1) with (x + y); [ idtac | simpl in |- *; ring ]. +rewrite tech5. +cut (forall p:nat, C p p = 1). +cut (forall p:nat, C p 0 = 1). +intros; rewrite H0; rewrite <- minus_n_n; rewrite Rmult_1_l. +replace (y ^ 0) with 1; [ rewrite Rmult_1_r | simpl in |- *; reflexivity ]. +induction n as [| n Hrecn0]. +simpl in |- *; do 2 rewrite H; ring. (* N >= 1 *) -Pose N := (S n). -Rewrite Rmult_Rplus_distr. -Replace (Rmult (sum_f_R0 ([i:nat]``(C N i)*(pow x i)*(pow y (minus N i))``) N) x) with (sum_f_R0 [i:nat]``(C N i)*(pow x (S i))*(pow y (minus N i))`` N). -Replace (Rmult (sum_f_R0 ([i:nat]``(C N i)*(pow x i)*(pow y (minus N i))``) N) y) with (sum_f_R0 [i:nat]``(C N i)*(pow x i)*(pow y (minus (S N) i))`` N). -Rewrite (decomp_sum [i:nat]``(C (S N) i)*(pow x i)*(pow y (minus (S N) i))`` N). -Rewrite H; Replace (pow x O) with R1; [Idtac | Reflexivity]. -Do 2 Rewrite Rmult_1l. -Replace (minus (S N) O) with (S N); [Idtac | Reflexivity]. -Pose An := [i:nat]``(C N i)*(pow x (S i))*(pow y (minus N i))``. -Pose Bn := [i:nat]``(C N (S i))*(pow x (S i))*(pow y (minus N i))``. -Replace (pred N) with n. -Replace (sum_f_R0 ([i:nat]``(C (S N) (S i))*(pow x (S i))*(pow y (minus (S N) (S i)))``) n) with (sum_f_R0 [i:nat]``(An i)+(Bn i)`` n). -Rewrite plus_sum. -Replace (pow x (S N)) with (An (S n)). -Rewrite (Rplus_sym (sum_f_R0 An n)). -Repeat Rewrite Rplus_assoc. -Rewrite <- tech5. -Fold N. -Pose Cn := [i:nat]``(C N i)*(pow x i)*(pow y (minus (S N) i))``. -Cut (i:nat) (lt i N)-> (Cn (S i))==(Bn i). -Intro; Replace (sum_f_R0 Bn n) with (sum_f_R0 [i:nat](Cn (S i)) n). -Replace (pow y (S N)) with (Cn O). -Rewrite <- Rplus_assoc; Rewrite (decomp_sum Cn N). -Replace (pred N) with n. -Ring. -Unfold N; Simpl; Reflexivity. -Unfold N; Apply lt_O_Sn. -Unfold Cn; Rewrite H; Simpl; Ring. -Apply sum_eq. -Intros; Apply H1. -Unfold N; Apply le_lt_trans with n; [Assumption | Apply lt_n_Sn]. -Intros; Unfold Bn Cn. -Replace (minus (S N) (S i)) with (minus N i); Reflexivity. -Unfold An; Fold N; Rewrite <- minus_n_n; Rewrite H0; Simpl; Ring. -Apply sum_eq. -Intros; Unfold An Bn; Replace (minus (S N) (S i)) with (minus N i); [Idtac | Reflexivity]. -Rewrite <- pascal; [Ring | Apply le_lt_trans with n; [Assumption | Unfold N; Apply lt_n_Sn]]. -Unfold N; Reflexivity. -Unfold N; Apply lt_O_Sn. -Rewrite <- (Rmult_sym y); Rewrite scal_sum; Apply sum_eq. -Intros; Replace (minus (S N) i) with (S (minus N i)). -Replace (S (minus N i)) with (plus (minus N i) (1)); [Idtac | Ring]. -Rewrite pow_add; Replace (pow y (S O)) with y; [Idtac | Simpl; Ring]; Ring. -Apply minus_Sn_m; Assumption. -Rewrite <- (Rmult_sym x); Rewrite scal_sum; Apply sum_eq. -Intros; Replace (S i) with (plus i (1)); [Idtac | Ring]; Rewrite pow_add; Replace (pow x (S O)) with x; [Idtac | Simpl; Ring]; Ring. -Intro; Unfold C. -Replace (INR (fact O)) with R1; [Idtac | Reflexivity]. -Replace (minus p O) with p; [Idtac | Apply minus_n_O]. -Rewrite Rmult_1l; Unfold Rdiv; Rewrite <- Rinv_r_sym; [Reflexivity | Apply INR_fact_neq_0]. -Intro; Unfold C. -Replace (minus p p) with O; [Idtac | Apply minus_n_n]. -Replace (INR (fact O)) with R1; [Idtac | Reflexivity]. -Rewrite Rmult_1r; Unfold Rdiv; Rewrite <- Rinv_r_sym; [Reflexivity | Apply INR_fact_neq_0]. -Qed. +pose (N := S n). +rewrite Rmult_plus_distr_l. +replace (sum_f_R0 (fun i:nat => C N i * x ^ i * y ^ (N - i)) N * x) with + (sum_f_R0 (fun i:nat => C N i * x ^ S i * y ^ (N - i)) N). +replace (sum_f_R0 (fun i:nat => C N i * x ^ i * y ^ (N - i)) N * y) with + (sum_f_R0 (fun i:nat => C N i * x ^ i * y ^ (S N - i)) N). +rewrite (decomp_sum (fun i:nat => C (S N) i * x ^ i * y ^ (S N - i)) N). +rewrite H; replace (x ^ 0) with 1; [ idtac | reflexivity ]. +do 2 rewrite Rmult_1_l. +replace (S N - 0)%nat with (S N); [ idtac | reflexivity ]. +pose (An := fun i:nat => C N i * x ^ S i * y ^ (N - i)). +pose (Bn := fun i:nat => C N (S i) * x ^ S i * y ^ (N - i)). +replace (pred N) with n. +replace (sum_f_R0 (fun i:nat => C (S N) (S i) * x ^ S i * y ^ (S N - S i)) n) + with (sum_f_R0 (fun i:nat => An i + Bn i) n). +rewrite plus_sum. +replace (x ^ S N) with (An (S n)). +rewrite (Rplus_comm (sum_f_R0 An n)). +repeat rewrite Rplus_assoc. +rewrite <- tech5. +fold N in |- *. +pose (Cn := fun i:nat => C N i * x ^ i * y ^ (S N - i)). +cut (forall i:nat, (i < N)%nat -> Cn (S i) = Bn i). +intro; replace (sum_f_R0 Bn n) with (sum_f_R0 (fun i:nat => Cn (S i)) n). +replace (y ^ S N) with (Cn 0%nat). +rewrite <- Rplus_assoc; rewrite (decomp_sum Cn N). +replace (pred N) with n. +ring. +unfold N in |- *; simpl in |- *; reflexivity. +unfold N in |- *; apply lt_O_Sn. +unfold Cn in |- *; rewrite H; simpl in |- *; ring. +apply sum_eq. +intros; apply H1. +unfold N in |- *; apply le_lt_trans with n; [ assumption | apply lt_n_Sn ]. +intros; unfold Bn, Cn in |- *. +replace (S N - S i)%nat with (N - i)%nat; reflexivity. +unfold An in |- *; fold N in |- *; rewrite <- minus_n_n; rewrite H0; + simpl in |- *; ring. +apply sum_eq. +intros; unfold An, Bn in |- *; replace (S N - S i)%nat with (N - i)%nat; + [ idtac | reflexivity ]. +rewrite <- pascal; + [ ring + | apply le_lt_trans with n; [ assumption | unfold N in |- *; apply lt_n_Sn ] ]. +unfold N in |- *; reflexivity. +unfold N in |- *; apply lt_O_Sn. +rewrite <- (Rmult_comm y); rewrite scal_sum; apply sum_eq. +intros; replace (S N - i)%nat with (S (N - i)). +replace (S (N - i)) with (N - i + 1)%nat; [ idtac | ring ]. +rewrite pow_add; replace (y ^ 1) with y; [ idtac | simpl in |- *; ring ]; + ring. +apply minus_Sn_m; assumption. +rewrite <- (Rmult_comm x); rewrite scal_sum; apply sum_eq. +intros; replace (S i) with (i + 1)%nat; [ idtac | ring ]; rewrite pow_add; + replace (x ^ 1) with x; [ idtac | simpl in |- *; ring ]; + ring. +intro; unfold C in |- *. +replace (INR (fact 0)) with 1; [ idtac | reflexivity ]. +replace (p - 0)%nat with p; [ idtac | apply minus_n_O ]. +rewrite Rmult_1_l; unfold Rdiv in |- *; rewrite <- Rinv_r_sym; + [ reflexivity | apply INR_fact_neq_0 ]. +intro; unfold C in |- *. +replace (p - p)%nat with 0%nat; [ idtac | apply minus_n_n ]. +replace (INR (fact 0)) with 1; [ idtac | reflexivity ]. +rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite <- Rinv_r_sym; + [ reflexivity | apply INR_fact_neq_0 ]. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Cauchy_prod.v b/theories/Reals/Cauchy_prod.v index a76307320..6cd5fa17f 100644 --- a/theories/Reals/Cauchy_prod.v +++ b/theories/Reals/Cauchy_prod.v @@ -8,340 +8,451 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Rseries. -Require PartSum. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import Rseries. +Require Import PartSum. Open Local Scope R_scope. (**********) -Lemma sum_N_predN : (An:nat->R;N:nat) (lt O N) -> (sum_f_R0 An N)==``(sum_f_R0 An (pred N)) + (An N)``. -Intros. -Replace N with (S (pred N)). -Rewrite tech5. -Reflexivity. -Symmetry; Apply S_pred with O; Assumption. +Lemma sum_N_predN : + forall (An:nat -> R) (N:nat), + (0 < N)%nat -> sum_f_R0 An N = sum_f_R0 An (pred N) + An N. +intros. +replace N with (S (pred N)). +rewrite tech5. +reflexivity. +symmetry in |- *; apply S_pred with 0%nat; assumption. Qed. (**********) -Lemma sum_plus : (An,Bn:nat->R;N:nat) (sum_f_R0 [l:nat]``(An l)+(Bn l)`` N)==``(sum_f_R0 An N)+(sum_f_R0 Bn N)``. -Intros. -Induction N. -Reflexivity. -Do 3 Rewrite tech5. -Rewrite HrecN; Ring. +Lemma sum_plus : + forall (An Bn:nat -> R) (N:nat), + sum_f_R0 (fun l:nat => An l + Bn l) N = sum_f_R0 An N + sum_f_R0 Bn N. +intros. +induction N as [| N HrecN]. +reflexivity. +do 3 rewrite tech5. +rewrite HrecN; ring. Qed. (* The main result *) -Theorem cauchy_finite : (An,Bn:nat->R;N:nat) (lt O N) -> (Rmult (sum_f_R0 An N) (sum_f_R0 Bn N)) == (Rplus (sum_f_R0 [k:nat](sum_f_R0 [p:nat]``(An p)*(Bn (minus k p))`` k) N) (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus N l))`` (pred (minus N k))) (pred N))). -Intros; Induction N. -Elim (lt_n_n ? H). -Cut N=O\/(lt O N). -Intro; Elim H0; Intro. -Rewrite H1; Simpl; Ring. -Replace (pred (S N)) with (S (pred N)). -Do 5 Rewrite tech5. -Rewrite Rmult_Rplus_distrl; Rewrite Rmult_Rplus_distr; Rewrite (HrecN H1). -Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. -Replace (pred (minus (S N) (S (pred N)))) with (O). -Rewrite Rmult_Rplus_distr; Replace (sum_f_R0 [l:nat]``(An (S (plus l (S (pred N)))))*(Bn (minus (S N) l))`` O) with ``(An (S N))*(Bn (S N))``. -Repeat Rewrite <- Rplus_assoc; Do 2 Rewrite <- (Rplus_sym ``(An (S N))*(Bn (S N))``); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. -Rewrite <- minus_n_n; Cut N=(1)\/(le (2) N). -Intro; Elim H2; Intro. -Rewrite H3; Simpl; Ring. -Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus N l))`` (pred (minus N k))) (pred N)) with (Rplus (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) (pred (pred N))) (sum_f_R0 [l:nat]``(An (S l))*(Bn (minus N l))`` (pred N))). -Replace (sum_f_R0 [p:nat]``(An p)*(Bn (minus (S N) p))`` N) with (Rplus (sum_f_R0 [l:nat]``(An (S l))*(Bn (minus N l))`` (pred N)) ``(An O)*(Bn (S N))``). -Repeat Rewrite <- Rplus_assoc; Rewrite <- (Rplus_sym (sum_f_R0 [l:nat]``(An (S l))*(Bn (minus N l))`` (pred N))); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. -Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus (S N) l))`` (pred (minus (S N) k))) (pred N)) with (Rplus (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) (pred N)) (Rmult (Bn (S N)) (sum_f_R0 [l:nat](An (S l)) (pred N)))). -Rewrite (decomp_sum An N H1); Rewrite Rmult_Rplus_distrl; Repeat Rewrite <- Rplus_assoc; Rewrite <- (Rplus_sym ``(An O)*(Bn (S N))``); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. -Repeat Rewrite <- Rplus_assoc; Rewrite <- (Rplus_sym (Rmult (sum_f_R0 [i:nat](An (S i)) (pred N)) (Bn (S N)))); Rewrite <- (Rplus_sym (Rmult (Bn (S N)) (sum_f_R0 [i:nat](An (S i)) (pred N)))); Rewrite (Rmult_sym (Bn (S N))); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. -Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) (pred N)) with (Rplus (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) (pred (pred N))) (Rmult (An (S N)) (sum_f_R0 [l:nat](Bn (S l)) (pred N)))). -Rewrite (decomp_sum Bn N H1); Rewrite Rmult_Rplus_distr. -Pose Z := (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) (pred (pred N))); Pose Z2 := (sum_f_R0 [i:nat](Bn (S i)) (pred N)); Ring. -Rewrite (sum_N_predN [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) (pred N)). -Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) (pred (pred N))) with (sum_f_R0 [k:nat](Rplus (sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) ``(An (S N))*(Bn (S k))``) (pred (pred N))). -Rewrite (sum_plus [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (pred (minus N k)))) [k:nat]``(An (S N))*(Bn (S k))`` (pred (pred N))). -Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. -Replace (pred (minus N (pred N))) with O. -Simpl; Rewrite <- minus_n_O. -Replace (S (pred N)) with N. -Replace (sum_f_R0 [k:nat]``(An (S N))*(Bn (S k))`` (pred (pred N))) with (sum_f_R0 [k:nat]``(Bn (S k))*(An (S N))`` (pred (pred N))). -Rewrite <- (scal_sum [l:nat](Bn (S l)) (pred (pred N)) (An (S N))); Rewrite (sum_N_predN [l:nat](Bn (S l)) (pred N)). -Replace (S (pred N)) with N. -Ring. -Apply S_pred with O; Assumption. -Apply lt_pred; Apply lt_le_trans with (2); [Apply lt_n_Sn | Assumption]. -Apply sum_eq; Intros; Apply Rmult_sym. -Apply S_pred with O; Assumption. -Replace (minus N (pred N)) with (1). -Reflexivity. -Pattern 1 N; Replace N with (S (pred N)). -Rewrite <- minus_Sn_m. -Rewrite <- minus_n_n; Reflexivity. -Apply le_n. -Symmetry; Apply S_pred with O; Assumption. -Apply sum_eq; Intros; Rewrite (sum_N_predN [l:nat]``(An (S (S (plus l i))))*(Bn (minus N l))`` (pred (minus N i))). -Replace (S (S (plus (pred (minus N i)) i))) with (S N). -Replace (minus N (pred (minus N i))) with (S i). -Ring. -Rewrite pred_of_minus; Apply INR_eq; Repeat Rewrite minus_INR. -Rewrite S_INR; Ring. -Apply le_trans with (pred (pred N)). -Assumption. -Apply le_trans with (pred N); Apply le_pred_n. -Apply INR_le; Rewrite minus_INR. -Apply Rle_anti_compatibility with ``(INR i)-1``. -Replace ``(INR i)-1+(INR (S O))`` with (INR i); [Idtac | Ring]. -Replace ``(INR i)-1+((INR N)-(INR i))`` with ``(INR N)-(INR (S O))``; [Idtac | Ring]. -Rewrite <- minus_INR. -Apply le_INR; Apply le_trans with (pred (pred N)). -Assumption. -Rewrite <- pred_of_minus; Apply le_pred_n. -Apply le_trans with (2). -Apply le_n_Sn. -Assumption. -Apply le_trans with (pred (pred N)). -Assumption. -Apply le_trans with (pred N); Apply le_pred_n. -Rewrite <- pred_of_minus. -Apply le_trans with (pred N). -Apply le_S_n. -Replace (S (pred N)) with N. -Replace (S (pred (minus N i))) with (minus N i). -Apply simpl_le_plus_l with i; Rewrite le_plus_minus_r. -Apply le_plus_r. -Apply le_trans with (pred (pred N)); [Assumption | Apply le_trans with (pred N); Apply le_pred_n]. -Apply S_pred with O. -Apply simpl_lt_plus_l with i; Rewrite le_plus_minus_r. -Replace (plus i O) with i; [Idtac | Ring]. -Apply le_lt_trans with (pred (pred N)); [Assumption | Apply lt_trans with (pred N); Apply lt_pred_n_n]. -Apply lt_S_n. -Replace (S (pred N)) with N. -Apply lt_le_trans with (2). -Apply lt_n_Sn. -Assumption. -Apply S_pred with O; Assumption. -Assumption. -Apply le_trans with (pred (pred N)). -Assumption. -Apply le_trans with (pred N); Apply le_pred_n. -Apply S_pred with O; Assumption. -Apply le_pred_n. -Apply INR_eq; Rewrite pred_of_minus; Do 3 Rewrite S_INR; Rewrite plus_INR; Repeat Rewrite minus_INR. -Ring. -Apply le_trans with (pred (pred N)). -Assumption. -Apply le_trans with (pred N); Apply le_pred_n. -Apply INR_le. -Rewrite minus_INR. -Apply Rle_anti_compatibility with ``(INR i)-1``. -Replace ``(INR i)-1+(INR (S O))`` with (INR i); [Idtac | Ring]. -Replace ``(INR i)-1+((INR N)-(INR i))`` with ``(INR N)-(INR (S O))``; [Idtac | Ring]. -Rewrite <- minus_INR. -Apply le_INR. -Apply le_trans with (pred (pred N)). -Assumption. -Rewrite <- pred_of_minus. -Apply le_pred_n. -Apply le_trans with (2). -Apply le_n_Sn. -Assumption. -Apply le_trans with (pred (pred N)). -Assumption. -Apply le_trans with (pred N); Apply le_pred_n. -Apply lt_le_trans with (1). -Apply lt_O_Sn. -Apply INR_le. -Rewrite pred_of_minus. -Repeat Rewrite minus_INR. -Apply Rle_anti_compatibility with ``(INR i)-1``. -Replace ``(INR i)-1+(INR (S O))`` with (INR i); [Idtac | Ring]. -Replace ``(INR i)-1+((INR N)-(INR i)-(INR (S O)))`` with ``(INR N)-(INR (S O)) -(INR (S O))``. -Repeat Rewrite <- minus_INR. -Apply le_INR. -Apply le_trans with (pred (pred N)). -Assumption. -Do 2 Rewrite <- pred_of_minus. -Apply le_n. -Apply simpl_le_plus_l with (1). -Rewrite le_plus_minus_r. -Simpl; Assumption. -Apply le_trans with (2); [Apply le_n_Sn | Assumption]. -Apply le_trans with (2); [Apply le_n_Sn | Assumption]. -Ring. -Apply le_trans with (pred (pred N)). -Assumption. -Apply le_trans with (pred N); Apply le_pred_n. -Apply simpl_le_plus_l with i. -Rewrite le_plus_minus_r. -Replace (plus i (1)) with (S i). -Replace N with (S (pred N)). -Apply le_n_S. -Apply le_trans with (pred (pred N)). -Assumption. -Apply le_pred_n. -Symmetry; Apply S_pred with O; Assumption. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Reflexivity. -Apply le_trans with (pred (pred N)). -Assumption. -Apply le_trans with (pred N); Apply le_pred_n. -Apply lt_le_trans with (1). -Apply lt_O_Sn. -Apply le_S_n. -Replace (S (pred N)) with N. -Assumption. -Apply S_pred with O; Assumption. -Replace (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus (S N) l))`` (pred (minus (S N) k))) (pred N)) with (sum_f_R0 [k:nat](Rplus (sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) ``(An (S k))*(Bn (S N))``) (pred N)). -Rewrite (sum_plus [k:nat](sum_f_R0 [l:nat]``(An (S (S (plus l k))))*(Bn (minus N l))`` (pred (minus N k))) [k:nat]``(An (S k))*(Bn (S N))``). -Apply Rplus_plus_r. -Rewrite scal_sum; Reflexivity. -Apply sum_eq; Intros; Rewrite Rplus_sym; Rewrite (decomp_sum [l:nat]``(An (S (plus l i)))*(Bn (minus (S N) l))`` (pred (minus (S N) i))). -Replace (plus O i) with i; [Idtac | Ring]. -Rewrite <- minus_n_O; Apply Rplus_plus_r. -Replace (pred (pred (minus (S N) i))) with (pred (minus N i)). -Apply sum_eq; Intros. -Replace (minus (S N) (S i0)) with (minus N i0); [Idtac | Reflexivity]. -Replace (plus (S i0) i) with (S (plus i0 i)). -Reflexivity. -Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite S_INR; Ring. -Cut (minus N i)=(pred (minus (S N) i)). -Intro; Rewrite H5; Reflexivity. -Rewrite pred_of_minus. -Apply INR_eq; Repeat Rewrite minus_INR. -Rewrite S_INR; Ring. -Apply le_trans with N. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_n_Sn. -Apply simpl_le_plus_l with i. -Rewrite le_plus_minus_r. -Replace (plus i (1)) with (S i). -Apply le_n_S. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Ring. -Apply le_trans with N. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_n_Sn. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Replace (pred (minus (S N) i)) with (minus (S N) (S i)). -Replace (minus (S N) (S i)) with (minus N i); [Idtac | Reflexivity]. -Apply simpl_lt_plus_l with i. -Rewrite le_plus_minus_r. -Replace (plus i O) with i; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Assumption. -Apply lt_pred_n_n. -Assumption. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Rewrite pred_of_minus. -Apply INR_eq; Repeat Rewrite minus_INR. -Repeat Rewrite S_INR; Ring. -Apply le_trans with N. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_n_Sn. -Apply simpl_le_plus_l with i. -Rewrite le_plus_minus_r. -Replace (plus i (1)) with (S i). -Apply le_n_S. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Ring. -Apply le_trans with N. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_n_Sn. -Apply le_n_S. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Rewrite Rplus_sym. -Rewrite (decomp_sum [p:nat]``(An p)*(Bn (minus (S N) p))`` N). -Rewrite <- minus_n_O. -Apply Rplus_plus_r. -Apply sum_eq; Intros. -Reflexivity. -Assumption. -Rewrite Rplus_sym. -Rewrite (decomp_sum [k:nat](sum_f_R0 [l:nat]``(An (S (plus l k)))*(Bn (minus N l))`` (pred (minus N k))) (pred N)). -Rewrite <- minus_n_O. -Replace (sum_f_R0 [l:nat]``(An (S (plus l O)))*(Bn (minus N l))`` (pred N)) with (sum_f_R0 [l:nat]``(An (S l))*(Bn (minus N l))`` (pred N)). -Apply Rplus_plus_r. -Apply sum_eq; Intros. -Replace (pred (minus N (S i))) with (pred (pred (minus N i))). -Apply sum_eq; Intros. -Replace (plus i0 (S i)) with (S (plus i0 i)). -Reflexivity. -Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite S_INR; Ring. -Cut (pred (minus N i))=(minus N (S i)). -Intro; Rewrite H5; Reflexivity. -Rewrite pred_of_minus. -Apply INR_eq. -Repeat Rewrite minus_INR. -Repeat Rewrite S_INR; Ring. -Apply le_trans with (S (pred (pred N))). -Apply le_n_S; Assumption. -Replace (S (pred (pred N))) with (pred N). -Apply le_pred_n. -Apply S_pred with O. -Apply lt_S_n. -Replace (S (pred N)) with N. -Apply lt_le_trans with (2). -Apply lt_n_Sn. -Assumption. -Apply S_pred with O; Assumption. -Apply le_trans with (pred (pred N)). -Assumption. -Apply le_trans with (pred N); Apply le_pred_n. -Apply simpl_le_plus_l with i. -Rewrite le_plus_minus_r. -Replace (plus i (1)) with (S i). -Replace N with (S (pred N)). -Apply le_n_S. -Apply le_trans with (pred (pred N)). -Assumption. -Apply le_pred_n. -Symmetry; Apply S_pred with O; Assumption. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Ring. -Apply le_trans with (pred (pred N)). -Assumption. -Apply le_trans with (pred N); Apply le_pred_n. -Apply sum_eq; Intros. -Replace (plus i O) with i; [Reflexivity | Trivial]. -Apply lt_S_n. -Replace (S (pred N)) with N. -Apply lt_le_trans with (2); [Apply lt_n_Sn | Assumption]. -Apply S_pred with O; Assumption. -Inversion H1. -Left; Reflexivity. -Right; Apply le_n_S; Assumption. -Simpl. -Replace (S (pred N)) with N. -Reflexivity. -Apply S_pred with O; Assumption. -Simpl. -Cut (minus N (pred N))=(1). -Intro; Rewrite H2; Reflexivity. -Rewrite pred_of_minus. -Apply INR_eq; Repeat Rewrite minus_INR. -Ring. -Apply lt_le_S; Assumption. -Rewrite <- pred_of_minus; Apply le_pred_n. -Simpl; Symmetry; Apply S_pred with O; Assumption. -Inversion H. -Left; Reflexivity. -Right; Apply lt_le_trans with (1); [Apply lt_n_Sn | Exact H1]. -Qed. +Theorem cauchy_finite : + forall (An Bn:nat -> R) (N:nat), + (0 < N)%nat -> + sum_f_R0 An N * sum_f_R0 Bn N = + sum_f_R0 (fun k:nat => sum_f_R0 (fun p:nat => An p * Bn (k - p)%nat) k) N + + sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (l + k)) * Bn (N - l)%nat) + (pred (N - k))) (pred N). +intros; induction N as [| N HrecN]. +elim (lt_irrefl _ H). +cut (N = 0%nat \/ (0 < N)%nat). +intro; elim H0; intro. +rewrite H1; simpl in |- *; ring. +replace (pred (S N)) with (S (pred N)). +do 5 rewrite tech5. +rewrite Rmult_plus_distr_r; rewrite Rmult_plus_distr_l; rewrite (HrecN H1). +repeat rewrite Rplus_assoc; apply Rplus_eq_compat_l. +replace (pred (S N - S (pred N))) with 0%nat. +rewrite Rmult_plus_distr_l; + replace + (sum_f_R0 (fun l:nat => An (S (l + S (pred N))) * Bn (S N - l)%nat) 0) with + (An (S N) * Bn (S N)). +repeat rewrite <- Rplus_assoc; + do 2 rewrite <- (Rplus_comm (An (S N) * Bn (S N))); + repeat rewrite Rplus_assoc; apply Rplus_eq_compat_l. +rewrite <- minus_n_n; cut (N = 1%nat \/ (2 <= N)%nat). +intro; elim H2; intro. +rewrite H3; simpl in |- *; ring. +replace + (sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (l + k)) * Bn (N - l)%nat) (pred (N - k))) + (pred N)) with + (sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (S (l + k))) * Bn (N - l)%nat) + (pred (pred (N - k)))) (pred (pred N)) + + sum_f_R0 (fun l:nat => An (S l) * Bn (N - l)%nat) (pred N)). +replace (sum_f_R0 (fun p:nat => An p * Bn (S N - p)%nat) N) with + (sum_f_R0 (fun l:nat => An (S l) * Bn (N - l)%nat) (pred N) + + An 0%nat * Bn (S N)). +repeat rewrite <- Rplus_assoc; + rewrite <- + (Rplus_comm (sum_f_R0 (fun l:nat => An (S l) * Bn (N - l)%nat) (pred N))) + ; repeat rewrite Rplus_assoc; apply Rplus_eq_compat_l. +replace + (sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (l + k)) * Bn (S N - l)%nat) + (pred (S N - k))) (pred N)) with + (sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (S (l + k))) * Bn (N - l)%nat) + (pred (N - k))) (pred N) + + Bn (S N) * sum_f_R0 (fun l:nat => An (S l)) (pred N)). +rewrite (decomp_sum An N H1); rewrite Rmult_plus_distr_r; + repeat rewrite <- Rplus_assoc; rewrite <- (Rplus_comm (An 0%nat * Bn (S N))); + repeat rewrite Rplus_assoc; apply Rplus_eq_compat_l. +repeat rewrite <- Rplus_assoc; + rewrite <- + (Rplus_comm (sum_f_R0 (fun i:nat => An (S i)) (pred N) * Bn (S N))) + ; + rewrite <- + (Rplus_comm (Bn (S N) * sum_f_R0 (fun i:nat => An (S i)) (pred N))) + ; rewrite (Rmult_comm (Bn (S N))); repeat rewrite Rplus_assoc; + apply Rplus_eq_compat_l. +replace + (sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (S (l + k))) * Bn (N - l)%nat) + (pred (N - k))) (pred N)) with + (sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (S (l + k))) * Bn (N - l)%nat) + (pred (pred (N - k)))) (pred (pred N)) + + An (S N) * sum_f_R0 (fun l:nat => Bn (S l)) (pred N)). +rewrite (decomp_sum Bn N H1); rewrite Rmult_plus_distr_l. +pose + (Z := + sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (S (l + k))) * Bn (N - l)%nat) + (pred (pred (N - k)))) (pred (pred N))); + pose (Z2 := sum_f_R0 (fun i:nat => Bn (S i)) (pred N)); + ring. +rewrite + (sum_N_predN + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (S (l + k))) * Bn (N - l)%nat) + (pred (N - k))) (pred N)). +replace + (sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (S (l + k))) * Bn (N - l)%nat) + (pred (N - k))) (pred (pred N))) with + (sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (S (l + k))) * Bn (N - l)%nat) + (pred (pred (N - k))) + An (S N) * Bn (S k)) ( + pred (pred N))). +rewrite + (sum_plus + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (S (l + k))) * Bn (N - l)%nat) + (pred (pred (N - k)))) (fun k:nat => An (S N) * Bn (S k)) + (pred (pred N))). +repeat rewrite Rplus_assoc; apply Rplus_eq_compat_l. +replace (pred (N - pred N)) with 0%nat. +simpl in |- *; rewrite <- minus_n_O. +replace (S (pred N)) with N. +replace (sum_f_R0 (fun k:nat => An (S N) * Bn (S k)) (pred (pred N))) with + (sum_f_R0 (fun k:nat => Bn (S k) * An (S N)) (pred (pred N))). +rewrite <- (scal_sum (fun l:nat => Bn (S l)) (pred (pred N)) (An (S N))); + rewrite (sum_N_predN (fun l:nat => Bn (S l)) (pred N)). +replace (S (pred N)) with N. +ring. +apply S_pred with 0%nat; assumption. +apply lt_pred; apply lt_le_trans with 2%nat; [ apply lt_n_Sn | assumption ]. +apply sum_eq; intros; apply Rmult_comm. +apply S_pred with 0%nat; assumption. +replace (N - pred N)%nat with 1%nat. +reflexivity. +pattern N at 1 in |- *; replace N with (S (pred N)). +rewrite <- minus_Sn_m. +rewrite <- minus_n_n; reflexivity. +apply le_n. +symmetry in |- *; apply S_pred with 0%nat; assumption. +apply sum_eq; intros; + rewrite + (sum_N_predN (fun l:nat => An (S (S (l + i))) * Bn (N - l)%nat) + (pred (N - i))). +replace (S (S (pred (N - i) + i))) with (S N). +replace (N - pred (N - i))%nat with (S i). +ring. +rewrite pred_of_minus; apply INR_eq; repeat rewrite minus_INR. +rewrite S_INR; ring. +apply le_trans with (pred (pred N)). +assumption. +apply le_trans with (pred N); apply le_pred_n. +apply INR_le; rewrite minus_INR. +apply Rplus_le_reg_l with (INR i - 1). +replace (INR i - 1 + INR 1) with (INR i); [ idtac | ring ]. +replace (INR i - 1 + (INR N - INR i)) with (INR N - INR 1); [ idtac | ring ]. +rewrite <- minus_INR. +apply le_INR; apply le_trans with (pred (pred N)). +assumption. +rewrite <- pred_of_minus; apply le_pred_n. +apply le_trans with 2%nat. +apply le_n_Sn. +assumption. +apply le_trans with (pred (pred N)). +assumption. +apply le_trans with (pred N); apply le_pred_n. +rewrite <- pred_of_minus. +apply le_trans with (pred N). +apply le_S_n. +replace (S (pred N)) with N. +replace (S (pred (N - i))) with (N - i)%nat. +apply (fun p n m:nat => plus_le_reg_l n m p) with i; rewrite le_plus_minus_r. +apply le_plus_r. +apply le_trans with (pred (pred N)); + [ assumption | apply le_trans with (pred N); apply le_pred_n ]. +apply S_pred with 0%nat. +apply plus_lt_reg_l with i; rewrite le_plus_minus_r. +replace (i + 0)%nat with i; [ idtac | ring ]. +apply le_lt_trans with (pred (pred N)); + [ assumption | apply lt_trans with (pred N); apply lt_pred_n_n ]. +apply lt_S_n. +replace (S (pred N)) with N. +apply lt_le_trans with 2%nat. +apply lt_n_Sn. +assumption. +apply S_pred with 0%nat; assumption. +assumption. +apply le_trans with (pred (pred N)). +assumption. +apply le_trans with (pred N); apply le_pred_n. +apply S_pred with 0%nat; assumption. +apply le_pred_n. +apply INR_eq; rewrite pred_of_minus; do 3 rewrite S_INR; rewrite plus_INR; + repeat rewrite minus_INR. +ring. +apply le_trans with (pred (pred N)). +assumption. +apply le_trans with (pred N); apply le_pred_n. +apply INR_le. +rewrite minus_INR. +apply Rplus_le_reg_l with (INR i - 1). +replace (INR i - 1 + INR 1) with (INR i); [ idtac | ring ]. +replace (INR i - 1 + (INR N - INR i)) with (INR N - INR 1); [ idtac | ring ]. +rewrite <- minus_INR. +apply le_INR. +apply le_trans with (pred (pred N)). +assumption. +rewrite <- pred_of_minus. +apply le_pred_n. +apply le_trans with 2%nat. +apply le_n_Sn. +assumption. +apply le_trans with (pred (pred N)). +assumption. +apply le_trans with (pred N); apply le_pred_n. +apply lt_le_trans with 1%nat. +apply lt_O_Sn. +apply INR_le. +rewrite pred_of_minus. +repeat rewrite minus_INR. +apply Rplus_le_reg_l with (INR i - 1). +replace (INR i - 1 + INR 1) with (INR i); [ idtac | ring ]. +replace (INR i - 1 + (INR N - INR i - INR 1)) with (INR N - INR 1 - INR 1). +repeat rewrite <- minus_INR. +apply le_INR. +apply le_trans with (pred (pred N)). +assumption. +do 2 rewrite <- pred_of_minus. +apply le_n. +apply (fun p n m:nat => plus_le_reg_l n m p) with 1%nat. +rewrite le_plus_minus_r. +simpl in |- *; assumption. +apply le_trans with 2%nat; [ apply le_n_Sn | assumption ]. +apply le_trans with 2%nat; [ apply le_n_Sn | assumption ]. +ring. +apply le_trans with (pred (pred N)). +assumption. +apply le_trans with (pred N); apply le_pred_n. +apply (fun p n m:nat => plus_le_reg_l n m p) with i. +rewrite le_plus_minus_r. +replace (i + 1)%nat with (S i). +replace N with (S (pred N)). +apply le_n_S. +apply le_trans with (pred (pred N)). +assumption. +apply le_pred_n. +symmetry in |- *; apply S_pred with 0%nat; assumption. +apply INR_eq; rewrite S_INR; rewrite plus_INR; reflexivity. +apply le_trans with (pred (pred N)). +assumption. +apply le_trans with (pred N); apply le_pred_n. +apply lt_le_trans with 1%nat. +apply lt_O_Sn. +apply le_S_n. +replace (S (pred N)) with N. +assumption. +apply S_pred with 0%nat; assumption. +replace + (sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (l + k)) * Bn (S N - l)%nat) + (pred (S N - k))) (pred N)) with + (sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (S (l + k))) * Bn (N - l)%nat) + (pred (N - k)) + An (S k) * Bn (S N)) (pred N)). +rewrite + (sum_plus + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (S (l + k))) * Bn (N - l)%nat) + (pred (N - k))) (fun k:nat => An (S k) * Bn (S N))) + . +apply Rplus_eq_compat_l. +rewrite scal_sum; reflexivity. +apply sum_eq; intros; rewrite Rplus_comm; + rewrite + (decomp_sum (fun l:nat => An (S (l + i)) * Bn (S N - l)%nat) + (pred (S N - i))). +replace (0 + i)%nat with i; [ idtac | ring ]. +rewrite <- minus_n_O; apply Rplus_eq_compat_l. +replace (pred (pred (S N - i))) with (pred (N - i)). +apply sum_eq; intros. +replace (S N - S i0)%nat with (N - i0)%nat; [ idtac | reflexivity ]. +replace (S i0 + i)%nat with (S (i0 + i)). +reflexivity. +apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; rewrite S_INR; ring. +cut ((N - i)%nat = pred (S N - i)). +intro; rewrite H5; reflexivity. +rewrite pred_of_minus. +apply INR_eq; repeat rewrite minus_INR. +rewrite S_INR; ring. +apply le_trans with N. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply (fun p n m:nat => plus_le_reg_l n m p) with i. +rewrite le_plus_minus_r. +replace (i + 1)%nat with (S i). +apply le_n_S. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply INR_eq; rewrite S_INR; rewrite plus_INR; ring. +apply le_trans with N. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +replace (pred (S N - i)) with (S N - S i)%nat. +replace (S N - S i)%nat with (N - i)%nat; [ idtac | reflexivity ]. +apply plus_lt_reg_l with i. +rewrite le_plus_minus_r. +replace (i + 0)%nat with i; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n. +assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +rewrite pred_of_minus. +apply INR_eq; repeat rewrite minus_INR. +repeat rewrite S_INR; ring. +apply le_trans with N. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply (fun p n m:nat => plus_le_reg_l n m p) with i. +rewrite le_plus_minus_r. +replace (i + 1)%nat with (S i). +apply le_n_S. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply INR_eq; rewrite S_INR; rewrite plus_INR; ring. +apply le_trans with N. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply le_n_S. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +rewrite Rplus_comm. +rewrite (decomp_sum (fun p:nat => An p * Bn (S N - p)%nat) N). +rewrite <- minus_n_O. +apply Rplus_eq_compat_l. +apply sum_eq; intros. +reflexivity. +assumption. +rewrite Rplus_comm. +rewrite + (decomp_sum + (fun k:nat => + sum_f_R0 (fun l:nat => An (S (l + k)) * Bn (N - l)%nat) (pred (N - k))) + (pred N)). +rewrite <- minus_n_O. +replace (sum_f_R0 (fun l:nat => An (S (l + 0)) * Bn (N - l)%nat) (pred N)) + with (sum_f_R0 (fun l:nat => An (S l) * Bn (N - l)%nat) (pred N)). +apply Rplus_eq_compat_l. +apply sum_eq; intros. +replace (pred (N - S i)) with (pred (pred (N - i))). +apply sum_eq; intros. +replace (i0 + S i)%nat with (S (i0 + i)). +reflexivity. +apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; rewrite S_INR; ring. +cut (pred (N - i) = (N - S i)%nat). +intro; rewrite H5; reflexivity. +rewrite pred_of_minus. +apply INR_eq. +repeat rewrite minus_INR. +repeat rewrite S_INR; ring. +apply le_trans with (S (pred (pred N))). +apply le_n_S; assumption. +replace (S (pred (pred N))) with (pred N). +apply le_pred_n. +apply S_pred with 0%nat. +apply lt_S_n. +replace (S (pred N)) with N. +apply lt_le_trans with 2%nat. +apply lt_n_Sn. +assumption. +apply S_pred with 0%nat; assumption. +apply le_trans with (pred (pred N)). +assumption. +apply le_trans with (pred N); apply le_pred_n. +apply (fun p n m:nat => plus_le_reg_l n m p) with i. +rewrite le_plus_minus_r. +replace (i + 1)%nat with (S i). +replace N with (S (pred N)). +apply le_n_S. +apply le_trans with (pred (pred N)). +assumption. +apply le_pred_n. +symmetry in |- *; apply S_pred with 0%nat; assumption. +apply INR_eq; rewrite S_INR; rewrite plus_INR; ring. +apply le_trans with (pred (pred N)). +assumption. +apply le_trans with (pred N); apply le_pred_n. +apply sum_eq; intros. +replace (i + 0)%nat with i; [ reflexivity | trivial ]. +apply lt_S_n. +replace (S (pred N)) with N. +apply lt_le_trans with 2%nat; [ apply lt_n_Sn | assumption ]. +apply S_pred with 0%nat; assumption. +inversion H1. +left; reflexivity. +right; apply le_n_S; assumption. +simpl in |- *. +replace (S (pred N)) with N. +reflexivity. +apply S_pred with 0%nat; assumption. +simpl in |- *. +cut ((N - pred N)%nat = 1%nat). +intro; rewrite H2; reflexivity. +rewrite pred_of_minus. +apply INR_eq; repeat rewrite minus_INR. +ring. +apply lt_le_S; assumption. +rewrite <- pred_of_minus; apply le_pred_n. +simpl in |- *; symmetry in |- *; apply S_pred with 0%nat; assumption. +inversion H. +left; reflexivity. +right; apply lt_le_trans with 1%nat; [ apply lt_n_Sn | exact H1 ]. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Cos_plus.v b/theories/Reals/Cos_plus.v index 41815fc20..d29193ad7 100644 --- a/theories/Reals/Cos_plus.v +++ b/theories/Reals/Cos_plus.v @@ -8,1010 +8,1054 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo_def. -Require Cos_rel. -Require Max. -V7only [Import nat_scope.]. Open Local Scope nat_scope. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo_def. +Require Import Cos_rel. +Require Import Max. Open Local Scope nat_scope. Open Local Scope R_scope. -Definition Majxy [x,y:R] : nat->R := [n:nat](Rdiv (pow (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (4) (S n))) (INR (fact n))). +Definition Majxy (x y:R) (n:nat) : R := + Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S n) / INR (fact n). -Lemma Majxy_cv_R0 : (x,y:R) (Un_cv (Majxy x y) R0). -Intros. -Pose C := (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))). -Pose C0 := (pow C (4)). -Cut ``0<C``. -Intro. -Cut ``0<C0``. -Intro. -Assert H1 := (cv_speed_pow_fact C0). -Unfold Un_cv in H1; Unfold R_dist in H1. -Unfold Un_cv; Unfold R_dist; Intros. -Cut ``0<eps/C0``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Assumption]]. -Elim (H1 ``eps/C0`` H3); Intros N0 H4. -Exists N0; Intros. -Replace (Majxy x y n) with ``(pow C0 (S n))/(INR (fact n))``. -Simpl. -Apply Rlt_monotony_contra with ``(Rabsolu (/C0))``. -Apply Rabsolu_pos_lt. -Apply Rinv_neq_R0. -Red; Intro; Rewrite H6 in H0; Elim (Rlt_antirefl ? H0). -Rewrite <- Rabsolu_mult. -Unfold Rminus; Rewrite Rmult_Rplus_distr. -Rewrite Ropp_O; Rewrite Rmult_Or. -Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Rewrite (Rabsolu_right ``/C0``). -Rewrite <- (Rmult_sym eps). -Replace ``(pow C0 n)*/(INR (fact n))+0`` with ``(pow C0 n)*/(INR (fact n))-0``; [Idtac | Ring]. -Unfold Rdiv in H4; Apply H4; Assumption. -Apply Rle_sym1; Left; Apply Rlt_Rinv; Assumption. -Red; Intro; Rewrite H6 in H0; Elim (Rlt_antirefl ? H0). -Unfold Majxy. -Unfold C0. -Rewrite pow_mult. -Unfold C; Reflexivity. -Unfold C0; Apply pow_lt; Assumption. -Apply Rlt_le_trans with R1. -Apply Rlt_R0_R1. -Unfold C. -Apply RmaxLess1. +Lemma Majxy_cv_R0 : forall x y:R, Un_cv (Majxy x y) 0. +intros. +pose (C := Rmax 1 (Rmax (Rabs x) (Rabs y))). +pose (C0 := C ^ 4). +cut (0 < C). +intro. +cut (0 < C0). +intro. +assert (H1 := cv_speed_pow_fact C0). +unfold Un_cv in H1; unfold R_dist in H1. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +cut (0 < eps / C0); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; assumption ] ]. +elim (H1 (eps / C0) H3); intros N0 H4. +exists N0; intros. +replace (Majxy x y n) with (C0 ^ S n / INR (fact n)). +simpl in |- *. +apply Rmult_lt_reg_l with (Rabs (/ C0)). +apply Rabs_pos_lt. +apply Rinv_neq_0_compat. +red in |- *; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0). +rewrite <- Rabs_mult. +unfold Rminus in |- *; rewrite Rmult_plus_distr_l. +rewrite Ropp_0; rewrite Rmult_0_r. +unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +rewrite (Rabs_right (/ C0)). +rewrite <- (Rmult_comm eps). +replace (C0 ^ n * / INR (fact n) + 0) with (C0 ^ n * / INR (fact n) - 0); + [ idtac | ring ]. +unfold Rdiv in H4; apply H4; assumption. +apply Rle_ge; left; apply Rinv_0_lt_compat; assumption. +red in |- *; intro; rewrite H6 in H0; elim (Rlt_irrefl _ H0). +unfold Majxy in |- *. +unfold C0 in |- *. +rewrite pow_mult. +unfold C in |- *; reflexivity. +unfold C0 in |- *; apply pow_lt; assumption. +apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *. +apply RmaxLess1. Qed. -Lemma reste1_maj : (x,y:R;N:nat) (lt O N) -> ``(Rabsolu (Reste1 x y N))<=(Majxy x y (pred N))``. -Intros. -Pose C := (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))). -Unfold Reste1. -Apply Rle_trans with (sum_f_R0 - [k:nat] - (Rabsolu (sum_f_R0 - [l:nat] - ``(pow ( -1) (S (plus l k)))/ - (INR (fact (mult (S (S O)) (S (plus l k)))))* - (pow x (mult (S (S O)) (S (plus l k))))* - (pow ( -1) (minus N l))/ - (INR (fact (mult (S (S O)) (minus N l))))* - (pow y (mult (S (S O)) (minus N l)))`` (pred (minus N k)))) - (pred N)). -Apply (sum_Rabsolu [k:nat] - (sum_f_R0 - [l:nat] - ``(pow ( -1) (S (plus l k)))/ - (INR (fact (mult (S (S O)) (S (plus l k)))))* - (pow x (mult (S (S O)) (S (plus l k))))* - (pow ( -1) (minus N l))/ - (INR (fact (mult (S (S O)) (minus N l))))* - (pow y (mult (S (S O)) (minus N l)))`` (pred (minus N k))) (pred N)). -Apply Rle_trans with (sum_f_R0 - [k:nat] - (sum_f_R0 - [l:nat] - (Rabsolu (``(pow ( -1) (S (plus l k)))/ - (INR (fact (mult (S (S O)) (S (plus l k)))))* - (pow x (mult (S (S O)) (S (plus l k))))* - (pow ( -1) (minus N l))/ - (INR (fact (mult (S (S O)) (minus N l))))* - (pow y (mult (S (S O)) (minus N l)))``)) (pred (minus N k))) - (pred N)). -Apply sum_Rle. -Intros. -Apply (sum_Rabsolu [l:nat] - ``(pow ( -1) (S (plus l n)))/ - (INR (fact (mult (S (S O)) (S (plus l n)))))* - (pow x (mult (S (S O)) (S (plus l n))))* - (pow ( -1) (minus N l))/ - (INR (fact (mult (S (S O)) (minus N l))))* - (pow y (mult (S (S O)) (minus N l)))`` (pred (minus N n))). -Apply Rle_trans with (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``/(INR (mult (fact (mult (S (S O)) (S (plus l k)))) (fact (mult (S (S O)) (minus N l)))))*(pow C (mult (S (S O)) (S (plus N k))))`` (pred (minus N k))) (pred N)). -Apply sum_Rle; Intros. -Apply sum_Rle; Intros. -Unfold Rdiv; Repeat Rewrite Rabsolu_mult. -Do 2 Rewrite pow_1_abs. -Do 2 Rewrite Rmult_1l. -Rewrite (Rabsolu_right ``/(INR (fact (mult (S (S O)) (S (plus n0 n)))))``). -Rewrite (Rabsolu_right ``/(INR (fact (mult (S (S O)) (minus N n0))))``). -Rewrite mult_INR. -Rewrite Rinv_Rmult. -Repeat Rewrite Rmult_assoc. -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Rewrite <- Rmult_assoc. -Rewrite <- (Rmult_sym ``/(INR (fact (mult (S (S O)) (minus N n0))))``). -Rewrite Rmult_assoc. -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Do 2 Rewrite <- Pow_Rabsolu. -Apply Rle_trans with ``(pow (Rabsolu x) (mult (S (S O)) (S (plus n0 n))))*(pow C (mult (S (S O)) (minus N n0)))``. -Apply Rle_monotony. -Apply pow_le; Apply Rabsolu_pos. -Apply pow_incr. -Split. -Apply Rabsolu_pos. -Unfold C. -Apply Rle_trans with (Rmax (Rabsolu x) (Rabsolu y)); Apply RmaxLess2. -Apply Rle_trans with ``(pow C (mult (S (S O)) (S (plus n0 n))))*(pow C (mult (S (S O)) (minus N n0)))``. -Do 2 Rewrite <- (Rmult_sym ``(pow C (mult (S (S O)) (minus N n0)))``). -Apply Rle_monotony. -Apply pow_le. -Apply Rle_trans with R1. -Left; Apply Rlt_R0_R1. -Unfold C; Apply RmaxLess1. -Apply pow_incr. -Split. -Apply Rabsolu_pos. -Unfold C; Apply Rle_trans with (Rmax (Rabsolu x) (Rabsolu y)). -Apply RmaxLess1. -Apply RmaxLess2. -Right. -Replace (mult (2) (S (plus N n))) with (plus (mult (2) (minus N n0)) (mult (2) (S (plus n0 n)))). -Rewrite pow_add. -Apply Rmult_sym. -Apply INR_eq; Rewrite plus_INR; Do 3 Rewrite mult_INR. -Rewrite minus_INR. -Repeat Rewrite S_INR; Do 2 Rewrite plus_INR; Ring. -Apply le_trans with (pred (minus N n)). -Exact H1. -Apply le_S_n. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_trans with N. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_n_Sn. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n O) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Assumption. -Apply lt_pred_n_n; Assumption. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply Rle_trans with (sum_f_R0 - [k:nat] - (sum_f_R0 - [l:nat] - ``/(INR - (mult (fact (mult (S (S O)) (S (plus l k)))) - (fact (mult (S (S O)) (minus N l)))))* - (pow C (mult (S (S (S (S O)))) N))`` (pred (minus N k))) - (pred N)). -Apply sum_Rle; Intros. -Apply sum_Rle; Intros. -Apply Rle_monotony. -Left; Apply Rlt_Rinv. -Rewrite mult_INR; Apply Rmult_lt_pos; Apply INR_fact_lt_0. -Apply Rle_pow. -Unfold C; Apply RmaxLess1. -Replace (mult (4) N) with (mult (2) (mult (2) N)); [Idtac | Ring]. -Apply mult_le. -Replace (mult (2) N) with (S (plus N (pred N))). -Apply le_n_S. -Apply le_reg_l; Assumption. -Rewrite pred_of_minus. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Rewrite minus_INR. -Repeat Rewrite S_INR; Ring. -Apply lt_le_S; Assumption. -Apply Rle_trans with (sum_f_R0 - [k:nat] - (sum_f_R0 - [l:nat] - ``(pow C (mult (S (S (S (S O)))) N))*(Rsqr (/(INR (fact (S (plus N k))))))`` (pred (minus N k))) - (pred N)). -Apply sum_Rle; Intros. -Apply sum_Rle; Intros. -Rewrite <- (Rmult_sym ``(pow C (mult (S (S (S (S O)))) N))``). -Apply Rle_monotony. -Apply pow_le. -Left; Apply Rlt_le_trans with R1. -Apply Rlt_R0_R1. -Unfold C; Apply RmaxLess1. -Replace ``/(INR - (mult (fact (mult (S (S O)) (S (plus n0 n)))) - (fact (mult (S (S O)) (minus N n0)))))`` with ``(Binomial.C (mult (S (S O)) (S (plus N n))) (mult (S (S O)) (S (plus n0 n))))/(INR (fact (mult (S (S O)) (S (plus N n)))))``. -Apply Rle_trans with ``(Binomial.C (mult (S (S O)) (S (plus N n))) (S (plus N n)))/(INR (fact (mult (S (S O)) (S (plus N n)))))``. -Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/(INR (fact (mult (S (S O)) (S (plus N n)))))``). -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply C_maj. -Apply mult_le. -Apply le_n_S. -Apply le_reg_r. -Apply le_trans with (pred (minus N n)). -Assumption. -Apply le_S_n. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_trans with N. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_n_Sn. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n O) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Assumption. -Apply lt_pred_n_n; Assumption. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Right. -Unfold Rdiv; Rewrite Rmult_sym. -Unfold Binomial.C. -Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Replace (minus (mult (2) (S (plus N n))) (S (plus N n))) with (S (plus N n)). -Rewrite Rinv_Rmult. -Unfold Rsqr; Reflexivity. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_eq; Rewrite S_INR; Rewrite minus_INR. -Rewrite mult_INR; Repeat Rewrite S_INR; Rewrite plus_INR; Ring. -Apply le_n_2n. -Apply INR_fact_neq_0. -Unfold Rdiv; Rewrite Rmult_sym. -Unfold Binomial.C. -Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Replace (minus (mult (2) (S (plus N n))) (mult (2) (S (plus n0 n)))) with (mult (2) (minus N n0)). -Rewrite mult_INR. -Reflexivity. -Apply INR_eq; Rewrite minus_INR. -Do 3 Rewrite mult_INR; Repeat Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite minus_INR. -Ring. -Apply le_trans with (pred (minus N n)). -Assumption. -Apply le_S_n. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_trans with N. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_n_Sn. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n O) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Assumption. -Apply lt_pred_n_n; Assumption. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply mult_le. -Apply le_n_S. -Apply le_reg_r. -Apply le_trans with (pred (minus N n)). -Assumption. -Apply le_S_n. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_trans with N. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_n_Sn. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n O) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Assumption. -Apply lt_pred_n_n; Assumption. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply INR_fact_neq_0. -Apply Rle_trans with (sum_f_R0 [k:nat]``(INR N)/(INR (fact (S N)))*(pow C (mult (S (S (S (S O)))) N))`` (pred N)). -Apply sum_Rle; Intros. -Rewrite <- (scal_sum [_:nat]``(pow C (mult (S (S (S (S O)))) N))`` (pred (minus N n)) ``(Rsqr (/(INR (fact (S (plus N n))))))``). -Rewrite sum_cte. -Rewrite <- Rmult_assoc. -Do 2 Rewrite <- (Rmult_sym ``(pow C (mult (S (S (S (S O)))) N))``). -Rewrite Rmult_assoc. -Apply Rle_monotony. -Apply pow_le. -Left; Apply Rlt_le_trans with R1. -Apply Rlt_R0_R1. -Unfold C; Apply RmaxLess1. -Apply Rle_trans with ``(Rsqr (/(INR (fact (S (plus N n))))))*(INR N)``. -Apply Rle_monotony. -Apply pos_Rsqr. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_INR. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n O) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Assumption. -Apply lt_pred_n_n; Assumption. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Rewrite Rmult_sym; Unfold Rdiv; Apply Rle_monotony. -Apply pos_INR. -Apply Rle_trans with ``/(INR (fact (S (plus N n))))``. -Pattern 2 ``/(INR (fact (S (plus N n))))``; Rewrite <- Rmult_1r. -Unfold Rsqr. -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply Rle_monotony_contra with ``(INR (fact (S (plus N n))))``. -Apply INR_fact_lt_0. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r. -Replace R1 with (INR (S O)). -Apply le_INR. -Apply lt_le_S. -Apply INR_lt; Apply INR_fact_lt_0. -Reflexivity. -Apply INR_fact_neq_0. -Apply Rle_monotony_contra with ``(INR (fact (S (plus N n))))``. -Apply INR_fact_lt_0. -Rewrite <- Rinv_r_sym. -Apply Rle_monotony_contra with ``(INR (fact (S N)))``. -Apply INR_fact_lt_0. -Rewrite Rmult_1r. -Rewrite (Rmult_sym (INR (fact (S N)))). -Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Apply le_INR. -Apply fact_growing. -Apply le_n_S. -Apply le_plus_l. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Rewrite sum_cte. -Apply Rle_trans with ``(pow C (mult (S (S (S (S O)))) N))/(INR (fact (pred N)))``. -Rewrite <- (Rmult_sym ``(pow C (mult (S (S (S (S O)))) N))``). -Unfold Rdiv; Rewrite Rmult_assoc; Apply Rle_monotony. -Apply pow_le. -Left; Apply Rlt_le_trans with R1. -Apply Rlt_R0_R1. -Unfold C; Apply RmaxLess1. -Cut (S (pred N)) = N. -Intro; Rewrite H0. -Pattern 2 N; Rewrite <- H0. -Do 2 Rewrite fact_simpl. -Rewrite H0. -Repeat Rewrite mult_INR. -Repeat Rewrite Rinv_Rmult. -Rewrite (Rmult_sym ``/(INR (S N))``). -Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l. -Pattern 2 ``/(INR (fact (pred N)))``; Rewrite <- Rmult_1r. -Rewrite Rmult_assoc. -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply Rle_monotony_contra with (INR (S N)). -Apply lt_INR_0; Apply lt_O_Sn. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Rewrite Rmult_1l. -Apply le_INR; Apply le_n_Sn. -Apply not_O_INR; Discriminate. -Apply not_O_INR. -Red; Intro; Rewrite H1 in H; Elim (lt_n_n ? H). -Apply not_O_INR. -Red; Intro; Rewrite H1 in H; Elim (lt_n_n ? H). -Apply INR_fact_neq_0. -Apply not_O_INR; Discriminate. -Apply prod_neq_R0. -Apply not_O_INR. -Red; Intro; Rewrite H1 in H; Elim (lt_n_n ? H). -Apply INR_fact_neq_0. -Symmetry; Apply S_pred with O; Assumption. -Right. -Unfold Majxy. -Unfold C. -Replace (S (pred N)) with N. -Reflexivity. -Apply S_pred with O; Assumption. +Lemma reste1_maj : + forall (x y:R) (N:nat), + (0 < N)%nat -> Rabs (Reste1 x y N) <= Majxy x y (pred N). +intros. +pose (C := Rmax 1 (Rmax (Rabs x) (Rabs y))). +unfold Reste1 in |- *. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + Rabs + (sum_f_R0 + (fun l:nat => + (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) * + x ^ (2 * S (l + k)) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) * + y ^ (2 * (N - l))) (pred (N - k)))) ( + pred N)). +apply + (Rsum_abs + (fun k:nat => + sum_f_R0 + (fun l:nat => + (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) * + x ^ (2 * S (l + k)) * ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) * + y ^ (2 * (N - l))) (pred (N - k))) (pred N)). +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + Rabs + ((-1) ^ S (l + k) / INR (fact (2 * S (l + k))) * + x ^ (2 * S (l + k)) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) * + y ^ (2 * (N - l)))) (pred (N - k))) ( + pred N)). +apply sum_Rle. +intros. +apply + (Rsum_abs + (fun l:nat => + (-1) ^ S (l + n) / INR (fact (2 * S (l + n))) * x ^ (2 * S (l + n)) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) * + y ^ (2 * (N - l))) (pred (N - n))). +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + / INR (fact (2 * S (l + k)) * fact (2 * (N - l))) * + C ^ (2 * S (N + k))) (pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +unfold Rdiv in |- *; repeat rewrite Rabs_mult. +do 2 rewrite pow_1_abs. +do 2 rewrite Rmult_1_l. +rewrite (Rabs_right (/ INR (fact (2 * S (n0 + n))))). +rewrite (Rabs_right (/ INR (fact (2 * (N - n0))))). +rewrite mult_INR. +rewrite Rinv_mult_distr. +repeat rewrite Rmult_assoc. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +rewrite <- Rmult_assoc. +rewrite <- (Rmult_comm (/ INR (fact (2 * (N - n0))))). +rewrite Rmult_assoc. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +do 2 rewrite <- RPow_abs. +apply Rle_trans with (Rabs x ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0))). +apply Rmult_le_compat_l. +apply pow_le; apply Rabs_pos. +apply pow_incr. +split. +apply Rabs_pos. +unfold C in |- *. +apply Rle_trans with (Rmax (Rabs x) (Rabs y)); apply RmaxLess2. +apply Rle_trans with (C ^ (2 * S (n0 + n)) * C ^ (2 * (N - n0))). +do 2 rewrite <- (Rmult_comm (C ^ (2 * (N - n0)))). +apply Rmult_le_compat_l. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +apply pow_incr. +split. +apply Rabs_pos. +unfold C in |- *; apply Rle_trans with (Rmax (Rabs x) (Rabs y)). +apply RmaxLess1. +apply RmaxLess2. +right. +replace (2 * S (N + n))%nat with (2 * (N - n0) + 2 * S (n0 + n))%nat. +rewrite pow_add. +apply Rmult_comm. +apply INR_eq; rewrite plus_INR; do 3 rewrite mult_INR. +rewrite minus_INR. +repeat rewrite S_INR; do 2 rewrite plus_INR; ring. +apply le_trans with (pred (N - n)). +exact H1. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply Rle_ge; left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_ge; left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + / INR (fact (2 * S (l + k)) * fact (2 * (N - l))) * C ^ (4 * N)) + (pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat. +rewrite mult_INR; apply Rmult_lt_0_compat; apply INR_fact_lt_0. +apply Rle_pow. +unfold C in |- *; apply RmaxLess1. +replace (4 * N)%nat with (2 * (2 * N))%nat; [ idtac | ring ]. +apply (fun m n p:nat => mult_le_compat_l p n m). +replace (2 * N)%nat with (S (N + pred N)). +apply le_n_S. +apply plus_le_compat_l; assumption. +rewrite pred_of_minus. +apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR; + rewrite minus_INR. +repeat rewrite S_INR; ring. +apply lt_le_S; assumption. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => C ^ (4 * N) * Rsqr (/ INR (fact (S (N + k))))) + (pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +rewrite <- (Rmult_comm (C ^ (4 * N))). +apply Rmult_le_compat_l. +apply pow_le. +left; apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +replace (/ INR (fact (2 * S (n0 + n)) * fact (2 * (N - n0)))) with + (Binomial.C (2 * S (N + n)) (2 * S (n0 + n)) / INR (fact (2 * S (N + n)))). +apply Rle_trans with + (Binomial.C (2 * S (N + n)) (S (N + n)) / INR (fact (2 * S (N + n)))). +unfold Rdiv in |- *; + do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (N + n))))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply C_maj. +apply (fun m n p:nat => mult_le_compat_l p n m). +apply le_n_S. +apply plus_le_compat_r. +apply le_trans with (pred (N - n)). +assumption. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +right. +unfold Rdiv in |- *; rewrite Rmult_comm. +unfold Binomial.C in |- *. +unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +replace (2 * S (N + n) - S (N + n))%nat with (S (N + n)). +rewrite Rinv_mult_distr. +unfold Rsqr in |- *; reflexivity. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_eq; rewrite S_INR; rewrite minus_INR. +rewrite mult_INR; repeat rewrite S_INR; rewrite plus_INR; ring. +apply le_n_2n. +apply INR_fact_neq_0. +unfold Rdiv in |- *; rewrite Rmult_comm. +unfold Binomial.C in |- *. +unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +replace (2 * S (N + n) - 2 * S (n0 + n))%nat with (2 * (N - n0))%nat. +rewrite mult_INR. +reflexivity. +apply INR_eq; rewrite minus_INR. +do 3 rewrite mult_INR; repeat rewrite S_INR; do 2 rewrite plus_INR; + rewrite minus_INR. +ring. +apply le_trans with (pred (N - n)). +assumption. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply (fun m n p:nat => mult_le_compat_l p n m). +apply le_n_S. +apply plus_le_compat_r. +apply le_trans with (pred (N - n)). +assumption. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply INR_fact_neq_0. +apply Rle_trans with + (sum_f_R0 (fun k:nat => INR N / INR (fact (S N)) * C ^ (4 * N)) (pred N)). +apply sum_Rle; intros. +rewrite <- + (scal_sum (fun _:nat => C ^ (4 * N)) (pred (N - n)) + (Rsqr (/ INR (fact (S (N + n)))))). +rewrite sum_cte. +rewrite <- Rmult_assoc. +do 2 rewrite <- (Rmult_comm (C ^ (4 * N))). +rewrite Rmult_assoc. +apply Rmult_le_compat_l. +apply pow_le. +left; apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +apply Rle_trans with (Rsqr (/ INR (fact (S (N + n)))) * INR N). +apply Rmult_le_compat_l. +apply Rle_0_sqr. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_INR. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +rewrite Rmult_comm; unfold Rdiv in |- *; apply Rmult_le_compat_l. +apply pos_INR. +apply Rle_trans with (/ INR (fact (S (N + n)))). +pattern (/ INR (fact (S (N + n)))) at 2 in |- *; rewrite <- Rmult_1_r. +unfold Rsqr in |- *. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rmult_le_reg_l with (INR (fact (S (N + n)))). +apply INR_fact_lt_0. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r. +replace 1 with (INR 1). +apply le_INR. +apply lt_le_S. +apply INR_lt; apply INR_fact_lt_0. +reflexivity. +apply INR_fact_neq_0. +apply Rmult_le_reg_l with (INR (fact (S (N + n)))). +apply INR_fact_lt_0. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with (INR (fact (S N))). +apply INR_fact_lt_0. +rewrite Rmult_1_r. +rewrite (Rmult_comm (INR (fact (S N)))). +rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +apply le_INR. +apply fact_le. +apply le_n_S. +apply le_plus_l. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +rewrite sum_cte. +apply Rle_trans with (C ^ (4 * N) / INR (fact (pred N))). +rewrite <- (Rmult_comm (C ^ (4 * N))). +unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l. +apply pow_le. +left; apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +cut (S (pred N) = N). +intro; rewrite H0. +pattern N at 2 in |- *; rewrite <- H0. +do 2 rewrite fact_simpl. +rewrite H0. +repeat rewrite mult_INR. +repeat rewrite Rinv_mult_distr. +rewrite (Rmult_comm (/ INR (S N))). +repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_l. +pattern (/ INR (fact (pred N))) at 2 in |- *; rewrite <- Rmult_1_r. +rewrite Rmult_assoc. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rmult_le_reg_l with (INR (S N)). +apply lt_INR_0; apply lt_O_Sn. +rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; rewrite Rmult_1_l. +apply le_INR; apply le_n_Sn. +apply not_O_INR; discriminate. +apply not_O_INR. +red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H). +apply not_O_INR. +red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H). +apply INR_fact_neq_0. +apply not_O_INR; discriminate. +apply prod_neq_R0. +apply not_O_INR. +red in |- *; intro; rewrite H1 in H; elim (lt_irrefl _ H). +apply INR_fact_neq_0. +symmetry in |- *; apply S_pred with 0%nat; assumption. +right. +unfold Majxy in |- *. +unfold C in |- *. +replace (S (pred N)) with N. +reflexivity. +apply S_pred with 0%nat; assumption. Qed. -Lemma reste2_maj : (x,y:R;N:nat) (lt O N) -> ``(Rabsolu (Reste2 x y N))<=(Majxy x y N)``. -Intros. -Pose C := (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))). -Unfold Reste2. -Apply Rle_trans with (sum_f_R0 - [k:nat] - (Rabsolu (sum_f_R0 - [l:nat] - ``(pow ( -1) (S (plus l k)))/ - (INR (fact (plus (mult (S (S O)) (S (plus l k))) (S O))))* - (pow x (plus (mult (S (S O)) (S (plus l k))) (S O)))* - (pow ( -1) (minus N l))/ - (INR (fact (plus (mult (S (S O)) (minus N l)) (S O))))* - (pow y (plus (mult (S (S O)) (minus N l)) (S O)))`` (pred (minus N k)))) - (pred N)). -Apply (sum_Rabsolu [k:nat] - (sum_f_R0 - [l:nat] - ``(pow ( -1) (S (plus l k)))/ - (INR (fact (plus (mult (S (S O)) (S (plus l k))) (S O))))* - (pow x (plus (mult (S (S O)) (S (plus l k))) (S O)))* - (pow ( -1) (minus N l))/ - (INR (fact (plus (mult (S (S O)) (minus N l)) (S O))))* - (pow y (plus (mult (S (S O)) (minus N l)) (S O)))`` (pred (minus N k))) (pred N)). -Apply Rle_trans with (sum_f_R0 - [k:nat] - (sum_f_R0 - [l:nat] - (Rabsolu (``(pow ( -1) (S (plus l k)))/ - (INR (fact (plus (mult (S (S O)) (S (plus l k))) (S O))))* - (pow x (plus (mult (S (S O)) (S (plus l k))) (S O)))* - (pow ( -1) (minus N l))/ - (INR (fact (plus (mult (S (S O)) (minus N l)) (S O))))* - (pow y (plus (mult (S (S O)) (minus N l)) (S O)))``)) (pred (minus N k))) - (pred N)). -Apply sum_Rle. -Intros. -Apply (sum_Rabsolu [l:nat] - ``(pow ( -1) (S (plus l n)))/ - (INR (fact (plus (mult (S (S O)) (S (plus l n))) (S O))))* - (pow x (plus (mult (S (S O)) (S (plus l n))) (S O)))* - (pow ( -1) (minus N l))/ - (INR (fact (plus (mult (S (S O)) (minus N l)) (S O))))* - (pow y (plus (mult (S (S O)) (minus N l)) (S O)))`` (pred (minus N n))). -Apply Rle_trans with (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``/(INR (mult (fact (plus (mult (S (S O)) (S (plus l k))) (S O))) (fact (plus (mult (S (S O)) (minus N l)) (S O)))))*(pow C (mult (S (S O)) (S (S (plus N k)))))`` (pred (minus N k))) (pred N)). -Apply sum_Rle; Intros. -Apply sum_Rle; Intros. -Unfold Rdiv; Repeat Rewrite Rabsolu_mult. -Do 2 Rewrite pow_1_abs. -Do 2 Rewrite Rmult_1l. -Rewrite (Rabsolu_right ``/(INR (fact (plus (mult (S (S O)) (S (plus n0 n))) (S O))))``). -Rewrite (Rabsolu_right ``/(INR (fact (plus (mult (S (S O)) (minus N n0)) (S O))))``). -Rewrite mult_INR. -Rewrite Rinv_Rmult. -Repeat Rewrite Rmult_assoc. -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Rewrite <- Rmult_assoc. -Rewrite <- (Rmult_sym ``/(INR (fact (plus (mult (S (S O)) (minus N n0)) (S O))))``). -Rewrite Rmult_assoc. -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Do 2 Rewrite <- Pow_Rabsolu. -Apply Rle_trans with ``(pow (Rabsolu x) (plus (mult (S (S O)) (S (plus n0 n))) (S O)))*(pow C (plus (mult (S (S O)) (minus N n0)) (S O)))``. -Apply Rle_monotony. -Apply pow_le; Apply Rabsolu_pos. -Apply pow_incr. -Split. -Apply Rabsolu_pos. -Unfold C. -Apply Rle_trans with (Rmax (Rabsolu x) (Rabsolu y)); Apply RmaxLess2. -Apply Rle_trans with ``(pow C (plus (mult (S (S O)) (S (plus n0 n))) (S O)))*(pow C (plus (mult (S (S O)) (minus N n0)) (S O)))``. -Do 2 Rewrite <- (Rmult_sym ``(pow C (plus (mult (S (S O)) (minus N n0)) (S O)))``). -Apply Rle_monotony. -Apply pow_le. -Apply Rle_trans with R1. -Left; Apply Rlt_R0_R1. -Unfold C; Apply RmaxLess1. -Apply pow_incr. -Split. -Apply Rabsolu_pos. -Unfold C; Apply Rle_trans with (Rmax (Rabsolu x) (Rabsolu y)). -Apply RmaxLess1. -Apply RmaxLess2. -Right. -Replace (mult (2) (S (S (plus N n)))) with (plus (plus (mult (2) (minus N n0)) (S O)) (plus (mult (2) (S (plus n0 n))) (S O))). -Repeat Rewrite pow_add. -Ring. -Apply INR_eq; Repeat Rewrite plus_INR; Do 3 Rewrite mult_INR. -Rewrite minus_INR. -Repeat Rewrite S_INR; Do 2 Rewrite plus_INR; Ring. -Apply le_trans with (pred (minus N n)). -Exact H1. -Apply le_S_n. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_trans with N. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_n_Sn. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n O) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Assumption. -Apply lt_pred_n_n; Assumption. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply Rle_sym1; Left; Apply Rlt_Rinv. -Apply INR_fact_lt_0. -Apply Rle_sym1; Left; Apply Rlt_Rinv. -Apply INR_fact_lt_0. -Apply Rle_trans with (sum_f_R0 - [k:nat] - (sum_f_R0 - [l:nat] - ``/(INR - (mult (fact (plus (mult (S (S O)) (S (plus l k))) (S O))) - (fact (plus (mult (S (S O)) (minus N l)) (S O)))))* - (pow C (mult (S (S (S (S O)))) (S N)))`` (pred (minus N k))) - (pred N)). -Apply sum_Rle; Intros. -Apply sum_Rle; Intros. -Apply Rle_monotony. -Left; Apply Rlt_Rinv. -Rewrite mult_INR; Apply Rmult_lt_pos; Apply INR_fact_lt_0. -Apply Rle_pow. -Unfold C; Apply RmaxLess1. -Replace (mult (4) (S N)) with (mult (2) (mult (2) (S N))); [Idtac | Ring]. -Apply mult_le. -Replace (mult (2) (S N)) with (S (S (plus N N))). -Repeat Apply le_n_S. -Apply le_reg_l. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply INR_eq; Do 2Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR. -Repeat Rewrite S_INR; Ring. -Apply Rle_trans with (sum_f_R0 - [k:nat] - (sum_f_R0 - [l:nat] - ``(pow C (mult (S (S (S (S O)))) (S N)))*(Rsqr (/(INR (fact (S (S (plus N k)))))))`` (pred (minus N k))) - (pred N)). -Apply sum_Rle; Intros. -Apply sum_Rle; Intros. -Rewrite <- (Rmult_sym ``(pow C (mult (S (S (S (S O)))) (S N)))``). -Apply Rle_monotony. -Apply pow_le. -Left; Apply Rlt_le_trans with R1. -Apply Rlt_R0_R1. -Unfold C; Apply RmaxLess1. -Replace ``/(INR - (mult (fact (plus (mult (S (S O)) (S (plus n0 n))) (S O))) - (fact (plus (mult (S (S O)) (minus N n0)) (S O)))))`` with ``(Binomial.C (mult (S (S O)) (S (S (plus N n)))) (plus (mult (S (S O)) (S (plus n0 n))) (S O)))/(INR (fact (mult (S (S O)) (S (S (plus N n))))))``. -Apply Rle_trans with ``(Binomial.C (mult (S (S O)) (S (S (plus N n)))) (S (S (plus N n))))/(INR (fact (mult (S (S O)) (S (S (plus N n))))))``. -Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/(INR (fact (mult (S (S O)) (S (S (plus N n))))))``). -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply C_maj. -Apply le_trans with (mult (2) (S (S (plus n0 n)))). -Replace (mult (2) (S (S (plus n0 n)))) with (S (plus (mult (2) (S (plus n0 n))) (1))). -Apply le_n_Sn. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Rewrite plus_INR; Ring. -Apply mult_le. -Repeat Apply le_n_S. -Apply le_reg_r. -Apply le_trans with (pred (minus N n)). -Assumption. -Apply le_S_n. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_trans with N. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_n_Sn. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n O) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Assumption. -Apply lt_pred_n_n; Assumption. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Right. -Unfold Rdiv; Rewrite Rmult_sym. -Unfold Binomial.C. -Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Replace (minus (mult (2) (S (S (plus N n)))) (S (S (plus N n)))) with (S (S (plus N n))). -Rewrite Rinv_Rmult. -Unfold Rsqr; Reflexivity. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_eq; Do 2 Rewrite S_INR; Rewrite minus_INR. -Rewrite mult_INR; Repeat Rewrite S_INR; Rewrite plus_INR; Ring. -Apply le_n_2n. -Apply INR_fact_neq_0. -Unfold Rdiv; Rewrite Rmult_sym. -Unfold Binomial.C. -Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Replace (minus (mult (2) (S (S (plus N n)))) (plus (mult (2) (S (plus n0 n))) (S O))) with (plus (mult (2) (minus N n0)) (S O)). -Rewrite mult_INR. -Reflexivity. -Apply INR_eq; Rewrite minus_INR. -Do 2 Rewrite plus_INR; Do 3 Rewrite mult_INR; Repeat Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite minus_INR. -Ring. -Apply le_trans with (pred (minus N n)). -Assumption. -Apply le_S_n. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_trans with N. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_n_Sn. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n O) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Assumption. -Apply lt_pred_n_n; Assumption. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_trans with (mult (2) (S (S (plus n0 n)))). -Replace (mult (2) (S (S (plus n0 n)))) with (S (plus (mult (2) (S (plus n0 n))) (1))). -Apply le_n_Sn. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Rewrite plus_INR; Ring. -Apply mult_le. -Repeat Apply le_n_S. -Apply le_reg_r. -Apply le_trans with (pred (minus N n)). -Assumption. -Apply le_S_n. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_trans with N. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply le_n_Sn. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n O) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Assumption. -Apply lt_pred_n_n; Assumption. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply INR_fact_neq_0. -Apply Rle_trans with (sum_f_R0 [k:nat]``(INR N)/(INR (fact (S (S N))))*(pow C (mult (S (S (S (S O)))) (S N)))`` (pred N)). -Apply sum_Rle; Intros. -Rewrite <- (scal_sum [_:nat]``(pow C (mult (S (S (S (S O)))) (S N)))`` (pred (minus N n)) ``(Rsqr (/(INR (fact (S (S (plus N n)))))))``). -Rewrite sum_cte. -Rewrite <- Rmult_assoc. -Do 2 Rewrite <- (Rmult_sym ``(pow C (mult (S (S (S (S O)))) (S N)))``). -Rewrite Rmult_assoc. -Apply Rle_monotony. -Apply pow_le. -Left; Apply Rlt_le_trans with R1. -Apply Rlt_R0_R1. -Unfold C; Apply RmaxLess1. -Apply Rle_trans with ``(Rsqr (/(INR (fact (S (S (plus N n)))))))*(INR N)``. -Apply Rle_monotony. -Apply pos_Rsqr. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_INR. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n O) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Assumption. -Apply lt_pred_n_n; Assumption. -Apply le_trans with (pred N). -Assumption. -Apply le_pred_n. -Rewrite Rmult_sym; Unfold Rdiv; Apply Rle_monotony. -Apply pos_INR. -Apply Rle_trans with ``/(INR (fact (S (S (plus N n)))))``. -Pattern 2 ``/(INR (fact (S (S (plus N n)))))``; Rewrite <- Rmult_1r. -Unfold Rsqr. -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply Rle_monotony_contra with ``(INR (fact (S (S (plus N n)))))``. -Apply INR_fact_lt_0. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r. -Replace R1 with (INR (S O)). -Apply le_INR. -Apply lt_le_S. -Apply INR_lt; Apply INR_fact_lt_0. -Reflexivity. -Apply INR_fact_neq_0. -Apply Rle_monotony_contra with ``(INR (fact (S (S (plus N n)))))``. -Apply INR_fact_lt_0. -Rewrite <- Rinv_r_sym. -Apply Rle_monotony_contra with ``(INR (fact (S (S N))))``. -Apply INR_fact_lt_0. -Rewrite Rmult_1r. -Rewrite (Rmult_sym (INR (fact (S (S N))))). -Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Apply le_INR. -Apply fact_growing. -Repeat Apply le_n_S. -Apply le_plus_l. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Rewrite sum_cte. -Apply Rle_trans with ``(pow C (mult (S (S (S (S O)))) (S N)))/(INR (fact N))``. -Rewrite <- (Rmult_sym ``(pow C (mult (S (S (S (S O)))) (S N)))``). -Unfold Rdiv; Rewrite Rmult_assoc; Apply Rle_monotony. -Apply pow_le. -Left; Apply Rlt_le_trans with R1. -Apply Rlt_R0_R1. -Unfold C; Apply RmaxLess1. -Cut (S (pred N)) = N. -Intro; Rewrite H0. -Do 2 Rewrite fact_simpl. -Repeat Rewrite mult_INR. -Repeat Rewrite Rinv_Rmult. -Apply Rle_trans with ``(INR (S (S N)))*(/(INR (S (S N)))*(/(INR (S N))*/(INR (fact N))))* - (INR N)``. -Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym (INR N)). -Rewrite (Rmult_sym (INR (S (S N)))). -Apply Rle_monotony. -Repeat Apply Rmult_le_pos. -Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply lt_O_Sn. -Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply lt_O_Sn. -Left; Apply Rlt_Rinv. -Apply INR_fact_lt_0. -Apply pos_INR. -Apply le_INR. -Apply le_trans with (S N); Apply le_n_Sn. -Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l. -Apply Rle_trans with ``/(INR (S N))*/(INR (fact N))*(INR (S N))``. -Repeat Rewrite Rmult_assoc. -Repeat Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply lt_O_Sn. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply le_INR; Apply le_n_Sn. -Rewrite (Rmult_sym ``/(INR (S N))``). -Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Right; Reflexivity. -Apply not_O_INR; Discriminate. -Apply not_O_INR; Discriminate. -Apply not_O_INR; Discriminate. -Apply INR_fact_neq_0. -Apply not_O_INR; Discriminate. -Apply prod_neq_R0; [Apply not_O_INR; Discriminate | Apply INR_fact_neq_0]. -Symmetry; Apply S_pred with O; Assumption. -Right. -Unfold Majxy. -Unfold C. -Reflexivity. +Lemma reste2_maj : + forall (x y:R) (N:nat), (0 < N)%nat -> Rabs (Reste2 x y N) <= Majxy x y N. +intros. +pose (C := Rmax 1 (Rmax (Rabs x) (Rabs y))). +unfold Reste2 in |- *. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + Rabs + (sum_f_R0 + (fun l:nat => + (-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) * + x ^ (2 * S (l + k) + 1) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) * + y ^ (2 * (N - l) + 1)) (pred (N - k)))) ( + pred N)). +apply + (Rsum_abs + (fun k:nat => + sum_f_R0 + (fun l:nat => + (-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) * + x ^ (2 * S (l + k) + 1) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) * + y ^ (2 * (N - l) + 1)) (pred (N - k))) ( + pred N)). +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + Rabs + ((-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) * + x ^ (2 * S (l + k) + 1) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) * + y ^ (2 * (N - l) + 1))) (pred (N - k))) ( + pred N)). +apply sum_Rle. +intros. +apply + (Rsum_abs + (fun l:nat => + (-1) ^ S (l + n) / INR (fact (2 * S (l + n) + 1)) * + x ^ (2 * S (l + n) + 1) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) * + y ^ (2 * (N - l) + 1)) (pred (N - n))). +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + / INR (fact (2 * S (l + k) + 1) * fact (2 * (N - l) + 1)) * + C ^ (2 * S (S (N + k)))) (pred (N - k))) ( + pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +unfold Rdiv in |- *; repeat rewrite Rabs_mult. +do 2 rewrite pow_1_abs. +do 2 rewrite Rmult_1_l. +rewrite (Rabs_right (/ INR (fact (2 * S (n0 + n) + 1)))). +rewrite (Rabs_right (/ INR (fact (2 * (N - n0) + 1)))). +rewrite mult_INR. +rewrite Rinv_mult_distr. +repeat rewrite Rmult_assoc. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +rewrite <- Rmult_assoc. +rewrite <- (Rmult_comm (/ INR (fact (2 * (N - n0) + 1)))). +rewrite Rmult_assoc. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +do 2 rewrite <- RPow_abs. +apply Rle_trans with (Rabs x ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1)). +apply Rmult_le_compat_l. +apply pow_le; apply Rabs_pos. +apply pow_incr. +split. +apply Rabs_pos. +unfold C in |- *. +apply Rle_trans with (Rmax (Rabs x) (Rabs y)); apply RmaxLess2. +apply Rle_trans with (C ^ (2 * S (n0 + n) + 1) * C ^ (2 * (N - n0) + 1)). +do 2 rewrite <- (Rmult_comm (C ^ (2 * (N - n0) + 1))). +apply Rmult_le_compat_l. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +apply pow_incr. +split. +apply Rabs_pos. +unfold C in |- *; apply Rle_trans with (Rmax (Rabs x) (Rabs y)). +apply RmaxLess1. +apply RmaxLess2. +right. +replace (2 * S (S (N + n)))%nat with + (2 * (N - n0) + 1 + (2 * S (n0 + n) + 1))%nat. +repeat rewrite pow_add. +ring. +apply INR_eq; repeat rewrite plus_INR; do 3 rewrite mult_INR. +rewrite minus_INR. +repeat rewrite S_INR; do 2 rewrite plus_INR; ring. +apply le_trans with (pred (N - n)). +exact H1. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply Rle_ge; left; apply Rinv_0_lt_compat. +apply INR_fact_lt_0. +apply Rle_ge; left; apply Rinv_0_lt_compat. +apply INR_fact_lt_0. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + / INR (fact (2 * S (l + k) + 1) * fact (2 * (N - l) + 1)) * + C ^ (4 * S N)) (pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat. +rewrite mult_INR; apply Rmult_lt_0_compat; apply INR_fact_lt_0. +apply Rle_pow. +unfold C in |- *; apply RmaxLess1. +replace (4 * S N)%nat with (2 * (2 * S N))%nat; [ idtac | ring ]. +apply (fun m n p:nat => mult_le_compat_l p n m). +replace (2 * S N)%nat with (S (S (N + N))). +repeat apply le_n_S. +apply plus_le_compat_l. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply INR_eq; do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR. +repeat rewrite S_INR; ring. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => C ^ (4 * S N) * Rsqr (/ INR (fact (S (S (N + k)))))) + (pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +rewrite <- (Rmult_comm (C ^ (4 * S N))). +apply Rmult_le_compat_l. +apply pow_le. +left; apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +replace (/ INR (fact (2 * S (n0 + n) + 1) * fact (2 * (N - n0) + 1))) with + (Binomial.C (2 * S (S (N + n))) (2 * S (n0 + n) + 1) / + INR (fact (2 * S (S (N + n))))). +apply Rle_trans with + (Binomial.C (2 * S (S (N + n))) (S (S (N + n))) / + INR (fact (2 * S (S (N + n))))). +unfold Rdiv in |- *; + do 2 rewrite <- (Rmult_comm (/ INR (fact (2 * S (S (N + n)))))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply C_maj. +apply le_trans with (2 * S (S (n0 + n)))%nat. +replace (2 * S (S (n0 + n)))%nat with (S (2 * S (n0 + n) + 1)). +apply le_n_Sn. +apply INR_eq; rewrite S_INR; rewrite plus_INR; do 2 rewrite mult_INR; + repeat rewrite S_INR; rewrite plus_INR; ring. +apply (fun m n p:nat => mult_le_compat_l p n m). +repeat apply le_n_S. +apply plus_le_compat_r. +apply le_trans with (pred (N - n)). +assumption. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +right. +unfold Rdiv in |- *; rewrite Rmult_comm. +unfold Binomial.C in |- *. +unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +replace (2 * S (S (N + n)) - S (S (N + n)))%nat with (S (S (N + n))). +rewrite Rinv_mult_distr. +unfold Rsqr in |- *; reflexivity. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_eq; do 2 rewrite S_INR; rewrite minus_INR. +rewrite mult_INR; repeat rewrite S_INR; rewrite plus_INR; ring. +apply le_n_2n. +apply INR_fact_neq_0. +unfold Rdiv in |- *; rewrite Rmult_comm. +unfold Binomial.C in |- *. +unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +replace (2 * S (S (N + n)) - (2 * S (n0 + n) + 1))%nat with + (2 * (N - n0) + 1)%nat. +rewrite mult_INR. +reflexivity. +apply INR_eq; rewrite minus_INR. +do 2 rewrite plus_INR; do 3 rewrite mult_INR; repeat rewrite S_INR; + do 2 rewrite plus_INR; rewrite minus_INR. +ring. +apply le_trans with (pred (N - n)). +assumption. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_trans with (2 * S (S (n0 + n)))%nat. +replace (2 * S (S (n0 + n)))%nat with (S (2 * S (n0 + n) + 1)). +apply le_n_Sn. +apply INR_eq; rewrite S_INR; rewrite plus_INR; do 2 rewrite mult_INR; + repeat rewrite S_INR; rewrite plus_INR; ring. +apply (fun m n p:nat => mult_le_compat_l p n m). +repeat apply le_n_S. +apply plus_le_compat_r. +apply le_trans with (pred (N - n)). +assumption. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply INR_fact_neq_0. +apply Rle_trans with + (sum_f_R0 (fun k:nat => INR N / INR (fact (S (S N))) * C ^ (4 * S N)) + (pred N)). +apply sum_Rle; intros. +rewrite <- + (scal_sum (fun _:nat => C ^ (4 * S N)) (pred (N - n)) + (Rsqr (/ INR (fact (S (S (N + n))))))). +rewrite sum_cte. +rewrite <- Rmult_assoc. +do 2 rewrite <- (Rmult_comm (C ^ (4 * S N))). +rewrite Rmult_assoc. +apply Rmult_le_compat_l. +apply pow_le. +left; apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +apply Rle_trans with (Rsqr (/ INR (fact (S (S (N + n))))) * INR N). +apply Rmult_le_compat_l. +apply Rle_0_sqr. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_INR. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +assumption. +apply lt_pred_n_n; assumption. +apply le_trans with (pred N). +assumption. +apply le_pred_n. +rewrite Rmult_comm; unfold Rdiv in |- *; apply Rmult_le_compat_l. +apply pos_INR. +apply Rle_trans with (/ INR (fact (S (S (N + n))))). +pattern (/ INR (fact (S (S (N + n))))) at 2 in |- *; rewrite <- Rmult_1_r. +unfold Rsqr in |- *. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rmult_le_reg_l with (INR (fact (S (S (N + n))))). +apply INR_fact_lt_0. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r. +replace 1 with (INR 1). +apply le_INR. +apply lt_le_S. +apply INR_lt; apply INR_fact_lt_0. +reflexivity. +apply INR_fact_neq_0. +apply Rmult_le_reg_l with (INR (fact (S (S (N + n))))). +apply INR_fact_lt_0. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with (INR (fact (S (S N)))). +apply INR_fact_lt_0. +rewrite Rmult_1_r. +rewrite (Rmult_comm (INR (fact (S (S N))))). +rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +apply le_INR. +apply fact_le. +repeat apply le_n_S. +apply le_plus_l. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +rewrite sum_cte. +apply Rle_trans with (C ^ (4 * S N) / INR (fact N)). +rewrite <- (Rmult_comm (C ^ (4 * S N))). +unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l. +apply pow_le. +left; apply Rlt_le_trans with 1. +apply Rlt_0_1. +unfold C in |- *; apply RmaxLess1. +cut (S (pred N) = N). +intro; rewrite H0. +do 2 rewrite fact_simpl. +repeat rewrite mult_INR. +repeat rewrite Rinv_mult_distr. +apply Rle_trans with + (INR (S (S N)) * (/ INR (S (S N)) * (/ INR (S N) * / INR (fact N))) * INR N). +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (INR N)). +rewrite (Rmult_comm (INR (S (S N)))). +apply Rmult_le_compat_l. +repeat apply Rmult_le_pos. +left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. +left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. +left; apply Rinv_0_lt_compat. +apply INR_fact_lt_0. +apply pos_INR. +apply le_INR. +apply le_trans with (S N); apply le_n_Sn. +repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_l. +apply Rle_trans with (/ INR (S N) * / INR (fact N) * INR (S N)). +repeat rewrite Rmult_assoc. +repeat apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply le_INR; apply le_n_Sn. +rewrite (Rmult_comm (/ INR (S N))). +rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; right; reflexivity. +apply not_O_INR; discriminate. +apply not_O_INR; discriminate. +apply not_O_INR; discriminate. +apply INR_fact_neq_0. +apply not_O_INR; discriminate. +apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ]. +symmetry in |- *; apply S_pred with 0%nat; assumption. +right. +unfold Majxy in |- *. +unfold C in |- *. +reflexivity. Qed. -Lemma reste1_cv_R0 : (x,y:R) (Un_cv (Reste1 x y) R0). -Intros. -Assert H := (Majxy_cv_R0 x y). -Unfold Un_cv in H; Unfold R_dist in H. -Unfold Un_cv; Unfold R_dist; Intros. -Elim (H eps H0); Intros N0 H1. -Exists (S N0); Intros. -Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or. -Apply Rle_lt_trans with (Rabsolu (Majxy x y (pred n))). -Rewrite (Rabsolu_right (Majxy x y (pred n))). -Apply reste1_maj. -Apply lt_le_trans with (S N0). -Apply lt_O_Sn. -Assumption. -Apply Rle_sym1. -Unfold Majxy. -Unfold Rdiv; Apply Rmult_le_pos. -Apply pow_le. -Apply Rle_trans with R1. -Left; Apply Rlt_R0_R1. -Apply RmaxLess1. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Replace (Majxy x y (pred n)) with ``(Majxy x y (pred n))-0``; [Idtac | Ring]. -Apply H1. -Unfold ge; Apply le_S_n. -Replace (S (pred n)) with n. -Assumption. -Apply S_pred with O. -Apply lt_le_trans with (S N0); [Apply lt_O_Sn | Assumption]. +Lemma reste1_cv_R0 : forall x y:R, Un_cv (Reste1 x y) 0. +intros. +assert (H := Majxy_cv_R0 x y). +unfold Un_cv in H; unfold R_dist in H. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H eps H0); intros N0 H1. +exists (S N0); intros. +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r. +apply Rle_lt_trans with (Rabs (Majxy x y (pred n))). +rewrite (Rabs_right (Majxy x y (pred n))). +apply reste1_maj. +apply lt_le_trans with (S N0). +apply lt_O_Sn. +assumption. +apply Rle_ge. +unfold Majxy in |- *. +unfold Rdiv in |- *; apply Rmult_le_pos. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +apply RmaxLess1. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +replace (Majxy x y (pred n)) with (Majxy x y (pred n) - 0); [ idtac | ring ]. +apply H1. +unfold ge in |- *; apply le_S_n. +replace (S (pred n)) with n. +assumption. +apply S_pred with 0%nat. +apply lt_le_trans with (S N0); [ apply lt_O_Sn | assumption ]. Qed. -Lemma reste2_cv_R0 : (x,y:R) (Un_cv (Reste2 x y) R0). -Intros. -Assert H := (Majxy_cv_R0 x y). -Unfold Un_cv in H; Unfold R_dist in H. -Unfold Un_cv; Unfold R_dist; Intros. -Elim (H eps H0); Intros N0 H1. -Exists (S N0); Intros. -Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or. -Apply Rle_lt_trans with (Rabsolu (Majxy x y n)). -Rewrite (Rabsolu_right (Majxy x y n)). -Apply reste2_maj. -Apply lt_le_trans with (S N0). -Apply lt_O_Sn. -Assumption. -Apply Rle_sym1. -Unfold Majxy. -Unfold Rdiv; Apply Rmult_le_pos. -Apply pow_le. -Apply Rle_trans with R1. -Left; Apply Rlt_R0_R1. -Apply RmaxLess1. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Replace (Majxy x y n) with ``(Majxy x y n)-0``; [Idtac | Ring]. -Apply H1. -Unfold ge; Apply le_trans with (S N0). -Apply le_n_Sn. -Exact H2. +Lemma reste2_cv_R0 : forall x y:R, Un_cv (Reste2 x y) 0. +intros. +assert (H := Majxy_cv_R0 x y). +unfold Un_cv in H; unfold R_dist in H. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H eps H0); intros N0 H1. +exists (S N0); intros. +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r. +apply Rle_lt_trans with (Rabs (Majxy x y n)). +rewrite (Rabs_right (Majxy x y n)). +apply reste2_maj. +apply lt_le_trans with (S N0). +apply lt_O_Sn. +assumption. +apply Rle_ge. +unfold Majxy in |- *. +unfold Rdiv in |- *; apply Rmult_le_pos. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +apply RmaxLess1. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +replace (Majxy x y n) with (Majxy x y n - 0); [ idtac | ring ]. +apply H1. +unfold ge in |- *; apply le_trans with (S N0). +apply le_n_Sn. +exact H2. Qed. -Lemma reste_cv_R0 : (x,y:R) (Un_cv (Reste x y) R0). -Intros. -Unfold Reste. -Pose An := [n:nat](Reste2 x y n). -Pose Bn := [n:nat](Reste1 x y (S n)). -Cut (Un_cv [n:nat]``(An n)-(Bn n)`` ``0-0``) -> (Un_cv [N:nat]``(Reste2 x y N)-(Reste1 x y (S N))`` ``0``). -Intro. -Apply H. -Apply CV_minus. -Unfold An. -Replace [n:nat](Reste2 x y n) with (Reste2 x y). -Apply reste2_cv_R0. -Reflexivity. -Unfold Bn. -Assert H0 := (reste1_cv_R0 x y). -Unfold Un_cv in H0; Unfold R_dist in H0. -Unfold Un_cv; Unfold R_dist; Intros. -Elim (H0 eps H1); Intros N0 H2. -Exists N0; Intros. -Apply H2. -Unfold ge; Apply le_trans with (S N0). -Apply le_n_Sn. -Apply le_n_S; Assumption. -Unfold An Bn. -Intro. -Replace R0 with ``0-0``; [Idtac | Ring]. -Exact H. +Lemma reste_cv_R0 : forall x y:R, Un_cv (Reste x y) 0. +intros. +unfold Reste in |- *. +pose (An := fun n:nat => Reste2 x y n). +pose (Bn := fun n:nat => Reste1 x y (S n)). +cut + (Un_cv (fun n:nat => An n - Bn n) (0 - 0) -> + Un_cv (fun N:nat => Reste2 x y N - Reste1 x y (S N)) 0). +intro. +apply H. +apply CV_minus. +unfold An in |- *. +replace (fun n:nat => Reste2 x y n) with (Reste2 x y). +apply reste2_cv_R0. +reflexivity. +unfold Bn in |- *. +assert (H0 := reste1_cv_R0 x y). +unfold Un_cv in H0; unfold R_dist in H0. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H0 eps H1); intros N0 H2. +exists N0; intros. +apply H2. +unfold ge in |- *; apply le_trans with (S N0). +apply le_n_Sn. +apply le_n_S; assumption. +unfold An, Bn in |- *. +intro. +replace 0 with (0 - 0); [ idtac | ring ]. +exact H. Qed. -Theorem cos_plus : (x,y:R) ``(cos (x+y))==(cos x)*(cos y)-(sin x)*(sin y)``. -Intros. -Cut (Un_cv (C1 x y) ``(cos x)*(cos y)-(sin x)*(sin y)``). -Cut (Un_cv (C1 x y) ``(cos (x+y))``). -Intros. -Apply UL_sequence with (C1 x y); Assumption. -Apply C1_cvg. -Unfold Un_cv; Unfold R_dist. -Intros. -Assert H0 := (A1_cvg x). -Assert H1 := (A1_cvg y). -Assert H2 := (B1_cvg x). -Assert H3 := (B1_cvg y). -Assert H4 := (CV_mult ? ? ? ? H0 H1). -Assert H5 := (CV_mult ? ? ? ? H2 H3). -Assert H6 := (reste_cv_R0 x y). -Unfold Un_cv in H4; Unfold Un_cv in H5; Unfold Un_cv in H6. -Unfold R_dist in H4; Unfold R_dist in H5; Unfold R_dist in H6. -Cut ``0<eps/3``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. -Elim (H4 ``eps/3`` H7); Intros N1 H8. -Elim (H5 ``eps/3`` H7); Intros N2 H9. -Elim (H6 ``eps/3`` H7); Intros N3 H10. -Pose N := (S (S (max (max N1 N2) N3))). -Exists N. -Intros. -Cut n = (S (pred n)). -Intro; Rewrite H12. -Rewrite <- cos_plus_form. -Rewrite <- H12. -Apply Rle_lt_trans with ``(Rabsolu ((A1 x n)*(A1 y n)-(cos x)*(cos y)))+(Rabsolu ((sin x)*(sin y)-(B1 x (pred n))*(B1 y (pred n))+(Reste x y (pred n))))``. -Replace ``(A1 x n)*(A1 y n)-(B1 x (pred n))*(B1 y (pred n))+ - (Reste x y (pred n))-((cos x)*(cos y)-(sin x)*(sin y))`` with ``((A1 x n)*(A1 y n)-(cos x)*(cos y))+((sin x)*(sin y)-(B1 x (pred n))*(B1 y (pred n))+(Reste x y (pred n)))``; [Apply Rabsolu_triang | Ring]. -Replace ``eps`` with ``eps/3+(eps/3+eps/3)``. -Apply Rplus_lt. -Apply H8. -Unfold ge; Apply le_trans with N. -Unfold N. -Apply le_trans with (max N1 N2). -Apply le_max_l. -Apply le_trans with (max (max N1 N2) N3). -Apply le_max_l. -Apply le_trans with (S (max (max N1 N2) N3)); Apply le_n_Sn. -Assumption. -Apply Rle_lt_trans with ``(Rabsolu ((sin x)*(sin y)-(B1 x (pred n))*(B1 y (pred n))))+(Rabsolu (Reste x y (pred n)))``. -Apply Rabsolu_triang. -Apply Rplus_lt. -Rewrite <- Rabsolu_Ropp. -Rewrite Ropp_distr2. -Apply H9. -Unfold ge; Apply le_trans with (max N1 N2). -Apply le_max_r. -Apply le_S_n. -Rewrite <- H12. -Apply le_trans with N. -Unfold N. -Apply le_n_S. -Apply le_trans with (max (max N1 N2) N3). -Apply le_max_l. -Apply le_n_Sn. -Assumption. -Replace (Reste x y (pred n)) with ``(Reste x y (pred n))-0``. -Apply H10. -Unfold ge. -Apply le_S_n. -Rewrite <- H12. -Apply le_trans with N. -Unfold N. -Apply le_n_S. -Apply le_trans with (max (max N1 N2) N3). -Apply le_max_r. -Apply le_n_Sn. -Assumption. -Ring. -Pattern 4 eps; Replace eps with ``3*eps/3``. -Ring. -Unfold Rdiv. -Rewrite <- Rmult_assoc. -Apply Rinv_r_simpl_m. -DiscrR. -Apply lt_le_trans with (pred N). -Unfold N; Simpl; Apply lt_O_Sn. -Apply le_S_n. -Rewrite <- H12. -Replace (S (pred N)) with N. -Assumption. -Unfold N; Simpl; Reflexivity. -Cut (lt O N). -Intro. -Cut (lt O n). -Intro. -Apply S_pred with O; Assumption. -Apply lt_le_trans with N; Assumption. -Unfold N; Apply lt_O_Sn. -Qed. +Theorem cos_plus : forall x y:R, cos (x + y) = cos x * cos y - sin x * sin y. +intros. +cut (Un_cv (C1 x y) (cos x * cos y - sin x * sin y)). +cut (Un_cv (C1 x y) (cos (x + y))). +intros. +apply UL_sequence with (C1 x y); assumption. +apply C1_cvg. +unfold Un_cv in |- *; unfold R_dist in |- *. +intros. +assert (H0 := A1_cvg x). +assert (H1 := A1_cvg y). +assert (H2 := B1_cvg x). +assert (H3 := B1_cvg y). +assert (H4 := CV_mult _ _ _ _ H0 H1). +assert (H5 := CV_mult _ _ _ _ H2 H3). +assert (H6 := reste_cv_R0 x y). +unfold Un_cv in H4; unfold Un_cv in H5; unfold Un_cv in H6. +unfold R_dist in H4; unfold R_dist in H5; unfold R_dist in H6. +cut (0 < eps / 3); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H4 (eps / 3) H7); intros N1 H8. +elim (H5 (eps / 3) H7); intros N2 H9. +elim (H6 (eps / 3) H7); intros N3 H10. +pose (N := S (S (max (max N1 N2) N3))). +exists N. +intros. +cut (n = S (pred n)). +intro; rewrite H12. +rewrite <- cos_plus_form. +rewrite <- H12. +apply Rle_lt_trans with + (Rabs (A1 x n * A1 y n - cos x * cos y) + + Rabs (sin x * sin y - B1 x (pred n) * B1 y (pred n) + Reste x y (pred n))). +replace + (A1 x n * A1 y n - B1 x (pred n) * B1 y (pred n) + Reste x y (pred n) - + (cos x * cos y - sin x * sin y)) with + (A1 x n * A1 y n - cos x * cos y + + (sin x * sin y - B1 x (pred n) * B1 y (pred n) + Reste x y (pred n))); + [ apply Rabs_triang | ring ]. +replace eps with (eps / 3 + (eps / 3 + eps / 3)). +apply Rplus_lt_compat. +apply H8. +unfold ge in |- *; apply le_trans with N. +unfold N in |- *. +apply le_trans with (max N1 N2). +apply le_max_l. +apply le_trans with (max (max N1 N2) N3). +apply le_max_l. +apply le_trans with (S (max (max N1 N2) N3)); apply le_n_Sn. +assumption. +apply Rle_lt_trans with + (Rabs (sin x * sin y - B1 x (pred n) * B1 y (pred n)) + + Rabs (Reste x y (pred n))). +apply Rabs_triang. +apply Rplus_lt_compat. +rewrite <- Rabs_Ropp. +rewrite Ropp_minus_distr. +apply H9. +unfold ge in |- *; apply le_trans with (max N1 N2). +apply le_max_r. +apply le_S_n. +rewrite <- H12. +apply le_trans with N. +unfold N in |- *. +apply le_n_S. +apply le_trans with (max (max N1 N2) N3). +apply le_max_l. +apply le_n_Sn. +assumption. +replace (Reste x y (pred n)) with (Reste x y (pred n) - 0). +apply H10. +unfold ge in |- *. +apply le_S_n. +rewrite <- H12. +apply le_trans with N. +unfold N in |- *. +apply le_n_S. +apply le_trans with (max (max N1 N2) N3). +apply le_max_r. +apply le_n_Sn. +assumption. +ring. +pattern eps at 4 in |- *; replace eps with (3 * (eps / 3)). +ring. +unfold Rdiv in |- *. +rewrite <- Rmult_assoc. +apply Rinv_r_simpl_m. +discrR. +apply lt_le_trans with (pred N). +unfold N in |- *; simpl in |- *; apply lt_O_Sn. +apply le_S_n. +rewrite <- H12. +replace (S (pred N)) with N. +assumption. +unfold N in |- *; simpl in |- *; reflexivity. +cut (0 < N)%nat. +intro. +cut (0 < n)%nat. +intro. +apply S_pred with 0%nat; assumption. +apply lt_le_trans with N; assumption. +unfold N in |- *; apply lt_O_Sn. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Cos_rel.v b/theories/Reals/Cos_rel.v index 0bc58169c..5e9d26001 100644 --- a/theories/Reals/Cos_rel.v +++ b/theories/Reals/Cos_rel.v @@ -8,353 +8,413 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo_def. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo_def. Open Local Scope R_scope. -Definition A1 [x:R] : nat->R := [N:nat](sum_f_R0 [k:nat]``(pow (-1) k)/(INR (fact (mult (S (S O)) k)))*(pow x (mult (S (S O)) k))`` N). +Definition A1 (x:R) (N:nat) : R := + sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k)) N. -Definition B1 [x:R] : nat->R := [N:nat](sum_f_R0 [k:nat]``(pow (-1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow x (plus (mult (S (S O)) k) (S O)))`` N). +Definition B1 (x:R) (N:nat) : R := + sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1)) + N. -Definition C1 [x,y:R] : nat -> R := [N:nat](sum_f_R0 [k:nat]``(pow (-1) k)/(INR (fact (mult (S (S O)) k)))*(pow (x+y) (mult (S (S O)) k))`` N). +Definition C1 (x y:R) (N:nat) : R := + sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * (x + y) ^ (2 * k)) N. -Definition Reste1 [x,y:R] : nat -> R := [N:nat](sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(pow (-1) (S (plus l k)))/(INR (fact (mult (S (S O)) (S (plus l k)))))*(pow x (mult (S (S O)) (S (plus l k))))*(pow (-1) (minus N l))/(INR (fact (mult (S (S O)) (minus N l))))*(pow y (mult (S (S O)) (minus N l)))`` (pred (minus N k))) (pred N)). +Definition Reste1 (x y:R) (N:nat) : R := + sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) * + x ^ (2 * S (l + k)) * ((-1) ^ (N - l) / INR (fact (2 * (N - l)))) * + y ^ (2 * (N - l))) (pred (N - k))) (pred N). -Definition Reste2 [x,y:R] : nat -> R := [N:nat](sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(pow (-1) (S (plus l k)))/(INR (fact (plus (mult (S (S O)) (S (plus l k))) (S O))))*(pow x (plus (mult (S (S O)) (S (plus l k))) (S O)))*(pow (-1) (minus N l))/(INR (fact (plus (mult (S (S O)) (minus N l)) (S O))))*(pow y (plus (mult (S (S O)) (minus N l)) (S O)))`` (pred (minus N k))) (pred N)). +Definition Reste2 (x y:R) (N:nat) : R := + sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + (-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) * + x ^ (2 * S (l + k) + 1) * + ((-1) ^ (N - l) / INR (fact (2 * (N - l) + 1))) * + y ^ (2 * (N - l) + 1)) (pred (N - k))) ( + pred N). -Definition Reste [x,y:R] : nat -> R := [N:nat]``(Reste2 x y N)-(Reste1 x y (S N))``. +Definition Reste (x y:R) (N:nat) : R := Reste2 x y N - Reste1 x y (S N). (* Here is the main result that will be used to prove that (cos (x+y))=(cos x)(cos y)-(sin x)(sin y) *) -Theorem cos_plus_form : (x,y:R;n:nat) (lt O n) -> ``(A1 x (S n))*(A1 y (S n))-(B1 x n)*(B1 y n)+(Reste x y n)``==(C1 x y (S n)). -Intros. -Unfold A1 B1. -Rewrite (cauchy_finite [k:nat] - ``(pow ( -1) k)/(INR (fact (mult (S (S O)) k)))* - (pow x (mult (S (S O)) k))`` [k:nat] - ``(pow ( -1) k)/(INR (fact (mult (S (S O)) k)))* - (pow y (mult (S (S O)) k))`` (S n)). -Rewrite (cauchy_finite [k:nat] - ``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))* - (pow x (plus (mult (S (S O)) k) (S O)))`` [k:nat] - ``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))* - (pow y (plus (mult (S (S O)) k) (S O)))`` n H). -Unfold Reste. -Replace (sum_f_R0 - [k:nat] - (sum_f_R0 - [l:nat] - ``(pow ( -1) (S (plus l k)))/ - (INR (fact (mult (S (S O)) (S (plus l k)))))* - (pow x (mult (S (S O)) (S (plus l k))))* - ((pow ( -1) (minus (S n) l))/ - (INR (fact (mult (S (S O)) (minus (S n) l))))* - (pow y (mult (S (S O)) (minus (S n) l))))`` - (pred (minus (S n) k))) (pred (S n))) with (Reste1 x y (S n)). -Replace (sum_f_R0 - [k:nat] - (sum_f_R0 - [l:nat] - ``(pow ( -1) (S (plus l k)))/ - (INR (fact (plus (mult (S (S O)) (S (plus l k))) (S O))))* - (pow x (plus (mult (S (S O)) (S (plus l k))) (S O)))* - ((pow ( -1) (minus n l))/ - (INR (fact (plus (mult (S (S O)) (minus n l)) (S O))))* - (pow y (plus (mult (S (S O)) (minus n l)) (S O))))`` - (pred (minus n k))) (pred n)) with (Reste2 x y n). -Ring. -Replace (sum_f_R0 - [k:nat] - (sum_f_R0 - [p:nat] - ``(pow ( -1) p)/(INR (fact (mult (S (S O)) p)))* - (pow x (mult (S (S O)) p))*((pow ( -1) (minus k p))/ - (INR (fact (mult (S (S O)) (minus k p))))* - (pow y (mult (S (S O)) (minus k p))))`` k) (S n)) with (sum_f_R0 [k:nat](Rmult ``(pow (-1) k)/(INR (fact (mult (S (S O)) k)))`` (sum_f_R0 [l:nat]``(C (mult (S (S O)) k) (mult (S (S O)) l))*(pow x (mult (S (S O)) l))*(pow y (mult (S (S O)) (minus k l)))`` k)) (S n)). -Pose sin_nnn := [n:nat]Cases n of O => R0 | (S p) => (Rmult ``(pow (-1) (S p))/(INR (fact (mult (S (S O)) (S p))))`` (sum_f_R0 [l:nat]``(C (mult (S (S O)) (S p)) (S (mult (S (S O)) l)))*(pow x (S (mult (S (S O)) l)))*(pow y (S (mult (S (S O)) (minus p l))))`` p)) end. -Replace (Ropp (sum_f_R0 - [k:nat] - (sum_f_R0 - [p:nat] - ``(pow ( -1) p)/ - (INR (fact (plus (mult (S (S O)) p) (S O))))* - (pow x (plus (mult (S (S O)) p) (S O)))* - ((pow ( -1) (minus k p))/ - (INR (fact (plus (mult (S (S O)) (minus k p)) (S O))))* - (pow y (plus (mult (S (S O)) (minus k p)) (S O))))`` k) - n)) with (sum_f_R0 sin_nnn (S n)). -Rewrite <- sum_plus. -Unfold C1. -Apply sum_eq; Intros. -Induction i. -Simpl. -Rewrite Rplus_Ol. -Replace (C O O) with R1. -Unfold Rdiv; Rewrite Rinv_R1. -Ring. -Unfold C. -Rewrite <- minus_n_n. -Simpl. -Unfold Rdiv; Rewrite Rmult_1r; Rewrite Rinv_R1; Ring. -Unfold sin_nnn. -Rewrite <- Rmult_Rplus_distr. -Apply Rmult_mult_r. -Rewrite binomial. -Pose Wn := [i0:nat]``(C (mult (S (S O)) (S i)) i0)*(pow x i0)* - (pow y (minus (mult (S (S O)) (S i)) i0))``. -Replace (sum_f_R0 - [l:nat] - ``(C (mult (S (S O)) (S i)) (mult (S (S O)) l))* - (pow x (mult (S (S O)) l))* - (pow y (mult (S (S O)) (minus (S i) l)))`` (S i)) with (sum_f_R0 [l:nat](Wn (mult (2) l)) (S i)). -Replace (sum_f_R0 - [l:nat] - ``(C (mult (S (S O)) (S i)) (S (mult (S (S O)) l)))* - (pow x (S (mult (S (S O)) l)))* - (pow y (S (mult (S (S O)) (minus i l))))`` i) with (sum_f_R0 [l:nat](Wn (S (mult (2) l))) i). -Rewrite Rplus_sym. -Apply sum_decomposition. -Apply sum_eq; Intros. -Unfold Wn. -Apply Rmult_mult_r. -Replace (minus (mult (2) (S i)) (S (mult (2) i0))) with (S (mult (2) (minus i i0))). -Reflexivity. -Apply INR_eq. -Rewrite S_INR; Rewrite mult_INR. -Repeat Rewrite minus_INR. -Rewrite mult_INR; Repeat Rewrite S_INR. -Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Replace (mult (2) (S i)) with (S (S (mult (2) i))). -Apply le_n_S. -Apply le_trans with (mult (2) i). -Apply mult_le; Assumption. -Apply le_n_Sn. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Assumption. -Apply sum_eq; Intros. -Unfold Wn. -Apply Rmult_mult_r. -Replace (minus (mult (2) (S i)) (mult (2) i0)) with (mult (2) (minus (S i) i0)). -Reflexivity. -Apply INR_eq. -Rewrite mult_INR. -Repeat Rewrite minus_INR. -Rewrite mult_INR; Repeat Rewrite S_INR. -Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Apply mult_le; Assumption. -Assumption. -Rewrite <- (Ropp_Ropp (sum_f_R0 sin_nnn (S n))). -Apply eq_Ropp. -Replace ``-(sum_f_R0 sin_nnn (S n))`` with ``-1*(sum_f_R0 sin_nnn (S n))``; [Idtac | Ring]. -Rewrite scal_sum. -Rewrite decomp_sum. -Replace (sin_nnn O) with R0. -Rewrite Rmult_Ol; Rewrite Rplus_Ol. -Replace (pred (S n)) with n; [Idtac | Reflexivity]. -Apply sum_eq; Intros. -Rewrite Rmult_sym. -Unfold sin_nnn. -Rewrite scal_sum. -Rewrite scal_sum. -Apply sum_eq; Intros. -Unfold Rdiv. -Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym ``/(INR (fact (mult (S (S O)) (S i))))``). -Repeat Rewrite <- Rmult_assoc. -Rewrite <- (Rmult_sym ``/(INR (fact (mult (S (S O)) (S i))))``). -Repeat Rewrite <- Rmult_assoc. -Replace ``/(INR (fact (mult (S (S O)) (S i))))* - (C (mult (S (S O)) (S i)) (S (mult (S (S O)) i0)))`` with ``/(INR (fact (plus (mult (S (S O)) i0) (S O))))*/(INR (fact (plus (mult (S (S O)) (minus i i0)) (S O))))``. -Replace (S (mult (2) i0)) with (plus (mult (2) i0) (1)); [Idtac | Ring]. -Replace (S (mult (2) (minus i i0))) with (plus (mult (2) (minus i i0)) (1)); [Idtac | Ring]. -Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i0)*(pow (-1) (minus i i0))``. -Ring. -Simpl. -Pattern 2 i; Replace i with (plus i0 (minus i i0)). -Rewrite pow_add. -Ring. -Symmetry; Apply le_plus_minus; Assumption. -Unfold C. -Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Rewrite Rinv_Rmult. -Replace (S (mult (S (S O)) i0)) with (plus (mult (2) i0) (1)); [Apply Rmult_mult_r | Ring]. -Replace (minus (mult (2) (S i)) (plus (mult (2) i0) (1))) with (plus (mult (2) (minus i i0)) (1)). -Reflexivity. -Apply INR_eq. -Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite minus_INR. -Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Replace (plus (mult (2) i0) (1)) with (S (mult (2) i0)). -Replace (mult (2) (S i)) with (S (S (mult (2) i))). -Apply le_n_S. -Apply le_trans with (mult (2) i). -Apply mult_le; Assumption. -Apply le_n_Sn. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Assumption. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Reflexivity. -Apply lt_O_Sn. -Apply sum_eq; Intros. -Rewrite scal_sum. -Apply sum_eq; Intros. -Unfold Rdiv. -Repeat Rewrite <- Rmult_assoc. -Rewrite <- (Rmult_sym ``/(INR (fact (mult (S (S O)) i)))``). -Repeat Rewrite <- Rmult_assoc. -Replace ``/(INR (fact (mult (S (S O)) i)))* - (C (mult (S (S O)) i) (mult (S (S O)) i0))`` with ``/(INR (fact (mult (S (S O)) i0)))*/(INR (fact (mult (S (S O)) (minus i i0))))``. -Replace ``(pow (-1) i)`` with ``(pow (-1) i0)*(pow (-1) (minus i i0))``. -Ring. -Pattern 2 i; Replace i with (plus i0 (minus i i0)). -Rewrite pow_add. -Ring. -Symmetry; Apply le_plus_minus; Assumption. -Unfold C. -Unfold Rdiv; Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Rewrite Rinv_Rmult. -Replace (minus (mult (2) i) (mult (2) i0)) with (mult (2) (minus i i0)). -Reflexivity. -Apply INR_eq. -Rewrite mult_INR; Repeat Rewrite minus_INR. -Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Apply mult_le; Assumption. -Assumption. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Unfold Reste2; Apply sum_eq; Intros. -Apply sum_eq; Intros. -Unfold Rdiv; Ring. -Unfold Reste1; Apply sum_eq; Intros. -Apply sum_eq; Intros. -Unfold Rdiv; Ring. -Apply lt_O_Sn. +Theorem cos_plus_form : + forall (x y:R) (n:nat), + (0 < n)%nat -> + A1 x (S n) * A1 y (S n) - B1 x n * B1 y n + Reste x y n = C1 x y (S n). +intros. +unfold A1, B1 in |- *. +rewrite + (cauchy_finite (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k)) + (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * y ^ (2 * k)) ( + S n)). +rewrite + (cauchy_finite + (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1)) + (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * y ^ (2 * k + 1)) n H) + . +unfold Reste in |- *. +replace + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + (-1) ^ S (l + k) / INR (fact (2 * S (l + k))) * + x ^ (2 * S (l + k)) * + ((-1) ^ (S n - l) / INR (fact (2 * (S n - l))) * + y ^ (2 * (S n - l)))) (pred (S n - k))) ( + pred (S n))) with (Reste1 x y (S n)). +replace + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + (-1) ^ S (l + k) / INR (fact (2 * S (l + k) + 1)) * + x ^ (2 * S (l + k) + 1) * + ((-1) ^ (n - l) / INR (fact (2 * (n - l) + 1)) * + y ^ (2 * (n - l) + 1))) (pred (n - k))) ( + pred n)) with (Reste2 x y n). +ring. +replace + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun p:nat => + (-1) ^ p / INR (fact (2 * p)) * x ^ (2 * p) * + ((-1) ^ (k - p) / INR (fact (2 * (k - p))) * y ^ (2 * (k - p)))) + k) (S n)) with + (sum_f_R0 + (fun k:nat => + (-1) ^ k / INR (fact (2 * k)) * + sum_f_R0 + (fun l:nat => C (2 * k) (2 * l) * x ^ (2 * l) * y ^ (2 * (k - l))) k) + (S n)). +pose + (sin_nnn := + fun n:nat => + match n with + | O => 0 + | S p => + (-1) ^ S p / INR (fact (2 * S p)) * + sum_f_R0 + (fun l:nat => + C (2 * S p) (S (2 * l)) * x ^ S (2 * l) * y ^ S (2 * (p - l))) p + end). +replace + (- + sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun p:nat => + (-1) ^ p / INR (fact (2 * p + 1)) * x ^ (2 * p + 1) * + ((-1) ^ (k - p) / INR (fact (2 * (k - p) + 1)) * + y ^ (2 * (k - p) + 1))) k) n) with (sum_f_R0 sin_nnn (S n)). +rewrite <- sum_plus. +unfold C1 in |- *. +apply sum_eq; intros. +induction i as [| i Hreci]. +simpl in |- *. +rewrite Rplus_0_l. +replace (C 0 0) with 1. +unfold Rdiv in |- *; rewrite Rinv_1. +ring. +unfold C in |- *. +rewrite <- minus_n_n. +simpl in |- *. +unfold Rdiv in |- *; rewrite Rmult_1_r; rewrite Rinv_1; ring. +unfold sin_nnn in |- *. +rewrite <- Rmult_plus_distr_l. +apply Rmult_eq_compat_l. +rewrite binomial. +pose (Wn := fun i0:nat => C (2 * S i) i0 * x ^ i0 * y ^ (2 * S i - i0)). +replace + (sum_f_R0 + (fun l:nat => C (2 * S i) (2 * l) * x ^ (2 * l) * y ^ (2 * (S i - l))) + (S i)) with (sum_f_R0 (fun l:nat => Wn (2 * l)%nat) (S i)). +replace + (sum_f_R0 + (fun l:nat => + C (2 * S i) (S (2 * l)) * x ^ S (2 * l) * y ^ S (2 * (i - l))) i) with + (sum_f_R0 (fun l:nat => Wn (S (2 * l))) i). +rewrite Rplus_comm. +apply sum_decomposition. +apply sum_eq; intros. +unfold Wn in |- *. +apply Rmult_eq_compat_l. +replace (2 * S i - S (2 * i0))%nat with (S (2 * (i - i0))). +reflexivity. +apply INR_eq. +rewrite S_INR; rewrite mult_INR. +repeat rewrite minus_INR. +rewrite mult_INR; repeat rewrite S_INR. +rewrite mult_INR; repeat rewrite S_INR; ring. +replace (2 * S i)%nat with (S (S (2 * i))). +apply le_n_S. +apply le_trans with (2 * i)%nat. +apply (fun m n p:nat => mult_le_compat_l p n m); assumption. +apply le_n_Sn. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +assumption. +apply sum_eq; intros. +unfold Wn in |- *. +apply Rmult_eq_compat_l. +replace (2 * S i - 2 * i0)%nat with (2 * (S i - i0))%nat. +reflexivity. +apply INR_eq. +rewrite mult_INR. +repeat rewrite minus_INR. +rewrite mult_INR; repeat rewrite S_INR. +rewrite mult_INR; repeat rewrite S_INR; ring. +apply (fun m n p:nat => mult_le_compat_l p n m); assumption. +assumption. +rewrite <- (Ropp_involutive (sum_f_R0 sin_nnn (S n))). +apply Ropp_eq_compat. +replace (- sum_f_R0 sin_nnn (S n)) with (-1 * sum_f_R0 sin_nnn (S n)); + [ idtac | ring ]. +rewrite scal_sum. +rewrite decomp_sum. +replace (sin_nnn 0%nat) with 0. +rewrite Rmult_0_l; rewrite Rplus_0_l. +replace (pred (S n)) with n; [ idtac | reflexivity ]. +apply sum_eq; intros. +rewrite Rmult_comm. +unfold sin_nnn in |- *. +rewrite scal_sum. +rewrite scal_sum. +apply sum_eq; intros. +unfold Rdiv in |- *. +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (/ INR (fact (2 * S i)))). +repeat rewrite <- Rmult_assoc. +rewrite <- (Rmult_comm (/ INR (fact (2 * S i)))). +repeat rewrite <- Rmult_assoc. +replace (/ INR (fact (2 * S i)) * C (2 * S i) (S (2 * i0))) with + (/ INR (fact (2 * i0 + 1)) * / INR (fact (2 * (i - i0) + 1))). +replace (S (2 * i0)) with (2 * i0 + 1)%nat; [ idtac | ring ]. +replace (S (2 * (i - i0))) with (2 * (i - i0) + 1)%nat; [ idtac | ring ]. +replace ((-1) ^ S i) with (-1 * (-1) ^ i0 * (-1) ^ (i - i0)). +ring. +simpl in |- *. +pattern i at 2 in |- *; replace i with (i0 + (i - i0))%nat. +rewrite pow_add. +ring. +symmetry in |- *; apply le_plus_minus; assumption. +unfold C in |- *. +unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +rewrite Rinv_mult_distr. +replace (S (2 * i0)) with (2 * i0 + 1)%nat; + [ apply Rmult_eq_compat_l | ring ]. +replace (2 * S i - (2 * i0 + 1))%nat with (2 * (i - i0) + 1)%nat. +reflexivity. +apply INR_eq. +rewrite plus_INR; rewrite mult_INR; repeat rewrite minus_INR. +rewrite plus_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; ring. +replace (2 * i0 + 1)%nat with (S (2 * i0)). +replace (2 * S i)%nat with (S (S (2 * i))). +apply le_n_S. +apply le_trans with (2 * i)%nat. +apply (fun m n p:nat => mult_le_compat_l p n m); assumption. +apply le_n_Sn. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR; + repeat rewrite S_INR; ring. +assumption. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +reflexivity. +apply lt_O_Sn. +apply sum_eq; intros. +rewrite scal_sum. +apply sum_eq; intros. +unfold Rdiv in |- *. +repeat rewrite <- Rmult_assoc. +rewrite <- (Rmult_comm (/ INR (fact (2 * i)))). +repeat rewrite <- Rmult_assoc. +replace (/ INR (fact (2 * i)) * C (2 * i) (2 * i0)) with + (/ INR (fact (2 * i0)) * / INR (fact (2 * (i - i0)))). +replace ((-1) ^ i) with ((-1) ^ i0 * (-1) ^ (i - i0)). +ring. +pattern i at 2 in |- *; replace i with (i0 + (i - i0))%nat. +rewrite pow_add. +ring. +symmetry in |- *; apply le_plus_minus; assumption. +unfold C in |- *. +unfold Rdiv in |- *; repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +rewrite Rinv_mult_distr. +replace (2 * i - 2 * i0)%nat with (2 * (i - i0))%nat. +reflexivity. +apply INR_eq. +rewrite mult_INR; repeat rewrite minus_INR. +do 2 rewrite mult_INR; repeat rewrite S_INR; ring. +apply (fun m n p:nat => mult_le_compat_l p n m); assumption. +assumption. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +unfold Reste2 in |- *; apply sum_eq; intros. +apply sum_eq; intros. +unfold Rdiv in |- *; ring. +unfold Reste1 in |- *; apply sum_eq; intros. +apply sum_eq; intros. +unfold Rdiv in |- *; ring. +apply lt_O_Sn. Qed. -Lemma pow_sqr : (x:R;i:nat) (pow x (mult (2) i))==(pow ``x*x`` i). -Intros. -Assert H := (pow_Rsqr x i). -Unfold Rsqr in H; Exact H. +Lemma pow_sqr : forall (x:R) (i:nat), x ^ (2 * i) = (x * x) ^ i. +intros. +assert (H := pow_Rsqr x i). +unfold Rsqr in H; exact H. Qed. -Lemma A1_cvg : (x:R) (Un_cv (A1 x) (cos x)). -Intro. -Assert H := (exist_cos ``x*x``). -Elim H; Intros. -Assert p_i := p. -Unfold cos_in in p. -Unfold cos_n infinit_sum in p. -Unfold R_dist in p. -Cut ``(cos x)==x0``. -Intro. -Rewrite H0. -Unfold Un_cv; Unfold R_dist; Intros. -Elim (p eps H1); Intros. -Exists x1; Intros. -Unfold A1. -Replace (sum_f_R0 ([k:nat]``(pow ( -1) k)/(INR (fact (mult (S (S O)) k)))*(pow x (mult (S (S O)) k))``) n) with (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (mult (S (S O)) i)))*(pow (x*x) i)``) n). -Apply H2; Assumption. -Apply sum_eq. -Intros. -Replace ``(pow (x*x) i)`` with ``(pow x (mult (S (S O)) i))``. -Reflexivity. -Apply pow_sqr. -Unfold cos. -Case (exist_cos (Rsqr x)). -Unfold Rsqr; Intros. -Unfold cos_in in p_i. -Unfold cos_in in c. -Apply unicity_sum with [i:nat]``(cos_n i)*(pow (x*x) i)``; Assumption. +Lemma A1_cvg : forall x:R, Un_cv (A1 x) (cos x). +intro. +assert (H := exist_cos (x * x)). +elim H; intros. +assert (p_i := p). +unfold cos_in in p. +unfold cos_n, infinit_sum in p. +unfold R_dist in p. +cut (cos x = x0). +intro. +rewrite H0. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (p eps H1); intros. +exists x1; intros. +unfold A1 in |- *. +replace + (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * x ^ (2 * k)) n) with + (sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i)) * (x * x) ^ i) n). +apply H2; assumption. +apply sum_eq. +intros. +replace ((x * x) ^ i) with (x ^ (2 * i)). +reflexivity. +apply pow_sqr. +unfold cos in |- *. +case (exist_cos (Rsqr x)). +unfold Rsqr in |- *; intros. +unfold cos_in in p_i. +unfold cos_in in c. +apply uniqueness_sum with (fun i:nat => cos_n i * (x * x) ^ i); assumption. Qed. -Lemma C1_cvg : (x,y:R) (Un_cv (C1 x y) (cos (Rplus x y))). -Intros. -Assert H := (exist_cos ``(x+y)*(x+y)``). -Elim H; Intros. -Assert p_i := p. -Unfold cos_in in p. -Unfold cos_n infinit_sum in p. -Unfold R_dist in p. -Cut ``(cos (x+y))==x0``. -Intro. -Rewrite H0. -Unfold Un_cv; Unfold R_dist; Intros. -Elim (p eps H1); Intros. -Exists x1; Intros. -Unfold C1. -Replace (sum_f_R0 ([k:nat]``(pow ( -1) k)/(INR (fact (mult (S (S O)) k)))*(pow (x+y) (mult (S (S O)) k))``) n) with (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (mult (S (S O)) i)))*(pow ((x+y)*(x+y)) i)``) n). -Apply H2; Assumption. -Apply sum_eq. -Intros. -Replace ``(pow ((x+y)*(x+y)) i)`` with ``(pow (x+y) (mult (S (S O)) i))``. -Reflexivity. -Apply pow_sqr. -Unfold cos. -Case (exist_cos (Rsqr ``x+y``)). -Unfold Rsqr; Intros. -Unfold cos_in in p_i. -Unfold cos_in in c. -Apply unicity_sum with [i:nat]``(cos_n i)*(pow ((x+y)*(x+y)) i)``; Assumption. +Lemma C1_cvg : forall x y:R, Un_cv (C1 x y) (cos (x + y)). +intros. +assert (H := exist_cos ((x + y) * (x + y))). +elim H; intros. +assert (p_i := p). +unfold cos_in in p. +unfold cos_n, infinit_sum in p. +unfold R_dist in p. +cut (cos (x + y) = x0). +intro. +rewrite H0. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (p eps H1); intros. +exists x1; intros. +unfold C1 in |- *. +replace + (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k)) * (x + y) ^ (2 * k)) n) + with + (sum_f_R0 + (fun i:nat => (-1) ^ i / INR (fact (2 * i)) * ((x + y) * (x + y)) ^ i) n). +apply H2; assumption. +apply sum_eq. +intros. +replace (((x + y) * (x + y)) ^ i) with ((x + y) ^ (2 * i)). +reflexivity. +apply pow_sqr. +unfold cos in |- *. +case (exist_cos (Rsqr (x + y))). +unfold Rsqr in |- *; intros. +unfold cos_in in p_i. +unfold cos_in in c. +apply uniqueness_sum with (fun i:nat => cos_n i * ((x + y) * (x + y)) ^ i); + assumption. Qed. -Lemma B1_cvg : (x:R) (Un_cv (B1 x) (sin x)). -Intro. -Case (Req_EM x R0); Intro. -Rewrite H. -Rewrite sin_0. -Unfold B1. -Unfold Un_cv; Unfold R_dist; Intros; Exists O; Intros. -Replace (sum_f_R0 ([k:nat]``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow 0 (plus (mult (S (S O)) k) (S O)))``) n) with R0. -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Induction n. -Simpl; Ring. -Rewrite tech5; Rewrite <- Hrecn. -Simpl; Ring. -Unfold ge; Apply le_O_n. -Assert H0 := (exist_sin ``x*x``). -Elim H0; Intros. -Assert p_i := p. -Unfold sin_in in p. -Unfold sin_n infinit_sum in p. -Unfold R_dist in p. -Cut ``(sin x)==x*x0``. -Intro. -Rewrite H1. -Unfold Un_cv; Unfold R_dist; Intros. -Cut ``0<eps/(Rabsolu x)``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption]]. -Elim (p ``eps/(Rabsolu x)`` H3); Intros. -Exists x1; Intros. -Unfold B1. -Replace (sum_f_R0 ([k:nat]``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow x (plus (mult (S (S O)) k) (S O)))``) n) with (Rmult x (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (plus (mult (S (S O)) i) (S O))))*(pow (x*x) i)``) n)). -Replace (Rminus (Rmult x (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (plus (mult (S (S O)) i) (S O))))*(pow (x*x) i)``) n)) (Rmult x x0)) with (Rmult x (Rminus (sum_f_R0 ([i:nat]``(pow ( -1) i)/(INR (fact (plus (mult (S (S O)) i) (S O))))*(pow (x*x) i)``) n) x0)); [Idtac | Ring]. -Rewrite Rabsolu_mult. -Apply Rlt_monotony_contra with ``/(Rabsolu x)``. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. -Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Unfold Rdiv in H4; Apply H4; Assumption. -Apply Rabsolu_no_R0; Assumption. -Rewrite scal_sum. -Apply sum_eq. -Intros. -Rewrite pow_add. -Rewrite pow_sqr. -Simpl. -Ring. -Unfold sin. -Case (exist_sin (Rsqr x)). -Unfold Rsqr; Intros. -Unfold sin_in in p_i. -Unfold sin_in in s. -Assert H1 := (unicity_sum [i:nat]``(sin_n i)*(pow (x*x) i)`` x0 x1 p_i s). -Rewrite H1; Reflexivity. -Qed. +Lemma B1_cvg : forall x:R, Un_cv (B1 x) (sin x). +intro. +case (Req_dec x 0); intro. +rewrite H. +rewrite sin_0. +unfold B1 in |- *. +unfold Un_cv in |- *; unfold R_dist in |- *; intros; exists 0%nat; intros. +replace + (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * 0 ^ (2 * k + 1)) + n) with 0. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. +induction n as [| n Hrecn]. +simpl in |- *; ring. +rewrite tech5; rewrite <- Hrecn. +simpl in |- *; ring. +unfold ge in |- *; apply le_O_n. +assert (H0 := exist_sin (x * x)). +elim H0; intros. +assert (p_i := p). +unfold sin_in in p. +unfold sin_n, infinit_sum in p. +unfold R_dist in p. +cut (sin x = x * x0). +intro. +rewrite H1. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +cut (0 < eps / Rabs x); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ] ]. +elim (p (eps / Rabs x) H3); intros. +exists x1; intros. +unfold B1 in |- *. +replace + (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * x ^ (2 * k + 1)) + n) with + (x * + sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * (x * x) ^ i) n). +replace + (x * + sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * (x * x) ^ i) n - + x * x0) with + (x * + (sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * (x * x) ^ i) n - + x0)); [ idtac | ring ]. +rewrite Rabs_mult. +apply Rmult_lt_reg_l with (/ Rabs x). +apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. +rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); unfold Rdiv in H4; apply H4; + assumption. +apply Rabs_no_R0; assumption. +rewrite scal_sum. +apply sum_eq. +intros. +rewrite pow_add. +rewrite pow_sqr. +simpl in |- *. +ring. +unfold sin in |- *. +case (exist_sin (Rsqr x)). +unfold Rsqr in |- *; intros. +unfold sin_in in p_i. +unfold sin_in in s. +assert + (H1 := uniqueness_sum (fun i:nat => sin_n i * (x * x) ^ i) x0 x1 p_i s). +rewrite H1; reflexivity. +Qed.
\ No newline at end of file diff --git a/theories/Reals/DiscrR.v b/theories/Reals/DiscrR.v index 3f0986480..474451903 100644 --- a/theories/Reals/DiscrR.v +++ b/theories/Reals/DiscrR.v @@ -8,51 +8,90 @@ (*i $Id$ i*) -Require RIneq. -Require Omega. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import RIneq. +Require Import Omega. Open Local Scope R_scope. -Lemma Rlt_R0_R2 : ``0<2``. -Replace ``2`` with (INR (2)); [Apply lt_INR_0; Apply lt_O_Sn | Reflexivity]. +Lemma Rlt_R0_R2 : 0 < 2. +replace 2 with (INR 2); [ apply lt_INR_0; apply lt_O_Sn | reflexivity ]. Qed. -Lemma Rplus_lt_pos : (x,y:R) ``0<x`` -> ``0<y`` -> ``0<x+y``. -Intros. -Apply Rlt_trans with x. -Assumption. -Pattern 1 x; Rewrite <- Rplus_Or. -Apply Rlt_compatibility. -Assumption. +Lemma Rplus_lt_pos : forall x y:R, 0 < x -> 0 < y -> 0 < x + y. +intros. +apply Rlt_trans with x. +assumption. +pattern x at 1 in |- *; rewrite <- Rplus_0_r. +apply Rplus_lt_compat_l. +assumption. Qed. -Lemma IZR_eq : (z1,z2:Z) z1=z2 -> (IZR z1)==(IZR z2). -Intros; Rewrite H; Reflexivity. +Lemma IZR_eq : forall z1 z2:Z, z1 = z2 -> IZR z1 = IZR z2. +intros; rewrite H; reflexivity. Qed. -Lemma IZR_neq : (z1,z2:Z) `z1<>z2` -> ``(IZR z1)<>(IZR z2)``. -Intros; Red; Intro; Elim H; Apply eq_IZR; Assumption. +Lemma IZR_neq : forall z1 z2:Z, z1 <> z2 -> IZR z1 <> IZR z2. +intros; red in |- *; intro; elim H; apply eq_IZR; assumption. Qed. -Tactic Definition DiscrR := - Try Match Context With - | [ |- ~(?1==?2) ] -> Replace ``2`` with (IZR `2`); [Replace R1 with (IZR `1`); [Replace R0 with (IZR `0`); [Repeat Rewrite <- plus_IZR Orelse Rewrite <- mult_IZR Orelse Rewrite <- Ropp_Ropp_IZR Orelse Rewrite Z_R_minus; Apply IZR_neq; Try Discriminate | Reflexivity] | Reflexivity] | Reflexivity]. +Ltac discrR := + try + match goal with + | |- (?X1 <> ?X2) => + replace 2 with (IZR 2); + [ replace 1 with (IZR 1); + [ replace 0 with (IZR 0); + [ repeat + rewrite <- plus_IZR || + rewrite <- mult_IZR || + rewrite <- Ropp_Ropp_IZR || rewrite Z_R_minus; + apply IZR_neq; try discriminate + | reflexivity ] + | reflexivity ] + | reflexivity ] + end. -Recursive Tactic Definition Sup0 := - Match Context With - | [ |- ``0<1`` ] -> Apply Rlt_R0_R1 - | [ |- ``0<?1`` ] -> Repeat (Apply Rmult_lt_pos Orelse Apply Rplus_lt_pos; Try Apply Rlt_R0_R1 Orelse Apply Rlt_R0_R2) - | [ |- ``?1>0`` ] -> Change ``0<?1``; Sup0. +Ltac prove_sup0 := + match goal with + | |- (0 < 1) => apply Rlt_0_1 + | |- (0 < ?X1) => + repeat + (apply Rmult_lt_0_compat || apply Rplus_lt_pos; + try apply Rlt_0_1 || apply Rlt_R0_R2) + | |- (?X1 > 0) => change (0 < X1) in |- *; prove_sup0 + end. -Tactic Definition SupOmega := Replace ``2`` with (IZR `2`); [Replace R1 with (IZR `1`); [Replace R0 with (IZR `0`); [Repeat Rewrite <- plus_IZR Orelse Rewrite <- mult_IZR Orelse Rewrite <- Ropp_Ropp_IZR Orelse Rewrite Z_R_minus; Apply IZR_lt; Omega | Reflexivity] | Reflexivity] | Reflexivity]. +Ltac omega_sup := + replace 2 with (IZR 2); + [ replace 1 with (IZR 1); + [ replace 0 with (IZR 0); + [ repeat + rewrite <- plus_IZR || + rewrite <- mult_IZR || + rewrite <- Ropp_Ropp_IZR || rewrite Z_R_minus; + apply IZR_lt; omega + | reflexivity ] + | reflexivity ] + | reflexivity ]. -Recursive Tactic Definition Sup := - Match Context With - | [ |- (Rgt ?1 ?2) ] -> Change ``?2<?1``; Sup - | [ |- ``0<?1`` ] -> Sup0 - | [ |- (Rlt (Ropp ?1) R0) ] -> Rewrite <- Ropp_O; Sup - | [ |- (Rlt (Ropp ?1) (Ropp ?2)) ] -> Apply Rlt_Ropp; Sup - | [ |- (Rlt (Ropp ?1) ?2) ] -> Apply Rlt_trans with ``0``; Sup - | [ |- (Rlt ?1 ?2) ] -> SupOmega - | _ -> Idtac. - -Tactic Definition RCompute := Replace ``2`` with (IZR `2`); [Replace R1 with (IZR `1`); [Replace R0 with (IZR `0`); [Repeat Rewrite <- plus_IZR Orelse Rewrite <- mult_IZR Orelse Rewrite <- Ropp_Ropp_IZR Orelse Rewrite Z_R_minus; Apply IZR_eq; Try Reflexivity | Reflexivity] | Reflexivity] | Reflexivity]. +Ltac prove_sup := + match goal with + | |- (?X1 > ?X2) => change (X2 < X1) in |- *; prove_sup + | |- (0 < ?X1) => prove_sup0 + | |- (- ?X1 < 0) => rewrite <- Ropp_0; prove_sup + | |- (- ?X1 < - ?X2) => apply Ropp_lt_gt_contravar; prove_sup + | |- (- ?X1 < ?X2) => apply Rlt_trans with 0; prove_sup + | |- (?X1 < ?X2) => omega_sup + | _ => idtac + end. + +Ltac Rcompute := + replace 2 with (IZR 2); + [ replace 1 with (IZR 1); + [ replace 0 with (IZR 0); + [ repeat + rewrite <- plus_IZR || + rewrite <- mult_IZR || + rewrite <- Ropp_Ropp_IZR || rewrite Z_R_minus; + apply IZR_eq; try reflexivity + | reflexivity ] + | reflexivity ] + | reflexivity ].
\ No newline at end of file diff --git a/theories/Reals/Exp_prop.v b/theories/Reals/Exp_prop.v index 5c06af34a..c424b9e14 100644 --- a/theories/Reals/Exp_prop.v +++ b/theories/Reals/Exp_prop.v @@ -8,883 +8,1004 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo. -Require Ranalysis1. -Require PSeries_reg. -Require Div2. -Require Even. -Require Max. -V7only [Import R_scope.]. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import Ranalysis1. +Require Import PSeries_reg. +Require Import Div2. +Require Import Even. +Require Import Max. Open Local Scope nat_scope. -V7only [Import nat_scope.]. Open Local Scope R_scope. -Definition E1 [x:R] : nat->R := [N:nat](sum_f_R0 [k:nat]``/(INR (fact k))*(pow x k)`` N). +Definition E1 (x:R) (N:nat) : R := + sum_f_R0 (fun k:nat => / INR (fact k) * x ^ k) N. -Lemma E1_cvg : (x:R) (Un_cv (E1 x) (exp x)). -Intro; Unfold exp; Unfold projT1. -Case (exist_exp x); Intro. -Unfold exp_in Un_cv; Unfold infinit_sum E1; Trivial. +Lemma E1_cvg : forall x:R, Un_cv (E1 x) (exp x). +intro; unfold exp in |- *; unfold projT1 in |- *. +case (exist_exp x); intro. +unfold exp_in, Un_cv in |- *; unfold infinit_sum, E1 in |- *; trivial. Qed. -Definition Reste_E [x,y:R] : nat->R := [N:nat](sum_f_R0 [k:nat](sum_f_R0 [l:nat]``/(INR (fact (S (plus l k))))*(pow x (S (plus l k)))*(/(INR (fact (minus N l)))*(pow y (minus N l)))`` (pred (minus N k))) (pred N)). +Definition Reste_E (x y:R) (N:nat) : R := + sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + / INR (fact (S (l + k))) * x ^ S (l + k) * + (/ INR (fact (N - l)) * y ^ (N - l))) ( + pred (N - k))) (pred N). -Lemma exp_form : (x,y:R;n:nat) (lt O n) -> ``(E1 x n)*(E1 y n)-(Reste_E x y n)==(E1 (x+y) n)``. -Intros; Unfold E1. -Rewrite cauchy_finite. -Unfold Reste_E; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Apply sum_eq; Intros. -Rewrite binomial. -Rewrite scal_sum; Apply sum_eq; Intros. -Unfold C; Unfold Rdiv; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym (INR (fact i))); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite Rinv_Rmult. -Ring. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply H. +Lemma exp_form : + forall (x y:R) (n:nat), + (0 < n)%nat -> E1 x n * E1 y n - Reste_E x y n = E1 (x + y) n. +intros; unfold E1 in |- *. +rewrite cauchy_finite. +unfold Reste_E in |- *; unfold Rminus in |- *; rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; apply sum_eq; + intros. +rewrite binomial. +rewrite scal_sum; apply sum_eq; intros. +unfold C in |- *; unfold Rdiv in |- *; repeat rewrite Rmult_assoc; + rewrite (Rmult_comm (INR (fact i))); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite Rinv_mult_distr. +ring. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply H. Qed. -Definition maj_Reste_E [x,y:R] : nat->R := [N:nat]``4*(pow (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S O)) N))/(Rsqr (INR (fact (div2 (pred N)))))``. +Definition maj_Reste_E (x y:R) (N:nat) : R := + 4 * + (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * N) / + Rsqr (INR (fact (div2 (pred N))))). -Lemma Rle_Rinv : (x,y:R) ``0<x`` -> ``0<y`` -> ``x<=y`` -> ``/y<=/x``. -Intros; Apply Rle_monotony_contra with x. -Apply H. -Rewrite <- Rinv_r_sym. -Apply Rle_monotony_contra with y. -Apply H0. -Rewrite Rmult_1r; Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Apply H1. -Red; Intro; Rewrite H2 in H0; Elim (Rlt_antirefl ? H0). -Red; Intro; Rewrite H2 in H; Elim (Rlt_antirefl ? H). +Lemma Rle_Rinv : forall x y:R, 0 < x -> 0 < y -> x <= y -> / y <= / x. +intros; apply Rmult_le_reg_l with x. +apply H. +rewrite <- Rinv_r_sym. +apply Rmult_le_reg_l with y. +apply H0. +rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; apply H1. +red in |- *; intro; rewrite H2 in H0; elim (Rlt_irrefl _ H0). +red in |- *; intro; rewrite H2 in H; elim (Rlt_irrefl _ H). Qed. (**********) -Lemma div2_double : (N:nat) (div2 (mult (2) N))=N. -Intro; Induction N. -Reflexivity. -Replace (mult (2) (S N)) with (S (S (mult (2) N))). -Simpl; Simpl in HrecN; Rewrite HrecN; Reflexivity. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Lemma div2_double : forall N:nat, div2 (2 * N) = N. +intro; induction N as [| N HrecN]. +reflexivity. +replace (2 * S N)%nat with (S (S (2 * N))). +simpl in |- *; simpl in HrecN; rewrite HrecN; reflexivity. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. Qed. -Lemma div2_S_double : (N:nat) (div2 (S (mult (2) N)))=N. -Intro; Induction N. -Reflexivity. -Replace (mult (2) (S N)) with (S (S (mult (2) N))). -Simpl; Simpl in HrecN; Rewrite HrecN; Reflexivity. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. +Lemma div2_S_double : forall N:nat, div2 (S (2 * N)) = N. +intro; induction N as [| N HrecN]. +reflexivity. +replace (2 * S N)%nat with (S (S (2 * N))). +simpl in |- *; simpl in HrecN; rewrite HrecN; reflexivity. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. Qed. -Lemma div2_not_R0 : (N:nat) (lt (1) N) -> (lt O (div2 N)). -Intros; Induction N. -Elim (lt_n_O ? H). -Cut (lt (1) N)\/N=(1). -Intro; Elim H0; Intro. -Assert H2 := (even_odd_dec N). -Elim H2; Intro. -Rewrite <- (even_div2 ? a); Apply HrecN; Assumption. -Rewrite <- (odd_div2 ? b); Apply lt_O_Sn. -Rewrite H1; Simpl; Apply lt_O_Sn. -Inversion H. -Right; Reflexivity. -Left; Apply lt_le_trans with (2); [Apply lt_n_Sn | Apply H1]. +Lemma div2_not_R0 : forall N:nat, (1 < N)%nat -> (0 < div2 N)%nat. +intros; induction N as [| N HrecN]. +elim (lt_n_O _ H). +cut ((1 < N)%nat \/ N = 1%nat). +intro; elim H0; intro. +assert (H2 := even_odd_dec N). +elim H2; intro. +rewrite <- (even_div2 _ a); apply HrecN; assumption. +rewrite <- (odd_div2 _ b); apply lt_O_Sn. +rewrite H1; simpl in |- *; apply lt_O_Sn. +inversion H. +right; reflexivity. +left; apply lt_le_trans with 2%nat; [ apply lt_n_Sn | apply H1 ]. Qed. -Lemma Reste_E_maj : (x,y:R;N:nat) (lt O N) -> ``(Rabsolu (Reste_E x y N))<=(maj_Reste_E x y N)``. -Intros; Pose M := (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))). -Apply Rle_trans with (Rmult (pow M (mult (2) N)) (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``/(Rsqr (INR (fact (div2 (S N)))))`` (pred (minus N k))) (pred N))). -Unfold Reste_E. -Apply Rle_trans with (sum_f_R0 [k:nat](Rabsolu (sum_f_R0 [l:nat]``/(INR (fact (S (plus l k))))*(pow x (S (plus l k)))*(/(INR (fact (minus N l)))*(pow y (minus N l)))`` (pred (minus N k)))) (pred N)). -Apply (sum_Rabsolu [k:nat](sum_f_R0 [l:nat]``/(INR (fact (S (plus l k))))*(pow x (S (plus l k)))*(/(INR (fact (minus N l)))*(pow y (minus N l)))`` (pred (minus N k))) (pred N)). -Apply Rle_trans with (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(Rabsolu (/(INR (fact (S (plus l k))))*(pow x (S (plus l k)))*(/(INR (fact (minus N l)))*(pow y (minus N l)))))`` (pred (minus N k))) (pred N)). -Apply sum_Rle; Intros. -Apply (sum_Rabsolu [l:nat]``/(INR (fact (S (plus l n))))*(pow x (S (plus l n)))*(/(INR (fact (minus N l)))*(pow y (minus N l)))``). -Apply Rle_trans with (sum_f_R0 [k:nat](sum_f_R0 [l:nat]``(pow M (mult (S (S O)) N))*/(INR (fact (S l)))*/(INR (fact (minus N l)))`` (pred (minus N k))) (pred N)). -Apply sum_Rle; Intros. -Apply sum_Rle; Intros. -Repeat Rewrite Rabsolu_mult. -Do 2 Rewrite <- Pow_Rabsolu. -Rewrite (Rabsolu_right ``/(INR (fact (S (plus n0 n))))``). -Rewrite (Rabsolu_right ``/(INR (fact (minus N n0)))``). -Replace ``/(INR (fact (S (plus n0 n))))*(pow (Rabsolu x) (S (plus n0 n)))* - (/(INR (fact (minus N n0)))*(pow (Rabsolu y) (minus N n0)))`` with ``/(INR (fact (minus N n0)))*/(INR (fact (S (plus n0 n))))*(pow (Rabsolu x) (S (plus n0 n)))*(pow (Rabsolu y) (minus N n0))``; [Idtac | Ring]. -Rewrite <- (Rmult_sym ``/(INR (fact (minus N n0)))``). -Repeat Rewrite Rmult_assoc. -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply Rle_trans with ``/(INR (fact (S n0)))*(pow (Rabsolu x) (S (plus n0 n)))*(pow (Rabsolu y) (minus N n0))``. -Rewrite (Rmult_sym ``/(INR (fact (S (plus n0 n))))``); Rewrite (Rmult_sym ``/(INR (fact (S n0)))``); Repeat Rewrite Rmult_assoc; Apply Rle_monotony. -Apply pow_le; Apply Rabsolu_pos. -Rewrite (Rmult_sym ``/(INR (fact (S n0)))``); Apply Rle_monotony. -Apply pow_le; Apply Rabsolu_pos. -Apply Rle_Rinv. -Apply INR_fact_lt_0. -Apply INR_fact_lt_0. -Apply le_INR; Apply fact_growing; Apply le_n_S. -Apply le_plus_l. -Rewrite (Rmult_sym ``(pow M (mult (S (S O)) N))``); Rewrite Rmult_assoc; Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply Rle_trans with ``(pow M (S (plus n0 n)))*(pow (Rabsolu y) (minus N n0))``. -Do 2 Rewrite <- (Rmult_sym ``(pow (Rabsolu y) (minus N n0))``). -Apply Rle_monotony. -Apply pow_le; Apply Rabsolu_pos. -Apply pow_incr; Split. -Apply Rabsolu_pos. -Apply Rle_trans with (Rmax (Rabsolu x) (Rabsolu y)). -Apply RmaxLess1. -Unfold M; Apply RmaxLess2. -Apply Rle_trans with ``(pow M (S (plus n0 n)))*(pow M (minus N n0))``. -Apply Rle_monotony. -Apply pow_le; Apply Rle_trans with R1. -Left; Apply Rlt_R0_R1. -Unfold M; Apply RmaxLess1. -Apply pow_incr; Split. -Apply Rabsolu_pos. -Apply Rle_trans with (Rmax (Rabsolu x) (Rabsolu y)). -Apply RmaxLess2. -Unfold M; Apply RmaxLess2. -Rewrite <- pow_add; Replace (plus (S (plus n0 n)) (minus N n0)) with (plus N (S n)). -Apply Rle_pow. -Unfold M; Apply RmaxLess1. -Replace (mult (2) N) with (plus N N); [Idtac | Ring]. -Apply le_reg_l. -Replace N with (S (pred N)). -Apply le_n_S; Apply H0. -Symmetry; Apply S_pred with O; Apply H. -Apply INR_eq; Do 2 Rewrite plus_INR; Do 2 Rewrite S_INR; Rewrite plus_INR; Rewrite minus_INR. -Ring. -Apply le_trans with (pred (minus N n)). -Apply H1. -Apply le_S_n. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_trans with N. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Apply H0. -Apply le_pred_n. -Apply le_n_Sn. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n (0)) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Apply H0. -Apply lt_pred_n_n. -Apply H. -Apply le_trans with (pred N). -Apply H0. -Apply le_pred_n. -Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Rewrite scal_sum. -Apply sum_Rle; Intros. -Rewrite <- Rmult_sym. -Rewrite scal_sum. -Apply sum_Rle; Intros. -Rewrite (Rmult_sym ``/(Rsqr (INR (fact (div2 (S N)))))``). -Rewrite Rmult_assoc; Apply Rle_monotony. -Apply pow_le. -Apply Rle_trans with R1. -Left; Apply Rlt_R0_R1. -Unfold M; Apply RmaxLess1. -Assert H2 := (even_odd_cor N). -Elim H2; Intros N0 H3. -Elim H3; Intro. -Apply Rle_trans with ``/(INR (fact n0))*/(INR (fact (minus N n0)))``. -Do 2 Rewrite <- (Rmult_sym ``/(INR (fact (minus N n0)))``). -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply Rle_Rinv. -Apply INR_fact_lt_0. -Apply INR_fact_lt_0. -Apply le_INR. -Apply fact_growing. -Apply le_n_Sn. -Replace ``/(INR (fact n0))*/(INR (fact (minus N n0)))`` with ``(C N n0)/(INR (fact N))``. -Pattern 1 N; Rewrite H4. -Apply Rle_trans with ``(C N N0)/(INR (fact N))``. -Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/(INR (fact N))``). -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Rewrite H4. -Apply C_maj. -Rewrite <- H4; Apply le_trans with (pred (minus N n)). -Apply H1. -Apply le_S_n. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_trans with N. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Apply H0. -Apply le_pred_n. -Apply le_n_Sn. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n (0)) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Apply H0. -Apply lt_pred_n_n. -Apply H. -Apply le_trans with (pred N). -Apply H0. -Apply le_pred_n. -Replace ``(C N N0)/(INR (fact N))`` with ``/(Rsqr (INR (fact N0)))``. -Rewrite H4; Rewrite div2_S_double; Right; Reflexivity. -Unfold Rsqr C Rdiv. -Repeat Rewrite Rinv_Rmult. -Rewrite (Rmult_sym (INR (fact N))). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Replace (minus N N0) with N0. -Ring. -Replace N with (plus N0 N0). -Symmetry; Apply minus_plus. -Rewrite H4. -Apply INR_eq; Rewrite plus_INR; Rewrite mult_INR; Do 2 Rewrite S_INR; Ring. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Unfold C Rdiv. -Rewrite (Rmult_sym (INR (fact N))). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_r_sym. -Rewrite Rinv_Rmult. -Rewrite Rmult_1r; Ring. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Replace ``/(INR (fact (S n0)))*/(INR (fact (minus N n0)))`` with ``(C (S N) (S n0))/(INR (fact (S N)))``. -Apply Rle_trans with ``(C (S N) (S N0))/(INR (fact (S N)))``. -Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/(INR (fact (S N)))``). -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Cut (S N) = (mult (2) (S N0)). -Intro; Rewrite H5; Apply C_maj. -Rewrite <- H5; Apply le_n_S. -Apply le_trans with (pred (minus N n)). -Apply H1. -Apply le_S_n. -Replace (S (pred (minus N n))) with (minus N n). -Apply le_trans with N. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Apply H0. -Apply le_pred_n. -Apply le_n_Sn. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n (0)) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Apply H0. -Apply lt_pred_n_n. -Apply H. -Apply le_trans with (pred N). -Apply H0. -Apply le_pred_n. -Apply INR_eq; Rewrite H4. -Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Cut (S N) = (mult (2) (S N0)). -Intro. -Replace ``(C (S N) (S N0))/(INR (fact (S N)))`` with ``/(Rsqr (INR (fact (S N0))))``. -Rewrite H5; Rewrite div2_double. -Right; Reflexivity. -Unfold Rsqr C Rdiv. -Repeat Rewrite Rinv_Rmult. -Replace (minus (S N) (S N0)) with (S N0). -Rewrite (Rmult_sym (INR (fact (S N)))). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Reflexivity. -Apply INR_fact_neq_0. -Replace (S N) with (plus (S N0) (S N0)). -Symmetry; Apply minus_plus. -Rewrite H5; Ring. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_eq; Rewrite H4; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Unfold C Rdiv. -Rewrite (Rmult_sym (INR (fact (S N)))). -Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Rewrite Rinv_Rmult. -Reflexivity. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Unfold maj_Reste_E. -Unfold Rdiv; Rewrite (Rmult_sym ``4``). -Rewrite Rmult_assoc. -Apply Rle_monotony. -Apply pow_le. -Apply Rle_trans with R1. -Left; Apply Rlt_R0_R1. -Apply RmaxLess1. -Apply Rle_trans with (sum_f_R0 [k:nat]``(INR (minus N k))*/(Rsqr (INR (fact (div2 (S N)))))`` (pred N)). -Apply sum_Rle; Intros. -Rewrite sum_cte. -Replace (S (pred (minus N n))) with (minus N n). -Right; Apply Rmult_sym. -Apply S_pred with O. -Apply simpl_lt_plus_l with n. -Rewrite <- le_plus_minus. -Replace (plus n (0)) with n; [Idtac | Ring]. -Apply le_lt_trans with (pred N). -Apply H0. -Apply lt_pred_n_n. -Apply H. -Apply le_trans with (pred N). -Apply H0. -Apply le_pred_n. -Apply Rle_trans with (sum_f_R0 [k:nat]``(INR N)*/(Rsqr (INR (fact (div2 (S N)))))`` (pred N)). -Apply sum_Rle; Intros. -Do 2 Rewrite <- (Rmult_sym ``/(Rsqr (INR (fact (div2 (S N)))))``). -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply Rsqr_pos_lt. -Apply INR_fact_neq_0. -Apply le_INR. -Apply simpl_le_plus_l with n. -Rewrite <- le_plus_minus. -Apply le_plus_r. -Apply le_trans with (pred N). -Apply H0. -Apply le_pred_n. -Rewrite sum_cte; Replace (S (pred N)) with N. -Cut (div2 (S N)) = (S (div2 (pred N))). -Intro; Rewrite H0. -Rewrite fact_simpl; Rewrite mult_sym; Rewrite mult_INR; Rewrite Rsqr_times. -Rewrite Rinv_Rmult. -Rewrite (Rmult_sym (INR N)); Repeat Rewrite Rmult_assoc; Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply Rsqr_pos_lt; Apply INR_fact_neq_0. -Rewrite <- H0. -Cut ``(INR N)<=(INR (mult (S (S O)) (div2 (S N))))``. -Intro; Apply Rle_monotony_contra with ``(Rsqr (INR (div2 (S N))))``. -Apply Rsqr_pos_lt. -Apply not_O_INR; Red; Intro. -Cut (lt (1) (S N)). -Intro; Assert H4 := (div2_not_R0 ? H3). -Rewrite H2 in H4; Elim (lt_n_O ? H4). -Apply lt_n_S; Apply H. -Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l. -Replace ``(INR N)*(INR N)`` with (Rsqr (INR N)); [Idtac | Reflexivity]. -Rewrite Rmult_assoc. -Rewrite Rmult_sym. -Replace ``4`` with (Rsqr ``2``); [Idtac | SqRing]. -Rewrite <- Rsqr_times. -Apply Rsqr_incr_1. -Replace ``2`` with (INR (2)). -Rewrite <- mult_INR; Apply H1. -Reflexivity. -Left; Apply lt_INR_0; Apply H. -Left; Apply Rmult_lt_pos. -Sup0. -Apply lt_INR_0; Apply div2_not_R0. -Apply lt_n_S; Apply H. -Cut (lt (1) (S N)). -Intro; Unfold Rsqr; Apply prod_neq_R0; Apply not_O_INR; Intro; Assert H4 := (div2_not_R0 ? H2); Rewrite H3 in H4; Elim (lt_n_O ? H4). -Apply lt_n_S; Apply H. -Assert H1 := (even_odd_cor N). -Elim H1; Intros N0 H2. -Elim H2; Intro. -Pattern 2 N; Rewrite H3. -Rewrite div2_S_double. -Right; Rewrite H3; Reflexivity. -Pattern 2 N; Rewrite H3. -Replace (S (S (mult (2) N0))) with (mult (2) (S N0)). -Rewrite div2_double. -Rewrite H3. -Rewrite S_INR; Do 2 Rewrite mult_INR. -Rewrite (S_INR N0). -Rewrite Rmult_Rplus_distr. -Apply Rle_compatibility. -Rewrite Rmult_1r. -Simpl. -Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Apply Rlt_R0_R1. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Unfold Rsqr; Apply prod_neq_R0; Apply INR_fact_neq_0. -Unfold Rsqr; Apply prod_neq_R0; Apply not_O_INR; Discriminate. -Assert H0 := (even_odd_cor N). -Elim H0; Intros N0 H1. -Elim H1; Intro. -Cut (lt O N0). -Intro; Rewrite H2. -Rewrite div2_S_double. -Replace (mult (2) N0) with (S (S (mult (2) (pred N0)))). -Replace (pred (S (S (mult (2) (pred N0))))) with (S (mult (2) (pred N0))). -Rewrite div2_S_double. -Apply S_pred with O; Apply H3. -Reflexivity. -Replace N0 with (S (pred N0)). -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Symmetry; Apply S_pred with O; Apply H3. -Rewrite H2 in H. -Apply neq_O_lt. -Red; Intro. -Rewrite <- H3 in H. -Simpl in H. -Elim (lt_n_O ? H). -Rewrite H2. -Replace (pred (S (mult (2) N0))) with (mult (2) N0); [Idtac | Reflexivity]. -Replace (S (S (mult (2) N0))) with (mult (2) (S N0)). -Do 2 Rewrite div2_double. -Reflexivity. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Apply S_pred with O; Apply H. +Lemma Reste_E_maj : + forall (x y:R) (N:nat), + (0 < N)%nat -> Rabs (Reste_E x y N) <= maj_Reste_E x y N. +intros; pose (M := Rmax 1 (Rmax (Rabs x) (Rabs y))). +apply Rle_trans with + (M ^ (2 * N) * + sum_f_R0 + (fun k:nat => + sum_f_R0 (fun l:nat => / Rsqr (INR (fact (div2 (S N))))) + (pred (N - k))) (pred N)). +unfold Reste_E in |- *. +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + Rabs + (sum_f_R0 + (fun l:nat => + / INR (fact (S (l + k))) * x ^ S (l + k) * + (/ INR (fact (N - l)) * y ^ (N - l))) ( + pred (N - k)))) (pred N)). +apply + (Rsum_abs + (fun k:nat => + sum_f_R0 + (fun l:nat => + / INR (fact (S (l + k))) * x ^ S (l + k) * + (/ INR (fact (N - l)) * y ^ (N - l))) ( + pred (N - k))) (pred N)). +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + Rabs + (/ INR (fact (S (l + k))) * x ^ S (l + k) * + (/ INR (fact (N - l)) * y ^ (N - l)))) ( + pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply + (Rsum_abs + (fun l:nat => + / INR (fact (S (l + n))) * x ^ S (l + n) * + (/ INR (fact (N - l)) * y ^ (N - l)))). +apply Rle_trans with + (sum_f_R0 + (fun k:nat => + sum_f_R0 + (fun l:nat => + M ^ (2 * N) * / INR (fact (S l)) * / INR (fact (N - l))) + (pred (N - k))) (pred N)). +apply sum_Rle; intros. +apply sum_Rle; intros. +repeat rewrite Rabs_mult. +do 2 rewrite <- RPow_abs. +rewrite (Rabs_right (/ INR (fact (S (n0 + n))))). +rewrite (Rabs_right (/ INR (fact (N - n0)))). +replace + (/ INR (fact (S (n0 + n))) * Rabs x ^ S (n0 + n) * + (/ INR (fact (N - n0)) * Rabs y ^ (N - n0))) with + (/ INR (fact (N - n0)) * / INR (fact (S (n0 + n))) * Rabs x ^ S (n0 + n) * + Rabs y ^ (N - n0)); [ idtac | ring ]. +rewrite <- (Rmult_comm (/ INR (fact (N - n0)))). +repeat rewrite Rmult_assoc. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_trans with + (/ INR (fact (S n0)) * Rabs x ^ S (n0 + n) * Rabs y ^ (N - n0)). +rewrite (Rmult_comm (/ INR (fact (S (n0 + n))))); + rewrite (Rmult_comm (/ INR (fact (S n0)))); repeat rewrite Rmult_assoc; + apply Rmult_le_compat_l. +apply pow_le; apply Rabs_pos. +rewrite (Rmult_comm (/ INR (fact (S n0)))); apply Rmult_le_compat_l. +apply pow_le; apply Rabs_pos. +apply Rle_Rinv. +apply INR_fact_lt_0. +apply INR_fact_lt_0. +apply le_INR; apply fact_le; apply le_n_S. +apply le_plus_l. +rewrite (Rmult_comm (M ^ (2 * N))); rewrite Rmult_assoc; + apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_trans with (M ^ S (n0 + n) * Rabs y ^ (N - n0)). +do 2 rewrite <- (Rmult_comm (Rabs y ^ (N - n0))). +apply Rmult_le_compat_l. +apply pow_le; apply Rabs_pos. +apply pow_incr; split. +apply Rabs_pos. +apply Rle_trans with (Rmax (Rabs x) (Rabs y)). +apply RmaxLess1. +unfold M in |- *; apply RmaxLess2. +apply Rle_trans with (M ^ S (n0 + n) * M ^ (N - n0)). +apply Rmult_le_compat_l. +apply pow_le; apply Rle_trans with 1. +left; apply Rlt_0_1. +unfold M in |- *; apply RmaxLess1. +apply pow_incr; split. +apply Rabs_pos. +apply Rle_trans with (Rmax (Rabs x) (Rabs y)). +apply RmaxLess2. +unfold M in |- *; apply RmaxLess2. +rewrite <- pow_add; replace (S (n0 + n) + (N - n0))%nat with (N + S n)%nat. +apply Rle_pow. +unfold M in |- *; apply RmaxLess1. +replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ]. +apply plus_le_compat_l. +replace N with (S (pred N)). +apply le_n_S; apply H0. +symmetry in |- *; apply S_pred with 0%nat; apply H. +apply INR_eq; do 2 rewrite plus_INR; do 2 rewrite S_INR; rewrite plus_INR; + rewrite minus_INR. +ring. +apply le_trans with (pred (N - n)). +apply H1. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +apply H0. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +apply H0. +apply lt_pred_n_n. +apply H. +apply le_trans with (pred N). +apply H0. +apply le_pred_n. +apply Rle_ge; left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_ge; left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +rewrite scal_sum. +apply sum_Rle; intros. +rewrite <- Rmult_comm. +rewrite scal_sum. +apply sum_Rle; intros. +rewrite (Rmult_comm (/ Rsqr (INR (fact (div2 (S N)))))). +rewrite Rmult_assoc; apply Rmult_le_compat_l. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +unfold M in |- *; apply RmaxLess1. +assert (H2 := even_odd_cor N). +elim H2; intros N0 H3. +elim H3; intro. +apply Rle_trans with (/ INR (fact n0) * / INR (fact (N - n0))). +do 2 rewrite <- (Rmult_comm (/ INR (fact (N - n0)))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_Rinv. +apply INR_fact_lt_0. +apply INR_fact_lt_0. +apply le_INR. +apply fact_le. +apply le_n_Sn. +replace (/ INR (fact n0) * / INR (fact (N - n0))) with + (C N n0 / INR (fact N)). +pattern N at 1 in |- *; rewrite H4. +apply Rle_trans with (C N N0 / INR (fact N)). +unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ INR (fact N))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +rewrite H4. +apply C_maj. +rewrite <- H4; apply le_trans with (pred (N - n)). +apply H1. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +apply H0. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +apply H0. +apply lt_pred_n_n. +apply H. +apply le_trans with (pred N). +apply H0. +apply le_pred_n. +replace (C N N0 / INR (fact N)) with (/ Rsqr (INR (fact N0))). +rewrite H4; rewrite div2_S_double; right; reflexivity. +unfold Rsqr, C, Rdiv in |- *. +repeat rewrite Rinv_mult_distr. +rewrite (Rmult_comm (INR (fact N))). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; replace (N - N0)%nat with N0. +ring. +replace N with (N0 + N0)%nat. +symmetry in |- *; apply minus_plus. +rewrite H4. +apply INR_eq; rewrite plus_INR; rewrite mult_INR; do 2 rewrite S_INR; ring. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +unfold C, Rdiv in |- *. +rewrite (Rmult_comm (INR (fact N))). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rinv_mult_distr. +rewrite Rmult_1_r; ring. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +replace (/ INR (fact (S n0)) * / INR (fact (N - n0))) with + (C (S N) (S n0) / INR (fact (S N))). +apply Rle_trans with (C (S N) (S N0) / INR (fact (S N))). +unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ INR (fact (S N)))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +cut (S N = (2 * S N0)%nat). +intro; rewrite H5; apply C_maj. +rewrite <- H5; apply le_n_S. +apply le_trans with (pred (N - n)). +apply H1. +apply le_S_n. +replace (S (pred (N - n))) with (N - n)%nat. +apply le_trans with N. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +apply H0. +apply le_pred_n. +apply le_n_Sn. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +apply H0. +apply lt_pred_n_n. +apply H. +apply le_trans with (pred N). +apply H0. +apply le_pred_n. +apply INR_eq; rewrite H4. +do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; ring. +cut (S N = (2 * S N0)%nat). +intro. +replace (C (S N) (S N0) / INR (fact (S N))) with (/ Rsqr (INR (fact (S N0)))). +rewrite H5; rewrite div2_double. +right; reflexivity. +unfold Rsqr, C, Rdiv in |- *. +repeat rewrite Rinv_mult_distr. +replace (S N - S N0)%nat with (S N0). +rewrite (Rmult_comm (INR (fact (S N)))). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; reflexivity. +apply INR_fact_neq_0. +replace (S N) with (S N0 + S N0)%nat. +symmetry in |- *; apply minus_plus. +rewrite H5; ring. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_eq; rewrite H4; do 2 rewrite S_INR; do 2 rewrite mult_INR; + repeat rewrite S_INR; ring. +unfold C, Rdiv in |- *. +rewrite (Rmult_comm (INR (fact (S N)))). +rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; rewrite Rinv_mult_distr. +reflexivity. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +unfold maj_Reste_E in |- *. +unfold Rdiv in |- *; rewrite (Rmult_comm 4). +rewrite Rmult_assoc. +apply Rmult_le_compat_l. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +apply RmaxLess1. +apply Rle_trans with + (sum_f_R0 (fun k:nat => INR (N - k) * / Rsqr (INR (fact (div2 (S N))))) + (pred N)). +apply sum_Rle; intros. +rewrite sum_cte. +replace (S (pred (N - n))) with (N - n)%nat. +right; apply Rmult_comm. +apply S_pred with 0%nat. +apply plus_lt_reg_l with n. +rewrite <- le_plus_minus. +replace (n + 0)%nat with n; [ idtac | ring ]. +apply le_lt_trans with (pred N). +apply H0. +apply lt_pred_n_n. +apply H. +apply le_trans with (pred N). +apply H0. +apply le_pred_n. +apply Rle_trans with + (sum_f_R0 (fun k:nat => INR N * / Rsqr (INR (fact (div2 (S N))))) (pred N)). +apply sum_Rle; intros. +do 2 rewrite <- (Rmult_comm (/ Rsqr (INR (fact (div2 (S N)))))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt. +apply INR_fact_neq_0. +apply le_INR. +apply (fun p n m:nat => plus_le_reg_l n m p) with n. +rewrite <- le_plus_minus. +apply le_plus_r. +apply le_trans with (pred N). +apply H0. +apply le_pred_n. +rewrite sum_cte; replace (S (pred N)) with N. +cut (div2 (S N) = S (div2 (pred N))). +intro; rewrite H0. +rewrite fact_simpl; rewrite mult_comm; rewrite mult_INR; rewrite Rsqr_mult. +rewrite Rinv_mult_distr. +rewrite (Rmult_comm (INR N)); repeat rewrite Rmult_assoc; + apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt; apply INR_fact_neq_0. +rewrite <- H0. +cut (INR N <= INR (2 * div2 (S N))). +intro; apply Rmult_le_reg_l with (Rsqr (INR (div2 (S N)))). +apply Rsqr_pos_lt. +apply not_O_INR; red in |- *; intro. +cut (1 < S N)%nat. +intro; assert (H4 := div2_not_R0 _ H3). +rewrite H2 in H4; elim (lt_n_O _ H4). +apply lt_n_S; apply H. +repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_l. +replace (INR N * INR N) with (Rsqr (INR N)); [ idtac | reflexivity ]. +rewrite Rmult_assoc. +rewrite Rmult_comm. +replace 4 with (Rsqr 2); [ idtac | ring_Rsqr ]. +rewrite <- Rsqr_mult. +apply Rsqr_incr_1. +replace 2 with (INR 2). +rewrite <- mult_INR; apply H1. +reflexivity. +left; apply lt_INR_0; apply H. +left; apply Rmult_lt_0_compat. +prove_sup0. +apply lt_INR_0; apply div2_not_R0. +apply lt_n_S; apply H. +cut (1 < S N)%nat. +intro; unfold Rsqr in |- *; apply prod_neq_R0; apply not_O_INR; intro; + assert (H4 := div2_not_R0 _ H2); rewrite H3 in H4; + elim (lt_n_O _ H4). +apply lt_n_S; apply H. +assert (H1 := even_odd_cor N). +elim H1; intros N0 H2. +elim H2; intro. +pattern N at 2 in |- *; rewrite H3. +rewrite div2_S_double. +right; rewrite H3; reflexivity. +pattern N at 2 in |- *; rewrite H3. +replace (S (S (2 * N0))) with (2 * S N0)%nat. +rewrite div2_double. +rewrite H3. +rewrite S_INR; do 2 rewrite mult_INR. +rewrite (S_INR N0). +rewrite Rmult_plus_distr_l. +apply Rplus_le_compat_l. +rewrite Rmult_1_r. +simpl in |- *. +pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + apply Rlt_0_1. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +unfold Rsqr in |- *; apply prod_neq_R0; apply INR_fact_neq_0. +unfold Rsqr in |- *; apply prod_neq_R0; apply not_O_INR; discriminate. +assert (H0 := even_odd_cor N). +elim H0; intros N0 H1. +elim H1; intro. +cut (0 < N0)%nat. +intro; rewrite H2. +rewrite div2_S_double. +replace (2 * N0)%nat with (S (S (2 * pred N0))). +replace (pred (S (S (2 * pred N0)))) with (S (2 * pred N0)). +rewrite div2_S_double. +apply S_pred with 0%nat; apply H3. +reflexivity. +replace N0 with (S (pred N0)). +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +symmetry in |- *; apply S_pred with 0%nat; apply H3. +rewrite H2 in H. +apply neq_O_lt. +red in |- *; intro. +rewrite <- H3 in H. +simpl in H. +elim (lt_n_O _ H). +rewrite H2. +replace (pred (S (2 * N0))) with (2 * N0)%nat; [ idtac | reflexivity ]. +replace (S (S (2 * N0))) with (2 * S N0)%nat. +do 2 rewrite div2_double. +reflexivity. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +apply S_pred with 0%nat; apply H. Qed. -Lemma maj_Reste_cv_R0 : (x,y:R) (Un_cv (maj_Reste_E x y) ``0``). -Intros; Assert H := (Majxy_cv_R0 x y). -Unfold Un_cv in H; Unfold Un_cv; Intros. -Cut ``0<eps/4``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. -Elim (H ? H1); Intros N0 H2. -Exists (max (mult (2) (S N0)) (2)); Intros. -Unfold R_dist in H2; Unfold R_dist; Rewrite minus_R0; Unfold Majxy in H2; Unfold maj_Reste_E. -Rewrite Rabsolu_right. -Apply Rle_lt_trans with ``4*(pow (Rmax 1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S (S (S O)))) (S (div2 (pred n)))))/(INR (fact (div2 (pred n))))``. -Apply Rle_monotony. -Left; Sup0. -Unfold Rdiv Rsqr; Rewrite Rinv_Rmult. -Rewrite (Rmult_sym ``(pow (Rmax 1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S O)) n))``); Rewrite (Rmult_sym ``(pow (Rmax 1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S (S (S O)))) (S (div2 (pred n)))))``); Rewrite Rmult_assoc; Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply Rle_trans with ``(pow (Rmax 1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S O)) n))``. -Rewrite Rmult_sym; Pattern 2 (pow (Rmax R1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (2) n)); Rewrite <- Rmult_1r; Apply Rle_monotony. -Apply pow_le; Apply Rle_trans with R1. -Left; Apply Rlt_R0_R1. -Apply RmaxLess1. -Apply Rle_monotony_contra with ``(INR (fact (div2 (pred n))))``. -Apply INR_fact_lt_0. -Rewrite Rmult_1r; Rewrite <- Rinv_r_sym. -Replace R1 with (INR (1)); [Apply le_INR | Reflexivity]. -Apply lt_le_S. -Apply INR_lt. -Apply INR_fact_lt_0. -Apply INR_fact_neq_0. -Apply Rle_pow. -Apply RmaxLess1. -Assert H4 := (even_odd_cor n). -Elim H4; Intros N1 H5. -Elim H5; Intro. -Cut (lt O N1). -Intro. -Rewrite H6. -Replace (pred (mult (2) N1)) with (S (mult (2) (pred N1))). -Rewrite div2_S_double. -Replace (S (pred N1)) with N1. -Apply INR_le. -Right. -Do 3 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Apply S_pred with O; Apply H7. -Replace (mult (2) N1) with (S (S (mult (2) (pred N1)))). -Reflexivity. -Pattern 2 N1; Replace N1 with (S (pred N1)). -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Symmetry ; Apply S_pred with O; Apply H7. -Apply INR_lt. -Apply Rlt_monotony_contra with (INR (2)). -Simpl; Sup0. -Rewrite Rmult_Or; Rewrite <- mult_INR. -Apply lt_INR_0. -Rewrite <- H6. -Apply lt_le_trans with (2). -Apply lt_O_Sn. -Apply le_trans with (max (mult (2) (S N0)) (2)). -Apply le_max_r. -Apply H3. -Rewrite H6. -Replace (pred (S (mult (2) N1))) with (mult (2) N1). -Rewrite div2_double. -Replace (mult (4) (S N1)) with (mult (2) (mult (2) (S N1))). -Apply mult_le. -Replace (mult (2) (S N1)) with (S (S (mult (2) N1))). -Apply le_n_Sn. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Ring. -Reflexivity. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply Rlt_monotony_contra with ``/4``. -Apply Rlt_Rinv; Sup0. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Rewrite Rmult_sym. -Replace ``(pow (Rmax 1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S (S (S O)))) (S (div2 (pred n)))))/(INR (fact (div2 (pred n))))`` with ``(Rabsolu ((pow (Rmax 1 (Rmax (Rabsolu x) (Rabsolu y))) (mult (S (S (S (S O)))) (S (div2 (pred n)))))/(INR (fact (div2 (pred n))))-0))``. -Apply H2; Unfold ge. -Cut (le (mult (2) (S N0)) n). -Intro; Apply le_S_n. -Apply INR_le; Apply Rle_monotony_contra with (INR (2)). -Simpl; Sup0. -Do 2 Rewrite <- mult_INR; Apply le_INR. -Apply le_trans with n. -Apply H4. -Assert H5 := (even_odd_cor n). -Elim H5; Intros N1 H6. -Elim H6; Intro. -Cut (lt O N1). -Intro. -Rewrite H7. -Apply mult_le. -Replace (pred (mult (2) N1)) with (S (mult (2) (pred N1))). -Rewrite div2_S_double. -Replace (S (pred N1)) with N1. -Apply le_n. -Apply S_pred with O; Apply H8. -Replace (mult (2) N1) with (S (S (mult (2) (pred N1)))). -Reflexivity. -Pattern 2 N1; Replace N1 with (S (pred N1)). -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Symmetry; Apply S_pred with O; Apply H8. -Apply INR_lt. -Apply Rlt_monotony_contra with (INR (2)). -Simpl; Sup0. -Rewrite Rmult_Or; Rewrite <- mult_INR. -Apply lt_INR_0. -Rewrite <- H7. -Apply lt_le_trans with (2). -Apply lt_O_Sn. -Apply le_trans with (max (mult (2) (S N0)) (2)). -Apply le_max_r. -Apply H3. -Rewrite H7. -Replace (pred (S (mult (2) N1))) with (mult (2) N1). -Rewrite div2_double. -Replace (mult (2) (S N1)) with (S (S (mult (2) N1))). -Apply le_n_Sn. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Reflexivity. -Apply le_trans with (max (mult (2) (S N0)) (2)). -Apply le_max_l. -Apply H3. -Rewrite minus_R0; Apply Rabsolu_right. -Apply Rle_sym1. -Unfold Rdiv; Repeat Apply Rmult_le_pos. -Apply pow_le. -Apply Rle_trans with R1. -Left; Apply Rlt_R0_R1. -Apply RmaxLess1. -Left; Apply Rlt_Rinv; Apply INR_fact_lt_0. -DiscrR. -Apply Rle_sym1. -Unfold Rdiv; Apply Rmult_le_pos. -Left; Sup0. -Apply Rmult_le_pos. -Apply pow_le. -Apply Rle_trans with R1. -Left; Apply Rlt_R0_R1. -Apply RmaxLess1. -Left; Apply Rlt_Rinv; Apply Rsqr_pos_lt; Apply INR_fact_neq_0. +Lemma maj_Reste_cv_R0 : forall x y:R, Un_cv (maj_Reste_E x y) 0. +intros; assert (H := Majxy_cv_R0 x y). +unfold Un_cv in H; unfold Un_cv in |- *; intros. +cut (0 < eps / 4); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H _ H1); intros N0 H2. +exists (max (2 * S N0) 2); intros. +unfold R_dist in H2; unfold R_dist in |- *; rewrite Rminus_0_r; + unfold Majxy in H2; unfold maj_Reste_E in |- *. +rewrite Rabs_right. +apply Rle_lt_trans with + (4 * + (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))) / + INR (fact (div2 (pred n))))). +apply Rmult_le_compat_l. +left; prove_sup0. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr. +rewrite (Rmult_comm (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n))); + rewrite + (Rmult_comm (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))))) + ; rewrite Rmult_assoc; apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Rle_trans with (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)). +rewrite Rmult_comm; + pattern (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (2 * n)) at 2 in |- *; + rewrite <- Rmult_1_r; apply Rmult_le_compat_l. +apply pow_le; apply Rle_trans with 1. +left; apply Rlt_0_1. +apply RmaxLess1. +apply Rmult_le_reg_l with (INR (fact (div2 (pred n)))). +apply INR_fact_lt_0. +rewrite Rmult_1_r; rewrite <- Rinv_r_sym. +replace 1 with (INR 1); [ apply le_INR | reflexivity ]. +apply lt_le_S. +apply INR_lt. +apply INR_fact_lt_0. +apply INR_fact_neq_0. +apply Rle_pow. +apply RmaxLess1. +assert (H4 := even_odd_cor n). +elim H4; intros N1 H5. +elim H5; intro. +cut (0 < N1)%nat. +intro. +rewrite H6. +replace (pred (2 * N1)) with (S (2 * pred N1)). +rewrite div2_S_double. +replace (S (pred N1)) with N1. +apply INR_le. +right. +do 3 rewrite mult_INR; repeat rewrite S_INR; ring. +apply S_pred with 0%nat; apply H7. +replace (2 * N1)%nat with (S (S (2 * pred N1))). +reflexivity. +pattern N1 at 2 in |- *; replace N1 with (S (pred N1)). +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +symmetry in |- *; apply S_pred with 0%nat; apply H7. +apply INR_lt. +apply Rmult_lt_reg_l with (INR 2). +simpl in |- *; prove_sup0. +rewrite Rmult_0_r; rewrite <- mult_INR. +apply lt_INR_0. +rewrite <- H6. +apply lt_le_trans with 2%nat. +apply lt_O_Sn. +apply le_trans with (max (2 * S N0) 2). +apply le_max_r. +apply H3. +rewrite H6. +replace (pred (S (2 * N1))) with (2 * N1)%nat. +rewrite div2_double. +replace (4 * S N1)%nat with (2 * (2 * S N1))%nat. +apply (fun m n p:nat => mult_le_compat_l p n m). +replace (2 * S N1)%nat with (S (S (2 * N1))). +apply le_n_Sn. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +ring. +reflexivity. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply Rmult_lt_reg_l with (/ 4). +apply Rinv_0_lt_compat; prove_sup0. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite Rmult_comm. +replace + (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))) / + INR (fact (div2 (pred n)))) with + (Rabs + (Rmax 1 (Rmax (Rabs x) (Rabs y)) ^ (4 * S (div2 (pred n))) / + INR (fact (div2 (pred n))) - 0)). +apply H2; unfold ge in |- *. +cut (2 * S N0 <= n)%nat. +intro; apply le_S_n. +apply INR_le; apply Rmult_le_reg_l with (INR 2). +simpl in |- *; prove_sup0. +do 2 rewrite <- mult_INR; apply le_INR. +apply le_trans with n. +apply H4. +assert (H5 := even_odd_cor n). +elim H5; intros N1 H6. +elim H6; intro. +cut (0 < N1)%nat. +intro. +rewrite H7. +apply (fun m n p:nat => mult_le_compat_l p n m). +replace (pred (2 * N1)) with (S (2 * pred N1)). +rewrite div2_S_double. +replace (S (pred N1)) with N1. +apply le_n. +apply S_pred with 0%nat; apply H8. +replace (2 * N1)%nat with (S (S (2 * pred N1))). +reflexivity. +pattern N1 at 2 in |- *; replace N1 with (S (pred N1)). +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +symmetry in |- *; apply S_pred with 0%nat; apply H8. +apply INR_lt. +apply Rmult_lt_reg_l with (INR 2). +simpl in |- *; prove_sup0. +rewrite Rmult_0_r; rewrite <- mult_INR. +apply lt_INR_0. +rewrite <- H7. +apply lt_le_trans with 2%nat. +apply lt_O_Sn. +apply le_trans with (max (2 * S N0) 2). +apply le_max_r. +apply H3. +rewrite H7. +replace (pred (S (2 * N1))) with (2 * N1)%nat. +rewrite div2_double. +replace (2 * S N1)%nat with (S (S (2 * N1))). +apply le_n_Sn. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +reflexivity. +apply le_trans with (max (2 * S N0) 2). +apply le_max_l. +apply H3. +rewrite Rminus_0_r; apply Rabs_right. +apply Rle_ge. +unfold Rdiv in |- *; repeat apply Rmult_le_pos. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +apply RmaxLess1. +left; apply Rinv_0_lt_compat; apply INR_fact_lt_0. +discrR. +apply Rle_ge. +unfold Rdiv in |- *; apply Rmult_le_pos. +left; prove_sup0. +apply Rmult_le_pos. +apply pow_le. +apply Rle_trans with 1. +left; apply Rlt_0_1. +apply RmaxLess1. +left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt; apply INR_fact_neq_0. Qed. (**********) -Lemma Reste_E_cv : (x,y:R) (Un_cv (Reste_E x y) R0). -Intros; Assert H := (maj_Reste_cv_R0 x y). -Unfold Un_cv in H; Unfold Un_cv; Intros; Elim (H ? H0); Intros. -Exists (max x0 (1)); Intros. -Unfold R_dist; Rewrite minus_R0. -Apply Rle_lt_trans with (maj_Reste_E x y n). -Apply Reste_E_maj. -Apply lt_le_trans with (1). -Apply lt_O_Sn. -Apply le_trans with (max x0 (1)). -Apply le_max_r. -Apply H2. -Replace (maj_Reste_E x y n) with (R_dist (maj_Reste_E x y n) R0). -Apply H1. -Unfold ge; Apply le_trans with (max x0 (1)). -Apply le_max_l. -Apply H2. -Unfold R_dist; Rewrite minus_R0; Apply Rabsolu_right. -Apply Rle_sym1; Apply Rle_trans with (Rabsolu (Reste_E x y n)). -Apply Rabsolu_pos. -Apply Reste_E_maj. -Apply lt_le_trans with (1). -Apply lt_O_Sn. -Apply le_trans with (max x0 (1)). -Apply le_max_r. -Apply H2. +Lemma Reste_E_cv : forall x y:R, Un_cv (Reste_E x y) 0. +intros; assert (H := maj_Reste_cv_R0 x y). +unfold Un_cv in H; unfold Un_cv in |- *; intros; elim (H _ H0); intros. +exists (max x0 1); intros. +unfold R_dist in |- *; rewrite Rminus_0_r. +apply Rle_lt_trans with (maj_Reste_E x y n). +apply Reste_E_maj. +apply lt_le_trans with 1%nat. +apply lt_O_Sn. +apply le_trans with (max x0 1). +apply le_max_r. +apply H2. +replace (maj_Reste_E x y n) with (R_dist (maj_Reste_E x y n) 0). +apply H1. +unfold ge in |- *; apply le_trans with (max x0 1). +apply le_max_l. +apply H2. +unfold R_dist in |- *; rewrite Rminus_0_r; apply Rabs_right. +apply Rle_ge; apply Rle_trans with (Rabs (Reste_E x y n)). +apply Rabs_pos. +apply Reste_E_maj. +apply lt_le_trans with 1%nat. +apply lt_O_Sn. +apply le_trans with (max x0 1). +apply le_max_r. +apply H2. Qed. (**********) -Lemma exp_plus : (x,y:R) ``(exp (x+y))==(exp x)*(exp y)``. -Intros; Assert H0 := (E1_cvg x). -Assert H := (E1_cvg y). -Assert H1 := (E1_cvg ``x+y``). -EApply UL_sequence. -Apply H1. -Assert H2 := (CV_mult ? ? ? ? H0 H). -Assert H3 := (CV_minus ? ? ? ? H2 (Reste_E_cv x y)). -Unfold Un_cv; Unfold Un_cv in H3; Intros. -Elim (H3 ? H4); Intros. -Exists (S x0); Intros. -Rewrite <- (exp_form x y n). -Rewrite minus_R0 in H5. -Apply H5. -Unfold ge; Apply le_trans with (S x0). -Apply le_n_Sn. -Apply H6. -Apply lt_le_trans with (S x0). -Apply lt_O_Sn. -Apply H6. +Lemma exp_plus : forall x y:R, exp (x + y) = exp x * exp y. +intros; assert (H0 := E1_cvg x). +assert (H := E1_cvg y). +assert (H1 := E1_cvg (x + y)). +eapply UL_sequence. +apply H1. +assert (H2 := CV_mult _ _ _ _ H0 H). +assert (H3 := CV_minus _ _ _ _ H2 (Reste_E_cv x y)). +unfold Un_cv in |- *; unfold Un_cv in H3; intros. +elim (H3 _ H4); intros. +exists (S x0); intros. +rewrite <- (exp_form x y n). +rewrite Rminus_0_r in H5. +apply H5. +unfold ge in |- *; apply le_trans with (S x0). +apply le_n_Sn. +apply H6. +apply lt_le_trans with (S x0). +apply lt_O_Sn. +apply H6. Qed. (**********) -Lemma exp_pos_pos : (x:R) ``0<x`` -> ``0<(exp x)``. -Intros; Pose An := [N:nat]``/(INR (fact N))*(pow x N)``. -Cut (Un_cv [n:nat](sum_f_R0 An n) (exp x)). -Intro; Apply Rlt_le_trans with (sum_f_R0 An O). -Unfold An; Simpl; Rewrite Rinv_R1; Rewrite Rmult_1r; Apply Rlt_R0_R1. -Apply sum_incr. -Assumption. -Intro; Unfold An; Left; Apply Rmult_lt_pos. -Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply (pow_lt ? n H). -Unfold exp; Unfold projT1; Case (exist_exp x); Intro. -Unfold exp_in; Unfold infinit_sum Un_cv; Trivial. +Lemma exp_pos_pos : forall x:R, 0 < x -> 0 < exp x. +intros; pose (An := fun N:nat => / INR (fact N) * x ^ N). +cut (Un_cv (fun n:nat => sum_f_R0 An n) (exp x)). +intro; apply Rlt_le_trans with (sum_f_R0 An 0). +unfold An in |- *; simpl in |- *; rewrite Rinv_1; rewrite Rmult_1_r; + apply Rlt_0_1. +apply sum_incr. +assumption. +intro; unfold An in |- *; left; apply Rmult_lt_0_compat. +apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply (pow_lt _ n H). +unfold exp in |- *; unfold projT1 in |- *; case (exist_exp x); intro. +unfold exp_in in |- *; unfold infinit_sum, Un_cv in |- *; trivial. Qed. (**********) -Lemma exp_pos : (x:R) ``0<(exp x)``. -Intro; Case (total_order_T R0 x); Intro. -Elim s; Intro. -Apply (exp_pos_pos ? a). -Rewrite <- b; Rewrite exp_0; Apply Rlt_R0_R1. -Replace (exp x) with ``1/(exp (-x))``. -Unfold Rdiv; Apply Rmult_lt_pos. -Apply Rlt_R0_R1. -Apply Rlt_Rinv; Apply exp_pos_pos. -Apply (Rgt_RO_Ropp ? r). -Cut ``(exp (-x))<>0``. -Intro; Unfold Rdiv; Apply r_Rmult_mult with ``(exp (-x))``. -Rewrite Rmult_1l; Rewrite <- Rinv_r_sym. -Rewrite <- exp_plus. -Rewrite Rplus_Ropp_l; Rewrite exp_0; Reflexivity. -Apply H. -Apply H. -Assert H := (exp_plus x ``-x``). -Rewrite Rplus_Ropp_r in H; Rewrite exp_0 in H. -Red; Intro; Rewrite H0 in H. -Rewrite Rmult_Or in H. -Elim R1_neq_R0; Assumption. +Lemma exp_pos : forall x:R, 0 < exp x. +intro; case (total_order_T 0 x); intro. +elim s; intro. +apply (exp_pos_pos _ a). +rewrite <- b; rewrite exp_0; apply Rlt_0_1. +replace (exp x) with (1 / exp (- x)). +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply Rlt_0_1. +apply Rinv_0_lt_compat; apply exp_pos_pos. +apply (Ropp_0_gt_lt_contravar _ r). +cut (exp (- x) <> 0). +intro; unfold Rdiv in |- *; apply Rmult_eq_reg_l with (exp (- x)). +rewrite Rmult_1_l; rewrite <- Rinv_r_sym. +rewrite <- exp_plus. +rewrite Rplus_opp_l; rewrite exp_0; reflexivity. +apply H. +apply H. +assert (H := exp_plus x (- x)). +rewrite Rplus_opp_r in H; rewrite exp_0 in H. +red in |- *; intro; rewrite H0 in H. +rewrite Rmult_0_r in H. +elim R1_neq_R0; assumption. Qed. (* ((exp h)-1)/h -> 0 quand h->0 *) -Lemma derivable_pt_lim_exp_0 : (derivable_pt_lim exp ``0`` ``1``). -Unfold derivable_pt_lim; Intros. -Pose fn := [N:nat][x:R]``(pow x N)/(INR (fact (S N)))``. -Cut (CVN_R fn). -Intro; Cut (x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l)). -Intro cv; Cut ((n:nat)(continuity (fn n))). -Intro; Cut (continuity (SFL fn cv)). -Intro; Unfold continuity in H1. -Assert H2 := (H1 R0). -Unfold continuity_pt in H2; Unfold continue_in in H2; Unfold limit1_in in H2; Unfold limit_in in H2; Simpl in H2; Unfold R_dist in H2. -Elim (H2 ? H); Intros alp H3. -Elim H3; Intros. -Exists (mkposreal ? H4); Intros. -Rewrite Rplus_Ol; Rewrite exp_0. -Replace ``((exp h)-1)/h`` with (SFL fn cv h). -Replace R1 with (SFL fn cv R0). -Apply H5. -Split. -Unfold D_x no_cond; Split. -Trivial. -Apply (not_sym ? ? H6). -Rewrite minus_R0; Apply H7. -Unfold SFL. -Case (cv ``0``); Intros. -EApply UL_sequence. -Apply u. -Unfold Un_cv SP. -Intros; Exists (1); Intros. -Unfold R_dist; Rewrite decomp_sum. -Rewrite (Rplus_sym (fn O R0)). -Replace (fn O R0) with R1. -Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or. -Replace (sum_f_R0 [i:nat](fn (S i) ``0``) (pred n)) with R0. -Rewrite Rabsolu_R0; Apply H8. -Symmetry; Apply sum_eq_R0; Intros. -Unfold fn. -Simpl. -Unfold Rdiv; Do 2 Rewrite Rmult_Ol; Reflexivity. -Unfold fn; Simpl. -Unfold Rdiv; Rewrite Rinv_R1; Rewrite Rmult_1r; Reflexivity. -Apply lt_le_trans with (1); [Apply lt_n_Sn | Apply H9]. -Unfold SFL exp. -Unfold projT1. -Case (cv h); Case (exist_exp h); Intros. -EApply UL_sequence. -Apply u. -Unfold Un_cv; Intros. -Unfold exp_in in e. -Unfold infinit_sum in e. -Cut ``0<eps0*(Rabsolu h)``. -Intro; Elim (e ? H9); Intros N0 H10. -Exists N0; Intros. -Unfold R_dist. -Apply Rlt_monotony_contra with ``(Rabsolu h)``. -Apply Rabsolu_pos_lt; Assumption. -Rewrite <- Rabsolu_mult. -Rewrite Rminus_distr. -Replace ``h*(x-1)/h`` with ``(x-1)``. -Unfold R_dist in H10. -Replace ``h*(SP fn n h)-(x-1)`` with (Rminus (sum_f_R0 [i:nat]``/(INR (fact i))*(pow h i)`` (S n)) x). -Rewrite (Rmult_sym (Rabsolu h)). -Apply H10. -Unfold ge. -Apply le_trans with (S N0). -Apply le_n_Sn. -Apply le_n_S; Apply H11. -Rewrite decomp_sum. -Replace ``/(INR (fact O))*(pow h O)`` with R1. -Unfold Rminus. -Rewrite Ropp_distr1. -Rewrite Ropp_Ropp. -Rewrite <- (Rplus_sym ``-x``). -Rewrite <- (Rplus_sym ``-x+1``). -Rewrite Rplus_assoc; Repeat Apply Rplus_plus_r. -Replace (pred (S n)) with n; [Idtac | Reflexivity]. -Unfold SP. -Rewrite scal_sum. -Apply sum_eq; Intros. -Unfold fn. -Replace (pow h (S i)) with ``h*(pow h i)``. -Unfold Rdiv; Ring. -Simpl; Ring. -Simpl; Rewrite Rinv_R1; Rewrite Rmult_1r; Reflexivity. -Apply lt_O_Sn. -Unfold Rdiv. -Rewrite <- Rmult_assoc. -Symmetry; Apply Rinv_r_simpl_m. -Assumption. -Apply Rmult_lt_pos. -Apply H8. -Apply Rabsolu_pos_lt; Assumption. -Apply SFL_continuity; Assumption. -Intro; Unfold fn. -Replace [x:R]``(pow x n)/(INR (fact (S n)))`` with (div_fct (pow_fct n) (fct_cte (INR (fact (S n))))); [Idtac | Reflexivity]. -Apply continuity_div. -Apply derivable_continuous; Apply (derivable_pow n). -Apply derivable_continuous; Apply derivable_const. -Intro; Unfold fct_cte; Apply INR_fact_neq_0. -Apply (CVN_R_CVS ? X). -Assert H0 := Alembert_exp. -Unfold CVN_R. -Intro; Unfold CVN_r. -Apply Specif.existT with [N:nat]``(pow r N)/(INR (fact (S N)))``. -Cut (SigT ? [l:R](Un_cv [n:nat](sum_f_R0 [k:nat](Rabsolu ``(pow r k)/(INR (fact (S k)))``) n) l)). -Intro. -Elim X; Intros. -Exists x; Intros. -Split. -Apply p. -Unfold Boule; Intros. -Rewrite minus_R0 in H1. -Unfold fn. -Unfold Rdiv; Rewrite Rabsolu_mult. -Cut ``0<(INR (fact (S n)))``. -Intro. -Rewrite (Rabsolu_right ``/(INR (fact (S n)))``). -Do 2 Rewrite <- (Rmult_sym ``/(INR (fact (S n)))``). -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply H2. -Rewrite <- Pow_Rabsolu. -Apply pow_maj_Rabs. -Rewrite Rabsolu_Rabsolu; Left; Apply H1. -Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply H2. -Apply INR_fact_lt_0. -Cut (r::R)<>``0``. -Intro; Apply Alembert_C2. -Intro; Apply Rabsolu_no_R0. -Unfold Rdiv; Apply prod_neq_R0. -Apply pow_nonzero; Assumption. -Apply Rinv_neq_R0; Apply INR_fact_neq_0. -Unfold Un_cv in H0. -Unfold Un_cv; Intros. -Cut ``0<eps0/r``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply (cond_pos r)]]. -Elim (H0 ? H3); Intros N0 H4. -Exists N0; Intros. -Cut (ge (S n) N0). -Intro hyp_sn. -Assert H6 := (H4 ? hyp_sn). -Unfold R_dist in H6; Rewrite minus_R0 in H6. -Rewrite Rabsolu_Rabsolu in H6. -Unfold R_dist; Rewrite minus_R0. -Rewrite Rabsolu_Rabsolu. -Replace ``(Rabsolu ((pow r (S n))/(INR (fact (S (S n))))))/ - (Rabsolu ((pow r n)/(INR (fact (S n)))))`` with ``r*/(INR (fact (S (S n))))*//(INR (fact (S n)))``. -Rewrite Rmult_assoc; Rewrite Rabsolu_mult. -Rewrite (Rabsolu_right r). -Apply Rlt_monotony_contra with ``/r``. -Apply Rlt_Rinv; Apply (cond_pos r). -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps0). -Apply H6. -Assumption. -Apply Rle_sym1; Left; Apply (cond_pos r). -Unfold Rdiv. -Repeat Rewrite Rabsolu_mult. -Repeat Rewrite Rabsolu_Rinv. -Rewrite Rinv_Rmult. -Repeat Rewrite Rabsolu_right. -Rewrite Rinv_Rinv. -Rewrite (Rmult_sym r). -Rewrite (Rmult_sym (pow r (S n))). -Repeat Rewrite Rmult_assoc. -Apply Rmult_mult_r. -Rewrite (Rmult_sym r). -Rewrite <- Rmult_assoc; Rewrite <- (Rmult_sym (INR (fact (S n)))). -Apply Rmult_mult_r. -Simpl. -Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Ring. -Apply pow_nonzero; Assumption. -Apply INR_fact_neq_0. -Apply Rle_sym1; Left; Apply INR_fact_lt_0. -Apply Rle_sym1; Left; Apply pow_lt; Apply (cond_pos r). -Apply Rle_sym1; Left; Apply INR_fact_lt_0. -Apply Rle_sym1; Left; Apply pow_lt; Apply (cond_pos r). -Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. -Apply Rinv_neq_R0; Apply Rabsolu_no_R0; Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Unfold ge; Apply le_trans with n. -Apply H5. -Apply le_n_Sn. -Assert H1 := (cond_pos r); Red; Intro; Rewrite H2 in H1; Elim (Rlt_antirefl ? H1). +Lemma derivable_pt_lim_exp_0 : derivable_pt_lim exp 0 1. +unfold derivable_pt_lim in |- *; intros. +pose (fn := fun (N:nat) (x:R) => x ^ N / INR (fact (S N))). +cut (CVN_R fn). +intro; cut (forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)). +intro cv; cut (forall n:nat, continuity (fn n)). +intro; cut (continuity (SFL fn cv)). +intro; unfold continuity in H1. +assert (H2 := H1 0). +unfold continuity_pt in H2; unfold continue_in in H2; unfold limit1_in in H2; + unfold limit_in in H2; simpl in H2; unfold R_dist in H2. +elim (H2 _ H); intros alp H3. +elim H3; intros. +exists (mkposreal _ H4); intros. +rewrite Rplus_0_l; rewrite exp_0. +replace ((exp h - 1) / h) with (SFL fn cv h). +replace 1 with (SFL fn cv 0). +apply H5. +split. +unfold D_x, no_cond in |- *; split. +trivial. +apply (sym_not_eq H6). +rewrite Rminus_0_r; apply H7. +unfold SFL in |- *. +case (cv 0); intros. +eapply UL_sequence. +apply u. +unfold Un_cv, SP in |- *. +intros; exists 1%nat; intros. +unfold R_dist in |- *; rewrite decomp_sum. +rewrite (Rplus_comm (fn 0%nat 0)). +replace (fn 0%nat 0) with 1. +unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_r; + rewrite Rplus_0_r. +replace (sum_f_R0 (fun i:nat => fn (S i) 0) (pred n)) with 0. +rewrite Rabs_R0; apply H8. +symmetry in |- *; apply sum_eq_R0; intros. +unfold fn in |- *. +simpl in |- *. +unfold Rdiv in |- *; do 2 rewrite Rmult_0_l; reflexivity. +unfold fn in |- *; simpl in |- *. +unfold Rdiv in |- *; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity. +apply lt_le_trans with 1%nat; [ apply lt_n_Sn | apply H9 ]. +unfold SFL, exp in |- *. +unfold projT1 in |- *. +case (cv h); case (exist_exp h); intros. +eapply UL_sequence. +apply u. +unfold Un_cv in |- *; intros. +unfold exp_in in e. +unfold infinit_sum in e. +cut (0 < eps0 * Rabs h). +intro; elim (e _ H9); intros N0 H10. +exists N0; intros. +unfold R_dist in |- *. +apply Rmult_lt_reg_l with (Rabs h). +apply Rabs_pos_lt; assumption. +rewrite <- Rabs_mult. +rewrite Rmult_minus_distr_l. +replace (h * ((x - 1) / h)) with (x - 1). +unfold R_dist in H10. +replace (h * SP fn n h - (x - 1)) with + (sum_f_R0 (fun i:nat => / INR (fact i) * h ^ i) (S n) - x). +rewrite (Rmult_comm (Rabs h)). +apply H10. +unfold ge in |- *. +apply le_trans with (S N0). +apply le_n_Sn. +apply le_n_S; apply H11. +rewrite decomp_sum. +replace (/ INR (fact 0) * h ^ 0) with 1. +unfold Rminus in |- *. +rewrite Ropp_plus_distr. +rewrite Ropp_involutive. +rewrite <- (Rplus_comm (- x)). +rewrite <- (Rplus_comm (- x + 1)). +rewrite Rplus_assoc; repeat apply Rplus_eq_compat_l. +replace (pred (S n)) with n; [ idtac | reflexivity ]. +unfold SP in |- *. +rewrite scal_sum. +apply sum_eq; intros. +unfold fn in |- *. +replace (h ^ S i) with (h * h ^ i). +unfold Rdiv in |- *; ring. +simpl in |- *; ring. +simpl in |- *; rewrite Rinv_1; rewrite Rmult_1_r; reflexivity. +apply lt_O_Sn. +unfold Rdiv in |- *. +rewrite <- Rmult_assoc. +symmetry in |- *; apply Rinv_r_simpl_m. +assumption. +apply Rmult_lt_0_compat. +apply H8. +apply Rabs_pos_lt; assumption. +apply SFL_continuity; assumption. +intro; unfold fn in |- *. +replace (fun x:R => x ^ n / INR (fact (S n))) with + (pow_fct n / fct_cte (INR (fact (S n))))%F; [ idtac | reflexivity ]. +apply continuity_div. +apply derivable_continuous; apply (derivable_pow n). +apply derivable_continuous; apply derivable_const. +intro; unfold fct_cte in |- *; apply INR_fact_neq_0. +apply (CVN_R_CVS _ X). +assert (H0 := Alembert_exp). +unfold CVN_R in |- *. +intro; unfold CVN_r in |- *. +apply existT with (fun N:nat => r ^ N / INR (fact (S N))). +cut + (sigT + (fun l:R => + Un_cv + (fun n:nat => + sum_f_R0 (fun k:nat => Rabs (r ^ k / INR (fact (S k)))) n) l)). +intro. +elim X; intros. +exists x; intros. +split. +apply p. +unfold Boule in |- *; intros. +rewrite Rminus_0_r in H1. +unfold fn in |- *. +unfold Rdiv in |- *; rewrite Rabs_mult. +cut (0 < INR (fact (S n))). +intro. +rewrite (Rabs_right (/ INR (fact (S n)))). +do 2 rewrite <- (Rmult_comm (/ INR (fact (S n)))). +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply H2. +rewrite <- RPow_abs. +apply pow_maj_Rabs. +rewrite Rabs_Rabsolu; left; apply H1. +apply Rle_ge; left; apply Rinv_0_lt_compat; apply H2. +apply INR_fact_lt_0. +cut ((r:R) <> 0). +intro; apply Alembert_C2. +intro; apply Rabs_no_R0. +unfold Rdiv in |- *; apply prod_neq_R0. +apply pow_nonzero; assumption. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. +unfold Un_cv in H0. +unfold Un_cv in |- *; intros. +cut (0 < eps0 / r); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; apply (cond_pos r) ] ]. +elim (H0 _ H3); intros N0 H4. +exists N0; intros. +cut (S n >= N0)%nat. +intro hyp_sn. +assert (H6 := H4 _ hyp_sn). +unfold R_dist in H6; rewrite Rminus_0_r in H6. +rewrite Rabs_Rabsolu in H6. +unfold R_dist in |- *; rewrite Rminus_0_r. +rewrite Rabs_Rabsolu. +replace + (Rabs (r ^ S n / INR (fact (S (S n)))) / Rabs (r ^ n / INR (fact (S n)))) + with (r * / INR (fact (S (S n))) * / / INR (fact (S n))). +rewrite Rmult_assoc; rewrite Rabs_mult. +rewrite (Rabs_right r). +apply Rmult_lt_reg_l with (/ r). +apply Rinv_0_lt_compat; apply (cond_pos r). +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite <- (Rmult_comm eps0). +apply H6. +assumption. +apply Rle_ge; left; apply (cond_pos r). +unfold Rdiv in |- *. +repeat rewrite Rabs_mult. +repeat rewrite Rabs_Rinv. +rewrite Rinv_mult_distr. +repeat rewrite Rabs_right. +rewrite Rinv_involutive. +rewrite (Rmult_comm r). +rewrite (Rmult_comm (r ^ S n)). +repeat rewrite Rmult_assoc. +apply Rmult_eq_compat_l. +rewrite (Rmult_comm r). +rewrite <- Rmult_assoc; rewrite <- (Rmult_comm (INR (fact (S n)))). +apply Rmult_eq_compat_l. +simpl in |- *. +rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +ring. +apply pow_nonzero; assumption. +apply INR_fact_neq_0. +apply Rle_ge; left; apply INR_fact_lt_0. +apply Rle_ge; left; apply pow_lt; apply (cond_pos r). +apply Rle_ge; left; apply INR_fact_lt_0. +apply Rle_ge; left; apply pow_lt; apply (cond_pos r). +apply Rabs_no_R0; apply pow_nonzero; assumption. +apply Rinv_neq_0_compat; apply Rabs_no_R0; apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +unfold ge in |- *; apply le_trans with n. +apply H5. +apply le_n_Sn. +assert (H1 := cond_pos r); red in |- *; intro; rewrite H2 in H1; + elim (Rlt_irrefl _ H1). Qed. (**********) -Lemma derivable_pt_lim_exp : (x:R) (derivable_pt_lim exp x (exp x)). -Intro; Assert H0 := derivable_pt_lim_exp_0. -Unfold derivable_pt_lim in H0; Unfold derivable_pt_lim; Intros. -Cut ``0<eps/(exp x)``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Apply H | Apply Rlt_Rinv; Apply exp_pos]]. -Elim (H0 ? H1); Intros del H2. -Exists del; Intros. -Assert H5 := (H2 ? H3 H4). -Rewrite Rplus_Ol in H5; Rewrite exp_0 in H5. -Replace ``((exp (x+h))-(exp x))/h-(exp x)`` with ``(exp x)*(((exp h)-1)/h-1)``. -Rewrite Rabsolu_mult; Rewrite (Rabsolu_right (exp x)). -Apply Rlt_monotony_contra with ``/(exp x)``. -Apply Rlt_Rinv; Apply exp_pos. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps). -Apply H5. -Assert H6 := (exp_pos x); Red; Intro; Rewrite H7 in H6; Elim (Rlt_antirefl ? H6). -Apply Rle_sym1; Left; Apply exp_pos. -Rewrite Rminus_distr. -Rewrite Rmult_1r; Unfold Rdiv; Rewrite <- Rmult_assoc; Rewrite Rminus_distr. -Rewrite Rmult_1r; Rewrite exp_plus; Reflexivity. -Qed. +Lemma derivable_pt_lim_exp : forall x:R, derivable_pt_lim exp x (exp x). +intro; assert (H0 := derivable_pt_lim_exp_0). +unfold derivable_pt_lim in H0; unfold derivable_pt_lim in |- *; intros. +cut (0 < eps / exp x); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply H | apply Rinv_0_lt_compat; apply exp_pos ] ]. +elim (H0 _ H1); intros del H2. +exists del; intros. +assert (H5 := H2 _ H3 H4). +rewrite Rplus_0_l in H5; rewrite exp_0 in H5. +replace ((exp (x + h) - exp x) / h - exp x) with + (exp x * ((exp h - 1) / h - 1)). +rewrite Rabs_mult; rewrite (Rabs_right (exp x)). +apply Rmult_lt_reg_l with (/ exp x). +apply Rinv_0_lt_compat; apply exp_pos. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite <- (Rmult_comm eps). +apply H5. +assert (H6 := exp_pos x); red in |- *; intro; rewrite H7 in H6; + elim (Rlt_irrefl _ H6). +apply Rle_ge; left; apply exp_pos. +rewrite Rmult_minus_distr_l. +rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite <- Rmult_assoc; + rewrite Rmult_minus_distr_l. +rewrite Rmult_1_r; rewrite exp_plus; reflexivity. +Qed.
\ No newline at end of file diff --git a/theories/Reals/MVT.v b/theories/Reals/MVT.v index 330d53812..5eab01e5b 100644 --- a/theories/Reals/MVT.v +++ b/theories/Reals/MVT.v @@ -8,510 +8,692 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Ranalysis1. -Require Rtopology. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rfunctions. +Require Import Ranalysis1. +Require Import Rtopology. Open Local Scope R_scope. (* The Mean Value Theorem *) -Theorem MVT : (f,g:R->R;a,b:R;pr1:(c:R)``a<c<b``->(derivable_pt f c);pr2:(c:R)``a<c<b``->(derivable_pt g c)) ``a<b`` -> ((c:R)``a<=c<=b``->(continuity_pt f c)) -> ((c:R)``a<=c<=b``->(continuity_pt g c)) -> (EXT c : R | (EXT P : ``a<c<b`` | ``((g b)-(g a))*(derive_pt f c (pr1 c P))==((f b)-(f a))*(derive_pt g c (pr2 c P))``)). -Intros; Assert H2 := (Rlt_le ? ? H). -Pose h := [y:R]``((g b)-(g a))*(f y)-((f b)-(f a))*(g y)``. -Cut (c:R)``a<c<b``->(derivable_pt h c). -Intro; Cut ((c:R)``a<=c<=b``->(continuity_pt h c)). -Intro; Assert H4 := (continuity_ab_maj h a b H2 H3). -Assert H5 := (continuity_ab_min h a b H2 H3). -Elim H4; Intros Mx H6. -Elim H5; Intros mx H7. -Cut (h a)==(h b). -Intro; Pose M := (h Mx); Pose m := (h mx). -Cut (c:R;P:``a<c<b``) (derive_pt h c (X c P))==``((g b)-(g a))*(derive_pt f c (pr1 c P))-((f b)-(f a))*(derive_pt g c (pr2 c P))``. -Intro; Case (Req_EM (h a) M); Intro. -Case (Req_EM (h a) m); Intro. -Cut ((c:R)``a<=c<=b``->(h c)==M). -Intro; Cut ``a<(a+b)/2<b``. +Theorem MVT : + forall (f g:R -> R) (a b:R) (pr1:forall c:R, a < c < b -> derivable_pt f c) + (pr2:forall c:R, a < c < b -> derivable_pt g c), + a < b -> + (forall c:R, a <= c <= b -> continuity_pt f c) -> + (forall c:R, a <= c <= b -> continuity_pt g c) -> + exists c : R + | ( exists P : a < c < b + | (g b - g a) * derive_pt f c (pr1 c P) = + (f b - f a) * derive_pt g c (pr2 c P)). +intros; assert (H2 := Rlt_le _ _ H). +pose (h := fun y:R => (g b - g a) * f y - (f b - f a) * g y). +cut (forall c:R, a < c < b -> derivable_pt h c). +intro; cut (forall c:R, a <= c <= b -> continuity_pt h c). +intro; assert (H4 := continuity_ab_maj h a b H2 H3). +assert (H5 := continuity_ab_min h a b H2 H3). +elim H4; intros Mx H6. +elim H5; intros mx H7. +cut (h a = h b). +intro; pose (M := h Mx); pose (m := h mx). +cut + (forall (c:R) (P:a < c < b), + derive_pt h c (X c P) = + (g b - g a) * derive_pt f c (pr1 c P) - + (f b - f a) * derive_pt g c (pr2 c P)). +intro; case (Req_dec (h a) M); intro. +case (Req_dec (h a) m); intro. +cut (forall c:R, a <= c <= b -> h c = M). +intro; cut (a < (a + b) / 2 < b). (*** h constant ***) -Intro; Exists ``(a+b)/2``. -Exists H13. -Apply Rminus_eq; Rewrite <- H9; Apply deriv_constant2 with a b. -Elim H13; Intros; Assumption. -Elim H13; Intros; Assumption. -Intros; Rewrite (H12 ``(a+b)/2``). -Apply H12; Split; Left; Assumption. -Elim H13; Intros; Split; Left; Assumption. -Split. -Apply Rlt_monotony_contra with ``2``. -Sup0. -Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Apply H. -DiscrR. -Apply Rlt_monotony_contra with ``2``. -Sup0. -Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite Rplus_sym; Rewrite double; Apply Rlt_compatibility; Apply H. -DiscrR. -Intros; Elim H6; Intros H13 _. -Elim H7; Intros H14 _. -Apply Rle_antisym. -Apply H13; Apply H12. -Rewrite H10 in H11; Rewrite H11; Apply H14; Apply H12. -Cut ``a<mx<b``. +intro; exists ((a + b) / 2). +exists H13. +apply Rminus_diag_uniq; rewrite <- H9; apply deriv_constant2 with a b. +elim H13; intros; assumption. +elim H13; intros; assumption. +intros; rewrite (H12 ((a + b) / 2)). +apply H12; split; left; assumption. +elim H13; intros; split; left; assumption. +split. +apply Rmult_lt_reg_l with 2. +prove_sup0. +unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H. +discrR. +apply Rmult_lt_reg_l with 2. +prove_sup0. +unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite Rplus_comm; rewrite double; + apply Rplus_lt_compat_l; apply H. +discrR. +intros; elim H6; intros H13 _. +elim H7; intros H14 _. +apply Rle_antisym. +apply H13; apply H12. +rewrite H10 in H11; rewrite H11; apply H14; apply H12. +cut (a < mx < b). (*** h admet un minimum global sur [a,b] ***) -Intro; Exists mx. -Exists H12. -Apply Rminus_eq; Rewrite <- H9; Apply deriv_minimum with a b. -Elim H12; Intros; Assumption. -Elim H12; Intros; Assumption. -Intros; Elim H7; Intros. -Apply H15; Split; Left; Assumption. -Elim H7; Intros _ H12; Elim H12; Intros; Split. -Inversion H13. -Apply H15. -Rewrite H15 in H11; Elim H11; Reflexivity. -Inversion H14. -Apply H15. -Rewrite H8 in H11; Rewrite <- H15 in H11; Elim H11; Reflexivity. -Cut ``a<Mx<b``. +intro; exists mx. +exists H12. +apply Rminus_diag_uniq; rewrite <- H9; apply deriv_minimum with a b. +elim H12; intros; assumption. +elim H12; intros; assumption. +intros; elim H7; intros. +apply H15; split; left; assumption. +elim H7; intros _ H12; elim H12; intros; split. +inversion H13. +apply H15. +rewrite H15 in H11; elim H11; reflexivity. +inversion H14. +apply H15. +rewrite H8 in H11; rewrite <- H15 in H11; elim H11; reflexivity. +cut (a < Mx < b). (*** h admet un maximum global sur [a,b] ***) -Intro; Exists Mx. -Exists H11. -Apply Rminus_eq; Rewrite <- H9; Apply deriv_maximum with a b. -Elim H11; Intros; Assumption. -Elim H11; Intros; Assumption. -Intros; Elim H6; Intros; Apply H14. -Split; Left; Assumption. -Elim H6; Intros _ H11; Elim H11; Intros; Split. -Inversion H12. -Apply H14. -Rewrite H14 in H10; Elim H10; Reflexivity. -Inversion H13. -Apply H14. -Rewrite H8 in H10; Rewrite <- H14 in H10; Elim H10; Reflexivity. -Intros; Unfold h; Replace (derive_pt [y:R]``((g b)-(g a))*(f y)-((f b)-(f a))*(g y)`` c (X c P)) with (derive_pt (minus_fct (mult_fct (fct_cte ``(g b)-(g a)``) f) (mult_fct (fct_cte ``(f b)-(f a)``) g)) c (derivable_pt_minus ? ? ? (derivable_pt_mult ? ? ? (derivable_pt_const ``(g b)-(g a)`` c) (pr1 c P)) (derivable_pt_mult ? ? ? (derivable_pt_const ``(f b)-(f a)`` c) (pr2 c P)))); [Idtac | Apply pr_nu]. -Rewrite derive_pt_minus; Do 2 Rewrite derive_pt_mult; Do 2 Rewrite derive_pt_const; Do 2 Rewrite Rmult_Ol; Do 2 Rewrite Rplus_Ol; Reflexivity. -Unfold h; Ring. -Intros; Unfold h; Change (continuity_pt (minus_fct (mult_fct (fct_cte ``(g b)-(g a)``) f) (mult_fct (fct_cte ``(f b)-(f a)``) g)) c). -Apply continuity_pt_minus; Apply continuity_pt_mult. -Apply derivable_continuous_pt; Apply derivable_const. -Apply H0; Apply H3. -Apply derivable_continuous_pt; Apply derivable_const. -Apply H1; Apply H3. -Intros; Change (derivable_pt (minus_fct (mult_fct (fct_cte ``(g b)-(g a)``) f) (mult_fct (fct_cte ``(f b)-(f a)``) g)) c). -Apply derivable_pt_minus; Apply derivable_pt_mult. -Apply derivable_pt_const. -Apply (pr1 ? H3). -Apply derivable_pt_const. -Apply (pr2 ? H3). +intro; exists Mx. +exists H11. +apply Rminus_diag_uniq; rewrite <- H9; apply deriv_maximum with a b. +elim H11; intros; assumption. +elim H11; intros; assumption. +intros; elim H6; intros; apply H14. +split; left; assumption. +elim H6; intros _ H11; elim H11; intros; split. +inversion H12. +apply H14. +rewrite H14 in H10; elim H10; reflexivity. +inversion H13. +apply H14. +rewrite H8 in H10; rewrite <- H14 in H10; elim H10; reflexivity. +intros; unfold h in |- *; + replace + (derive_pt (fun y:R => (g b - g a) * f y - (f b - f a) * g y) c (X c P)) + with + (derive_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F) c + (derivable_pt_minus _ _ _ + (derivable_pt_mult _ _ _ (derivable_pt_const (g b - g a) c) (pr1 c P)) + (derivable_pt_mult _ _ _ (derivable_pt_const (f b - f a) c) (pr2 c P)))); + [ idtac | apply pr_nu ]. +rewrite derive_pt_minus; do 2 rewrite derive_pt_mult; + do 2 rewrite derive_pt_const; do 2 rewrite Rmult_0_l; + do 2 rewrite Rplus_0_l; reflexivity. +unfold h in |- *; ring. +intros; unfold h in |- *; + change + (continuity_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F) + c) in |- *. +apply continuity_pt_minus; apply continuity_pt_mult. +apply derivable_continuous_pt; apply derivable_const. +apply H0; apply H3. +apply derivable_continuous_pt; apply derivable_const. +apply H1; apply H3. +intros; + change + (derivable_pt ((fct_cte (g b - g a) * f)%F - (fct_cte (f b - f a) * g)%F) + c) in |- *. +apply derivable_pt_minus; apply derivable_pt_mult. +apply derivable_pt_const. +apply (pr1 _ H3). +apply derivable_pt_const. +apply (pr2 _ H3). Qed. (* Corollaries ... *) -Lemma MVT_cor1 : (f:(R->R); a,b:R; pr:(derivable f)) ``a < b``->(EXT c:R | ``(f b)-(f a) == (derive_pt f c (pr c))*(b-a)``/\``a < c < b``). -Intros f a b pr H; Cut (c:R)``a<c<b``->(derivable_pt f c); [Intro | Intros; Apply pr]. -Cut (c:R)``a<c<b``->(derivable_pt id c); [Intro | Intros; Apply derivable_pt_id]. -Cut ((c:R)``a<=c<=b``->(continuity_pt f c)); [Intro | Intros; Apply derivable_continuous_pt; Apply pr]. -Cut ((c:R)``a<=c<=b``->(continuity_pt id c)); [Intro | Intros; Apply derivable_continuous_pt; Apply derivable_id]. -Assert H2 := (MVT f id a b X X0 H H0 H1). -Elim H2; Intros c H3; Elim H3; Intros. -Exists c; Split. -Cut (derive_pt id c (X0 c x)) == (derive_pt id c (derivable_pt_id c)); [Intro | Apply pr_nu]. -Rewrite H5 in H4; Rewrite (derive_pt_id c) in H4; Rewrite Rmult_1r in H4; Rewrite <- H4; Replace (derive_pt f c (X c x)) with (derive_pt f c (pr c)); [Idtac | Apply pr_nu]; Apply Rmult_sym. -Apply x. +Lemma MVT_cor1 : + forall (f:R -> R) (a b:R) (pr:derivable f), + a < b -> + exists c : R | f b - f a = derive_pt f c (pr c) * (b - a) /\ a < c < b. +intros f a b pr H; cut (forall c:R, a < c < b -> derivable_pt f c); + [ intro | intros; apply pr ]. +cut (forall c:R, a < c < b -> derivable_pt id c); + [ intro | intros; apply derivable_pt_id ]. +cut (forall c:R, a <= c <= b -> continuity_pt f c); + [ intro | intros; apply derivable_continuous_pt; apply pr ]. +cut (forall c:R, a <= c <= b -> continuity_pt id c); + [ intro | intros; apply derivable_continuous_pt; apply derivable_id ]. +assert (H2 := MVT f id a b X X0 H H0 H1). +elim H2; intros c H3; elim H3; intros. +exists c; split. +cut (derive_pt id c (X0 c x) = derive_pt id c (derivable_pt_id c)); + [ intro | apply pr_nu ]. +rewrite H5 in H4; rewrite (derive_pt_id c) in H4; rewrite Rmult_1_r in H4; + rewrite <- H4; replace (derive_pt f c (X c x)) with (derive_pt f c (pr c)); + [ idtac | apply pr_nu ]; apply Rmult_comm. +apply x. Qed. -Theorem MVT_cor2 : (f,f':R->R;a,b:R) ``a<b`` -> ((c:R)``a<=c<=b``->(derivable_pt_lim f c (f' c))) -> (EXT c:R | ``(f b)-(f a)==(f' c)*(b-a)``/\``a<c<b``). -Intros f f' a b H H0; Cut ((c:R)``a<=c<=b``->(derivable_pt f c)). -Intro; Cut ((c:R)``a<c<b``->(derivable_pt f c)). -Intro; Cut ((c:R)``a<=c<=b``->(continuity_pt f c)). -Intro; Cut ((c:R)``a<=c<=b``->(derivable_pt id c)). -Intro; Cut ((c:R)``a<c<b``->(derivable_pt id c)). -Intro; Cut ((c:R)``a<=c<=b``->(continuity_pt id c)). -Intro; Elim (MVT f id a b X0 X2 H H1 H2); Intros; Elim H3; Clear H3; Intros; Exists x; Split. -Cut (derive_pt id x (X2 x x0))==R1. -Cut (derive_pt f x (X0 x x0))==(f' x). -Intros; Rewrite H4 in H3; Rewrite H5 in H3; Unfold id in H3; Rewrite Rmult_1r in H3; Rewrite Rmult_sym; Symmetry; Assumption. -Apply derive_pt_eq_0; Apply H0; Elim x0; Intros; Split; Left; Assumption. -Apply derive_pt_eq_0; Apply derivable_pt_lim_id. -Assumption. -Intros; Apply derivable_continuous_pt; Apply X1; Assumption. -Intros; Apply derivable_pt_id. -Intros; Apply derivable_pt_id. -Intros; Apply derivable_continuous_pt; Apply X; Assumption. -Intros; Elim H1; Intros; Apply X; Split; Left; Assumption. -Intros; Unfold derivable_pt; Apply Specif.existT with (f' c); Apply H0; Apply H1. +Theorem MVT_cor2 : + forall (f f':R -> R) (a b:R), + a < b -> + (forall c:R, a <= c <= b -> derivable_pt_lim f c (f' c)) -> + exists c : R | f b - f a = f' c * (b - a) /\ a < c < b. +intros f f' a b H H0; cut (forall c:R, a <= c <= b -> derivable_pt f c). +intro; cut (forall c:R, a < c < b -> derivable_pt f c). +intro; cut (forall c:R, a <= c <= b -> continuity_pt f c). +intro; cut (forall c:R, a <= c <= b -> derivable_pt id c). +intro; cut (forall c:R, a < c < b -> derivable_pt id c). +intro; cut (forall c:R, a <= c <= b -> continuity_pt id c). +intro; elim (MVT f id a b X0 X2 H H1 H2); intros; elim H3; clear H3; intros; + exists x; split. +cut (derive_pt id x (X2 x x0) = 1). +cut (derive_pt f x (X0 x x0) = f' x). +intros; rewrite H4 in H3; rewrite H5 in H3; unfold id in H3; + rewrite Rmult_1_r in H3; rewrite Rmult_comm; symmetry in |- *; + assumption. +apply derive_pt_eq_0; apply H0; elim x0; intros; split; left; assumption. +apply derive_pt_eq_0; apply derivable_pt_lim_id. +assumption. +intros; apply derivable_continuous_pt; apply X1; assumption. +intros; apply derivable_pt_id. +intros; apply derivable_pt_id. +intros; apply derivable_continuous_pt; apply X; assumption. +intros; elim H1; intros; apply X; split; left; assumption. +intros; unfold derivable_pt in |- *; apply existT with (f' c); apply H0; + apply H1. Qed. -Lemma MVT_cor3 : (f,f':(R->R); a,b:R) ``a < b`` -> ((x:R)``a <= x`` -> ``x <= b``->(derivable_pt_lim f x (f' x))) -> (EXT c:R | ``a<=c``/\``c<=b``/\``(f b)==(f a) + (f' c)*(b-a)``). -Intros f f' a b H H0; Assert H1 : (EXT c:R | ``(f b) -(f a) == (f' c)*(b-a)``/\``a<c<b``); [Apply MVT_cor2; [Apply H | Intros; Elim H1; Intros; Apply (H0 ? H2 H3)] | Elim H1; Intros; Exists x; Elim H2; Intros; Elim H4; Intros; Split; [Left; Assumption | Split; [Left; Assumption | Rewrite <- H3; Ring]]]. +Lemma MVT_cor3 : + forall (f f':R -> R) (a b:R), + a < b -> + (forall x:R, a <= x -> x <= b -> derivable_pt_lim f x (f' x)) -> + exists c : R | a <= c /\ c <= b /\ f b = f a + f' c * (b - a). +intros f f' a b H H0; + assert (H1 : exists c : R | f b - f a = f' c * (b - a) /\ a < c < b); + [ apply MVT_cor2; [ apply H | intros; elim H1; intros; apply (H0 _ H2 H3) ] + | elim H1; intros; exists x; elim H2; intros; elim H4; intros; split; + [ left; assumption | split; [ left; assumption | rewrite <- H3; ring ] ] ]. Qed. -Lemma Rolle : (f:R->R;a,b:R;pr:(x:R)``a<x<b``->(derivable_pt f x)) ((x:R)``a<=x<=b``->(continuity_pt f x)) -> ``a<b`` -> (f a)==(f b) -> (EXT c:R | (EXT P: ``a<c<b`` | ``(derive_pt f c (pr c P))==0``)). -Intros; Assert H2 : (x:R)``a<x<b``->(derivable_pt id x). -Intros; Apply derivable_pt_id. -Assert H3 := (MVT f id a b pr H2 H0 H); Assert H4 : (x:R)``a<=x<=b``->(continuity_pt id x). -Intros; Apply derivable_continuous; Apply derivable_id. -Elim (H3 H4); Intros; Elim H5; Intros; Exists x; Exists x0; Rewrite H1 in H6; Unfold id in H6; Unfold Rminus in H6; Rewrite Rplus_Ropp_r in H6; Rewrite Rmult_Ol in H6; Apply r_Rmult_mult with ``b-a``; [Rewrite Rmult_Or; Apply H6 | Apply Rminus_eq_contra; Red; Intro; Rewrite H7 in H0; Elim (Rlt_antirefl ? H0)]. +Lemma Rolle : + forall (f:R -> R) (a b:R) (pr:forall x:R, a < x < b -> derivable_pt f x), + (forall x:R, a <= x <= b -> continuity_pt f x) -> + a < b -> + f a = f b -> + exists c : R | ( exists P : a < c < b | derive_pt f c (pr c P) = 0). +intros; assert (H2 : forall x:R, a < x < b -> derivable_pt id x). +intros; apply derivable_pt_id. +assert (H3 := MVT f id a b pr H2 H0 H); + assert (H4 : forall x:R, a <= x <= b -> continuity_pt id x). +intros; apply derivable_continuous; apply derivable_id. +elim (H3 H4); intros; elim H5; intros; exists x; exists x0; rewrite H1 in H6; + unfold id in H6; unfold Rminus in H6; rewrite Rplus_opp_r in H6; + rewrite Rmult_0_l in H6; apply Rmult_eq_reg_l with (b - a); + [ rewrite Rmult_0_r; apply H6 + | apply Rminus_eq_contra; red in |- *; intro; rewrite H7 in H0; + elim (Rlt_irrefl _ H0) ]. Qed. (**********) -Lemma nonneg_derivative_1 : (f:R->R;pr:(derivable f)) ((x:R) ``0<=(derive_pt f x (pr x))``) -> (increasing f). -Intros. -Unfold increasing. -Intros. -Case (total_order_T x y); Intro. -Elim s; Intro. -Apply Rle_anti_compatibility with ``-(f x)``. -Rewrite Rplus_Ropp_l; Rewrite Rplus_sym. -Assert H1 := (MVT_cor1 f ? ? pr a). -Elim H1; Intros. -Elim H2; Intros. -Unfold Rminus in H3. -Rewrite H3. -Apply Rmult_le_pos. -Apply H. -Apply Rle_anti_compatibility with x. -Rewrite Rplus_Or; Replace ``x+(y+ -x)`` with y; [Assumption | Ring]. -Rewrite b; Right; Reflexivity. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 r)). +Lemma nonneg_derivative_1 : + forall (f:R -> R) (pr:derivable f), + (forall x:R, 0 <= derive_pt f x (pr x)) -> increasing f. +intros. +unfold increasing in |- *. +intros. +case (total_order_T x y); intro. +elim s; intro. +apply Rplus_le_reg_l with (- f x). +rewrite Rplus_opp_l; rewrite Rplus_comm. +assert (H1 := MVT_cor1 f _ _ pr a). +elim H1; intros. +elim H2; intros. +unfold Rminus in H3. +rewrite H3. +apply Rmult_le_pos. +apply H. +apply Rplus_le_reg_l with x. +rewrite Rplus_0_r; replace (x + (y + - x)) with y; [ assumption | ring ]. +rewrite b; right; reflexivity. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 r)). Qed. (**********) -Lemma nonpos_derivative_0 : (f:R->R;pr:(derivable f)) (decreasing f) -> ((x:R) ``(derive_pt f x (pr x))<=0``). -Intros f pr H x; Assert H0 :=H; Unfold decreasing in H0; Generalize (derivable_derive f x (pr 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_eq_1 f x l (pr x) 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; Sup0]. -Rewrite Rabsolu_right. -Unfold Rdiv; Apply Rlt_monotony_contra with ``2``. -Sup0. -Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite double; Pattern 1 (pos delta); Rewrite <- Rplus_Or. -Apply Rlt_compatibility; Apply (cond_pos delta). -DiscrR. -Apply Rle_sym1; Unfold Rdiv; Left; Apply Rmult_lt_pos. -Apply (cond_pos delta). -Apply Rlt_Rinv; Sup0. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply H4 | Apply Rlt_Rinv; Sup0]. +Lemma nonpos_derivative_0 : + forall (f:R -> R) (pr:derivable f), + decreasing f -> forall x:R, derive_pt f x (pr x) <= 0. +intros f pr H x; assert (H0 := H); unfold decreasing in H0; + generalize (derivable_derive f x (pr x)); intro; elim H1; + intros l H2. +rewrite H2; case (Rtotal_order l 0); intro. +left; assumption. +elim H3; intro. +right; assumption. +generalize (derive_pt_eq_1 f x l (pr x) H2); intros; cut (0 < l / 2). +intro; elim (H5 (l / 2) H6); intros delta H7; + cut (delta / 2 <> 0 /\ 0 < delta / 2 /\ Rabs (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 Rabs in |- *; + case (Rcase_abs ((f (x + delta / 2) - f x) / (delta / 2) - l)). +intros; + generalize + (Rplus_lt_compat_r (- l) (- ((f (x + delta / 2) - f x) / (delta / 2) - l)) + (l / 2) H14); unfold Rminus in |- *. +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 + (Ropp_lt_gt_contravar (- ((f (x + delta / 2) + - f x) / (delta / 2))) + (- (l / 2)) H15). +repeat rewrite Ropp_involutive. +intro. +generalize + (Rlt_trans 0 (l / 2) ((f (x + delta / 2) - f x) / (delta / 2)) H6 H16); + intro. +elim + (Rlt_irrefl 0 + (Rlt_le_trans 0 ((f (x + delta / 2) - f x) / (delta / 2)) 0 H17 H10)). +ring. +pattern l at 3 in |- *; rewrite double_var. +ring. +intros. +generalize + (Ropp_ge_le_contravar ((f (x + delta / 2) - f x) / (delta / 2) - l) 0 r). +rewrite Ropp_0. +intro. +elim + (Rlt_irrefl 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 in |- *. +apply Rplus_le_lt_0_compat. +unfold Rdiv in |- *; apply Rmult_le_pos. +cut (x <= x + delta * / 2). +intro; generalize (H0 x (x + delta * / 2) H13); intro; + generalize + (Rplus_le_compat_l (- f (x + delta / 2)) (f (x + delta / 2)) (f x) H14); + rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption. +pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + left; assumption. +left; apply Rinv_0_lt_compat; assumption. +assumption. +rewrite Ropp_minus_distr. +unfold Rminus in |- *. +rewrite (Rplus_comm l). +unfold Rdiv in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +rewrite Ropp_plus_distr. +rewrite Ropp_involutive. +rewrite (Rplus_comm (f x)). +reflexivity. +replace ((f (x + delta / 2) - f x) / (delta / 2)) with + (- ((f x - f (x + delta / 2)) / (delta / 2))). +rewrite <- Ropp_0. +apply Ropp_ge_le_contravar. +apply Rle_ge. +unfold Rdiv in |- *; apply Rmult_le_pos. +cut (x <= x + delta * / 2). +intro; generalize (H0 x (x + delta * / 2) H10); intro. +generalize + (Rplus_le_compat_l (- f (x + delta / 2)) (f (x + delta / 2)) (f x) H13); + rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption. +pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + left; assumption. +left; apply Rinv_0_lt_compat; assumption. +unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse. +rewrite Ropp_minus_distr. +reflexivity. +split. +unfold Rdiv in |- *; apply prod_neq_R0. +generalize (cond_pos delta); intro; red in |- *; intro H9; rewrite H9 in H8; + elim (Rlt_irrefl 0 H8). +apply Rinv_neq_0_compat; discrR. +split. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. +rewrite Rabs_right. +unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite double; pattern (pos delta) at 1 in |- *; + rewrite <- Rplus_0_r. +apply Rplus_lt_compat_l; apply (cond_pos delta). +discrR. +apply Rle_ge; unfold Rdiv in |- *; left; apply Rmult_lt_0_compat. +apply (cond_pos delta). +apply Rinv_0_lt_compat; prove_sup0. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply H4 | apply Rinv_0_lt_compat; prove_sup0 ]. 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. +Lemma increasing_decreasing_opp : + forall f:R -> R, increasing f -> decreasing (- f)%F. +unfold increasing, decreasing, opp_fct in |- *; intros; generalize (H x y H0); + intro; apply Ropp_ge_le_contravar; apply Rle_ge; assumption. Qed. (**********) -Lemma nonpos_derivative_1 : (f:R->R;pr:(derivable f)) ((x:R) ``(derive_pt f x (pr x))<=0``) -> (decreasing f). -Intros. -Cut (h:R)``-(-(f h))==(f h)``. -Intro. -Generalize (increasing_decreasing_opp (opp_fct f)). -Unfold decreasing. -Unfold opp_fct. -Intros. -Rewrite <- (H0 x); Rewrite <- (H0 y). -Apply H1. -Cut (x:R)``0<=(derive_pt (opp_fct f) x ((derivable_opp f pr) x))``. -Intros. -Replace [x:R]``-(f x)`` with (opp_fct f); [Idtac | Reflexivity]. -Apply (nonneg_derivative_1 (opp_fct f) (derivable_opp f pr) H3). -Intro. -Assert H3 := (derive_pt_opp f x0 (pr x0)). -Cut ``(derive_pt (opp_fct f) x0 (derivable_pt_opp f x0 (pr x0)))==(derive_pt (opp_fct f) x0 (derivable_opp f pr x0))``. -Intro. -Rewrite <- H4. -Rewrite H3. -Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Apply (H x0). -Apply pr_nu. -Assumption. -Intro; Ring. +Lemma nonpos_derivative_1 : + forall (f:R -> R) (pr:derivable f), + (forall x:R, derive_pt f x (pr x) <= 0) -> decreasing f. +intros. +cut (forall h:R, - - f h = f h). +intro. +generalize (increasing_decreasing_opp (- f)%F). +unfold decreasing in |- *. +unfold opp_fct in |- *. +intros. +rewrite <- (H0 x); rewrite <- (H0 y). +apply H1. +cut (forall x:R, 0 <= derive_pt (- f) x (derivable_opp f pr x)). +intros. +replace (fun x:R => - f x) with (- f)%F; [ idtac | reflexivity ]. +apply (nonneg_derivative_1 (- f)%F (derivable_opp f pr) H3). +intro. +assert (H3 := derive_pt_opp f x0 (pr x0)). +cut + (derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) = + derive_pt (- f) x0 (derivable_opp f pr x0)). +intro. +rewrite <- H4. +rewrite H3. +rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; apply (H x0). +apply pr_nu. +assumption. +intro; ring. Qed. (**********) -Lemma positive_derivative : (f:R->R;pr:(derivable f)) ((x:R) ``0<(derive_pt f x (pr x))``)->(strict_increasing f). -Intros. -Unfold strict_increasing. -Intros. -Apply Rlt_anti_compatibility with ``-(f x)``. -Rewrite Rplus_Ropp_l; Rewrite Rplus_sym. -Assert H1 := (MVT_cor1 f ? ? pr H0). -Elim H1; Intros. -Elim H2; Intros. -Unfold Rminus in H3. -Rewrite H3. -Apply Rmult_lt_pos. -Apply H. -Apply Rlt_anti_compatibility with x. -Rewrite Rplus_Or; Replace ``x+(y+ -x)`` with y; [Assumption | Ring]. +Lemma positive_derivative : + forall (f:R -> R) (pr:derivable f), + (forall x:R, 0 < derive_pt f x (pr x)) -> strict_increasing f. +intros. +unfold strict_increasing in |- *. +intros. +apply Rplus_lt_reg_r with (- f x). +rewrite Rplus_opp_l; rewrite Rplus_comm. +assert (H1 := MVT_cor1 f _ _ pr H0). +elim H1; intros. +elim H2; intros. +unfold Rminus in H3. +rewrite H3. +apply Rmult_lt_0_compat. +apply H. +apply Rplus_lt_reg_r with x. +rewrite Rplus_0_r; replace (x + (y + - x)) with y; [ assumption | ring ]. Qed. (**********) -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. +Lemma strictincreasing_strictdecreasing_opp : + forall f:R -> R, strict_increasing f -> strict_decreasing (- f)%F. +unfold strict_increasing, strict_decreasing, opp_fct in |- *; intros; + generalize (H x y H0); intro; apply Ropp_lt_gt_contravar; + assumption. Qed. (**********) -Lemma negative_derivative : (f:R->R;pr:(derivable f)) ((x:R) ``(derive_pt f x (pr x))<0``)->(strict_decreasing f). -Intros. -Cut (h:R)``- (-(f h))==(f h)``. -Intros. -Generalize (strictincreasing_strictdecreasing_opp (opp_fct f)). -Unfold strict_decreasing opp_fct. -Intros. -Rewrite <- (H0 x). -Rewrite <- (H0 y). -Apply H1; [Idtac | Assumption]. -Cut (x:R)``0<(derive_pt (opp_fct f) x (derivable_opp f pr x))``. -Intros; EApply positive_derivative; Apply H3. -Intro. -Assert H3 := (derive_pt_opp f x0 (pr x0)). -Cut ``(derive_pt (opp_fct f) x0 (derivable_pt_opp f x0 (pr x0)))==(derive_pt (opp_fct f) x0 (derivable_opp f pr x0))``. -Intro. -Rewrite <- H4; Rewrite H3. -Rewrite <- Ropp_O; Apply Rlt_Ropp; Apply (H x0). -Apply pr_nu. -Intro; Ring. +Lemma negative_derivative : + forall (f:R -> R) (pr:derivable f), + (forall x:R, derive_pt f x (pr x) < 0) -> strict_decreasing f. +intros. +cut (forall h:R, - - f h = f h). +intros. +generalize (strictincreasing_strictdecreasing_opp (- f)%F). +unfold strict_decreasing, opp_fct in |- *. +intros. +rewrite <- (H0 x). +rewrite <- (H0 y). +apply H1; [ idtac | assumption ]. +cut (forall x:R, 0 < derive_pt (- f) x (derivable_opp f pr x)). +intros; eapply positive_derivative; apply H3. +intro. +assert (H3 := derive_pt_opp f x0 (pr x0)). +cut + (derive_pt (- f) x0 (derivable_pt_opp f x0 (pr x0)) = + derive_pt (- f) x0 (derivable_opp f pr x0)). +intro. +rewrite <- H4; rewrite H3. +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; apply (H x0). +apply pr_nu. +intro; ring. Qed. (**********) -Lemma null_derivative_0 : (f:R->R;pr:(derivable f)) (constant f)->((x:R) ``(derive_pt f x (pr x))==0``). -Intros. -Unfold constant in H. -Apply derive_pt_eq_0. -Intros; Exists (mkposreal ``1`` Rlt_R0_R1); Simpl; 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. +Lemma null_derivative_0 : + forall (f:R -> R) (pr:derivable f), + constant f -> forall x:R, derive_pt f x (pr x) = 0. +intros. +unfold constant in H. +apply derive_pt_eq_0. +intros; exists (mkposreal 1 Rlt_0_1); simpl in |- *; intros. +rewrite (H x (x + h)); unfold Rminus in |- *; unfold Rdiv in |- *; + rewrite Rplus_opp_r; rewrite Rmult_0_l; rewrite Rplus_opp_r; + rewrite Rabs_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)). +Lemma increasing_decreasing : + forall f:R -> R, increasing f -> decreasing f -> constant f. +unfold increasing, decreasing, constant in |- *; intros; + case (Rtotal_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 in |- *; + apply (Rle_antisym (f y) (f x) (H y x H3) (H0 y x H3)). Qed. (**********) -Lemma null_derivative_1 : (f:R->R;pr:(derivable f)) ((x:R) ``(derive_pt f x (pr x))==0``)->(constant f). -Intros. -Cut (x:R)``(derive_pt f x (pr x)) <= 0``. -Cut (x:R)``0 <= (derive_pt f x (pr x))``. -Intros. -Assert H2 := (nonneg_derivative_1 f pr H0). -Assert H3 := (nonpos_derivative_1 f pr H1). -Apply increasing_decreasing; Assumption. -Intro; Right; Symmetry; Apply (H x). -Intro; Right; Apply (H x). +Lemma null_derivative_1 : + forall (f:R -> R) (pr:derivable f), + (forall x:R, derive_pt f x (pr x) = 0) -> constant f. +intros. +cut (forall x:R, derive_pt f x (pr x) <= 0). +cut (forall x:R, 0 <= derive_pt f x (pr x)). +intros. +assert (H2 := nonneg_derivative_1 f pr H0). +assert (H3 := nonpos_derivative_1 f pr H1). +apply increasing_decreasing; assumption. +intro; right; symmetry in |- *; apply (H x). +intro; right; apply (H x). Qed. (**********) -Lemma derive_increasing_interv_ax : (a,b:R;f:R->R;pr:(derivable f)) ``a<b``-> (((t:R) ``a<t<b`` -> ``0<(derive_pt f t (pr 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 (pr t))``) -> ((x,y:R) ``a<=x<=b``->``a<=y<=b``->``x<y``->``(f x)<=(f y)``)). -Intros. -Split; Intros. -Apply Rlt_anti_compatibility with ``-(f x)``. -Rewrite Rplus_Ropp_l; Rewrite Rplus_sym. -Assert H4 := (MVT_cor1 f ? ? pr H3). -Elim H4; Intros. -Elim H5; Intros. -Unfold Rminus in H6. -Rewrite H6. -Apply Rmult_lt_pos. -Apply H0. -Elim H7; Intros. -Split. -Elim H1; Intros. -Apply Rle_lt_trans with x; Assumption. -Elim H2; Intros. -Apply Rlt_le_trans with y; Assumption. -Apply Rlt_anti_compatibility with x. -Rewrite Rplus_Or; Replace ``x+(y+ -x)`` with y; [Assumption | Ring]. -Apply Rle_anti_compatibility with ``-(f x)``. -Rewrite Rplus_Ropp_l; Rewrite Rplus_sym. -Assert H4 := (MVT_cor1 f ? ? pr H3). -Elim H4; Intros. -Elim H5; Intros. -Unfold Rminus in H6. -Rewrite H6. -Apply Rmult_le_pos. -Apply H0. -Elim H7; Intros. -Split. -Elim H1; Intros. -Apply Rle_lt_trans with x; Assumption. -Elim H2; Intros. -Apply Rlt_le_trans with y; Assumption. -Apply Rle_anti_compatibility with x. -Rewrite Rplus_Or; Replace ``x+(y+ -x)`` with y; [Left; Assumption | Ring]. +Lemma derive_increasing_interv_ax : + forall (a b:R) (f:R -> R) (pr:derivable f), + a < b -> + ((forall t:R, a < t < b -> 0 < derive_pt f t (pr t)) -> + forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x < f y) /\ + ((forall t:R, a < t < b -> 0 <= derive_pt f t (pr t)) -> + forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x <= f y). +intros. +split; intros. +apply Rplus_lt_reg_r with (- f x). +rewrite Rplus_opp_l; rewrite Rplus_comm. +assert (H4 := MVT_cor1 f _ _ pr H3). +elim H4; intros. +elim H5; intros. +unfold Rminus in H6. +rewrite H6. +apply Rmult_lt_0_compat. +apply H0. +elim H7; intros. +split. +elim H1; intros. +apply Rle_lt_trans with x; assumption. +elim H2; intros. +apply Rlt_le_trans with y; assumption. +apply Rplus_lt_reg_r with x. +rewrite Rplus_0_r; replace (x + (y + - x)) with y; [ assumption | ring ]. +apply Rplus_le_reg_l with (- f x). +rewrite Rplus_opp_l; rewrite Rplus_comm. +assert (H4 := MVT_cor1 f _ _ pr H3). +elim H4; intros. +elim H5; intros. +unfold Rminus in H6. +rewrite H6. +apply Rmult_le_pos. +apply H0. +elim H7; intros. +split. +elim H1; intros. +apply Rle_lt_trans with x; assumption. +elim H2; intros. +apply Rlt_le_trans with y; assumption. +apply Rplus_le_reg_l with x. +rewrite Rplus_0_r; replace (x + (y + - x)) with y; + [ left; assumption | ring ]. Qed. (**********) -Lemma derive_increasing_interv : (a,b:R;f:R->R;pr:(derivable f)) ``a<b``-> ((t:R) ``a<t<b`` -> ``0<(derive_pt f t (pr 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 pr H); Intro. -Elim H4; Intros H5 _; Apply (H5 H0 x y H1 H2 H3). +Lemma derive_increasing_interv : + forall (a b:R) (f:R -> R) (pr:derivable f), + a < b -> + (forall t:R, a < t < b -> 0 < derive_pt f t (pr t)) -> + forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x < f y. +intros. +generalize (derive_increasing_interv_ax a b f pr 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;pr:(derivable f)) ``a<b``-> ((t:R) ``a<t<b`` -> ``0<=(derive_pt f t (pr t))``) -> ((x,y:R) ``a<=x<=b``->``a<=y<=b``->``x<y``->``(f x)<=(f y)``). -Intros a b f pr H H0 x y H1 H2 H3; Generalize (derive_increasing_interv_ax a b f pr H); Intro; Elim H4; Intros _ H5; Apply (H5 H0 x y H1 H2 H3). +Lemma derive_increasing_interv_var : + forall (a b:R) (f:R -> R) (pr:derivable f), + a < b -> + (forall t:R, a < t < b -> 0 <= derive_pt f t (pr t)) -> + forall x y:R, a <= x <= b -> a <= y <= b -> x < y -> f x <= f y. +intros a b f pr H H0 x y H1 H2 H3; + generalize (derive_increasing_interv_ax a b f pr H); + intro; elim H4; intros _ H5; apply (H5 H0 x y H1 H2 H3). Qed. (**********) (**********) -Theorem IAF : (f:R->R;a,b,k:R;pr:(derivable f)) ``a<=b`` -> ((c:R) ``a<=c<=b`` -> ``(derive_pt f c (pr c))<=k``) -> ``(f b)-(f a)<=k*(b-a)``. -Intros. -Case (total_order_T a b); Intro. -Elim s; Intro. -Assert H1 := (MVT_cor1 f ? ? pr a0). -Elim H1; Intros. -Elim H2; Intros. -Rewrite H3. -Do 2 Rewrite <- (Rmult_sym ``(b-a)``). -Apply Rle_monotony. -Apply Rle_anti_compatibility with ``a``; Rewrite Rplus_Or. -Replace ``a+(b-a)`` with b; [Assumption | Ring]. -Apply H0. -Elim H4; Intros. -Split; Left; Assumption. -Rewrite b0. -Unfold Rminus; Do 2 Rewrite Rplus_Ropp_r. -Rewrite Rmult_Or; Right; Reflexivity. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Theorem IAF : + forall (f:R -> R) (a b k:R) (pr:derivable f), + a <= b -> + (forall c:R, a <= c <= b -> derive_pt f c (pr c) <= k) -> + f b - f a <= k * (b - a). +intros. +case (total_order_T a b); intro. +elim s; intro. +assert (H1 := MVT_cor1 f _ _ pr a0). +elim H1; intros. +elim H2; intros. +rewrite H3. +do 2 rewrite <- (Rmult_comm (b - a)). +apply Rmult_le_compat_l. +apply Rplus_le_reg_l with a; rewrite Rplus_0_r. +replace (a + (b - a)) with b; [ assumption | ring ]. +apply H0. +elim H4; intros. +split; left; assumption. +rewrite b0. +unfold Rminus in |- *; do 2 rewrite Rplus_opp_r. +rewrite Rmult_0_r; right; reflexivity. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). Qed. -Lemma IAF_var : (f,g:R->R;a,b:R;pr1:(derivable f);pr2:(derivable g)) ``a<=b`` -> ((c:R) ``a<=c<=b`` -> ``(derive_pt g c (pr2 c))<=(derive_pt f c (pr1 c))``) -> ``(g b)-(g a)<=(f b)-(f a)``. -Intros. -Cut (derivable (minus_fct g f)). -Intro. -Cut (c:R)``a<=c<=b``->``(derive_pt (minus_fct g f) c (X c))<=0``. -Intro. -Assert H2 := (IAF (minus_fct g f) a b R0 X H H1). -Rewrite Rmult_Ol in H2; Unfold minus_fct in H2. -Apply Rle_anti_compatibility with ``-(f b)+(f a)``. -Replace ``-(f b)+(f a)+((f b)-(f a))`` with R0; [Idtac | Ring]. -Replace ``-(f b)+(f a)+((g b)-(g a))`` with ``(g b)-(f b)-((g a)-(f a))``; [Apply H2 | Ring]. -Intros. -Cut (derive_pt (minus_fct g f) c (X c))==(derive_pt (minus_fct g f) c (derivable_pt_minus ? ? ? (pr2 c) (pr1 c))). -Intro. -Rewrite H2. -Rewrite derive_pt_minus. -Apply Rle_anti_compatibility with (derive_pt f c (pr1 c)). -Rewrite Rplus_Or. -Replace ``(derive_pt f c (pr1 c))+((derive_pt g c (pr2 c))-(derive_pt f c (pr1 c)))`` with ``(derive_pt g c (pr2 c))``; [Idtac | Ring]. -Apply H0; Assumption. -Apply pr_nu. -Apply derivable_minus; Assumption. +Lemma IAF_var : + forall (f g:R -> R) (a b:R) (pr1:derivable f) (pr2:derivable g), + a <= b -> + (forall c:R, a <= c <= b -> derive_pt g c (pr2 c) <= derive_pt f c (pr1 c)) -> + g b - g a <= f b - f a. +intros. +cut (derivable (g - f)). +intro. +cut (forall c:R, a <= c <= b -> derive_pt (g - f) c (X c) <= 0). +intro. +assert (H2 := IAF (g - f)%F a b 0 X H H1). +rewrite Rmult_0_l in H2; unfold minus_fct in H2. +apply Rplus_le_reg_l with (- f b + f a). +replace (- f b + f a + (f b - f a)) with 0; [ idtac | ring ]. +replace (- f b + f a + (g b - g a)) with (g b - f b - (g a - f a)); + [ apply H2 | ring ]. +intros. +cut + (derive_pt (g - f) c (X c) = + derive_pt (g - f) c (derivable_pt_minus _ _ _ (pr2 c) (pr1 c))). +intro. +rewrite H2. +rewrite derive_pt_minus. +apply Rplus_le_reg_l with (derive_pt f c (pr1 c)). +rewrite Rplus_0_r. +replace + (derive_pt f c (pr1 c) + (derive_pt g c (pr2 c) - derive_pt f c (pr1 c))) + with (derive_pt g c (pr2 c)); [ idtac | ring ]. +apply H0; assumption. +apply pr_nu. +apply derivable_minus; assumption. Qed. (* If f has a null derivative in ]a,b[ and is continue in [a,b], *) (* then f is constant on [a,b] *) -Lemma null_derivative_loc : (f:R->R;a,b:R;pr:(x:R)``a<x<b``->(derivable_pt f x)) ((x:R)``a<=x<=b``->(continuity_pt f x)) -> ((x:R;P:``a<x<b``)(derive_pt f x (pr x P))==R0) -> (constant_D_eq f [x:R]``a<=x<=b`` (f a)). -Intros; Unfold constant_D_eq; Intros; Case (total_order_T a b); Intro. -Elim s; Intro. -Assert H2 : (y:R)``a<y<x``->(derivable_pt id y). -Intros; Apply derivable_pt_id. -Assert H3 : (y:R)``a<=y<=x``->(continuity_pt id y). -Intros; Apply derivable_continuous; Apply derivable_id. -Assert H4 : (y:R)``a<y<x``->(derivable_pt f y). -Intros; Apply pr; Elim H4; Intros; Split. -Assumption. -Elim H1; Intros; Apply Rlt_le_trans with x; Assumption. -Assert H5 : (y:R)``a<=y<=x``->(continuity_pt f y). -Intros; Apply H; Elim H5; Intros; Split. -Assumption. -Elim H1; Intros; Apply Rle_trans with x; Assumption. -Elim H1; Clear H1; Intros; Elim H1; Clear H1; Intro. -Assert H7 := (MVT f id a x H4 H2 H1 H5 H3). -Elim H7; Intros; Elim H8; Intros; Assert H10 : ``a<x0<b``. -Elim x1; Intros; Split. -Assumption. -Apply Rlt_le_trans with x; Assumption. -Assert H11 : ``(derive_pt f x0 (H4 x0 x1))==0``. -Replace (derive_pt f x0 (H4 x0 x1)) with (derive_pt f x0 (pr x0 H10)); [Apply H0 | Apply pr_nu]. -Assert H12 : ``(derive_pt id x0 (H2 x0 x1))==1``. -Apply derive_pt_eq_0; Apply derivable_pt_lim_id. -Rewrite H11 in H9; Rewrite H12 in H9; Rewrite Rmult_Or in H9; Rewrite Rmult_1r in H9; Apply Rminus_eq; Symmetry; Assumption. -Rewrite H1; Reflexivity. -Assert H2 : x==a. -Rewrite <- b0 in H1; Elim H1; Intros; Apply Rle_antisym; Assumption. -Rewrite H2; Reflexivity. -Elim H1; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? (Rle_trans ? ? ? H2 H3) r)). +Lemma null_derivative_loc : + forall (f:R -> R) (a b:R) (pr:forall x:R, a < x < b -> derivable_pt f x), + (forall x:R, a <= x <= b -> continuity_pt f x) -> + (forall (x:R) (P:a < x < b), derive_pt f x (pr x P) = 0) -> + constant_D_eq f (fun x:R => a <= x <= b) (f a). +intros; unfold constant_D_eq in |- *; intros; case (total_order_T a b); intro. +elim s; intro. +assert (H2 : forall y:R, a < y < x -> derivable_pt id y). +intros; apply derivable_pt_id. +assert (H3 : forall y:R, a <= y <= x -> continuity_pt id y). +intros; apply derivable_continuous; apply derivable_id. +assert (H4 : forall y:R, a < y < x -> derivable_pt f y). +intros; apply pr; elim H4; intros; split. +assumption. +elim H1; intros; apply Rlt_le_trans with x; assumption. +assert (H5 : forall y:R, a <= y <= x -> continuity_pt f y). +intros; apply H; elim H5; intros; split. +assumption. +elim H1; intros; apply Rle_trans with x; assumption. +elim H1; clear H1; intros; elim H1; clear H1; intro. +assert (H7 := MVT f id a x H4 H2 H1 H5 H3). +elim H7; intros; elim H8; intros; assert (H10 : a < x0 < b). +elim x1; intros; split. +assumption. +apply Rlt_le_trans with x; assumption. +assert (H11 : derive_pt f x0 (H4 x0 x1) = 0). +replace (derive_pt f x0 (H4 x0 x1)) with (derive_pt f x0 (pr x0 H10)); + [ apply H0 | apply pr_nu ]. +assert (H12 : derive_pt id x0 (H2 x0 x1) = 1). +apply derive_pt_eq_0; apply derivable_pt_lim_id. +rewrite H11 in H9; rewrite H12 in H9; rewrite Rmult_0_r in H9; + rewrite Rmult_1_r in H9; apply Rminus_diag_uniq; symmetry in |- *; + assumption. +rewrite H1; reflexivity. +assert (H2 : x = a). +rewrite <- b0 in H1; elim H1; intros; apply Rle_antisym; assumption. +rewrite H2; reflexivity. +elim H1; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (Rle_trans _ _ _ H2 H3) r)). Qed. (* Unicity of the antiderivative *) -Lemma antiderivative_Ucte : (f,g1,g2:R->R;a,b:R) (antiderivative f g1 a b) -> (antiderivative f g2 a b) -> (EXT c:R | (x:R)``a<=x<=b``->``(g1 x)==(g2 x)+c``). -Unfold antiderivative; Intros; Elim H; Clear H; Intros; Elim H0; Clear H0; Intros H0 _; Exists ``(g1 a)-(g2 a)``; Intros; Assert H3 : (x:R)``a<=x<=b``->(derivable_pt g1 x). -Intros; Unfold derivable_pt; Apply Specif.existT with (f x0); Elim (H x0 H3); Intros; EApply derive_pt_eq_1; Symmetry; Apply H4. -Assert H4 : (x:R)``a<=x<=b``->(derivable_pt g2 x). -Intros; Unfold derivable_pt; Apply Specif.existT with (f x0); Elim (H0 x0 H4); Intros; EApply derive_pt_eq_1; Symmetry; Apply H5. -Assert H5 : (x:R)``a<x<b``->(derivable_pt (minus_fct g1 g2) x). -Intros; Elim H5; Intros; Apply derivable_pt_minus; [Apply H3; Split; Left; Assumption | Apply H4; Split; Left; Assumption]. -Assert H6 : (x:R)``a<=x<=b``->(continuity_pt (minus_fct g1 g2) x). -Intros; Apply derivable_continuous_pt; Apply derivable_pt_minus; [Apply H3 | Apply H4]; Assumption. -Assert H7 : (x:R;P:``a<x<b``)(derive_pt (minus_fct g1 g2) x (H5 x P))==``0``. -Intros; Elim P; Intros; Apply derive_pt_eq_0; Replace R0 with ``(f x0)-(f x0)``; [Idtac | Ring]. -Assert H9 : ``a<=x0<=b``. -Split; Left; Assumption. -Apply derivable_pt_lim_minus; [Elim (H ? H9) | Elim (H0 ? H9)]; Intros; EApply derive_pt_eq_1; Symmetry; Apply H10. -Assert H8 := (null_derivative_loc (minus_fct g1 g2) a b H5 H6 H7); Unfold constant_D_eq in H8; Assert H9 := (H8 ? H2); Unfold minus_fct in H9; Rewrite <- H9; Ring. -Qed. +Lemma antiderivative_Ucte : + forall (f g1 g2:R -> R) (a b:R), + antiderivative f g1 a b -> + antiderivative f g2 a b -> + exists c : R | (forall x:R, a <= x <= b -> g1 x = g2 x + c). +unfold antiderivative in |- *; intros; elim H; clear H; intros; elim H0; + clear H0; intros H0 _; exists (g1 a - g2 a); intros; + assert (H3 : forall x:R, a <= x <= b -> derivable_pt g1 x). +intros; unfold derivable_pt in |- *; apply existT with (f x0); elim (H x0 H3); + intros; eapply derive_pt_eq_1; symmetry in |- *; + apply H4. +assert (H4 : forall x:R, a <= x <= b -> derivable_pt g2 x). +intros; unfold derivable_pt in |- *; apply existT with (f x0); + elim (H0 x0 H4); intros; eapply derive_pt_eq_1; symmetry in |- *; + apply H5. +assert (H5 : forall x:R, a < x < b -> derivable_pt (g1 - g2) x). +intros; elim H5; intros; apply derivable_pt_minus; + [ apply H3; split; left; assumption | apply H4; split; left; assumption ]. +assert (H6 : forall x:R, a <= x <= b -> continuity_pt (g1 - g2) x). +intros; apply derivable_continuous_pt; apply derivable_pt_minus; + [ apply H3 | apply H4 ]; assumption. +assert (H7 : forall (x:R) (P:a < x < b), derive_pt (g1 - g2) x (H5 x P) = 0). +intros; elim P; intros; apply derive_pt_eq_0; replace 0 with (f x0 - f x0); + [ idtac | ring ]. +assert (H9 : a <= x0 <= b). +split; left; assumption. +apply derivable_pt_lim_minus; [ elim (H _ H9) | elim (H0 _ H9) ]; intros; + eapply derive_pt_eq_1; symmetry in |- *; apply H10. +assert (H8 := null_derivative_loc (g1 - g2)%F a b H5 H6 H7); + unfold constant_D_eq in H8; assert (H9 := H8 _ H2); + unfold minus_fct in H9; rewrite <- H9; ring. +Qed.
\ No newline at end of file diff --git a/theories/Reals/NewtonInt.v b/theories/Reals/NewtonInt.v index 961f8bf0a..e2080827b 100644 --- a/theories/Reals/NewtonInt.v +++ b/theories/Reals/NewtonInt.v @@ -8,593 +8,781 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo. -Require Ranalysis. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import Ranalysis. Open Local Scope R_scope. (*******************************************) (* Newton's Integral *) (*******************************************) -Definition Newton_integrable [f:R->R;a,b:R] : Type := (sigTT ? [g:R->R](antiderivative f g a b)\/(antiderivative f g b a)). +Definition Newton_integrable (f:R -> R) (a b:R) : Type := + sigT (fun g:R -> R => antiderivative f g a b \/ antiderivative f g b a). -Definition NewtonInt [f:R->R;a,b:R;pr:(Newton_integrable f a b)] : R := let g = Cases pr of (existTT a b) => a end in ``(g b)-(g a)``. +Definition NewtonInt (f:R -> R) (a b:R) (pr:Newton_integrable f a b) : R := + let g := match pr with + | existT a b => a + end in g b - g a. (* If f is differentiable, then f' is Newton integrable (Tautology ?) *) -Lemma FTCN_step1 : (f:Differential;a,b:R) (Newton_integrable [x:R](derive_pt f x (cond_diff f x)) a b). -Intros f a b; Unfold Newton_integrable; Apply existTT with (d1 f); Unfold antiderivative; Intros; Case (total_order_Rle a b); Intro; [Left; Split; [Intros; Exists (cond_diff f x); Reflexivity | Assumption] | Right; Split; [Intros; Exists (cond_diff f x); Reflexivity | Auto with real]]. +Lemma FTCN_step1 : + forall (f:Differential) (a b:R), + Newton_integrable (fun x:R => derive_pt f x (cond_diff f x)) a b. +intros f a b; unfold Newton_integrable in |- *; apply existT with (d1 f); + unfold antiderivative in |- *; intros; case (Rle_dec a b); + intro; + [ left; split; [ intros; exists (cond_diff f x); reflexivity | assumption ] + | right; split; + [ intros; exists (cond_diff f x); reflexivity | auto with real ] ]. Defined. (* By definition, we have the Fondamental Theorem of Calculus *) -Lemma FTC_Newton : (f:Differential;a,b:R) (NewtonInt [x:R](derive_pt f x (cond_diff f x)) a b (FTCN_step1 f a b))==``(f b)-(f a)``. -Intros; Unfold NewtonInt; Reflexivity. +Lemma FTC_Newton : + forall (f:Differential) (a b:R), + NewtonInt (fun x:R => derive_pt f x (cond_diff f x)) a b + (FTCN_step1 f a b) = f b - f a. +intros; unfold NewtonInt in |- *; reflexivity. Qed. (* $\int_a^a f$ exists forall a:R and f:R->R *) -Lemma NewtonInt_P1 : (f:R->R;a:R) (Newton_integrable f a a). -Intros f a; Unfold Newton_integrable; Apply existTT with (mult_fct (fct_cte (f a)) id); Left; Unfold antiderivative; Split. -Intros; Assert H1 : (derivable_pt (mult_fct (fct_cte (f a)) id) x). -Apply derivable_pt_mult. -Apply derivable_pt_const. -Apply derivable_pt_id. -Exists H1; Assert H2 : x==a. -Elim H; Intros; Apply Rle_antisym; Assumption. -Symmetry; Apply derive_pt_eq_0; Replace (f x) with ``0*(id x)+(fct_cte (f a) x)*1``; [Apply (derivable_pt_lim_mult (fct_cte (f a)) id x); [Apply derivable_pt_lim_const | Apply derivable_pt_lim_id] | Unfold id fct_cte; Rewrite H2; Ring]. -Right; Reflexivity. +Lemma NewtonInt_P1 : forall (f:R -> R) (a:R), Newton_integrable f a a. +intros f a; unfold Newton_integrable in |- *; + apply existT with (fct_cte (f a) * id)%F; left; + unfold antiderivative in |- *; split. +intros; assert (H1 : derivable_pt (fct_cte (f a) * id) x). +apply derivable_pt_mult. +apply derivable_pt_const. +apply derivable_pt_id. +exists H1; assert (H2 : x = a). +elim H; intros; apply Rle_antisym; assumption. +symmetry in |- *; apply derive_pt_eq_0; + replace (f x) with (0 * id x + fct_cte (f a) x * 1); + [ apply (derivable_pt_lim_mult (fct_cte (f a)) id x); + [ apply derivable_pt_lim_const | apply derivable_pt_lim_id ] + | unfold id, fct_cte in |- *; rewrite H2; ring ]. +right; reflexivity. Defined. (* $\int_a^a f = 0$ *) -Lemma NewtonInt_P2 : (f:R->R;a:R) ``(NewtonInt f a a (NewtonInt_P1 f a))==0``. -Intros; Unfold NewtonInt; Simpl; Unfold mult_fct fct_cte id; Ring. +Lemma NewtonInt_P2 : + forall (f:R -> R) (a:R), NewtonInt f a a (NewtonInt_P1 f a) = 0. +intros; unfold NewtonInt in |- *; simpl in |- *; + unfold mult_fct, fct_cte, id in |- *; ring. Qed. (* If $\int_a^b f$ exists, then $\int_b^a f$ exists too *) -Lemma NewtonInt_P3 : (f:R->R;a,b:R;X:(Newton_integrable f a b)) (Newton_integrable f b a). -Unfold Newton_integrable; Intros; Elim X; Intros g H; Apply existTT with g; Tauto. +Lemma NewtonInt_P3 : + forall (f:R -> R) (a b:R) (X:Newton_integrable f a b), + Newton_integrable f b a. +unfold Newton_integrable in |- *; intros; elim X; intros g H; + apply existT with g; tauto. Defined. (* $\int_a^b f = -\int_b^a f$ *) -Lemma NewtonInt_P4 : (f:R->R;a,b:R;pr:(Newton_integrable f a b)) ``(NewtonInt f a b pr)==-(NewtonInt f b a (NewtonInt_P3 f a b pr))``. -Intros; Unfold Newton_integrable in pr; Elim pr; Intros; Elim p; Intro. -Unfold NewtonInt; Case (NewtonInt_P3 f a b (existTT R->R [g:(R->R)](antiderivative f g a b)\/(antiderivative f g b a) x p)). -Intros; Elim o; Intro. -Unfold antiderivative in H0; Elim H0; Intros; Elim H2; Intro. -Unfold antiderivative in H; Elim H; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H5 H3)). -Rewrite H3; Ring. -Assert H1 := (antiderivative_Ucte f x x0 a b H H0); Elim H1; Intros; Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. -Assert H3 : ``a<=a<=b``. -Split; [Right; Reflexivity | Assumption]. -Assert H4 : ``a<=b<=b``. -Split; [Assumption | Right; Reflexivity]. -Assert H5 := (H2 ? H3); Assert H6 := (H2 ? H4); Rewrite H5; Rewrite H6; Ring. -Unfold NewtonInt; Case (NewtonInt_P3 f a b (existTT R->R [g:(R->R)](antiderivative f g a b)\/(antiderivative f g b a) x p)); Intros; Elim o; Intro. -Assert H1 := (antiderivative_Ucte f x x0 b a H H0); Elim H1; Intros; Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. -Assert H3 : ``b<=a<=a``. -Split; [Assumption | Right; Reflexivity]. -Assert H4 : ``b<=b<=a``. -Split; [Right; Reflexivity | Assumption]. -Assert H5 := (H2 ? H3); Assert H6 := (H2 ? H4); Rewrite H5; Rewrite H6; Ring. -Unfold antiderivative in H0; Elim H0; Intros; Elim H2; Intro. -Unfold antiderivative in H; Elim H; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H5 H3)). -Rewrite H3; Ring. +Lemma NewtonInt_P4 : + forall (f:R -> R) (a b:R) (pr:Newton_integrable f a b), + NewtonInt f a b pr = - NewtonInt f b a (NewtonInt_P3 f a b pr). +intros; unfold Newton_integrable in pr; elim pr; intros; elim p; intro. +unfold NewtonInt in |- *; + case + (NewtonInt_P3 f a b + (existT + (fun g:R -> R => antiderivative f g a b \/ antiderivative f g b a) x + p)). +intros; elim o; intro. +unfold antiderivative in H0; elim H0; intros; elim H2; intro. +unfold antiderivative in H; elim H; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H5 H3)). +rewrite H3; ring. +assert (H1 := antiderivative_Ucte f x x0 a b H H0); elim H1; intros; + unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +assert (H3 : a <= a <= b). +split; [ right; reflexivity | assumption ]. +assert (H4 : a <= b <= b). +split; [ assumption | right; reflexivity ]. +assert (H5 := H2 _ H3); assert (H6 := H2 _ H4); rewrite H5; rewrite H6; ring. +unfold NewtonInt in |- *; + case + (NewtonInt_P3 f a b + (existT + (fun g:R -> R => antiderivative f g a b \/ antiderivative f g b a) x + p)); intros; elim o; intro. +assert (H1 := antiderivative_Ucte f x x0 b a H H0); elim H1; intros; + unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +assert (H3 : b <= a <= a). +split; [ assumption | right; reflexivity ]. +assert (H4 : b <= b <= a). +split; [ right; reflexivity | assumption ]. +assert (H5 := H2 _ H3); assert (H6 := H2 _ H4); rewrite H5; rewrite H6; ring. +unfold antiderivative in H0; elim H0; intros; elim H2; intro. +unfold antiderivative in H; elim H; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H5 H3)). +rewrite H3; ring. Qed. (* The set of Newton integrable functions is a vectorial space *) -Lemma NewtonInt_P5 : (f,g:R->R;l,a,b:R) (Newton_integrable f a b) -> (Newton_integrable g a b) -> (Newton_integrable [x:R]``l*(f x)+(g x)`` a b). -Unfold Newton_integrable; Intros; Elim X; Intros; Elim X0; Intros; Exists [y:R]``l*(x y)+(x0 y)``. -Elim p; Intro. -Elim p0; Intro. -Left; Unfold antiderivative; Unfold antiderivative in H H0; Elim H; Clear H; Intros; Elim H0; Clear H0; Intros H0 _. -Split. -Intros; Elim (H ? H2); Elim (H0 ? H2); Intros. -Assert H5 : (derivable_pt [y:R]``l*(x y)+(x0 y)`` x1). -Reg. -Exists H5; Symmetry; Reg; Rewrite <- H3; Rewrite <- H4; Reflexivity. -Assumption. -Unfold antiderivative in H H0; Elim H; Elim H0; Intros; Elim H4; Intro. -Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H5 H2)). -Left; Rewrite <- H5; Unfold antiderivative; Split. -Intros; Elim H6; Intros; Assert H9 : ``x1==a``. -Apply Rle_antisym; Assumption. -Assert H10 : ``a<=x1<=b``. -Split; Right; [Symmetry; Assumption | Rewrite <- H5; Assumption]. -Assert H11 : ``b<=x1<=a``. -Split; Right; [Rewrite <- H5; Symmetry; Assumption | Assumption]. -Assert H12 : (derivable_pt x x1). -Unfold derivable_pt; Exists (f x1); Elim (H3 ? H10); Intros; EApply derive_pt_eq_1; Symmetry; Apply H12. -Assert H13 : (derivable_pt x0 x1). -Unfold derivable_pt; Exists (g x1); Elim (H1 ? H11); Intros; EApply derive_pt_eq_1; Symmetry; Apply H13. -Assert H14 : (derivable_pt [y:R]``l*(x y)+(x0 y)`` x1). -Reg. -Exists H14; Symmetry; Reg. -Assert H15 : ``(derive_pt x0 x1 H13)==(g x1)``. -Elim (H1 ? H11); Intros; Rewrite H15; Apply pr_nu. -Assert H16 : ``(derive_pt x x1 H12)==(f x1)``. -Elim (H3 ? H10); Intros; Rewrite H16; Apply pr_nu. -Rewrite H15; Rewrite H16; Ring. -Right; Reflexivity. -Elim p0; Intro. -Unfold antiderivative in H H0; Elim H; Elim H0; Intros; Elim H4; Intro. -Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H5 H2)). -Left; Rewrite H5; Unfold antiderivative; Split. -Intros; Elim H6; Intros; Assert H9 : ``x1==a``. -Apply Rle_antisym; Assumption. -Assert H10 : ``a<=x1<=b``. -Split; Right; [Symmetry; Assumption | Rewrite H5; Assumption]. -Assert H11 : ``b<=x1<=a``. -Split; Right; [Rewrite H5; Symmetry; Assumption | Assumption]. -Assert H12 : (derivable_pt x x1). -Unfold derivable_pt; Exists (f x1); Elim (H3 ? H11); Intros; EApply derive_pt_eq_1; Symmetry; Apply H12. -Assert H13 : (derivable_pt x0 x1). -Unfold derivable_pt; Exists (g x1); Elim (H1 ? H10); Intros; EApply derive_pt_eq_1; Symmetry; Apply H13. -Assert H14 : (derivable_pt [y:R]``l*(x y)+(x0 y)`` x1). -Reg. -Exists H14; Symmetry; Reg. -Assert H15 : ``(derive_pt x0 x1 H13)==(g x1)``. -Elim (H1 ? H10); Intros; Rewrite H15; Apply pr_nu. -Assert H16 : ``(derive_pt x x1 H12)==(f x1)``. -Elim (H3 ? H11); Intros; Rewrite H16; Apply pr_nu. -Rewrite H15; Rewrite H16; Ring. -Right; Reflexivity. -Right; Unfold antiderivative; Unfold antiderivative in H H0; Elim H; Clear H; Intros; Elim H0; Clear H0; Intros H0 _; Split. -Intros; Elim (H ? H2); Elim (H0 ? H2); Intros. -Assert H5 : (derivable_pt [y:R]``l*(x y)+(x0 y)`` x1). -Reg. -Exists H5; Symmetry; Reg; Rewrite <- H3; Rewrite <- H4; Reflexivity. -Assumption. +Lemma NewtonInt_P5 : + forall (f g:R -> R) (l a b:R), + Newton_integrable f a b -> + Newton_integrable g a b -> + Newton_integrable (fun x:R => l * f x + g x) a b. +unfold Newton_integrable in |- *; intros; elim X; intros; elim X0; intros; + exists (fun y:R => l * x y + x0 y). +elim p; intro. +elim p0; intro. +left; unfold antiderivative in |- *; unfold antiderivative in H, H0; elim H; + clear H; intros; elim H0; clear H0; intros H0 _. +split. +intros; elim (H _ H2); elim (H0 _ H2); intros. +assert (H5 : derivable_pt (fun y:R => l * x y + x0 y) x1). +reg. +exists H5; symmetry in |- *; reg; rewrite <- H3; rewrite <- H4; reflexivity. +assumption. +unfold antiderivative in H, H0; elim H; elim H0; intros; elim H4; intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H5 H2)). +left; rewrite <- H5; unfold antiderivative in |- *; split. +intros; elim H6; intros; assert (H9 : x1 = a). +apply Rle_antisym; assumption. +assert (H10 : a <= x1 <= b). +split; right; [ symmetry in |- *; assumption | rewrite <- H5; assumption ]. +assert (H11 : b <= x1 <= a). +split; right; [ rewrite <- H5; symmetry in |- *; assumption | assumption ]. +assert (H12 : derivable_pt x x1). +unfold derivable_pt in |- *; exists (f x1); elim (H3 _ H10); intros; + eapply derive_pt_eq_1; symmetry in |- *; apply H12. +assert (H13 : derivable_pt x0 x1). +unfold derivable_pt in |- *; exists (g x1); elim (H1 _ H11); intros; + eapply derive_pt_eq_1; symmetry in |- *; apply H13. +assert (H14 : derivable_pt (fun y:R => l * x y + x0 y) x1). +reg. +exists H14; symmetry in |- *; reg. +assert (H15 : derive_pt x0 x1 H13 = g x1). +elim (H1 _ H11); intros; rewrite H15; apply pr_nu. +assert (H16 : derive_pt x x1 H12 = f x1). +elim (H3 _ H10); intros; rewrite H16; apply pr_nu. +rewrite H15; rewrite H16; ring. +right; reflexivity. +elim p0; intro. +unfold antiderivative in H, H0; elim H; elim H0; intros; elim H4; intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H5 H2)). +left; rewrite H5; unfold antiderivative in |- *; split. +intros; elim H6; intros; assert (H9 : x1 = a). +apply Rle_antisym; assumption. +assert (H10 : a <= x1 <= b). +split; right; [ symmetry in |- *; assumption | rewrite H5; assumption ]. +assert (H11 : b <= x1 <= a). +split; right; [ rewrite H5; symmetry in |- *; assumption | assumption ]. +assert (H12 : derivable_pt x x1). +unfold derivable_pt in |- *; exists (f x1); elim (H3 _ H11); intros; + eapply derive_pt_eq_1; symmetry in |- *; apply H12. +assert (H13 : derivable_pt x0 x1). +unfold derivable_pt in |- *; exists (g x1); elim (H1 _ H10); intros; + eapply derive_pt_eq_1; symmetry in |- *; apply H13. +assert (H14 : derivable_pt (fun y:R => l * x y + x0 y) x1). +reg. +exists H14; symmetry in |- *; reg. +assert (H15 : derive_pt x0 x1 H13 = g x1). +elim (H1 _ H10); intros; rewrite H15; apply pr_nu. +assert (H16 : derive_pt x x1 H12 = f x1). +elim (H3 _ H11); intros; rewrite H16; apply pr_nu. +rewrite H15; rewrite H16; ring. +right; reflexivity. +right; unfold antiderivative in |- *; unfold antiderivative in H, H0; elim H; + clear H; intros; elim H0; clear H0; intros H0 _; split. +intros; elim (H _ H2); elim (H0 _ H2); intros. +assert (H5 : derivable_pt (fun y:R => l * x y + x0 y) x1). +reg. +exists H5; symmetry in |- *; reg; rewrite <- H3; rewrite <- H4; reflexivity. +assumption. Defined. (**********) -Lemma antiderivative_P1 : (f,g,F,G:R->R;l,a,b:R) (antiderivative f F a b) -> (antiderivative g G a b) -> (antiderivative [x:R]``l*(f x)+(g x)`` [x:R]``l*(F x)+(G x)`` a b). -Unfold antiderivative; Intros; Elim H; Elim H0; Clear H H0; Intros; Split. -Intros; Elim (H ? H3); Elim (H1 ? H3); Intros. -Assert H6 : (derivable_pt [x:R]``l*(F x)+(G x)`` x). -Reg. -Exists H6; Symmetry; Reg; Rewrite <- H4; Rewrite <- H5; Ring. -Assumption. +Lemma antiderivative_P1 : + forall (f g F G:R -> R) (l a b:R), + antiderivative f F a b -> + antiderivative g G a b -> + antiderivative (fun x:R => l * f x + g x) (fun x:R => l * F x + G x) a b. +unfold antiderivative in |- *; intros; elim H; elim H0; clear H H0; intros; + split. +intros; elim (H _ H3); elim (H1 _ H3); intros. +assert (H6 : derivable_pt (fun x:R => l * F x + G x) x). +reg. +exists H6; symmetry in |- *; reg; rewrite <- H4; rewrite <- H5; ring. +assumption. Qed. (* $\int_a^b \lambda f + g = \lambda \int_a^b f + \int_a^b f *) -Lemma NewtonInt_P6 : (f,g:R->R;l,a,b:R;pr1:(Newton_integrable f a b);pr2:(Newton_integrable g a b)) (NewtonInt [x:R]``l*(f x)+(g x)`` a b (NewtonInt_P5 f g l a b pr1 pr2))==``l*(NewtonInt f a b pr1)+(NewtonInt g a b pr2)``. -Intros f g l a b pr1 pr2; Unfold NewtonInt; Case (NewtonInt_P5 f g l a b pr1 pr2); Intros; Case pr1; Intros; Case pr2; Intros; Case (total_order_T a b); Intro. -Elim s; Intro. -Elim o; Intro. -Elim o0; Intro. -Elim o1; Intro. -Assert H2 := (antiderivative_P1 f g x0 x1 l a b H0 H1); Assert H3 := (antiderivative_Ucte ? ? ? ? ? H H2); Elim H3; Intros; Assert H5 : ``a<=a<=b``. -Split; [Right; Reflexivity | Left; Assumption]. -Assert H6 : ``a<=b<=b``. -Split; [Left; Assumption | Right; Reflexivity]. -Assert H7 := (H4 ? H5); Assert H8 := (H4 ? H6); Rewrite H7; Rewrite H8; Ring. -Unfold antiderivative in H1; Elim H1; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H3 a0)). -Unfold antiderivative in H0; Elim H0; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 a0)). -Unfold antiderivative in H; Elim H; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 a0)). -Rewrite b0; Ring. -Elim o; Intro. -Unfold antiderivative in H; Elim H; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 r)). -Elim o0; Intro. -Unfold antiderivative in H0; Elim H0; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 r)). -Elim o1; Intro. -Unfold antiderivative in H1; Elim H1; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H3 r)). -Assert H2 := (antiderivative_P1 f g x0 x1 l b a H0 H1); Assert H3 := (antiderivative_Ucte ? ? ? ? ? H H2); Elim H3; Intros; Assert H5 : ``b<=a<=a``. -Split; [Left; Assumption | Right; Reflexivity]. -Assert H6 : ``b<=b<=a``. -Split; [Right; Reflexivity | Left; Assumption]. -Assert H7 := (H4 ? H5); Assert H8 := (H4 ? H6); Rewrite H7; Rewrite H8; Ring. +Lemma NewtonInt_P6 : + forall (f g:R -> R) (l a b:R) (pr1:Newton_integrable f a b) + (pr2:Newton_integrable g a b), + NewtonInt (fun x:R => l * f x + g x) a b (NewtonInt_P5 f g l a b pr1 pr2) = + l * NewtonInt f a b pr1 + NewtonInt g a b pr2. +intros f g l a b pr1 pr2; unfold NewtonInt in |- *; + case (NewtonInt_P5 f g l a b pr1 pr2); intros; case pr1; + intros; case pr2; intros; case (total_order_T a b); + intro. +elim s; intro. +elim o; intro. +elim o0; intro. +elim o1; intro. +assert (H2 := antiderivative_P1 f g x0 x1 l a b H0 H1); + assert (H3 := antiderivative_Ucte _ _ _ _ _ H H2); + elim H3; intros; assert (H5 : a <= a <= b). +split; [ right; reflexivity | left; assumption ]. +assert (H6 : a <= b <= b). +split; [ left; assumption | right; reflexivity ]. +assert (H7 := H4 _ H5); assert (H8 := H4 _ H6); rewrite H7; rewrite H8; ring. +unfold antiderivative in H1; elim H1; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H3 a0)). +unfold antiderivative in H0; elim H0; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a0)). +unfold antiderivative in H; elim H; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 a0)). +rewrite b0; ring. +elim o; intro. +unfold antiderivative in H; elim H; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 r)). +elim o0; intro. +unfold antiderivative in H0; elim H0; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 r)). +elim o1; intro. +unfold antiderivative in H1; elim H1; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H3 r)). +assert (H2 := antiderivative_P1 f g x0 x1 l b a H0 H1); + assert (H3 := antiderivative_Ucte _ _ _ _ _ H H2); + elim H3; intros; assert (H5 : b <= a <= a). +split; [ left; assumption | right; reflexivity ]. +assert (H6 : b <= b <= a). +split; [ right; reflexivity | left; assumption ]. +assert (H7 := H4 _ H5); assert (H8 := H4 _ H6); rewrite H7; rewrite H8; ring. Qed. -Lemma antiderivative_P2 : (f,F0,F1:R->R;a,b,c:R) (antiderivative f F0 a b) -> (antiderivative f F1 b c) -> (antiderivative f [x:R](Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end) a c). -Unfold antiderivative; Intros; Elim H; Clear H; Intros; Elim H0; Clear H0; Intros; Split. -2:Apply Rle_trans with b; Assumption. -Intros; Elim H3; Clear H3; Intros; Case (total_order_T x b); Intro. -Elim s; Intro. -Assert H5 : ``a<=x<=b``. -Split; [Assumption | Left; Assumption]. -Assert H6 := (H ? H5); Elim H6; Clear H6; Intros; Assert H7 : (derivable_pt_lim [x:R](Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end) x (f x)). -Unfold derivable_pt_lim; Assert H7 : ``(derive_pt F0 x x0)==(f x)``. -Symmetry; Assumption. -Assert H8 := (derive_pt_eq_1 F0 x (f x) x0 H7); Unfold derivable_pt_lim in H8; Intros; Elim (H8 ? H9); Intros; Pose D := (Rmin x1 ``b-x``). -Assert H11 : ``0<D``. -Unfold D; Unfold Rmin; Case (total_order_Rle x1 ``b-x``); Intro. -Apply (cond_pos x1). -Apply Rlt_Rminus; Assumption. -Exists (mkposreal ? H11); Intros; Case (total_order_Rle x b); Intro. -Case (total_order_Rle ``x+h`` b); Intro. -Apply H10. -Assumption. -Apply Rlt_le_trans with D; [Assumption | Unfold D; Apply Rmin_l]. -Elim n; Left; Apply Rlt_le_trans with ``x+D``. -Apply Rlt_compatibility; Apply Rle_lt_trans with (Rabsolu h). -Apply Rle_Rabsolu. -Apply H13. -Apply Rle_anti_compatibility with ``-x``; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite Rplus_sym; Unfold D; Apply Rmin_r. -Elim n; Left; Assumption. -Assert H8 : (derivable_pt [x:R]Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end x). -Unfold derivable_pt; Apply Specif.existT with (f x); Apply H7. -Exists H8; Symmetry; Apply derive_pt_eq_0; Apply H7. -Assert H5 : ``a<=x<=b``. -Split; [Assumption | Right; Assumption]. -Assert H6 : ``b<=x<=c``. -Split; [Right; Symmetry; Assumption | Assumption]. -Elim (H ? H5); Elim (H0 ? H6); Intros; Assert H9 : (derive_pt F0 x x1)==(f x). -Symmetry; Assumption. -Assert H10 : (derive_pt F1 x x0)==(f x). -Symmetry; Assumption. -Assert H11 := (derive_pt_eq_1 F0 x (f x) x1 H9); Assert H12 := (derive_pt_eq_1 F1 x (f x) x0 H10); Assert H13 : (derivable_pt_lim [x:R]Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end x (f x)). -Unfold derivable_pt_lim; Unfold derivable_pt_lim in H11 H12; Intros; Elim (H11 ? H13); Elim (H12 ? H13); Intros; Pose D := (Rmin x2 x3); Assert H16 : ``0<D``. -Unfold D; Unfold Rmin; Case (total_order_Rle x2 x3); Intro. -Apply (cond_pos x2). -Apply (cond_pos x3). -Exists (mkposreal ? H16); Intros; Case (total_order_Rle x b); Intro. -Case (total_order_Rle ``x+h`` b); Intro. -Apply H15. -Assumption. -Apply Rlt_le_trans with D; [Assumption | Unfold D; Apply Rmin_r]. -Replace ``(F1 (x+h))+((F0 b)-(F1 b))-(F0 x)`` with ``(F1 (x+h))-(F1 x)``. -Apply H14. -Assumption. -Apply Rlt_le_trans with D; [Assumption | Unfold D; Apply Rmin_l]. -Rewrite b0; Ring. -Elim n; Right; Assumption. -Assert H14 : (derivable_pt [x:R](Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end) x). -Unfold derivable_pt; Apply Specif.existT with (f x); Apply H13. -Exists H14; Symmetry; Apply derive_pt_eq_0; Apply H13. -Assert H5 : ``b<=x<=c``. -Split; [Left; Assumption | Assumption]. -Assert H6 := (H0 ? H5); Elim H6; Clear H6; Intros; Assert H7 : (derivable_pt_lim [x:R]Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end x (f x)). -Unfold derivable_pt_lim; Assert H7 : ``(derive_pt F1 x x0)==(f x)``. -Symmetry; Assumption. -Assert H8 := (derive_pt_eq_1 F1 x (f x) x0 H7); Unfold derivable_pt_lim in H8; Intros; Elim (H8 ? H9); Intros; Pose D := (Rmin x1 ``x-b``); Assert H11 : ``0<D``. -Unfold D; Unfold Rmin; Case (total_order_Rle x1 ``x-b``); Intro. -Apply (cond_pos x1). -Apply Rlt_Rminus; Assumption. -Exists (mkposreal ? H11); Intros; Case (total_order_Rle x b); Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 r)). -Case (total_order_Rle ``x+h`` b); Intro. -Cut ``b<x+h``. -Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 H14)). -Apply Rlt_anti_compatibility with ``-h-b``; Replace ``-h-b+b`` with ``-h``; [Idtac | Ring]; Replace ``-h-b+(x+h)`` with ``x-b``; [Idtac | Ring]; Apply Rle_lt_trans with (Rabsolu h). -Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. -Apply Rlt_le_trans with D. -Apply H13. -Unfold D; Apply Rmin_r. -Replace ``((F1 (x+h))+((F0 b)-(F1 b)))-((F1 x)+((F0 b)-(F1 b)))`` with ``(F1 (x+h))-(F1 x)``; [Idtac | Ring]; Apply H10. -Assumption. -Apply Rlt_le_trans with D. -Assumption. -Unfold D; Apply Rmin_l. -Assert H8 : (derivable_pt [x:R]Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end x). -Unfold derivable_pt; Apply Specif.existT with (f x); Apply H7. -Exists H8; Symmetry; Apply derive_pt_eq_0; Apply H7. +Lemma antiderivative_P2 : + forall (f F0 F1:R -> R) (a b c:R), + antiderivative f F0 a b -> + antiderivative f F1 b c -> + antiderivative f + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) a c. +unfold antiderivative in |- *; intros; elim H; clear H; intros; elim H0; + clear H0; intros; split. +2: apply Rle_trans with b; assumption. +intros; elim H3; clear H3; intros; case (total_order_T x b); intro. +elim s; intro. +assert (H5 : a <= x <= b). +split; [ assumption | left; assumption ]. +assert (H6 := H _ H5); elim H6; clear H6; intros; + assert + (H7 : + derivable_pt_lim + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) x (f x)). +unfold derivable_pt_lim in |- *; assert (H7 : derive_pt F0 x x0 = f x). +symmetry in |- *; assumption. +assert (H8 := derive_pt_eq_1 F0 x (f x) x0 H7); unfold derivable_pt_lim in H8; + intros; elim (H8 _ H9); intros; pose (D := Rmin x1 (b - x)). +assert (H11 : 0 < D). +unfold D in |- *; unfold Rmin in |- *; case (Rle_dec x1 (b - x)); intro. +apply (cond_pos x1). +apply Rlt_Rminus; assumption. +exists (mkposreal _ H11); intros; case (Rle_dec x b); intro. +case (Rle_dec (x + h) b); intro. +apply H10. +assumption. +apply Rlt_le_trans with D; [ assumption | unfold D in |- *; apply Rmin_l ]. +elim n; left; apply Rlt_le_trans with (x + D). +apply Rplus_lt_compat_l; apply Rle_lt_trans with (Rabs h). +apply RRle_abs. +apply H13. +apply Rplus_le_reg_l with (- x); rewrite <- Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_l; rewrite Rplus_comm; unfold D in |- *; + apply Rmin_r. +elim n; left; assumption. +assert + (H8 : + derivable_pt + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) x). +unfold derivable_pt in |- *; apply existT with (f x); apply H7. +exists H8; symmetry in |- *; apply derive_pt_eq_0; apply H7. +assert (H5 : a <= x <= b). +split; [ assumption | right; assumption ]. +assert (H6 : b <= x <= c). +split; [ right; symmetry in |- *; assumption | assumption ]. +elim (H _ H5); elim (H0 _ H6); intros; assert (H9 : derive_pt F0 x x1 = f x). +symmetry in |- *; assumption. +assert (H10 : derive_pt F1 x x0 = f x). +symmetry in |- *; assumption. +assert (H11 := derive_pt_eq_1 F0 x (f x) x1 H9); + assert (H12 := derive_pt_eq_1 F1 x (f x) x0 H10); + assert + (H13 : + derivable_pt_lim + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) x (f x)). +unfold derivable_pt_lim in |- *; unfold derivable_pt_lim in H11, H12; intros; + elim (H11 _ H13); elim (H12 _ H13); intros; pose (D := Rmin x2 x3); + assert (H16 : 0 < D). +unfold D in |- *; unfold Rmin in |- *; case (Rle_dec x2 x3); intro. +apply (cond_pos x2). +apply (cond_pos x3). +exists (mkposreal _ H16); intros; case (Rle_dec x b); intro. +case (Rle_dec (x + h) b); intro. +apply H15. +assumption. +apply Rlt_le_trans with D; [ assumption | unfold D in |- *; apply Rmin_r ]. +replace (F1 (x + h) + (F0 b - F1 b) - F0 x) with (F1 (x + h) - F1 x). +apply H14. +assumption. +apply Rlt_le_trans with D; [ assumption | unfold D in |- *; apply Rmin_l ]. +rewrite b0; ring. +elim n; right; assumption. +assert + (H14 : + derivable_pt + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) x). +unfold derivable_pt in |- *; apply existT with (f x); apply H13. +exists H14; symmetry in |- *; apply derive_pt_eq_0; apply H13. +assert (H5 : b <= x <= c). +split; [ left; assumption | assumption ]. +assert (H6 := H0 _ H5); elim H6; clear H6; intros; + assert + (H7 : + derivable_pt_lim + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) x (f x)). +unfold derivable_pt_lim in |- *; assert (H7 : derive_pt F1 x x0 = f x). +symmetry in |- *; assumption. +assert (H8 := derive_pt_eq_1 F1 x (f x) x0 H7); unfold derivable_pt_lim in H8; + intros; elim (H8 _ H9); intros; pose (D := Rmin x1 (x - b)); + assert (H11 : 0 < D). +unfold D in |- *; unfold Rmin in |- *; case (Rle_dec x1 (x - b)); intro. +apply (cond_pos x1). +apply Rlt_Rminus; assumption. +exists (mkposreal _ H11); intros; case (Rle_dec x b); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 r)). +case (Rle_dec (x + h) b); intro. +cut (b < x + h). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 H14)). +apply Rplus_lt_reg_r with (- h - b); replace (- h - b + b) with (- h); + [ idtac | ring ]; replace (- h - b + (x + h)) with (x - b); + [ idtac | ring ]; apply Rle_lt_trans with (Rabs h). +rewrite <- Rabs_Ropp; apply RRle_abs. +apply Rlt_le_trans with D. +apply H13. +unfold D in |- *; apply Rmin_r. +replace (F1 (x + h) + (F0 b - F1 b) - (F1 x + (F0 b - F1 b))) with + (F1 (x + h) - F1 x); [ idtac | ring ]; apply H10. +assumption. +apply Rlt_le_trans with D. +assumption. +unfold D in |- *; apply Rmin_l. +assert + (H8 : + derivable_pt + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) x). +unfold derivable_pt in |- *; apply existT with (f x); apply H7. +exists H8; symmetry in |- *; apply derive_pt_eq_0; apply H7. Qed. -Lemma antiderivative_P3 : (f,F0,F1:R->R;a,b,c:R) (antiderivative f F0 a b) -> (antiderivative f F1 c b) -> (antiderivative f F1 c a)\/(antiderivative f F0 a c). -Intros; Unfold antiderivative in H H0; Elim H; Clear H; Elim H0; Clear H0; Intros; Case (total_order_T a c); Intro. -Elim s; Intro. -Right; Unfold antiderivative; Split. -Intros; Apply H1; Elim H3; Intros; Split; [Assumption | Apply Rle_trans with c; Assumption]. -Left; Assumption. -Right; Unfold antiderivative; Split. -Intros; Apply H1; Elim H3; Intros; Split; [Assumption | Apply Rle_trans with c; Assumption]. -Right; Assumption. -Left; Unfold antiderivative; Split. -Intros; Apply H; Elim H3; Intros; Split; [Assumption | Apply Rle_trans with a; Assumption]. -Left; Assumption. +Lemma antiderivative_P3 : + forall (f F0 F1:R -> R) (a b c:R), + antiderivative f F0 a b -> + antiderivative f F1 c b -> + antiderivative f F1 c a \/ antiderivative f F0 a c. +intros; unfold antiderivative in H, H0; elim H; clear H; elim H0; clear H0; + intros; case (total_order_T a c); intro. +elim s; intro. +right; unfold antiderivative in |- *; split. +intros; apply H1; elim H3; intros; split; + [ assumption | apply Rle_trans with c; assumption ]. +left; assumption. +right; unfold antiderivative in |- *; split. +intros; apply H1; elim H3; intros; split; + [ assumption | apply Rle_trans with c; assumption ]. +right; assumption. +left; unfold antiderivative in |- *; split. +intros; apply H; elim H3; intros; split; + [ assumption | apply Rle_trans with a; assumption ]. +left; assumption. Qed. -Lemma antiderivative_P4 : (f,F0,F1:R->R;a,b,c:R) (antiderivative f F0 a b) -> (antiderivative f F1 a c) -> (antiderivative f F1 b c)\/(antiderivative f F0 c b). -Intros; Unfold antiderivative in H H0; Elim H; Clear H; Elim H0; Clear H0; Intros; Case (total_order_T c b); Intro. -Elim s; Intro. -Right; Unfold antiderivative; Split. -Intros; Apply H1; Elim H3; Intros; Split; [Apply Rle_trans with c; Assumption | Assumption]. -Left; Assumption. -Right; Unfold antiderivative; Split. -Intros; Apply H1; Elim H3; Intros; Split; [Apply Rle_trans with c; Assumption | Assumption]. -Right; Assumption. -Left; Unfold antiderivative; Split. -Intros; Apply H; Elim H3; Intros; Split; [Apply Rle_trans with b; Assumption | Assumption]. -Left; Assumption. +Lemma antiderivative_P4 : + forall (f F0 F1:R -> R) (a b c:R), + antiderivative f F0 a b -> + antiderivative f F1 a c -> + antiderivative f F1 b c \/ antiderivative f F0 c b. +intros; unfold antiderivative in H, H0; elim H; clear H; elim H0; clear H0; + intros; case (total_order_T c b); intro. +elim s; intro. +right; unfold antiderivative in |- *; split. +intros; apply H1; elim H3; intros; split; + [ apply Rle_trans with c; assumption | assumption ]. +left; assumption. +right; unfold antiderivative in |- *; split. +intros; apply H1; elim H3; intros; split; + [ apply Rle_trans with c; assumption | assumption ]. +right; assumption. +left; unfold antiderivative in |- *; split. +intros; apply H; elim H3; intros; split; + [ apply Rle_trans with b; assumption | assumption ]. +left; assumption. Qed. -Lemma NewtonInt_P7 : (f:R->R;a,b,c:R) ``a<b`` -> ``b<c`` -> (Newton_integrable f a b) -> (Newton_integrable f b c) -> (Newton_integrable f a c). -Unfold Newton_integrable; Intros f a b c Hab Hbc X X0; Elim X; Clear X; Intros F0 H0; Elim X0; Clear X0; Intros F1 H1; Pose g := [x:R](Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end); Apply existTT with g; Left; Unfold g; Apply antiderivative_P2. -Elim H0; Intro. -Assumption. -Unfold antiderivative in H; Elim H; Clear H; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 Hab)). -Elim H1; Intro. -Assumption. -Unfold antiderivative in H; Elim H; Clear H; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 Hbc)). +Lemma NewtonInt_P7 : + forall (f:R -> R) (a b c:R), + a < b -> + b < c -> + Newton_integrable f a b -> + Newton_integrable f b c -> Newton_integrable f a c. +unfold Newton_integrable in |- *; intros f a b c Hab Hbc X X0; elim X; + clear X; intros F0 H0; elim X0; clear X0; intros F1 H1; + pose + (g := + fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end); apply existT with g; left; unfold g in |- *; + apply antiderivative_P2. +elim H0; intro. +assumption. +unfold antiderivative in H; elim H; clear H; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hab)). +elim H1; intro. +assumption. +unfold antiderivative in H; elim H; clear H; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hbc)). Qed. -Lemma NewtonInt_P8 : (f:(R->R); a,b,c:R) (Newton_integrable f a b) -> (Newton_integrable f b c) -> (Newton_integrable f a c). -Intros. -Elim X; Intros F0 H0. -Elim X0; Intros F1 H1. -Case (total_order_T a b); Intro. -Elim s; Intro. -Case (total_order_T b c); Intro. -Elim s0; Intro. +Lemma NewtonInt_P8 : + forall (f:R -> R) (a b c:R), + Newton_integrable f a b -> + Newton_integrable f b c -> Newton_integrable f a c. +intros. +elim X; intros F0 H0. +elim X0; intros F1 H1. +case (total_order_T a b); intro. +elim s; intro. +case (total_order_T b c); intro. +elim s0; intro. (* a<b & b<c *) -Unfold Newton_integrable; Apply existTT with [x:R](Cases (total_order_Rle x b) of (leftT _) => (F0 x) | (rightT _) => ``(F1 x)+((F0 b)-(F1 b))`` end). -Elim H0; Intro. -Elim H1; Intro. -Left; Apply antiderivative_P2; Assumption. -Unfold antiderivative in H2; Elim H2; Clear H2; Intros _ H2. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 a1)). -Unfold antiderivative in H; Elim H; Clear H; Intros _ H. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H a0)). +unfold Newton_integrable in |- *; + apply existT with + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end). +elim H0; intro. +elim H1; intro. +left; apply antiderivative_P2; assumption. +unfold antiderivative in H2; elim H2; clear H2; intros _ H2. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a1)). +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)). (* a<b & b=c *) -Rewrite b0 in X; Apply X. +rewrite b0 in X; apply X. (* a<b & b>c *) -Case (total_order_T a c); Intro. -Elim s0; Intro. -Unfold Newton_integrable; Apply existTT with F0. -Left. -Elim H1; Intro. -Unfold antiderivative in H; Elim H; Clear H; Intros _ H. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). -Elim H0; Intro. -Assert H3 := (antiderivative_P3 f F0 F1 a b c H2 H). -Elim H3; Intro. -Unfold antiderivative in H4; Elim H4; Clear H4; Intros _ H4. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H4 a1)). -Assumption. -Unfold antiderivative in H2; Elim H2; Clear H2; Intros _ H2. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 a0)). -Rewrite b0; Apply NewtonInt_P1. -Unfold Newton_integrable; Apply existTT with F1. -Right. -Elim H1; Intro. -Unfold antiderivative in H; Elim H; Clear H; Intros _ H. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). -Elim H0; Intro. -Assert H3 := (antiderivative_P3 f F0 F1 a b c H2 H). -Elim H3; Intro. -Assumption. -Unfold antiderivative in H4; Elim H4; Clear H4; Intros _ H4. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H4 r0)). -Unfold antiderivative in H2; Elim H2; Clear H2; Intros _ H2. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 a0)). +case (total_order_T a c); intro. +elim s0; intro. +unfold Newton_integrable in |- *; apply existT with F0. +left. +elim H1; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim H0; intro. +assert (H3 := antiderivative_P3 f F0 F1 a b c H2 H). +elim H3; intro. +unfold antiderivative in H4; elim H4; clear H4; intros _ H4. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 a1)). +assumption. +unfold antiderivative in H2; elim H2; clear H2; intros _ H2. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a0)). +rewrite b0; apply NewtonInt_P1. +unfold Newton_integrable in |- *; apply existT with F1. +right. +elim H1; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim H0; intro. +assert (H3 := antiderivative_P3 f F0 F1 a b c H2 H). +elim H3; intro. +assumption. +unfold antiderivative in H4; elim H4; clear H4; intros _ H4. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 r0)). +unfold antiderivative in H2; elim H2; clear H2; intros _ H2. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a0)). (* a=b *) -Rewrite b0; Apply X0. -Case (total_order_T b c); Intro. -Elim s; Intro. +rewrite b0; apply X0. +case (total_order_T b c); intro. +elim s; intro. (* a>b & b<c *) -Case (total_order_T a c); Intro. -Elim s0; Intro. -Unfold Newton_integrable; Apply existTT with F1. -Left. -Elim H1; Intro. +case (total_order_T a c); intro. +elim s0; intro. +unfold Newton_integrable in |- *; apply existT with F1. +left. +elim H1; intro. (*****************) -Elim H0; Intro. -Unfold antiderivative in H2; Elim H2; Clear H2; Intros _ H2. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 r)). -Assert H3 := (antiderivative_P4 f F0 F1 b a c H2 H). -Elim H3; Intro. -Assumption. -Unfold antiderivative in H4; Elim H4; Clear H4; Intros _ H4. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H4 a1)). -Unfold antiderivative in H; Elim H; Clear H; Intros _ H. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H a0)). -Rewrite b0; Apply NewtonInt_P1. -Unfold Newton_integrable; Apply existTT with F0. -Right. -Elim H0; Intro. -Unfold antiderivative in H; Elim H; Clear H; Intros _ H. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). -Elim H1; Intro. -Assert H3 := (antiderivative_P4 f F0 F1 b a c H H2). -Elim H3; Intro. -Unfold antiderivative in H4; Elim H4; Clear H4; Intros _ H4. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H4 r0)). -Assumption. -Unfold antiderivative in H2; Elim H2; Clear H2; Intros _ H2. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H2 a0)). +elim H0; intro. +unfold antiderivative in H2; elim H2; clear H2; intros _ H2. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 r)). +assert (H3 := antiderivative_P4 f F0 F1 b a c H2 H). +elim H3; intro. +assumption. +unfold antiderivative in H4; elim H4; clear H4; intros _ H4. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 a1)). +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)). +rewrite b0; apply NewtonInt_P1. +unfold Newton_integrable in |- *; apply existT with F0. +right. +elim H0; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim H1; intro. +assert (H3 := antiderivative_P4 f F0 F1 b a c H H2). +elim H3; intro. +unfold antiderivative in H4; elim H4; clear H4; intros _ H4. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 r0)). +assumption. +unfold antiderivative in H2; elim H2; clear H2; intros _ H2. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a0)). (* a>b & b=c *) -Rewrite b0 in X; Apply X. +rewrite b0 in X; apply X. (* a>b & b>c *) -Assert X1 := (NewtonInt_P3 f a b X). -Assert X2 := (NewtonInt_P3 f b c X0). -Apply NewtonInt_P3. -Apply NewtonInt_P7 with b; Assumption. +assert (X1 := NewtonInt_P3 f a b X). +assert (X2 := NewtonInt_P3 f b c X0). +apply NewtonInt_P3. +apply NewtonInt_P7 with b; assumption. Defined. (* Chasles' relation *) -Lemma NewtonInt_P9 : (f:R->R;a,b,c:R;pr1:(Newton_integrable f a b);pr2:(Newton_integrable f b c)) ``(NewtonInt f a c (NewtonInt_P8 f a b c pr1 pr2))==(NewtonInt f a b pr1)+(NewtonInt f b c pr2)``. -Intros; Unfold NewtonInt. -Case (NewtonInt_P8 f a b c pr1 pr2); Intros. -Case pr1; Intros. -Case pr2; Intros. -Case (total_order_T a b); Intro. -Elim s; Intro. -Case (total_order_T b c); Intro. -Elim s0; Intro. +Lemma NewtonInt_P9 : + forall (f:R -> R) (a b c:R) (pr1:Newton_integrable f a b) + (pr2:Newton_integrable f b c), + NewtonInt f a c (NewtonInt_P8 f a b c pr1 pr2) = + NewtonInt f a b pr1 + NewtonInt f b c pr2. +intros; unfold NewtonInt in |- *. +case (NewtonInt_P8 f a b c pr1 pr2); intros. +case pr1; intros. +case pr2; intros. +case (total_order_T a b); intro. +elim s; intro. +case (total_order_T b c); intro. +elim s0; intro. (* a<b & b<c *) -Elim o0; Intro. -Elim o1; Intro. -Elim o; Intro. -Assert H2 := (antiderivative_P2 f x0 x1 a b c H H0). -Assert H3 := (antiderivative_Ucte f x [x:R] - Cases (total_order_Rle x b) of - (leftT _) => (x0 x) - | (rightT _) => ``(x1 x)+((x0 b)-(x1 b))`` - end a c H1 H2). -Elim H3; Intros. -Assert H5 : ``a<=a<=c``. -Split; [Right; Reflexivity | Left; Apply Rlt_trans with b; Assumption]. -Assert H6 : ``a<=c<=c``. -Split; [Left; Apply Rlt_trans with b; Assumption | Right; Reflexivity]. -Rewrite (H4 ? H5); Rewrite (H4 ? H6). -Case (total_order_Rle a b); Intro. -Case (total_order_Rle c b); Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 a1)). -Ring. -Elim n; Left; Assumption. -Unfold antiderivative in H1; Elim H1; Clear H1; Intros _ H1. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 (Rlt_trans ? ? ? a0 a1))). -Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 a1)). -Unfold antiderivative in H; Elim H; Clear H; Intros _ H. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H a0)). +elim o0; intro. +elim o1; intro. +elim o; intro. +assert (H2 := antiderivative_P2 f x0 x1 a b c H H0). +assert + (H3 := + antiderivative_Ucte f x + (fun x:R => + match Rle_dec x b with + | left _ => x0 x + | right _ => x1 x + (x0 b - x1 b) + end) a c H1 H2). +elim H3; intros. +assert (H5 : a <= a <= c). +split; [ right; reflexivity | left; apply Rlt_trans with b; assumption ]. +assert (H6 : a <= c <= c). +split; [ left; apply Rlt_trans with b; assumption | right; reflexivity ]. +rewrite (H4 _ H5); rewrite (H4 _ H6). +case (Rle_dec a b); intro. +case (Rle_dec c b); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 a1)). +ring. +elim n; left; assumption. +unfold antiderivative in H1; elim H1; clear H1; intros _ H1. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 (Rlt_trans _ _ _ a0 a1))). +unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 a1)). +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)). (* a<b & b=c *) -Rewrite <- b0. -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or. -Rewrite <- b0 in o. -Elim o0; Intro. -Elim o; Intro. -Assert H1 := (antiderivative_Ucte f x x0 a b H0 H). -Elim H1; Intros. -Rewrite (H2 b). -Rewrite (H2 a). -Ring. -Split; [Right; Reflexivity | Left; Assumption]. -Split; [Left; Assumption | Right; Reflexivity]. -Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 a0)). -Unfold antiderivative in H; Elim H; Clear H; Intros _ H. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H a0)). +rewrite <- b0. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r. +rewrite <- b0 in o. +elim o0; intro. +elim o; intro. +assert (H1 := antiderivative_Ucte f x x0 a b H0 H). +elim H1; intros. +rewrite (H2 b). +rewrite (H2 a). +ring. +split; [ right; reflexivity | left; assumption ]. +split; [ left; assumption | right; reflexivity ]. +unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 a0)). +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)). (* a<b & b>c *) -Elim o1; Intro. -Unfold antiderivative in H; Elim H; Clear H; Intros _ H. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). -Elim o0; Intro. -Elim o; Intro. -Assert H2 := (antiderivative_P2 f x x1 a c b H1 H). -Assert H3 := (antiderivative_Ucte ? ? ? a b H0 H2). -Elim H3; Intros. -Rewrite (H4 a). -Rewrite (H4 b). -Case (total_order_Rle b c); Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 r)). -Case (total_order_Rle a c); Intro. -Ring. -Elim n0; Unfold antiderivative in H1; Elim H1; Intros; Assumption. -Split; [Left; Assumption | Right; Reflexivity]. -Split; [Right; Reflexivity | Left; Assumption]. -Assert H2 := (antiderivative_P2 ? ? ? ? ? ? H1 H0). -Assert H3 := (antiderivative_Ucte ? ? ? c b H H2). -Elim H3; Intros. -Rewrite (H4 c). -Rewrite (H4 b). -Case (total_order_Rle b a); Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 a0)). -Case (total_order_Rle c a); Intro. -Ring. -Elim n0; Unfold antiderivative in H1; Elim H1; Intros; Assumption. -Split; [Left; Assumption | Right; Reflexivity]. -Split; [Right; Reflexivity | Left; Assumption]. -Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 a0)). +elim o1; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim o0; intro. +elim o; intro. +assert (H2 := antiderivative_P2 f x x1 a c b H1 H). +assert (H3 := antiderivative_Ucte _ _ _ a b H0 H2). +elim H3; intros. +rewrite (H4 a). +rewrite (H4 b). +case (Rle_dec b c); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 r)). +case (Rle_dec a c); intro. +ring. +elim n0; unfold antiderivative in H1; elim H1; intros; assumption. +split; [ left; assumption | right; reflexivity ]. +split; [ right; reflexivity | left; assumption ]. +assert (H2 := antiderivative_P2 _ _ _ _ _ _ H1 H0). +assert (H3 := antiderivative_Ucte _ _ _ c b H H2). +elim H3; intros. +rewrite (H4 c). +rewrite (H4 b). +case (Rle_dec b a); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 a0)). +case (Rle_dec c a); intro. +ring. +elim n0; unfold antiderivative in H1; elim H1; intros; assumption. +split; [ left; assumption | right; reflexivity ]. +split; [ right; reflexivity | left; assumption ]. +unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 a0)). (* a=b *) -Rewrite b0 in o; Rewrite b0. -Elim o; Intro. -Elim o1; Intro. -Assert H1 := (antiderivative_Ucte ? ? ? b c H H0). -Elim H1; Intros. -Assert H3 : ``b<=c``. -Unfold antiderivative in H; Elim H; Intros; Assumption. -Rewrite (H2 b). -Rewrite (H2 c). -Ring. -Split; [Assumption | Right; Reflexivity]. -Split; [Right; Reflexivity | Assumption]. -Assert H1 : ``b==c``. -Unfold antiderivative in H H0; Elim H; Elim H0; Intros; Apply Rle_antisym; Assumption. -Rewrite H1; Ring. -Elim o1; Intro. -Assert H1 : ``b==c``. -Unfold antiderivative in H H0; Elim H; Elim H0; Intros; Apply Rle_antisym; Assumption. -Rewrite H1; Ring. -Assert H1 := (antiderivative_Ucte ? ? ? c b H H0). -Elim H1; Intros. -Assert H3 : ``c<=b``. -Unfold antiderivative in H; Elim H; Intros; Assumption. -Rewrite (H2 c). -Rewrite (H2 b). -Ring. -Split; [Assumption | Right; Reflexivity]. -Split; [Right; Reflexivity | Assumption]. +rewrite b0 in o; rewrite b0. +elim o; intro. +elim o1; intro. +assert (H1 := antiderivative_Ucte _ _ _ b c H H0). +elim H1; intros. +assert (H3 : b <= c). +unfold antiderivative in H; elim H; intros; assumption. +rewrite (H2 b). +rewrite (H2 c). +ring. +split; [ assumption | right; reflexivity ]. +split; [ right; reflexivity | assumption ]. +assert (H1 : b = c). +unfold antiderivative in H, H0; elim H; elim H0; intros; apply Rle_antisym; + assumption. +rewrite H1; ring. +elim o1; intro. +assert (H1 : b = c). +unfold antiderivative in H, H0; elim H; elim H0; intros; apply Rle_antisym; + assumption. +rewrite H1; ring. +assert (H1 := antiderivative_Ucte _ _ _ c b H H0). +elim H1; intros. +assert (H3 : c <= b). +unfold antiderivative in H; elim H; intros; assumption. +rewrite (H2 c). +rewrite (H2 b). +ring. +split; [ assumption | right; reflexivity ]. +split; [ right; reflexivity | assumption ]. (* a>b & b<c *) -Case (total_order_T b c); Intro. -Elim s; Intro. -Elim o0; Intro. -Unfold antiderivative in H; Elim H; Clear H; Intros _ H. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). -Elim o1; Intro. -Elim o; Intro. -Assert H2 := (antiderivative_P2 ? ? ? ? ? ? H H1). -Assert H3 := (antiderivative_Ucte ? ? ? b c H0 H2). -Elim H3; Intros. -Rewrite (H4 b). -Rewrite (H4 c). -Case (total_order_Rle b a); Intro. -Case (total_order_Rle c a); Intro. -Assert H5 : ``a==c``. -Unfold antiderivative in H1; Elim H1; Intros; Apply Rle_antisym; Assumption. -Rewrite H5; Ring. -Ring. -Elim n; Left; Assumption. -Split; [Left; Assumption | Right; Reflexivity]. -Split; [Right; Reflexivity | Left; Assumption]. -Assert H2 := (antiderivative_P2 ? ? ? ? ? ? H0 H1). -Assert H3 := (antiderivative_Ucte ? ? ? b a H H2). -Elim H3; Intros. -Rewrite (H4 a). -Rewrite (H4 b). -Case (total_order_Rle b c); Intro. -Case (total_order_Rle a c); Intro. -Assert H5 : ``a==c``. -Unfold antiderivative in H1; Elim H1; Intros; Apply Rle_antisym; Assumption. -Rewrite H5; Ring. -Ring. -Elim n; Left; Assumption. -Split; [Right; Reflexivity | Left; Assumption]. -Split; [Left; Assumption | Right; Reflexivity]. -Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 a0)). +case (total_order_T b c); intro. +elim s; intro. +elim o0; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim o1; intro. +elim o; intro. +assert (H2 := antiderivative_P2 _ _ _ _ _ _ H H1). +assert (H3 := antiderivative_Ucte _ _ _ b c H0 H2). +elim H3; intros. +rewrite (H4 b). +rewrite (H4 c). +case (Rle_dec b a); intro. +case (Rle_dec c a); intro. +assert (H5 : a = c). +unfold antiderivative in H1; elim H1; intros; apply Rle_antisym; assumption. +rewrite H5; ring. +ring. +elim n; left; assumption. +split; [ left; assumption | right; reflexivity ]. +split; [ right; reflexivity | left; assumption ]. +assert (H2 := antiderivative_P2 _ _ _ _ _ _ H0 H1). +assert (H3 := antiderivative_Ucte _ _ _ b a H H2). +elim H3; intros. +rewrite (H4 a). +rewrite (H4 b). +case (Rle_dec b c); intro. +case (Rle_dec a c); intro. +assert (H5 : a = c). +unfold antiderivative in H1; elim H1; intros; apply Rle_antisym; assumption. +rewrite H5; ring. +ring. +elim n; left; assumption. +split; [ right; reflexivity | left; assumption ]. +split; [ left; assumption | right; reflexivity ]. +unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 a0)). (* a>b & b=c *) -Rewrite <- b0. -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or. -Rewrite <- b0 in o. -Elim o0; Intro. -Unfold antiderivative in H; Elim H; Clear H; Intros _ H. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). -Elim o; Intro. -Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 r)). -Assert H1 := (antiderivative_Ucte f x x0 b a H0 H). -Elim H1; Intros. -Rewrite (H2 b). -Rewrite (H2 a). -Ring. -Split; [Left; Assumption | Right; Reflexivity]. -Split; [Right; Reflexivity | Left; Assumption]. +rewrite <- b0. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r. +rewrite <- b0 in o. +elim o0; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim o; intro. +unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 r)). +assert (H1 := antiderivative_Ucte f x x0 b a H0 H). +elim H1; intros. +rewrite (H2 b). +rewrite (H2 a). +ring. +split; [ left; assumption | right; reflexivity ]. +split; [ right; reflexivity | left; assumption ]. (* a>b & b>c *) -Elim o0; Intro. -Unfold antiderivative in H; Elim H; Clear H; Intros _ H. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). -Elim o1; Intro. -Unfold antiderivative in H0; Elim H0; Clear H0; Intros _ H0. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 r0)). -Elim o; Intro. -Unfold antiderivative in H1; Elim H1; Clear H1; Intros _ H1. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 (Rlt_trans ? ? ? r0 r))). -Assert H2 := (antiderivative_P2 ? ? ? ? ? ? H0 H). -Assert H3 := (antiderivative_Ucte ? ? ? c a H1 H2). -Elim H3; Intros. -Assert H5 : ``c<=a``. -Unfold antiderivative in H1; Elim H1; Intros; Assumption. -Rewrite (H4 c). -Rewrite (H4 a). -Case (total_order_Rle a b); Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r1 r)). -Case (total_order_Rle c b); Intro. -Ring. -Elim n0; Left; Assumption. -Split; [Assumption | Right; Reflexivity]. -Split; [Right; Reflexivity | Assumption]. +elim o0; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim o1; intro. +unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 r0)). +elim o; intro. +unfold antiderivative in H1; elim H1; clear H1; intros _ H1. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 (Rlt_trans _ _ _ r0 r))). +assert (H2 := antiderivative_P2 _ _ _ _ _ _ H0 H). +assert (H3 := antiderivative_Ucte _ _ _ c a H1 H2). +elim H3; intros. +assert (H5 : c <= a). +unfold antiderivative in H1; elim H1; intros; assumption. +rewrite (H4 c). +rewrite (H4 a). +case (Rle_dec a b); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r1 r)). +case (Rle_dec c b); intro. +ring. +elim n0; left; assumption. +split; [ assumption | right; reflexivity ]. +split; [ right; reflexivity | assumption ]. Qed. - diff --git a/theories/Reals/PSeries_reg.v b/theories/Reals/PSeries_reg.v index 2576d9275..4111377b7 100644 --- a/theories/Reals/PSeries_reg.v +++ b/theories/Reals/PSeries_reg.v @@ -8,187 +8,252 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Ranalysis1. -Require Max. -Require Even. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Ranalysis1. +Require Import Max. +Require Import Even. Open Local Scope R_scope. -Definition Boule [x:R;r:posreal] : R -> Prop := [y:R]``(Rabsolu (y-x))<r``. +Definition Boule (x:R) (r:posreal) (y:R) : Prop := Rabs (y - x) < r. (* Uniform convergence *) -Definition CVU [fn:nat->R->R;f:R->R;x:R;r:posreal] : Prop := (eps:R)``0<eps``->(EX N:nat | (n:nat;y:R) (le N n)->(Boule x r y)->``(Rabsolu ((f y)-(fn n y)))<eps``). +Definition CVU (fn:nat -> R -> R) (f:R -> R) (x:R) + (r:posreal) : Prop := + forall eps:R, + 0 < eps -> + exists N : nat + | (forall (n:nat) (y:R), + (N <= n)%nat -> Boule x r y -> Rabs (f y - fn n y) < eps). (* Normal convergence *) -Definition CVN_r [fn:nat->R->R;r:posreal] : Type := (SigT ? [An:nat->R](sigTT R [l:R]((Un_cv [n:nat](sum_f_R0 [k:nat](Rabsolu (An k)) n) l)/\((n:nat)(y:R)(Boule R0 r y)->(Rle (Rabsolu (fn n y)) (An n)))))). +Definition CVN_r (fn:nat -> R -> R) (r:posreal) : Type := + sigT + (fun An:nat -> R => + sigT + (fun l:R => + Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (An k)) n) l /\ + (forall (n:nat) (y:R), Boule 0 r y -> Rabs (fn n y) <= An n))). -Definition CVN_R [fn:nat->R->R] : Type := (r:posreal) (CVN_r fn r). +Definition CVN_R (fn:nat -> R -> R) : Type := forall r:posreal, CVN_r fn r. -Definition SFL [fn:nat->R->R;cv:(x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l))] : R-> R := [y:R](Cases (cv y) of (existTT a b) => a end). +Definition SFL (fn:nat -> R -> R) + (cv:forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)) + (y:R) : R := match cv y with + | existT a b => a + end. (* In a complete space, normal convergence implies uniform convergence *) -Lemma CVN_CVU : (fn:nat->R->R;cv:(x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l));r:posreal) (CVN_r fn r) -> (CVU [n:nat](SP fn n) (SFL fn cv) ``0`` r). -Intros; Unfold CVU; Intros. -Unfold CVN_r in X. -Elim X; Intros An X0. -Elim X0; Intros s H0. -Elim H0; Intros. -Cut (Un_cv [n:nat](Rminus (sum_f_R0 [k:nat]``(Rabsolu (An k))`` n) s) R0). -Intro; Unfold Un_cv in H3. -Elim (H3 eps H); Intros N0 H4. -Exists N0; Intros. -Apply Rle_lt_trans with (Rabsolu (Rminus (sum_f_R0 [k:nat]``(Rabsolu (An k))`` n) s)). -Rewrite <- (Rabsolu_Ropp (Rminus (sum_f_R0 [k:nat]``(Rabsolu (An k))`` n) s)); Rewrite Ropp_distr3; Rewrite (Rabsolu_right (Rminus s (sum_f_R0 [k:nat]``(Rabsolu (An k))`` n))). -EApply sum_maj1. -Unfold SFL; Case (cv y); Intro. -Trivial. -Apply H1. -Intro; Elim H0; Intros. -Rewrite (Rabsolu_right (An n0)). -Apply H8; Apply H6. -Apply Rle_sym1; Apply Rle_trans with (Rabsolu (fn n0 y)). -Apply Rabsolu_pos. -Apply H8; Apply H6. -Apply Rle_sym1; Apply Rle_anti_compatibility with (sum_f_R0 [k:nat](Rabsolu (An k)) n). -Rewrite Rplus_Or; Unfold Rminus; Rewrite (Rplus_sym s); Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Ol; Apply sum_incr. -Apply H1. -Intro; Apply Rabsolu_pos. -Unfold R_dist in H4; Unfold Rminus in H4; Rewrite Ropp_O in H4. -Assert H7 := (H4 n H5). -Rewrite Rplus_Or in H7; Apply H7. -Unfold Un_cv in H1; Unfold Un_cv; Intros. -Elim (H1? H3); Intros. -Exists x; Intros. -Unfold R_dist; Unfold R_dist in H4. -Rewrite minus_R0; Apply H4; Assumption. +Lemma CVN_CVU : + forall (fn:nat -> R -> R) + (cv:forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)) + (r:posreal), CVN_r fn r -> CVU (fun n:nat => SP fn n) (SFL fn cv) 0 r. +intros; unfold CVU in |- *; intros. +unfold CVN_r in X. +elim X; intros An X0. +elim X0; intros s H0. +elim H0; intros. +cut (Un_cv (fun n:nat => sum_f_R0 (fun k:nat => Rabs (An k)) n - s) 0). +intro; unfold Un_cv in H3. +elim (H3 eps H); intros N0 H4. +exists N0; intros. +apply Rle_lt_trans with (Rabs (sum_f_R0 (fun k:nat => Rabs (An k)) n - s)). +rewrite <- (Rabs_Ropp (sum_f_R0 (fun k:nat => Rabs (An k)) n - s)); + rewrite Ropp_minus_distr'; + rewrite (Rabs_right (s - sum_f_R0 (fun k:nat => Rabs (An k)) n)). +eapply sum_maj1. +unfold SFL in |- *; case (cv y); intro. +trivial. +apply H1. +intro; elim H0; intros. +rewrite (Rabs_right (An n0)). +apply H8; apply H6. +apply Rle_ge; apply Rle_trans with (Rabs (fn n0 y)). +apply Rabs_pos. +apply H8; apply H6. +apply Rle_ge; + apply Rplus_le_reg_l with (sum_f_R0 (fun k:nat => Rabs (An k)) n). +rewrite Rplus_0_r; unfold Rminus in |- *; rewrite (Rplus_comm s); + rewrite <- Rplus_assoc; rewrite Rplus_opp_r; rewrite Rplus_0_l; + apply sum_incr. +apply H1. +intro; apply Rabs_pos. +unfold R_dist in H4; unfold Rminus in H4; rewrite Ropp_0 in H4. +assert (H7 := H4 n H5). +rewrite Rplus_0_r in H7; apply H7. +unfold Un_cv in H1; unfold Un_cv in |- *; intros. +elim (H1 _ H3); intros. +exists x; intros. +unfold R_dist in |- *; unfold R_dist in H4. +rewrite Rminus_0_r; apply H4; assumption. Qed. (* Each limit of a sequence of functions which converges uniformly is continue *) -Lemma CVU_continuity : (fn:nat->R->R;f:R->R;x:R;r:posreal) (CVU fn f x r) -> ((n:nat)(y:R) (Boule x r y)->(continuity_pt (fn n) y)) -> ((y:R) (Boule x r y) -> (continuity_pt f y)). -Intros; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. -Unfold CVU in H. -Cut ``0<eps/3``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. -Elim (H ? H3); Intros N0 H4. -Assert H5 := (H0 N0 y H1). -Cut (EXT del : posreal | (h:R) ``(Rabsolu h)<del`` -> (Boule x r ``y+h``) ). -Intro. -Elim H6; Intros del1 H7. -Unfold continuity_pt in H5; Unfold continue_in in H5; Unfold limit1_in in H5; Unfold limit_in in H5; Simpl in H5; Unfold R_dist in H5. -Elim (H5 ? H3); Intros del2 H8. -Pose del := (Rmin del1 del2). -Exists del; Intros. -Split. -Unfold del; Unfold Rmin; Case (total_order_Rle del1 del2); Intro. -Apply (cond_pos del1). -Elim H8; Intros; Assumption. -Intros; Apply Rle_lt_trans with ``(Rabsolu ((f x0)-(fn N0 x0)))+(Rabsolu ((fn N0 x0)-(f y)))``. -Replace ``(f x0)-(f y)`` with ``((f x0)-(fn N0 x0))+((fn N0 x0)-(f y))``; [Apply Rabsolu_triang | Ring]. -Apply Rle_lt_trans with ``(Rabsolu ((f x0)-(fn N0 x0)))+(Rabsolu ((fn N0 x0)-(fn N0 y)))+(Rabsolu ((fn N0 y)-(f y)))``. -Rewrite Rplus_assoc; Apply Rle_compatibility. -Replace ``(fn N0 x0)-(f y)`` with ``((fn N0 x0)-(fn N0 y))+((fn N0 y)-(f y))``; [Apply Rabsolu_triang | Ring]. -Replace ``eps`` with ``eps/3+eps/3+eps/3``. -Repeat Apply Rplus_lt. -Apply H4. -Apply le_n. -Replace x0 with ``y+(x0-y)``; [Idtac | Ring]; Apply H7. -Elim H9; Intros. -Apply Rlt_le_trans with del. -Assumption. -Unfold del; Apply Rmin_l. -Elim H8; Intros. -Apply H11. -Split. -Elim H9; Intros; Assumption. -Elim H9; Intros; Apply Rlt_le_trans with del. -Assumption. -Unfold del; Apply Rmin_r. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr3; Apply H4. -Apply le_n. -Assumption. -Apply r_Rmult_mult with ``3``. -Do 2 Rewrite Rmult_Rplus_distr; Unfold Rdiv; Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m. -Ring. -DiscrR. -DiscrR. -Cut ``0<r-(Rabsolu (x-y))``. -Intro; Exists (mkposreal ? H6). -Simpl; Intros. -Unfold Boule; Replace ``y+h-x`` with ``h+(y-x)``; [Idtac | Ring]; Apply Rle_lt_trans with ``(Rabsolu h)+(Rabsolu (y-x))``. -Apply Rabsolu_triang. -Apply Rlt_anti_compatibility with ``-(Rabsolu (x-y))``. -Rewrite <- (Rabsolu_Ropp ``y-x``); Rewrite Ropp_distr3. -Replace ``-(Rabsolu (x-y))+r`` with ``r-(Rabsolu (x-y))``. -Replace ``-(Rabsolu (x-y))+((Rabsolu h)+(Rabsolu (x-y)))`` with (Rabsolu h). -Apply H7. -Ring. -Ring. -Unfold Boule in H1; Rewrite <- (Rabsolu_Ropp ``x-y``); Rewrite Ropp_distr3; Apply Rlt_anti_compatibility with ``(Rabsolu (y-x))``. -Rewrite Rplus_Or; Replace ``(Rabsolu (y-x))+(r-(Rabsolu (y-x)))`` with ``(pos r)``; [Apply H1 | Ring]. +Lemma CVU_continuity : + forall (fn:nat -> R -> R) (f:R -> R) (x:R) (r:posreal), + CVU fn f x r -> + (forall (n:nat) (y:R), Boule x r y -> continuity_pt (fn n) y) -> + forall y:R, Boule x r y -> continuity_pt f y. +intros; unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +unfold CVU in H. +cut (0 < eps / 3); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H _ H3); intros N0 H4. +assert (H5 := H0 N0 y H1). +cut ( exists del : posreal | (forall h:R, Rabs h < del -> Boule x r (y + h))). +intro. +elim H6; intros del1 H7. +unfold continuity_pt in H5; unfold continue_in in H5; unfold limit1_in in H5; + unfold limit_in in H5; simpl in H5; unfold R_dist in H5. +elim (H5 _ H3); intros del2 H8. +pose (del := Rmin del1 del2). +exists del; intros. +split. +unfold del in |- *; unfold Rmin in |- *; case (Rle_dec del1 del2); intro. +apply (cond_pos del1). +elim H8; intros; assumption. +intros; + apply Rle_lt_trans with (Rabs (f x0 - fn N0 x0) + Rabs (fn N0 x0 - f y)). +replace (f x0 - f y) with (f x0 - fn N0 x0 + (fn N0 x0 - f y)); + [ apply Rabs_triang | ring ]. +apply Rle_lt_trans with + (Rabs (f x0 - fn N0 x0) + Rabs (fn N0 x0 - fn N0 y) + Rabs (fn N0 y - f y)). +rewrite Rplus_assoc; apply Rplus_le_compat_l. +replace (fn N0 x0 - f y) with (fn N0 x0 - fn N0 y + (fn N0 y - f y)); + [ apply Rabs_triang | ring ]. +replace eps with (eps / 3 + eps / 3 + eps / 3). +repeat apply Rplus_lt_compat. +apply H4. +apply le_n. +replace x0 with (y + (x0 - y)); [ idtac | ring ]; apply H7. +elim H9; intros. +apply Rlt_le_trans with del. +assumption. +unfold del in |- *; apply Rmin_l. +elim H8; intros. +apply H11. +split. +elim H9; intros; assumption. +elim H9; intros; apply Rlt_le_trans with del. +assumption. +unfold del in |- *; apply Rmin_r. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply H4. +apply le_n. +assumption. +apply Rmult_eq_reg_l with 3. +do 2 rewrite Rmult_plus_distr_l; unfold Rdiv in |- *; rewrite <- Rmult_assoc; + rewrite Rinv_r_simpl_m. +ring. +discrR. +discrR. +cut (0 < r - Rabs (x - y)). +intro; exists (mkposreal _ H6). +simpl in |- *; intros. +unfold Boule in |- *; replace (y + h - x) with (h + (y - x)); + [ idtac | ring ]; apply Rle_lt_trans with (Rabs h + Rabs (y - x)). +apply Rabs_triang. +apply Rplus_lt_reg_r with (- Rabs (x - y)). +rewrite <- (Rabs_Ropp (y - x)); rewrite Ropp_minus_distr'. +replace (- Rabs (x - y) + r) with (r - Rabs (x - y)). +replace (- Rabs (x - y) + (Rabs h + Rabs (x - y))) with (Rabs h). +apply H7. +ring. +ring. +unfold Boule in H1; rewrite <- (Rabs_Ropp (x - y)); rewrite Ropp_minus_distr'; + apply Rplus_lt_reg_r with (Rabs (y - x)). +rewrite Rplus_0_r; replace (Rabs (y - x) + (r - Rabs (y - x))) with (pos r); + [ apply H1 | ring ]. Qed. (**********) -Lemma continuity_pt_finite_SF : (fn:nat->R->R;N:nat;x:R) ((n:nat)(le n N)->(continuity_pt (fn n) x)) -> (continuity_pt [y:R](sum_f_R0 [k:nat]``(fn k y)`` N) x). -Intros; Induction N. -Simpl; Apply (H O); Apply le_n. -Simpl; Replace [y:R](Rplus (sum_f_R0 [k:nat](fn k y) N) (fn (S N) y)) with (plus_fct [y:R](sum_f_R0 [k:nat](fn k y) N) [y:R](fn (S N) y)); [Idtac | Reflexivity]. -Apply continuity_pt_plus. -Apply HrecN. -Intros; Apply H. -Apply le_trans with N; [Assumption | Apply le_n_Sn]. -Apply (H (S N)); Apply le_n. +Lemma continuity_pt_finite_SF : + forall (fn:nat -> R -> R) (N:nat) (x:R), + (forall n:nat, (n <= N)%nat -> continuity_pt (fn n) x) -> + continuity_pt (fun y:R => sum_f_R0 (fun k:nat => fn k y) N) x. +intros; induction N as [| N HrecN]. +simpl in |- *; apply (H 0%nat); apply le_n. +simpl in |- *; + replace (fun y:R => sum_f_R0 (fun k:nat => fn k y) N + fn (S N) y) with + ((fun y:R => sum_f_R0 (fun k:nat => fn k y) N) + (fun y:R => fn (S N) y))%F; + [ idtac | reflexivity ]. +apply continuity_pt_plus. +apply HrecN. +intros; apply H. +apply le_trans with N; [ assumption | apply le_n_Sn ]. +apply (H (S N)); apply le_n. Qed. (* Continuity and normal convergence *) -Lemma SFL_continuity_pt : (fn:nat->R->R;cv:(x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l));r:posreal) (CVN_r fn r) -> ((n:nat)(y:R) (Boule ``0`` r y) -> (continuity_pt (fn n) y)) -> ((y:R) (Boule ``0`` r y) -> (continuity_pt (SFL fn cv) y)). -Intros; EApply CVU_continuity. -Apply CVN_CVU. -Apply X. -Intros; Unfold SP; Apply continuity_pt_finite_SF. -Intros; Apply H. -Apply H1. -Apply H0. +Lemma SFL_continuity_pt : + forall (fn:nat -> R -> R) + (cv:forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)) + (r:posreal), + CVN_r fn r -> + (forall (n:nat) (y:R), Boule 0 r y -> continuity_pt (fn n) y) -> + forall y:R, Boule 0 r y -> continuity_pt (SFL fn cv) y. +intros; eapply CVU_continuity. +apply CVN_CVU. +apply X. +intros; unfold SP in |- *; apply continuity_pt_finite_SF. +intros; apply H. +apply H1. +apply H0. Qed. -Lemma SFL_continuity : (fn:nat->R->R;cv:(x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l))) (CVN_R fn) -> ((n:nat)(continuity (fn n))) -> (continuity (SFL fn cv)). -Intros; Unfold continuity; Intro. -Cut ``0<(Rabsolu x)+1``; [Intro | Apply ge0_plus_gt0_is_gt0; [Apply Rabsolu_pos | Apply Rlt_R0_R1]]. -Cut (Boule ``0`` (mkposreal ? H0) x). -Intro; EApply SFL_continuity_pt with (mkposreal ? H0). -Apply X. -Intros; Apply (H n y). -Apply H1. -Unfold Boule; Simpl; Rewrite minus_R0; Pattern 1 (Rabsolu x); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1. +Lemma SFL_continuity : + forall (fn:nat -> R -> R) + (cv:forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)), + CVN_R fn -> (forall n:nat, continuity (fn n)) -> continuity (SFL fn cv). +intros; unfold continuity in |- *; intro. +cut (0 < Rabs x + 1); + [ intro | apply Rplus_le_lt_0_compat; [ apply Rabs_pos | apply Rlt_0_1 ] ]. +cut (Boule 0 (mkposreal _ H0) x). +intro; eapply SFL_continuity_pt with (mkposreal _ H0). +apply X. +intros; apply (H n y). +apply H1. +unfold Boule in |- *; simpl in |- *; rewrite Rminus_0_r; + pattern (Rabs x) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_lt_compat_l; apply Rlt_0_1. Qed. (* As R is complete, normal convergence implies that (fn) is simply-uniformly convergent *) -Lemma CVN_R_CVS : (fn:nat->R->R) (CVN_R fn) -> ((x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l))). -Intros; Apply R_complete. -Unfold SP; Pose An := [N:nat](fn N x). -Change (Cauchy_crit_series An). -Apply cauchy_abs. -Unfold Cauchy_crit_series; Apply CV_Cauchy. -Unfold CVN_R in X; Cut ``0<(Rabsolu x)+1``. -Intro; Assert H0 := (X (mkposreal ? H)). -Unfold CVN_r in H0; Elim H0; Intros Bn H1. -Elim H1; Intros l H2. -Elim H2; Intros. -Apply Rseries_CV_comp with Bn. -Intro; Split. -Apply Rabsolu_pos. -Unfold An; Apply H4; Unfold Boule; Simpl; Rewrite minus_R0. -Pattern 1 (Rabsolu x); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1. -Apply existTT with l. -Cut (n:nat)``0<=(Bn n)``. -Intro; Unfold Un_cv in H3; Unfold Un_cv; Intros. -Elim (H3 ? H6); Intros. -Exists x0; Intros. -Replace (sum_f_R0 Bn n) with (sum_f_R0 [k:nat](Rabsolu (Bn k)) n). -Apply H7; Assumption. -Apply sum_eq; Intros; Apply Rabsolu_right; Apply Rle_sym1; Apply H5. -Intro; Apply Rle_trans with (Rabsolu (An n)). -Apply Rabsolu_pos. -Unfold An; Apply H4; Unfold Boule; Simpl; Rewrite minus_R0; Pattern 1 (Rabsolu x); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1. -Apply ge0_plus_gt0_is_gt0; [Apply Rabsolu_pos | Apply Rlt_R0_R1]. -Qed. +Lemma CVN_R_CVS : + forall fn:nat -> R -> R, + CVN_R fn -> forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l). +intros; apply R_complete. +unfold SP in |- *; pose (An := fun N:nat => fn N x). +change (Cauchy_crit_series An) in |- *. +apply cauchy_abs. +unfold Cauchy_crit_series in |- *; apply CV_Cauchy. +unfold CVN_R in X; cut (0 < Rabs x + 1). +intro; assert (H0 := X (mkposreal _ H)). +unfold CVN_r in H0; elim H0; intros Bn H1. +elim H1; intros l H2. +elim H2; intros. +apply Rseries_CV_comp with Bn. +intro; split. +apply Rabs_pos. +unfold An in |- *; apply H4; unfold Boule in |- *; simpl in |- *; + rewrite Rminus_0_r. +pattern (Rabs x) at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + apply Rlt_0_1. +apply existT with l. +cut (forall n:nat, 0 <= Bn n). +intro; unfold Un_cv in H3; unfold Un_cv in |- *; intros. +elim (H3 _ H6); intros. +exists x0; intros. +replace (sum_f_R0 Bn n) with (sum_f_R0 (fun k:nat => Rabs (Bn k)) n). +apply H7; assumption. +apply sum_eq; intros; apply Rabs_right; apply Rle_ge; apply H5. +intro; apply Rle_trans with (Rabs (An n)). +apply Rabs_pos. +unfold An in |- *; apply H4; unfold Boule in |- *; simpl in |- *; + rewrite Rminus_0_r; pattern (Rabs x) at 1 in |- *; + rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; apply Rlt_0_1. +apply Rplus_le_lt_0_compat; [ apply Rabs_pos | apply Rlt_0_1 ]. +Qed.
\ No newline at end of file diff --git a/theories/Reals/PartSum.v b/theories/Reals/PartSum.v index 090680cf1..c12aea9df 100644 --- a/theories/Reals/PartSum.v +++ b/theories/Reals/PartSum.v @@ -8,469 +8,596 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Rseries. -Require Rcomplete. -Require Max. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import Rseries. +Require Import Rcomplete. +Require Import Max. Open Local Scope R_scope. -Lemma tech1 : (An:nat->R;N:nat) ((n:nat)``(le n N)``->``0<(An n)``) -> ``0 < (sum_f_R0 An N)``. -Intros; Induction N. -Simpl; Apply H; Apply le_n. -Simpl; Apply gt0_plus_gt0_is_gt0. -Apply HrecN; Intros; Apply H; Apply le_S; Assumption. -Apply H; Apply le_n. +Lemma tech1 : + forall (An:nat -> R) (N:nat), + (forall n:nat, (n <= N)%nat -> 0 < An n) -> 0 < sum_f_R0 An N. +intros; induction N as [| N HrecN]. +simpl in |- *; apply H; apply le_n. +simpl in |- *; apply Rplus_lt_0_compat. +apply HrecN; intros; apply H; apply le_S; assumption. +apply H; apply le_n. Qed. (* Chasles' relation *) -Lemma tech2 : (An:nat->R;m,n:nat) (lt m n) -> (sum_f_R0 An n) == (Rplus (sum_f_R0 An m) (sum_f_R0 [i:nat]``(An (plus (S m) i))`` (minus n (S m)))). -Intros; Induction n. -Elim (lt_n_O ? H). -Cut (lt m n)\/m=n. -Intro; Elim H0; Intro. -Replace (sum_f_R0 An (S n)) with ``(sum_f_R0 An n)+(An (S n))``; [Idtac | Reflexivity]. -Replace (minus (S n) (S m)) with (S (minus n (S m))). -Replace (sum_f_R0 [i:nat](An (plus (S m) i)) (S (minus n (S m)))) with (Rplus (sum_f_R0 [i:nat](An (plus (S m) i)) (minus n (S m))) (An (plus (S m) (S (minus n (S m)))))); [Idtac | Reflexivity]. -Replace (plus (S m) (S (minus n (S m)))) with (S n). -Rewrite (Hrecn H1). -Ring. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Do 2 Rewrite S_INR; Rewrite minus_INR. -Rewrite S_INR; Ring. -Apply lt_le_S; Assumption. -Apply INR_eq; Rewrite S_INR; Repeat Rewrite minus_INR. -Repeat Rewrite S_INR; Ring. -Apply le_n_S; Apply lt_le_weak; Assumption. -Apply lt_le_S; Assumption. -Rewrite H1; Rewrite <- minus_n_n; Simpl. -Replace (plus n O) with n; [Reflexivity | Ring]. -Inversion H. -Right; Reflexivity. -Left; Apply lt_le_trans with (S m); [Apply lt_n_Sn | Assumption]. +Lemma tech2 : + forall (An:nat -> R) (m n:nat), + (m < n)%nat -> + sum_f_R0 An n = + sum_f_R0 An m + sum_f_R0 (fun i:nat => An (S m + i)%nat) (n - S m). +intros; induction n as [| n Hrecn]. +elim (lt_n_O _ H). +cut ((m < n)%nat \/ m = n). +intro; elim H0; intro. +replace (sum_f_R0 An (S n)) with (sum_f_R0 An n + An (S n)); + [ idtac | reflexivity ]. +replace (S n - S m)%nat with (S (n - S m)). +replace (sum_f_R0 (fun i:nat => An (S m + i)%nat) (S (n - S m))) with + (sum_f_R0 (fun i:nat => An (S m + i)%nat) (n - S m) + + An (S m + S (n - S m))%nat); [ idtac | reflexivity ]. +replace (S m + S (n - S m))%nat with (S n). +rewrite (Hrecn H1). +ring. +apply INR_eq; rewrite S_INR; rewrite plus_INR; do 2 rewrite S_INR; + rewrite minus_INR. +rewrite S_INR; ring. +apply lt_le_S; assumption. +apply INR_eq; rewrite S_INR; repeat rewrite minus_INR. +repeat rewrite S_INR; ring. +apply le_n_S; apply lt_le_weak; assumption. +apply lt_le_S; assumption. +rewrite H1; rewrite <- minus_n_n; simpl in |- *. +replace (n + 0)%nat with n; [ reflexivity | ring ]. +inversion H. +right; reflexivity. +left; apply lt_le_trans with (S m); [ apply lt_n_Sn | assumption ]. Qed. (* Sum of geometric sequences *) -Lemma tech3 : (k:R;N:nat) ``k<>1`` -> (sum_f_R0 [i:nat](pow k i) N)==``(1-(pow k (S N)))/(1-k)``. -Intros; Cut ``1-k<>0``. -Intro; Induction N. -Simpl; Rewrite Rmult_1r; Unfold Rdiv; Rewrite <- Rinv_r_sym. -Reflexivity. -Apply H0. -Replace (sum_f_R0 ([i:nat](pow k i)) (S N)) with (Rplus (sum_f_R0 [i:nat](pow k i) N) (pow k (S N))); [Idtac | Reflexivity]; Rewrite HrecN; Replace ``(1-(pow k (S N)))/(1-k)+(pow k (S N))`` with ``((1-(pow k (S N)))+(1-k)*(pow k (S N)))/(1-k)``. -Apply r_Rmult_mult with ``1-k``. -Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/(1-k)``); Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [ Do 2 Rewrite Rmult_1l; Simpl; Ring | Apply H0]. -Apply H0. -Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Rewrite (Rmult_sym ``1-k``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Reflexivity. -Apply H0. -Apply Rminus_eq_contra; Red; Intro; Elim H; Symmetry; Assumption. +Lemma tech3 : + forall (k:R) (N:nat), + k <> 1 -> sum_f_R0 (fun i:nat => k ^ i) N = (1 - k ^ S N) / (1 - k). +intros; cut (1 - k <> 0). +intro; induction N as [| N HrecN]. +simpl in |- *; rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite <- Rinv_r_sym. +reflexivity. +apply H0. +replace (sum_f_R0 (fun i:nat => k ^ i) (S N)) with + (sum_f_R0 (fun i:nat => k ^ i) N + k ^ S N); [ idtac | reflexivity ]; + rewrite HrecN; + replace ((1 - k ^ S N) / (1 - k) + k ^ S N) with + ((1 - k ^ S N + (1 - k) * k ^ S N) / (1 - k)). +apply Rmult_eq_reg_l with (1 - k). +unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ (1 - k))); + repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; + [ do 2 rewrite Rmult_1_l; simpl in |- *; ring | apply H0 ]. +apply H0. +unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; rewrite (Rmult_comm (1 - k)); + repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; reflexivity. +apply H0. +apply Rminus_eq_contra; red in |- *; intro; elim H; symmetry in |- *; + assumption. Qed. -Lemma tech4 : (An:nat->R;k:R;N:nat) ``0<=k`` -> ((i:nat)``(An (S i))<k*(An i)``) -> ``(An N)<=(An O)*(pow k N)``. -Intros; Induction N. -Simpl; Right; Ring. -Apply Rle_trans with ``k*(An N)``. -Left; Apply (H0 N). -Replace (S N) with (plus N (1)); [Idtac | Ring]. -Rewrite pow_add; Simpl; Rewrite Rmult_1r; Replace ``(An O)*((pow k N)*k)`` with ``k*((An O)*(pow k N))``; [Idtac | Ring]; Apply Rle_monotony. -Assumption. -Apply HrecN. +Lemma tech4 : + forall (An:nat -> R) (k:R) (N:nat), + 0 <= k -> (forall i:nat, An (S i) < k * An i) -> An N <= An 0%nat * k ^ N. +intros; induction N as [| N HrecN]. +simpl in |- *; right; ring. +apply Rle_trans with (k * An N). +left; apply (H0 N). +replace (S N) with (N + 1)%nat; [ idtac | ring ]. +rewrite pow_add; simpl in |- *; rewrite Rmult_1_r; + replace (An 0%nat * (k ^ N * k)) with (k * (An 0%nat * k ^ N)); + [ idtac | ring ]; apply Rmult_le_compat_l. +assumption. +apply HrecN. Qed. -Lemma tech5 : (An:nat->R;N:nat) (sum_f_R0 An (S N))==``(sum_f_R0 An N)+(An (S N))``. -Intros; Reflexivity. +Lemma tech5 : + forall (An:nat -> R) (N:nat), sum_f_R0 An (S N) = sum_f_R0 An N + An (S N). +intros; reflexivity. Qed. -Lemma tech6 : (An:nat->R;k:R;N:nat) ``0<=k`` -> ((i:nat)``(An (S i))<k*(An i)``) -> (Rle (sum_f_R0 An N) (Rmult (An O) (sum_f_R0 [i:nat](pow k i) N))). -Intros; Induction N. -Simpl; Right; Ring. -Apply Rle_trans with (Rplus (Rmult (An O) (sum_f_R0 [i:nat](pow k i) N)) (An (S N))). -Rewrite tech5; Do 2 Rewrite <- (Rplus_sym (An (S N))); Apply Rle_compatibility. -Apply HrecN. -Rewrite tech5 ; Rewrite Rmult_Rplus_distr; Apply Rle_compatibility. -Apply tech4; Assumption. +Lemma tech6 : + forall (An:nat -> R) (k:R) (N:nat), + 0 <= k -> + (forall i:nat, An (S i) < k * An i) -> + sum_f_R0 An N <= An 0%nat * sum_f_R0 (fun i:nat => k ^ i) N. +intros; induction N as [| N HrecN]. +simpl in |- *; right; ring. +apply Rle_trans with (An 0%nat * sum_f_R0 (fun i:nat => k ^ i) N + An (S N)). +rewrite tech5; do 2 rewrite <- (Rplus_comm (An (S N))); + apply Rplus_le_compat_l. +apply HrecN. +rewrite tech5; rewrite Rmult_plus_distr_l; apply Rplus_le_compat_l. +apply tech4; assumption. Qed. -Lemma tech7 : (r1,r2:R) ``r1<>0`` -> ``r2<>0`` -> ``r1<>r2`` -> ``/r1<>/r2``. -Intros; Red; Intro. -Assert H3 := (Rmult_mult_r r1 ? ? H2). -Rewrite <- Rinv_r_sym in H3; [Idtac | Assumption]. -Assert H4 := (Rmult_mult_r r2 ? ? H3). -Rewrite Rmult_1r in H4; Rewrite <- Rmult_assoc in H4. -Rewrite Rinv_r_simpl_m in H4; [Idtac | Assumption]. -Elim H1; Symmetry; Assumption. +Lemma tech7 : forall r1 r2:R, r1 <> 0 -> r2 <> 0 -> r1 <> r2 -> / r1 <> / r2. +intros; red in |- *; intro. +assert (H3 := Rmult_eq_compat_l r1 _ _ H2). +rewrite <- Rinv_r_sym in H3; [ idtac | assumption ]. +assert (H4 := Rmult_eq_compat_l r2 _ _ H3). +rewrite Rmult_1_r in H4; rewrite <- Rmult_assoc in H4. +rewrite Rinv_r_simpl_m in H4; [ idtac | assumption ]. +elim H1; symmetry in |- *; assumption. Qed. -Lemma tech11 : (An,Bn,Cn:nat->R;N:nat) ((i:nat) (An i)==``(Bn i)-(Cn i)``) -> (sum_f_R0 An N)==``(sum_f_R0 Bn N)-(sum_f_R0 Cn N)``. -Intros; Induction N. -Simpl; Apply H. -Do 3 Rewrite tech5; Rewrite HrecN; Rewrite (H (S N)); Ring. +Lemma tech11 : + forall (An Bn Cn:nat -> R) (N:nat), + (forall i:nat, An i = Bn i - Cn i) -> + sum_f_R0 An N = sum_f_R0 Bn N - sum_f_R0 Cn N. +intros; induction N as [| N HrecN]. +simpl in |- *; apply H. +do 3 rewrite tech5; rewrite HrecN; rewrite (H (S N)); ring. Qed. -Lemma tech12 : (An:nat->R;x:R;l:R) (Un_cv [N:nat](sum_f_R0 [i:nat]``(An i)*(pow x i)`` N) l) -> (Pser An x l). -Intros; Unfold Pser; Unfold infinit_sum; Unfold Un_cv in H; Assumption. +Lemma tech12 : + forall (An:nat -> R) (x l:R), + Un_cv (fun N:nat => sum_f_R0 (fun i:nat => An i * x ^ i) N) l -> + Pser An x l. +intros; unfold Pser in |- *; unfold infinit_sum in |- *; unfold Un_cv in H; + assumption. Qed. -Lemma scal_sum : (An:nat->R;N:nat;x:R) (Rmult x (sum_f_R0 An N))==(sum_f_R0 [i:nat]``(An i)*x`` N). -Intros; Induction N. -Simpl; Ring. -Do 2 Rewrite tech5. -Rewrite Rmult_Rplus_distr; Rewrite <- HrecN; Ring. +Lemma scal_sum : + forall (An:nat -> R) (N:nat) (x:R), + x * sum_f_R0 An N = sum_f_R0 (fun i:nat => An i * x) N. +intros; induction N as [| N HrecN]. +simpl in |- *; ring. +do 2 rewrite tech5. +rewrite Rmult_plus_distr_l; rewrite <- HrecN; ring. Qed. -Lemma decomp_sum : (An:nat->R;N:nat) (lt O N) -> (sum_f_R0 An N)==(Rplus (An O) (sum_f_R0 [i:nat](An (S i)) (pred N))). -Intros; Induction N. -Elim (lt_n_n ? H). -Cut (lt O N)\/N=O. -Intro; Elim H0; Intro. -Cut (S (pred N))=(pred (S N)). -Intro; Rewrite <- H2. -Do 2 Rewrite tech5. -Replace (S (S (pred N))) with (S N). -Rewrite (HrecN H1); Ring. -Rewrite H2; Simpl; Reflexivity. -Assert H2 := (O_or_S N). -Elim H2; Intros. -Elim a; Intros. -Rewrite <- p. -Simpl; Reflexivity. -Rewrite <- b in H1; Elim (lt_n_n ? H1). -Rewrite H1; Simpl; Reflexivity. -Inversion H. -Right; Reflexivity. -Left; Apply lt_le_trans with (1); [Apply lt_O_Sn | Assumption]. +Lemma decomp_sum : + forall (An:nat -> R) (N:nat), + (0 < N)%nat -> + sum_f_R0 An N = An 0%nat + sum_f_R0 (fun i:nat => An (S i)) (pred N). +intros; induction N as [| N HrecN]. +elim (lt_irrefl _ H). +cut ((0 < N)%nat \/ N = 0%nat). +intro; elim H0; intro. +cut (S (pred N) = pred (S N)). +intro; rewrite <- H2. +do 2 rewrite tech5. +replace (S (S (pred N))) with (S N). +rewrite (HrecN H1); ring. +rewrite H2; simpl in |- *; reflexivity. +assert (H2 := O_or_S N). +elim H2; intros. +elim a; intros. +rewrite <- p. +simpl in |- *; reflexivity. +rewrite <- b in H1; elim (lt_irrefl _ H1). +rewrite H1; simpl in |- *; reflexivity. +inversion H. +right; reflexivity. +left; apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ]. Qed. -Lemma plus_sum : (An,Bn:nat->R;N:nat) (sum_f_R0 [i:nat]``(An i)+(Bn i)`` N)==``(sum_f_R0 An N)+(sum_f_R0 Bn N)``. -Intros; Induction N. -Simpl; Ring. -Do 3 Rewrite tech5; Rewrite HrecN; Ring. +Lemma plus_sum : + forall (An Bn:nat -> R) (N:nat), + sum_f_R0 (fun i:nat => An i + Bn i) N = sum_f_R0 An N + sum_f_R0 Bn N. +intros; induction N as [| N HrecN]. +simpl in |- *; ring. +do 3 rewrite tech5; rewrite HrecN; ring. Qed. -Lemma sum_eq : (An,Bn:nat->R;N:nat) ((i:nat)(le i N)->(An i)==(Bn i)) -> (sum_f_R0 An N)==(sum_f_R0 Bn N). -Intros; Induction N. -Simpl; Apply H; Apply le_n. -Do 2 Rewrite tech5; Rewrite HrecN. -Rewrite (H (S N)); [Reflexivity | Apply le_n]. -Intros; Apply H; Apply le_trans with N; [Assumption | Apply le_n_Sn]. +Lemma sum_eq : + forall (An Bn:nat -> R) (N:nat), + (forall i:nat, (i <= N)%nat -> An i = Bn i) -> + sum_f_R0 An N = sum_f_R0 Bn N. +intros; induction N as [| N HrecN]. +simpl in |- *; apply H; apply le_n. +do 2 rewrite tech5; rewrite HrecN. +rewrite (H (S N)); [ reflexivity | apply le_n ]. +intros; apply H; apply le_trans with N; [ assumption | apply le_n_Sn ]. Qed. (* Unicity of the limit defined by convergent series *) -Lemma unicity_sum : (An:nat->R;l1,l2:R) (infinit_sum An l1) -> (infinit_sum An l2) -> l1 == l2. -Unfold infinit_sum; Intros. -Case (Req_EM l1 l2); Intro. -Assumption. -Cut ``0<(Rabsolu ((l1-l2)/2))``; [Intro | Apply Rabsolu_pos_lt]. -Elim (H ``(Rabsolu ((l1-l2)/2))`` H2); Intros. -Elim (H0 ``(Rabsolu ((l1-l2)/2))`` H2); Intros. -Pose N := (max x0 x); Cut (ge N x0). -Cut (ge N x). -Intros; Assert H7 := (H3 N H5); Assert H8 := (H4 N H6). -Cut ``(Rabsolu (l1-l2)) <= (R_dist (sum_f_R0 An N) l1) + (R_dist (sum_f_R0 An N) l2)``. -Intro; Assert H10 := (Rplus_lt ? ? ? ? H7 H8); Assert H11 := (Rle_lt_trans ? ? ? H9 H10); Unfold Rdiv in H11; Rewrite Rabsolu_mult in H11. -Cut ``(Rabsolu (/2))==/2``. -Intro; Rewrite H12 in H11; Assert H13 := double_var; Unfold Rdiv in H13; Rewrite <- H13 in H11. -Elim (Rlt_antirefl ? H11). -Apply Rabsolu_right; Left; Change ``0</2``; Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H20; Generalize (lt_INR_0 (2) (neq_O_lt (2) H20)); Unfold INR; Intro; Assumption | Discriminate]. -Unfold R_dist; Rewrite <- (Rabsolu_Ropp ``(sum_f_R0 An N)-l1``); Rewrite Ropp_distr3. -Replace ``l1-l2`` with ``((l1-(sum_f_R0 An N)))+((sum_f_R0 An N)-l2)``; [Idtac | Ring]. -Apply Rabsolu_triang. -Unfold ge; Unfold N; Apply le_max_r. -Unfold ge; Unfold N; Apply le_max_l. -Unfold Rdiv; Apply prod_neq_R0. -Apply Rminus_eq_contra; Assumption. -Apply Rinv_neq_R0; DiscrR. +Lemma uniqueness_sum : + forall (An:nat -> R) (l1 l2:R), + infinit_sum An l1 -> infinit_sum An l2 -> l1 = l2. +unfold infinit_sum in |- *; intros. +case (Req_dec l1 l2); intro. +assumption. +cut (0 < Rabs ((l1 - l2) / 2)); [ intro | apply Rabs_pos_lt ]. +elim (H (Rabs ((l1 - l2) / 2)) H2); intros. +elim (H0 (Rabs ((l1 - l2) / 2)) H2); intros. +pose (N := max x0 x); cut (N >= x0)%nat. +cut (N >= x)%nat. +intros; assert (H7 := H3 N H5); assert (H8 := H4 N H6). +cut (Rabs (l1 - l2) <= R_dist (sum_f_R0 An N) l1 + R_dist (sum_f_R0 An N) l2). +intro; assert (H10 := Rplus_lt_compat _ _ _ _ H7 H8); + assert (H11 := Rle_lt_trans _ _ _ H9 H10); unfold Rdiv in H11; + rewrite Rabs_mult in H11. +cut (Rabs (/ 2) = / 2). +intro; rewrite H12 in H11; assert (H13 := double_var); unfold Rdiv in H13; + rewrite <- H13 in H11. +elim (Rlt_irrefl _ H11). +apply Rabs_right; left; change (0 < / 2) in |- *; apply Rinv_0_lt_compat; + cut (0%nat <> 2%nat); + [ intro H20; generalize (lt_INR_0 2 (neq_O_lt 2 H20)); unfold INR in |- *; + intro; assumption + | discriminate ]. +unfold R_dist in |- *; rewrite <- (Rabs_Ropp (sum_f_R0 An N - l1)); + rewrite Ropp_minus_distr'. +replace (l1 - l2) with (l1 - sum_f_R0 An N + (sum_f_R0 An N - l2)); + [ idtac | ring ]. +apply Rabs_triang. +unfold ge in |- *; unfold N in |- *; apply le_max_r. +unfold ge in |- *; unfold N in |- *; apply le_max_l. +unfold Rdiv in |- *; apply prod_neq_R0. +apply Rminus_eq_contra; assumption. +apply Rinv_neq_0_compat; discrR. Qed. -Lemma minus_sum : (An,Bn:nat->R;N:nat) (sum_f_R0 [i:nat]``(An i)-(Bn i)`` N)==``(sum_f_R0 An N)-(sum_f_R0 Bn N)``. -Intros; Induction N. -Simpl; Ring. -Do 3 Rewrite tech5; Rewrite HrecN; Ring. +Lemma minus_sum : + forall (An Bn:nat -> R) (N:nat), + sum_f_R0 (fun i:nat => An i - Bn i) N = sum_f_R0 An N - sum_f_R0 Bn N. +intros; induction N as [| N HrecN]. +simpl in |- *; ring. +do 3 rewrite tech5; rewrite HrecN; ring. Qed. -Lemma sum_decomposition : (An:nat->R;N:nat) (Rplus (sum_f_R0 [l:nat](An (mult (2) l)) (S N)) (sum_f_R0 [l:nat](An (S (mult (2) l))) N))==(sum_f_R0 An (mult (2) (S N))). -Intros. -Induction N. -Simpl; Ring. -Rewrite tech5. -Rewrite (tech5 [l:nat](An (S (mult (2) l))) N). -Replace (mult (2) (S (S N))) with (S (S (mult (2) (S N)))). -Rewrite (tech5 An (S (mult (2) (S N)))). -Rewrite (tech5 An (mult (2) (S N))). -Rewrite <- HrecN. -Ring. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR;Repeat Rewrite S_INR. -Ring. +Lemma sum_decomposition : + forall (An:nat -> R) (N:nat), + sum_f_R0 (fun l:nat => An (2 * l)%nat) (S N) + + sum_f_R0 (fun l:nat => An (S (2 * l))) N = sum_f_R0 An (2 * S N). +intros. +induction N as [| N HrecN]. +simpl in |- *; ring. +rewrite tech5. +rewrite (tech5 (fun l:nat => An (S (2 * l))) N). +replace (2 * S (S N))%nat with (S (S (2 * S N))). +rewrite (tech5 An (S (2 * S N))). +rewrite (tech5 An (2 * S N)). +rewrite <- HrecN. +ring. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR. +ring. Qed. -Lemma sum_Rle : (An,Bn:nat->R;N:nat) ((n:nat)(le n N)->``(An n)<=(Bn n)``) -> ``(sum_f_R0 An N)<=(sum_f_R0 Bn N)``. -Intros. -Induction N. -Simpl; Apply H. -Apply le_n. -Do 2 Rewrite tech5. -Apply Rle_trans with ``(sum_f_R0 An N)+(Bn (S N))``. -Apply Rle_compatibility. -Apply H. -Apply le_n. -Do 2 Rewrite <- (Rplus_sym ``(Bn (S N))``). -Apply Rle_compatibility. -Apply HrecN. -Intros; Apply H. -Apply le_trans with N; [Assumption | Apply le_n_Sn]. +Lemma sum_Rle : + forall (An Bn:nat -> R) (N:nat), + (forall n:nat, (n <= N)%nat -> An n <= Bn n) -> + sum_f_R0 An N <= sum_f_R0 Bn N. +intros. +induction N as [| N HrecN]. +simpl in |- *; apply H. +apply le_n. +do 2 rewrite tech5. +apply Rle_trans with (sum_f_R0 An N + Bn (S N)). +apply Rplus_le_compat_l. +apply H. +apply le_n. +do 2 rewrite <- (Rplus_comm (Bn (S N))). +apply Rplus_le_compat_l. +apply HrecN. +intros; apply H. +apply le_trans with N; [ assumption | apply le_n_Sn ]. Qed. -Lemma sum_Rabsolu : (An:nat->R;N:nat) (Rle (Rabsolu (sum_f_R0 An N)) (sum_f_R0 [l:nat](Rabsolu (An l)) N)). -Intros. -Induction N. -Simpl. -Right; Reflexivity. -Do 2 Rewrite tech5. -Apply Rle_trans with ``(Rabsolu (sum_f_R0 An N))+(Rabsolu (An (S N)))``. -Apply Rabsolu_triang. -Do 2 Rewrite <- (Rplus_sym (Rabsolu (An (S N)))). -Apply Rle_compatibility. -Apply HrecN. +Lemma Rsum_abs : + forall (An:nat -> R) (N:nat), + Rabs (sum_f_R0 An N) <= sum_f_R0 (fun l:nat => Rabs (An l)) N. +intros. +induction N as [| N HrecN]. +simpl in |- *. +right; reflexivity. +do 2 rewrite tech5. +apply Rle_trans with (Rabs (sum_f_R0 An N) + Rabs (An (S N))). +apply Rabs_triang. +do 2 rewrite <- (Rplus_comm (Rabs (An (S N)))). +apply Rplus_le_compat_l. +apply HrecN. Qed. -Lemma sum_cte : (x:R;N:nat) (sum_f_R0 [_:nat]x N) == ``x*(INR (S N))``. -Intros. -Induction N. -Simpl; Ring. -Rewrite tech5. -Rewrite HrecN; Repeat Rewrite S_INR; Ring. +Lemma sum_cte : + forall (x:R) (N:nat), sum_f_R0 (fun _:nat => x) N = x * INR (S N). +intros. +induction N as [| N HrecN]. +simpl in |- *; ring. +rewrite tech5. +rewrite HrecN; repeat rewrite S_INR; ring. Qed. (**********) -Lemma sum_growing : (An,Bn:nat->R;N:nat) ((n:nat)``(An n)<=(Bn n)``)->``(sum_f_R0 An N)<=(sum_f_R0 Bn N)``. -Intros. -Induction N. -Simpl; Apply H. -Do 2 Rewrite tech5. -Apply Rle_trans with ``(sum_f_R0 An N)+(Bn (S N))``. -Apply Rle_compatibility; Apply H. -Do 2 Rewrite <- (Rplus_sym (Bn (S N))). -Apply Rle_compatibility; Apply HrecN. +Lemma sum_growing : + forall (An Bn:nat -> R) (N:nat), + (forall n:nat, An n <= Bn n) -> sum_f_R0 An N <= sum_f_R0 Bn N. +intros. +induction N as [| N HrecN]. +simpl in |- *; apply H. +do 2 rewrite tech5. +apply Rle_trans with (sum_f_R0 An N + Bn (S N)). +apply Rplus_le_compat_l; apply H. +do 2 rewrite <- (Rplus_comm (Bn (S N))). +apply Rplus_le_compat_l; apply HrecN. Qed. (**********) -Lemma Rabsolu_triang_gen : (An:nat->R;N:nat) (Rle (Rabsolu (sum_f_R0 An N)) (sum_f_R0 [i:nat](Rabsolu (An i)) N)). -Intros. -Induction N. -Simpl. -Right; Reflexivity. -Do 2 Rewrite tech5. -Apply Rle_trans with ``(Rabsolu ((sum_f_R0 An N)))+(Rabsolu (An (S N)))``. -Apply Rabsolu_triang. -Do 2 Rewrite <- (Rplus_sym (Rabsolu (An (S N)))). -Apply Rle_compatibility; Apply HrecN. +Lemma Rabs_triang_gen : + forall (An:nat -> R) (N:nat), + Rabs (sum_f_R0 An N) <= sum_f_R0 (fun i:nat => Rabs (An i)) N. +intros. +induction N as [| N HrecN]. +simpl in |- *. +right; reflexivity. +do 2 rewrite tech5. +apply Rle_trans with (Rabs (sum_f_R0 An N) + Rabs (An (S N))). +apply Rabs_triang. +do 2 rewrite <- (Rplus_comm (Rabs (An (S N)))). +apply Rplus_le_compat_l; apply HrecN. Qed. (**********) -Lemma cond_pos_sum : (An:nat->R;N:nat) ((n:nat)``0<=(An n)``) -> ``0<=(sum_f_R0 An N)``. -Intros. -Induction N. -Simpl; Apply H. -Rewrite tech5. -Apply ge0_plus_ge0_is_ge0. -Apply HrecN. -Apply H. +Lemma cond_pos_sum : + forall (An:nat -> R) (N:nat), + (forall n:nat, 0 <= An n) -> 0 <= sum_f_R0 An N. +intros. +induction N as [| N HrecN]. +simpl in |- *; apply H. +rewrite tech5. +apply Rplus_le_le_0_compat. +apply HrecN. +apply H. Qed. (* Cauchy's criterion for series *) -Definition Cauchy_crit_series [An:nat->R] : Prop := (Cauchy_crit [N:nat](sum_f_R0 An N)). +Definition Cauchy_crit_series (An:nat -> R) : Prop := + Cauchy_crit (fun N:nat => sum_f_R0 An N). (* If (|An|) satisfies the Cauchy's criterion for series, then (An) too *) -Lemma cauchy_abs : (An:nat->R) (Cauchy_crit_series [i:nat](Rabsolu (An i))) -> (Cauchy_crit_series An). -Unfold Cauchy_crit_series; Unfold Cauchy_crit. -Intros. -Elim (H eps H0); Intros. -Exists x. -Intros. -Cut (Rle (R_dist (sum_f_R0 An n) (sum_f_R0 An m)) (R_dist (sum_f_R0 [i:nat](Rabsolu (An i)) n) (sum_f_R0 [i:nat](Rabsolu (An i)) m))). -Intro. -Apply Rle_lt_trans with (R_dist (sum_f_R0 [i:nat](Rabsolu (An i)) n) (sum_f_R0 [i:nat](Rabsolu (An i)) m)). -Assumption. -Apply H1; Assumption. -Assert H4 := (lt_eq_lt_dec n m). -Elim H4; Intro. -Elim a; Intro. -Rewrite (tech2 An n m); [Idtac | Assumption]. -Rewrite (tech2 [i:nat](Rabsolu (An i)) n m); [Idtac | Assumption]. -Unfold R_dist. -Unfold Rminus. -Do 2 Rewrite Ropp_distr1. -Do 2 Rewrite <- Rplus_assoc. -Do 2 Rewrite Rplus_Ropp_r. -Do 2 Rewrite Rplus_Ol. -Do 2 Rewrite Rabsolu_Ropp. -Rewrite (Rabsolu_right (sum_f_R0 [i:nat](Rabsolu (An (plus (S n) i))) (minus m (S n)))). -Pose Bn:=[i:nat](An (plus (S n) i)). -Replace [i:nat](Rabsolu (An (plus (S n) i))) with [i:nat](Rabsolu (Bn i)). -Apply Rabsolu_triang_gen. -Unfold Bn; Reflexivity. -Apply Rle_sym1. -Apply cond_pos_sum. -Intro; Apply Rabsolu_pos. -Rewrite b. -Unfold R_dist. -Unfold Rminus; Do 2 Rewrite Rplus_Ropp_r. -Rewrite Rabsolu_R0; Right; Reflexivity. -Rewrite (tech2 An m n); [Idtac | Assumption]. -Rewrite (tech2 [i:nat](Rabsolu (An i)) m n); [Idtac | Assumption]. -Unfold R_dist. -Unfold Rminus. -Do 2 Rewrite Rplus_assoc. -Rewrite (Rplus_sym (sum_f_R0 An m)). -Rewrite (Rplus_sym (sum_f_R0 [i:nat](Rabsolu (An i)) m)). -Do 2 Rewrite Rplus_assoc. -Do 2 Rewrite Rplus_Ropp_l. -Do 2 Rewrite Rplus_Or. -Rewrite (Rabsolu_right (sum_f_R0 [i:nat](Rabsolu (An (plus (S m) i))) (minus n (S m)))). -Pose Bn:=[i:nat](An (plus (S m) i)). -Replace [i:nat](Rabsolu (An (plus (S m) i))) with [i:nat](Rabsolu (Bn i)). -Apply Rabsolu_triang_gen. -Unfold Bn; Reflexivity. -Apply Rle_sym1. -Apply cond_pos_sum. -Intro; Apply Rabsolu_pos. +Lemma cauchy_abs : + forall An:nat -> R, + Cauchy_crit_series (fun i:nat => Rabs (An i)) -> Cauchy_crit_series An. +unfold Cauchy_crit_series in |- *; unfold Cauchy_crit in |- *. +intros. +elim (H eps H0); intros. +exists x. +intros. +cut + (R_dist (sum_f_R0 An n) (sum_f_R0 An m) <= + R_dist (sum_f_R0 (fun i:nat => Rabs (An i)) n) + (sum_f_R0 (fun i:nat => Rabs (An i)) m)). +intro. +apply Rle_lt_trans with + (R_dist (sum_f_R0 (fun i:nat => Rabs (An i)) n) + (sum_f_R0 (fun i:nat => Rabs (An i)) m)). +assumption. +apply H1; assumption. +assert (H4 := lt_eq_lt_dec n m). +elim H4; intro. +elim a; intro. +rewrite (tech2 An n m); [ idtac | assumption ]. +rewrite (tech2 (fun i:nat => Rabs (An i)) n m); [ idtac | assumption ]. +unfold R_dist in |- *. +unfold Rminus in |- *. +do 2 rewrite Ropp_plus_distr. +do 2 rewrite <- Rplus_assoc. +do 2 rewrite Rplus_opp_r. +do 2 rewrite Rplus_0_l. +do 2 rewrite Rabs_Ropp. +rewrite + (Rabs_right (sum_f_R0 (fun i:nat => Rabs (An (S n + i)%nat)) (m - S n))) + . +pose (Bn := fun i:nat => An (S n + i)%nat). +replace (fun i:nat => Rabs (An (S n + i)%nat)) with + (fun i:nat => Rabs (Bn i)). +apply Rabs_triang_gen. +unfold Bn in |- *; reflexivity. +apply Rle_ge. +apply cond_pos_sum. +intro; apply Rabs_pos. +rewrite b. +unfold R_dist in |- *. +unfold Rminus in |- *; do 2 rewrite Rplus_opp_r. +rewrite Rabs_R0; right; reflexivity. +rewrite (tech2 An m n); [ idtac | assumption ]. +rewrite (tech2 (fun i:nat => Rabs (An i)) m n); [ idtac | assumption ]. +unfold R_dist in |- *. +unfold Rminus in |- *. +do 2 rewrite Rplus_assoc. +rewrite (Rplus_comm (sum_f_R0 An m)). +rewrite (Rplus_comm (sum_f_R0 (fun i:nat => Rabs (An i)) m)). +do 2 rewrite Rplus_assoc. +do 2 rewrite Rplus_opp_l. +do 2 rewrite Rplus_0_r. +rewrite + (Rabs_right (sum_f_R0 (fun i:nat => Rabs (An (S m + i)%nat)) (n - S m))) + . +pose (Bn := fun i:nat => An (S m + i)%nat). +replace (fun i:nat => Rabs (An (S m + i)%nat)) with + (fun i:nat => Rabs (Bn i)). +apply Rabs_triang_gen. +unfold Bn in |- *; reflexivity. +apply Rle_ge. +apply cond_pos_sum. +intro; apply Rabs_pos. Qed. (**********) -Lemma cv_cauchy_1 : (An:nat->R) (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)) -> (Cauchy_crit_series An). -Intros. -Elim X; Intros. -Unfold Un_cv in p. -Unfold Cauchy_crit_series; Unfold Cauchy_crit. -Intros. -Cut ``0<eps/2``. -Intro. -Elim (p ``eps/2`` H0); Intros. -Exists x0. -Intros. -Apply Rle_lt_trans with ``(R_dist (sum_f_R0 An n) x)+(R_dist (sum_f_R0 An m) x)``. -Unfold R_dist. -Replace ``(sum_f_R0 An n)-(sum_f_R0 An m)`` with ``((sum_f_R0 An n)-x)+ -((sum_f_R0 An m)-x)``; [Idtac | Ring]. -Rewrite <- (Rabsolu_Ropp ``(sum_f_R0 An m)-x``). -Apply Rabsolu_triang. -Apply Rlt_le_trans with ``eps/2+eps/2``. -Apply Rplus_lt. -Apply H1; Assumption. -Apply H1; Assumption. -Right; Symmetry; Apply double_var. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Lemma cv_cauchy_1 : + forall An:nat -> R, + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l) -> + Cauchy_crit_series An. +intros. +elim X; intros. +unfold Un_cv in p. +unfold Cauchy_crit_series in |- *; unfold Cauchy_crit in |- *. +intros. +cut (0 < eps / 2). +intro. +elim (p (eps / 2) H0); intros. +exists x0. +intros. +apply Rle_lt_trans with (R_dist (sum_f_R0 An n) x + R_dist (sum_f_R0 An m) x). +unfold R_dist in |- *. +replace (sum_f_R0 An n - sum_f_R0 An m) with + (sum_f_R0 An n - x + - (sum_f_R0 An m - x)); [ idtac | ring ]. +rewrite <- (Rabs_Ropp (sum_f_R0 An m - x)). +apply Rabs_triang. +apply Rlt_le_trans with (eps / 2 + eps / 2). +apply Rplus_lt_compat. +apply H1; assumption. +apply H1; assumption. +right; symmetry in |- *; apply double_var. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. Qed. -Lemma cv_cauchy_2 : (An:nat->R) (Cauchy_crit_series An) -> (sigTT R [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). -Intros. -Apply R_complete. -Unfold Cauchy_crit_series in H. -Exact H. +Lemma cv_cauchy_2 : + forall An:nat -> R, + Cauchy_crit_series An -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l). +intros. +apply R_complete. +unfold Cauchy_crit_series in H. +exact H. Qed. (**********) -Lemma sum_eq_R0 : (An:nat->R;N:nat) ((n:nat)(le n N)->``(An n)==0``) -> (sum_f_R0 An N)==R0. -Intros; Induction N. -Simpl; Apply H; Apply le_n. -Rewrite tech5; Rewrite HrecN; [Rewrite Rplus_Ol; Apply H; Apply le_n | Intros; Apply H; Apply le_trans with N; [Assumption | Apply le_n_Sn]]. +Lemma sum_eq_R0 : + forall (An:nat -> R) (N:nat), + (forall n:nat, (n <= N)%nat -> An n = 0) -> sum_f_R0 An N = 0. +intros; induction N as [| N HrecN]. +simpl in |- *; apply H; apply le_n. +rewrite tech5; rewrite HrecN; + [ rewrite Rplus_0_l; apply H; apply le_n + | intros; apply H; apply le_trans with N; [ assumption | apply le_n_Sn ] ]. Qed. -Definition SP [fn:nat->R->R;N:nat] : R->R := [x:R](sum_f_R0 [k:nat]``(fn k x)`` N). +Definition SP (fn:nat -> R -> R) (N:nat) (x:R) : R := + sum_f_R0 (fun k:nat => fn k x) N. (**********) -Lemma sum_incr : (An:nat->R;N:nat;l:R) (Un_cv [n:nat](sum_f_R0 An n) l) -> ((n:nat)``0<=(An n)``) -> ``(sum_f_R0 An N)<=l``. -Intros; Case (total_order_T (sum_f_R0 An N) l); Intro. -Elim s; Intro. -Left; Apply a. -Right; Apply b. -Cut (Un_growing [n:nat](sum_f_R0 An n)). -Intro; Pose l1 := (sum_f_R0 An N). -Fold l1 in r. -Unfold Un_cv in H; Cut ``0<l1-l``. -Intro; Elim (H ? H2); Intros. -Pose N0 := (max x N); Cut (ge N0 x). -Intro; Assert H5 := (H3 N0 H4). -Cut ``l1<=(sum_f_R0 An N0)``. -Intro; Unfold R_dist in H5; Rewrite Rabsolu_right in H5. -Cut ``(sum_f_R0 An N0)<l1``. -Intro; Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H7 H6)). -Apply Rlt_anti_compatibility with ``-l``. -Do 2 Rewrite (Rplus_sym ``-l``). -Apply H5. -Apply Rle_sym1; Apply Rle_anti_compatibility with l. -Rewrite Rplus_Or; Replace ``l+((sum_f_R0 An N0)-l)`` with (sum_f_R0 An N0); [Idtac | Ring]; Apply Rle_trans with l1. -Left; Apply r. -Apply H6. -Unfold l1; Apply Rle_sym2; Apply (growing_prop [k:nat](sum_f_R0 An k)). -Apply H1. -Unfold ge N0; Apply le_max_r. -Unfold ge N0; Apply le_max_l. -Apply Rlt_anti_compatibility with l; Rewrite Rplus_Or; Replace ``l+(l1-l)`` with l1; [Apply r | Ring]. -Unfold Un_growing; Intro; Simpl; Pattern 1 (sum_f_R0 An n); Rewrite <- Rplus_Or; Apply Rle_compatibility; Apply H0. +Lemma sum_incr : + forall (An:nat -> R) (N:nat) (l:R), + Un_cv (fun n:nat => sum_f_R0 An n) l -> + (forall n:nat, 0 <= An n) -> sum_f_R0 An N <= l. +intros; case (total_order_T (sum_f_R0 An N) l); intro. +elim s; intro. +left; apply a. +right; apply b. +cut (Un_growing (fun n:nat => sum_f_R0 An n)). +intro; pose (l1 := sum_f_R0 An N). +fold l1 in r. +unfold Un_cv in H; cut (0 < l1 - l). +intro; elim (H _ H2); intros. +pose (N0 := max x N); cut (N0 >= x)%nat. +intro; assert (H5 := H3 N0 H4). +cut (l1 <= sum_f_R0 An N0). +intro; unfold R_dist in H5; rewrite Rabs_right in H5. +cut (sum_f_R0 An N0 < l1). +intro; elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H7 H6)). +apply Rplus_lt_reg_r with (- l). +do 2 rewrite (Rplus_comm (- l)). +apply H5. +apply Rle_ge; apply Rplus_le_reg_l with l. +rewrite Rplus_0_r; replace (l + (sum_f_R0 An N0 - l)) with (sum_f_R0 An N0); + [ idtac | ring ]; apply Rle_trans with l1. +left; apply r. +apply H6. +unfold l1 in |- *; apply Rge_le; + apply (growing_prop (fun k:nat => sum_f_R0 An k)). +apply H1. +unfold ge, N0 in |- *; apply le_max_r. +unfold ge, N0 in |- *; apply le_max_l. +apply Rplus_lt_reg_r with l; rewrite Rplus_0_r; + replace (l + (l1 - l)) with l1; [ apply r | ring ]. +unfold Un_growing in |- *; intro; simpl in |- *; + pattern (sum_f_R0 An n) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l; apply H0. Qed. (**********) -Lemma sum_cv_maj : (An:nat->R;fn:nat->R->R;x,l1,l2:R) (Un_cv [n:nat](SP fn n x) l1) -> (Un_cv [n:nat](sum_f_R0 An n) l2) -> ((n:nat)``(Rabsolu (fn n x))<=(An n)``) -> ``(Rabsolu l1)<=l2``. -Intros; Case (total_order_T (Rabsolu l1) l2); Intro. -Elim s; Intro. -Left; Apply a. -Right; Apply b. -Cut (n0:nat)``(Rabsolu (SP fn n0 x))<=(sum_f_R0 An n0)``. -Intro; Cut ``0<((Rabsolu l1)-l2)/2``. -Intro; Unfold Un_cv in H H0. -Elim (H ? H3); Intros Na H4. -Elim (H0 ? H3); Intros Nb H5. -Pose N := (max Na Nb). -Unfold R_dist in H4 H5. -Cut ``(Rabsolu ((sum_f_R0 An N)-l2))<((Rabsolu l1)-l2)/2``. -Intro; Cut ``(Rabsolu ((Rabsolu l1)-(Rabsolu (SP fn N x))))<((Rabsolu l1)-l2)/2``. -Intro; Cut ``(sum_f_R0 An N)<((Rabsolu l1)+l2)/2``. -Intro; Cut ``((Rabsolu l1)+l2)/2<(Rabsolu (SP fn N x))``. -Intro; Cut ``(sum_f_R0 An N)<(Rabsolu (SP fn N x))``. -Intro; Assert H11 := (H2 N). -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H11 H10)). -Apply Rlt_trans with ``((Rabsolu l1)+l2)/2``; Assumption. -Case (case_Rabsolu ``(Rabsolu l1)-(Rabsolu (SP fn N x))``); Intro. -Apply Rlt_trans with (Rabsolu l1). -Apply Rlt_monotony_contra with ``2``. -Sup0. -Unfold Rdiv; Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite double; Apply Rlt_compatibility; Apply r. -DiscrR. -Apply (Rminus_lt ? ? r0). -Rewrite (Rabsolu_right ? r0) in H7. -Apply Rlt_anti_compatibility with ``((Rabsolu l1)-l2)/2-(Rabsolu (SP fn N x))``. -Replace ``((Rabsolu l1)-l2)/2-(Rabsolu (SP fn N x))+((Rabsolu l1)+l2)/2`` with ``(Rabsolu l1)-(Rabsolu (SP fn N x))``. -Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply H7. -Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Rewrite <- (Rmult_sym ``/2``); Rewrite Rminus_distr; Repeat Rewrite (Rmult_sym ``/2``); Pattern 1 (Rabsolu l1); Rewrite double_var; Unfold Rdiv; Ring. -Case (case_Rabsolu ``(sum_f_R0 An N)-l2``); Intro. -Apply Rlt_trans with l2. -Apply (Rminus_lt ? ? r0). -Apply Rlt_monotony_contra with ``2``. -Sup0. -Rewrite (double l2); Unfold Rdiv; Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite (Rplus_sym (Rabsolu l1)); Apply Rlt_compatibility; Apply r. -DiscrR. -Rewrite (Rabsolu_right ? r0) in H6; Apply Rlt_anti_compatibility with ``-l2``. -Replace ``-l2+((Rabsolu l1)+l2)/2`` with ``((Rabsolu l1)-l2)/2``. -Rewrite Rplus_sym; Apply H6. -Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite Rminus_distr; Rewrite Rmult_Rplus_distrl; Pattern 2 l2; Rewrite double_var; Repeat Rewrite (Rmult_sym ``/2``); Rewrite Ropp_distr1; Unfold Rdiv; Ring. -Apply Rle_lt_trans with ``(Rabsolu ((SP fn N x)-l1))``. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr3; Apply Rabsolu_triang_inv2. -Apply H4; Unfold ge N; Apply le_max_l. -Apply H5; Unfold ge N; Apply le_max_r. -Unfold Rdiv; Apply Rmult_lt_pos. -Apply Rlt_anti_compatibility with l2. -Rewrite Rplus_Or; Replace ``l2+((Rabsolu l1)-l2)`` with (Rabsolu l1); [Apply r | Ring]. -Apply Rlt_Rinv; Sup0. -Intros; Induction n0. -Unfold SP; Simpl; Apply H1. -Unfold SP; Simpl. -Apply Rle_trans with (Rplus (Rabsolu (sum_f_R0 [k:nat](fn k x) n0)) (Rabsolu (fn (S n0) x))). -Apply Rabsolu_triang. -Apply Rle_trans with ``(sum_f_R0 An n0)+(Rabsolu (fn (S n0) x))``. -Do 2 Rewrite <- (Rplus_sym (Rabsolu (fn (S n0) x))). -Apply Rle_compatibility; Apply Hrecn0. -Apply Rle_compatibility; Apply H1. -Qed. +Lemma sum_cv_maj : + forall (An:nat -> R) (fn:nat -> R -> R) (x l1 l2:R), + Un_cv (fun n:nat => SP fn n x) l1 -> + Un_cv (fun n:nat => sum_f_R0 An n) l2 -> + (forall n:nat, Rabs (fn n x) <= An n) -> Rabs l1 <= l2. +intros; case (total_order_T (Rabs l1) l2); intro. +elim s; intro. +left; apply a. +right; apply b. +cut (forall n0:nat, Rabs (SP fn n0 x) <= sum_f_R0 An n0). +intro; cut (0 < (Rabs l1 - l2) / 2). +intro; unfold Un_cv in H, H0. +elim (H _ H3); intros Na H4. +elim (H0 _ H3); intros Nb H5. +pose (N := max Na Nb). +unfold R_dist in H4, H5. +cut (Rabs (sum_f_R0 An N - l2) < (Rabs l1 - l2) / 2). +intro; cut (Rabs (Rabs l1 - Rabs (SP fn N x)) < (Rabs l1 - l2) / 2). +intro; cut (sum_f_R0 An N < (Rabs l1 + l2) / 2). +intro; cut ((Rabs l1 + l2) / 2 < Rabs (SP fn N x)). +intro; cut (sum_f_R0 An N < Rabs (SP fn N x)). +intro; assert (H11 := H2 N). +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H11 H10)). +apply Rlt_trans with ((Rabs l1 + l2) / 2); assumption. +case (Rcase_abs (Rabs l1 - Rabs (SP fn N x))); intro. +apply Rlt_trans with (Rabs l1). +apply Rmult_lt_reg_l with 2. +prove_sup0. +unfold Rdiv in |- *; rewrite (Rmult_comm 2); rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite double; apply Rplus_lt_compat_l; apply r. +discrR. +apply (Rminus_lt _ _ r0). +rewrite (Rabs_right _ r0) in H7. +apply Rplus_lt_reg_r with ((Rabs l1 - l2) / 2 - Rabs (SP fn N x)). +replace ((Rabs l1 - l2) / 2 - Rabs (SP fn N x) + (Rabs l1 + l2) / 2) with + (Rabs l1 - Rabs (SP fn N x)). +unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_r; apply H7. +unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; + rewrite <- (Rmult_comm (/ 2)); rewrite Rmult_minus_distr_l; + repeat rewrite (Rmult_comm (/ 2)); pattern (Rabs l1) at 1 in |- *; + rewrite double_var; unfold Rdiv in |- *; ring. +case (Rcase_abs (sum_f_R0 An N - l2)); intro. +apply Rlt_trans with l2. +apply (Rminus_lt _ _ r0). +apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite (double l2); unfold Rdiv in |- *; rewrite (Rmult_comm 2); + rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite (Rplus_comm (Rabs l1)); apply Rplus_lt_compat_l; + apply r. +discrR. +rewrite (Rabs_right _ r0) in H6; apply Rplus_lt_reg_r with (- l2). +replace (- l2 + (Rabs l1 + l2) / 2) with ((Rabs l1 - l2) / 2). +rewrite Rplus_comm; apply H6. +unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + rewrite Rmult_minus_distr_l; rewrite Rmult_plus_distr_r; + pattern l2 at 2 in |- *; rewrite double_var; + repeat rewrite (Rmult_comm (/ 2)); rewrite Ropp_plus_distr; + unfold Rdiv in |- *; ring. +apply Rle_lt_trans with (Rabs (SP fn N x - l1)). +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply Rabs_triang_inv2. +apply H4; unfold ge, N in |- *; apply le_max_l. +apply H5; unfold ge, N in |- *; apply le_max_r. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply Rplus_lt_reg_r with l2. +rewrite Rplus_0_r; replace (l2 + (Rabs l1 - l2)) with (Rabs l1); + [ apply r | ring ]. +apply Rinv_0_lt_compat; prove_sup0. +intros; induction n0 as [| n0 Hrecn0]. +unfold SP in |- *; simpl in |- *; apply H1. +unfold SP in |- *; simpl in |- *. +apply Rle_trans with + (Rabs (sum_f_R0 (fun k:nat => fn k x) n0) + Rabs (fn (S n0) x)). +apply Rabs_triang. +apply Rle_trans with (sum_f_R0 An n0 + Rabs (fn (S n0) x)). +do 2 rewrite <- (Rplus_comm (Rabs (fn (S n0) x))). +apply Rplus_le_compat_l; apply Hrecn0. +apply Rplus_le_compat_l; apply H1. +Qed.
\ No newline at end of file diff --git a/theories/Reals/RIneq.v b/theories/Reals/RIneq.v index 12644ae37..5534cde45 100644 --- a/theories/Reals/RIneq.v +++ b/theories/Reals/RIneq.v @@ -14,83 +14,84 @@ Require Export Raxioms. Require Export ZArithRing. -Require Omega. +Require Import Omega. Require Export Field. Open Local Scope Z_scope. Open Local Scope R_scope. -Implicit Variable Type r:R. +Implicit Type r : R. (***************************************************************************) (** Instantiating Ring tactic on reals *) (***************************************************************************) -Lemma RTheory : (Ring_Theory Rplus Rmult R1 R0 Ropp [x,y:R]false). - Split. - Exact Rplus_sym. - Symmetry; Apply Rplus_assoc. - Exact Rmult_sym. - Symmetry; Apply Rmult_assoc. - Intro; Apply Rplus_Ol. - Intro; Apply Rmult_1l. - Exact Rplus_Ropp_r. - Intros. - Rewrite Rmult_sym. - Rewrite (Rmult_sym n p). - Rewrite (Rmult_sym m p). - Apply Rmult_Rplus_distr. - Intros; Contradiction. +Lemma RTheory : Ring_Theory Rplus Rmult 1 0 Ropp (fun x y:R => false). + split. + exact Rplus_comm. + symmetry in |- *; apply Rplus_assoc. + exact Rmult_comm. + symmetry in |- *; apply Rmult_assoc. + intro; apply Rplus_0_l. + intro; apply Rmult_1_l. + exact Rplus_opp_r. + intros. + rewrite Rmult_comm. + rewrite (Rmult_comm n p). + rewrite (Rmult_comm m p). + apply Rmult_plus_distr_l. + intros; contradiction. Defined. -Add Field R Rplus Rmult R1 R0 Ropp [x,y:R]false Rinv RTheory Rinv_l - with minus:=Rminus div:=Rdiv. +Add Field R Rplus Rmult 1 0 Ropp (fun x y:R => false) Rinv RTheory Rinv_l + with minus := Rminus div := Rdiv. (**************************************************************************) (** Relation between orders and equality *) (**************************************************************************) (**********) -Lemma Rlt_antirefl:(r:R)~``r<r``. - Generalize Rlt_antisym. Intuition EAuto. +Lemma Rlt_irrefl : forall r, ~ r < r. + generalize Rlt_asym. intuition eauto. Qed. -Hints Resolve Rlt_antirefl : real. +Hint Resolve Rlt_irrefl: real. -Lemma Rle_refl : (x:R) ``x<=x``. -Intro; Right; Reflexivity. +Lemma Rle_refl : forall r, r <= r. +intro; right; reflexivity. Qed. -Lemma Rlt_not_eq:(r1,r2:R)``r1<r2``->``r1<>r2``. - Red; Intros r1 r2 H H0; Apply (Rlt_antirefl r1). - Pattern 2 r1; Rewrite H0; Trivial. +Lemma Rlt_not_eq : forall r1 r2, r1 < r2 -> r1 <> r2. + red in |- *; intros r1 r2 H H0; apply (Rlt_irrefl r1). + pattern r1 at 2 in |- *; rewrite H0; trivial. Qed. -Lemma Rgt_not_eq:(r1,r2:R)``r1>r2``->``r1<>r2``. -Intros; Apply sym_not_eqT; Apply Rlt_not_eq; Auto with real. +Lemma Rgt_not_eq : forall r1 r2, r1 > r2 -> r1 <> r2. +intros; apply sym_not_eq; apply Rlt_not_eq; auto with real. Qed. (**********) -Lemma imp_not_Req:(r1,r2:R)(``r1<r2``\/ ``r1>r2``) -> ``r1<>r2``. -Generalize Rlt_not_eq Rgt_not_eq. Intuition EAuto. +Lemma Rlt_dichotomy_converse : forall r1 r2, r1 < r2 \/ r1 > r2 -> r1 <> r2. +generalize Rlt_not_eq Rgt_not_eq. intuition eauto. Qed. -Hints Resolve imp_not_Req : real. +Hint Resolve Rlt_dichotomy_converse: real. (** Reasoning by case on equalities and order *) (**********) -Lemma Req_EM:(r1,r2:R)(r1==r2)\/``r1<>r2``. -Intros ; Generalize (total_order_T r1 r2) imp_not_Req ; Intuition EAuto 3. +Lemma Req_dec : forall r1 r2, r1 = r2 \/ r1 <> r2. +intros; generalize (total_order_T r1 r2) Rlt_dichotomy_converse; + intuition eauto 3. Qed. -Hints Resolve Req_EM : real. +Hint Resolve Req_dec: real. (**********) -Lemma total_order:(r1,r2:R)``r1<r2``\/(r1==r2)\/``r1>r2``. -Intros;Generalize (total_order_T r1 r2);Tauto. +Lemma Rtotal_order : forall r1 r2, r1 < r2 \/ r1 = r2 \/ r1 > r2. +intros; generalize (total_order_T r1 r2); tauto. Qed. (**********) -Lemma not_Req:(r1,r2:R)``r1<>r2``->(``r1<r2``\/``r1>r2``). -Intros; Generalize (total_order_T r1 r2) ; Tauto. +Lemma Rdichotomy : forall r1 r2, r1 <> r2 -> r1 < r2 \/ r1 > r2. +intros; generalize (total_order_T r1 r2); tauto. Qed. @@ -99,152 +100,154 @@ Qed. (*********************************************************************************) (**********) -Lemma Rlt_le:(r1,r2:R)``r1<r2``-> ``r1<=r2``. -Intros ; Red ; Tauto. +Lemma Rlt_le : forall r1 r2, r1 < r2 -> r1 <= r2. +intros; red in |- *; tauto. Qed. -Hints Resolve Rlt_le : real. +Hint Resolve Rlt_le: real. (**********) -Lemma Rle_ge : (r1,r2:R)``r1<=r2`` -> ``r2>=r1``. -NewDestruct 1; Red; Auto with real. +Lemma Rle_ge : forall r1 r2, r1 <= r2 -> r2 >= r1. +destruct 1; red in |- *; auto with real. Qed. -Hints Immediate Rle_ge : real. +Hint Immediate Rle_ge: real. (**********) -Lemma Rge_le : (r1,r2:R)``r1>=r2`` -> ``r2<=r1``. -NewDestruct 1; Red; Auto with real. +Lemma Rge_le : forall r1 r2, r1 >= r2 -> r2 <= r1. +destruct 1; red in |- *; auto with real. Qed. -Hints Resolve Rge_le : real. +Hint Resolve Rge_le: real. (**********) -Lemma not_Rle:(r1,r2:R)~``r1<=r2`` -> ``r2<r1``. -Intros r1 r2 ; Generalize (total_order r1 r2) ; Unfold Rle; Tauto. +Lemma Rnot_le_lt : forall r1 r2, ~ r1 <= r2 -> r2 < r1. +intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rle in |- *; tauto. Qed. -Hints Immediate not_Rle : real. +Hint Immediate Rnot_le_lt: real. -Lemma not_Rge:(r1,r2:R)~``r1>=r2`` -> ``r1<r2``. -Intros; Apply not_Rle; Auto with real. +Lemma Rnot_ge_lt : forall r1 r2, ~ r1 >= r2 -> r1 < r2. +intros; apply Rnot_le_lt; auto with real. Qed. (**********) -Lemma Rlt_le_not:(r1,r2:R)``r2<r1`` -> ~``r1<=r2``. -Generalize Rlt_antisym imp_not_Req ; Unfold Rle. -Intuition EAuto 3. +Lemma Rlt_not_le : forall r1 r2, r2 < r1 -> ~ r1 <= r2. +generalize Rlt_asym Rlt_dichotomy_converse; unfold Rle in |- *. +intuition eauto 3. Qed. -Lemma Rle_not:(r1,r2:R)``r1>r2`` -> ~``r1<=r2``. -Proof Rlt_le_not. +Lemma Rgt_not_le : forall r1 r2, r1 > r2 -> ~ r1 <= r2. +Proof Rlt_not_le. -Hints Immediate Rlt_le_not : real. +Hint Immediate Rlt_not_le: real. -Lemma Rle_not_lt: (r1, r2:R) ``r2 <= r1`` -> ~``r1<r2``. -Intros r1 r2. Generalize (Rlt_antisym r1 r2) (imp_not_Req r1 r2). -Unfold Rle; Intuition. +Lemma Rle_not_lt : forall r1 r2, r2 <= r1 -> ~ r1 < r2. +intros r1 r2. generalize (Rlt_asym r1 r2) (Rlt_dichotomy_converse r1 r2). +unfold Rle in |- *; intuition. Qed. (**********) -Lemma Rlt_ge_not:(r1,r2:R)``r1<r2`` -> ~``r1>=r2``. -Generalize Rlt_le_not. Unfold Rle Rge. Intuition EAuto 3. +Lemma Rlt_not_ge : forall r1 r2, r1 < r2 -> ~ r1 >= r2. +generalize Rlt_not_le. unfold Rle, Rge in |- *. intuition eauto 3. Qed. -Hints Immediate Rlt_ge_not : real. +Hint Immediate Rlt_not_ge: real. (**********) -Lemma eq_Rle:(r1,r2:R)r1==r2->``r1<=r2``. -Unfold Rle; Tauto. +Lemma Req_le : forall r1 r2, r1 = r2 -> r1 <= r2. +unfold Rle in |- *; tauto. Qed. -Hints Immediate eq_Rle : real. +Hint Immediate Req_le: real. -Lemma eq_Rge:(r1,r2:R)r1==r2->``r1>=r2``. -Unfold Rge; Tauto. +Lemma Req_ge : forall r1 r2, r1 = r2 -> r1 >= r2. +unfold Rge in |- *; tauto. Qed. -Hints Immediate eq_Rge : real. +Hint Immediate Req_ge: real. -Lemma eq_Rle_sym:(r1,r2:R)r2==r1->``r1<=r2``. -Unfold Rle; Auto. +Lemma Req_le_sym : forall r1 r2, r2 = r1 -> r1 <= r2. +unfold Rle in |- *; auto. Qed. -Hints Immediate eq_Rle_sym : real. +Hint Immediate Req_le_sym: real. -Lemma eq_Rge_sym:(r1,r2:R)r2==r1->``r1>=r2``. -Unfold Rge; Auto. +Lemma Req_ge_sym : forall r1 r2, r2 = r1 -> r1 >= r2. +unfold Rge in |- *; auto. Qed. -Hints Immediate eq_Rge_sym : real. +Hint Immediate Req_ge_sym: real. -Lemma Rle_antisym : (r1,r2:R)``r1<=r2`` -> ``r2<=r1``-> r1==r2. -Intros r1 r2; Generalize (Rlt_antisym r1 r2) ; Unfold Rle ; Intuition. +Lemma Rle_antisym : forall r1 r2, r1 <= r2 -> r2 <= r1 -> r1 = r2. +intros r1 r2; generalize (Rlt_asym r1 r2); unfold Rle in |- *; intuition. Qed. -Hints Resolve Rle_antisym : real. +Hint Resolve Rle_antisym: real. (**********) -Lemma Rle_le_eq:(r1,r2:R)(``r1<=r2``/\``r2<=r1``)<->(r1==r2). -Intuition. +Lemma Rle_le_eq : forall r1 r2, r1 <= r2 /\ r2 <= r1 <-> r1 = r2. +intuition. Qed. -Lemma Rlt_rew : (x,x',y,y':R)``x==x'``->``x'<y'`` -> `` y' == y`` -> ``x < y``. -Intros x x' y y'; Intros; Replace x with x'; Replace y with y'; Assumption. +Lemma Rlt_eq_compat : + forall r1 r2 r3 r4, r1 = r2 -> r2 < r4 -> r4 = r3 -> r1 < r3. +intros x x' y y'; intros; replace x with x'; replace y with y'; assumption. Qed. (**********) -Lemma Rle_trans:(r1,r2,r3:R) ``r1<=r2``->``r2<=r3``->``r1<=r3``. -Generalize trans_eqT Rlt_trans Rlt_rew. -Unfold Rle. -Intuition EAuto 2. +Lemma Rle_trans : forall r1 r2 r3, r1 <= r2 -> r2 <= r3 -> r1 <= r3. +generalize trans_eq Rlt_trans Rlt_eq_compat. +unfold Rle in |- *. +intuition eauto 2. Qed. (**********) -Lemma Rle_lt_trans:(r1,r2,r3:R)``r1<=r2``->``r2<r3``->``r1<r3``. -Generalize Rlt_trans Rlt_rew. -Unfold Rle. -Intuition EAuto 2. +Lemma Rle_lt_trans : forall r1 r2 r3, r1 <= r2 -> r2 < r3 -> r1 < r3. +generalize Rlt_trans Rlt_eq_compat. +unfold Rle in |- *. +intuition eauto 2. Qed. (**********) -Lemma Rlt_le_trans:(r1,r2,r3:R)``r1<r2``->``r2<=r3``->``r1<r3``. -Generalize Rlt_trans Rlt_rew; Unfold Rle; Intuition EAuto 2. +Lemma Rlt_le_trans : forall r1 r2 r3, r1 < r2 -> r2 <= r3 -> r1 < r3. +generalize Rlt_trans Rlt_eq_compat; unfold Rle in |- *; intuition eauto 2. Qed. (** Decidability of the order *) -Lemma total_order_Rlt:(r1,r2:R)(sumboolT ``r1<r2`` ~(``r1<r2``)). -Intros;Generalize (total_order_T r1 r2) (imp_not_Req r1 r2) ; Intuition. +Lemma Rlt_dec : forall r1 r2, {r1 < r2} + {~ r1 < r2}. +intros; generalize (total_order_T r1 r2) (Rlt_dichotomy_converse r1 r2); + intuition. Qed. (**********) -Lemma total_order_Rle:(r1,r2:R)(sumboolT ``r1<=r2`` ~(``r1<=r2``)). -Intros r1 r2. -Generalize (total_order_T r1 r2) (imp_not_Req r1 r2). -Intuition EAuto 4 with real. +Lemma Rle_dec : forall r1 r2, {r1 <= r2} + {~ r1 <= r2}. +intros r1 r2. +generalize (total_order_T r1 r2) (Rlt_dichotomy_converse r1 r2). +intuition eauto 4 with real. Qed. (**********) -Lemma total_order_Rgt:(r1,r2:R)(sumboolT ``r1>r2`` ~(``r1>r2``)). -Intros;Unfold Rgt;Intros;Apply total_order_Rlt. +Lemma Rgt_dec : forall r1 r2, {r1 > r2} + {~ r1 > r2}. +intros; unfold Rgt in |- *; intros; apply Rlt_dec. Qed. (**********) -Lemma total_order_Rge:(r1,r2:R)(sumboolT (``r1>=r2``) ~(``r1>=r2``)). -Intros;Generalize (total_order_Rle r2 r1);Intuition. +Lemma Rge_dec : forall r1 r2, {r1 >= r2} + {~ r1 >= r2}. +intros; generalize (Rle_dec r2 r1); intuition. Qed. -Lemma total_order_Rlt_Rle:(r1,r2:R)(sumboolT ``r1<r2`` ``r2<=r1``). -Intros;Generalize (total_order_T r1 r2); Intuition. +Lemma Rlt_le_dec : forall r1 r2, {r1 < r2} + {r2 <= r1}. +intros; generalize (total_order_T r1 r2); intuition. Qed. -Lemma Rle_or_lt: (n, m:R)(Rle n m) \/ (Rlt m n). -Intros n m; Elim (total_order_Rlt_Rle m n);Auto with real. +Lemma Rle_or_lt : forall r1 r2, r1 <= r2 \/ r2 < r1. +intros n m; elim (Rlt_le_dec m n); auto with real. Qed. -Lemma total_order_Rle_Rlt_eq :(r1,r2:R)``r1<=r2``-> - (sumboolT ``r1<r2`` ``r1==r2``). -Intros r1 r2 H;Generalize (total_order_T r1 r2); Intuition. +Lemma Rle_lt_or_eq_dec : forall r1 r2, r1 <= r2 -> {r1 < r2} + {r1 = r2}. +intros r1 r2 H; generalize (total_order_T r1 r2); intuition. Qed. (**********) -Lemma inser_trans_R:(n,m,p,q:R)``n<=m<p``-> (sumboolT ``n<=m<q`` ``q<=m<p``). -Intros n m p q; Intros; Generalize (total_order_Rlt_Rle m q); Intuition. +Lemma inser_trans_R : + forall r1 r2 r3 r4, r1 <= r2 < r3 -> {r1 <= r2 < r4} + {r4 <= r2 < r3}. +intros n m p q; intros; generalize (Rlt_le_dec m q); intuition. Qed. (****************************************************************) @@ -255,53 +258,51 @@ Qed. (** Addition *) (*********************************************************) -Lemma Rplus_ne:(r:R)``r+0==r``/\``0+r==r``. -Intro;Split;Ring. +Lemma Rplus_ne : forall r, r + 0 = r /\ 0 + r = r. +intro; split; ring. Qed. -Hints Resolve Rplus_ne : real v62. +Hint Resolve Rplus_ne: real v62. -Lemma Rplus_Or:(r:R)``r+0==r``. -Intro; Ring. +Lemma Rplus_0_r : forall r, r + 0 = r. +intro; ring. Qed. -Hints Resolve Rplus_Or : real. +Hint Resolve Rplus_0_r: real. (**********) -Lemma Rplus_Ropp_l:(r:R)``(-r)+r==0``. - Intro; Ring. +Lemma Rplus_opp_l : forall r, - r + r = 0. + intro; ring. Qed. -Hints Resolve Rplus_Ropp_l : real. +Hint Resolve Rplus_opp_l: real. (**********) -Lemma Rplus_Ropp:(x,y:R)``x+y==0``->``y== -x``. - Intros x y H; Replace y with ``(-x+x)+y``; - [ Rewrite -> Rplus_assoc; Rewrite -> H; Ring - | Ring ]. +Lemma Rplus_opp_r_uniq : forall r1 r2, r1 + r2 = 0 -> r2 = - r1. + intros x y H; replace y with (- x + x + y); + [ rewrite Rplus_assoc; rewrite H; ring | ring ]. Qed. (*i New i*) -Hint eqT_R_congr : real := Resolve (congr_eqT R). +Hint Resolve (f_equal (A:=R)): real. -Lemma Rplus_plus_r:(r,r1,r2:R)(r1==r2)->``r+r1==r+r2``. - Auto with real. +Lemma Rplus_eq_compat_l : forall r r1 r2, r1 = r2 -> r + r1 = r + r2. + auto with real. Qed. -(*i Old i*)Hints Resolve Rplus_plus_r : v62. +(*i Old i*)Hint Resolve Rplus_eq_compat_l: v62. (**********) -Lemma r_Rplus_plus:(r,r1,r2:R)``r+r1==r+r2``->r1==r2. - Intros; Transitivity ``(-r+r)+r1``. - Ring. - Transitivity ``(-r+r)+r2``. - Repeat Rewrite -> Rplus_assoc; Rewrite <- H; Reflexivity. - Ring. +Lemma Rplus_eq_reg_l : forall r r1 r2, r + r1 = r + r2 -> r1 = r2. + intros; transitivity (- r + r + r1). + ring. + transitivity (- r + r + r2). + repeat rewrite Rplus_assoc; rewrite <- H; reflexivity. + ring. Qed. -Hints Resolve r_Rplus_plus : real. +Hint Resolve Rplus_eq_reg_l: real. (**********) -Lemma Rplus_ne_i:(r,b:R)``r+b==r`` -> ``b==0``. - Intros r b; Pattern 2 r; Replace r with ``r+0``; - EAuto with real. +Lemma Rplus_0_r_uniq : forall r r1, r + r1 = r -> r1 = 0. + intros r b; pattern r at 2 in |- *; replace r with (r + 0); eauto with real. Qed. (***********************************************************) @@ -309,119 +310,119 @@ Qed. (***********************************************************) (**********) -Lemma Rinv_r:(r:R)``r<>0``->``r* (/r)==1``. - Intros; Rewrite -> Rmult_sym; Auto with real. +Lemma Rinv_r : forall r, r <> 0 -> r * / r = 1. + intros; rewrite Rmult_comm; auto with real. Qed. -Hints Resolve Rinv_r : real. +Hint Resolve Rinv_r: real. -Lemma Rinv_l_sym:(r:R)``r<>0``->``1==(/r) * r``. - Symmetry; Auto with real. +Lemma Rinv_l_sym : forall r, r <> 0 -> 1 = / r * r. + symmetry in |- *; auto with real. Qed. -Lemma Rinv_r_sym:(r:R)``r<>0``->``1==r* (/r)``. - Symmetry; Auto with real. +Lemma Rinv_r_sym : forall r, r <> 0 -> 1 = r * / r. + symmetry in |- *; auto with real. Qed. -Hints Resolve Rinv_l_sym Rinv_r_sym : real. +Hint Resolve Rinv_l_sym Rinv_r_sym: real. (**********) -Lemma Rmult_Or :(r:R) ``r*0==0``. -Intro; Ring. +Lemma Rmult_0_r : forall r, r * 0 = 0. +intro; ring. Qed. -Hints Resolve Rmult_Or : real v62. +Hint Resolve Rmult_0_r: real v62. (**********) -Lemma Rmult_Ol:(r:R) ``0*r==0``. -Intro; Ring. +Lemma Rmult_0_l : forall r, 0 * r = 0. +intro; ring. Qed. -Hints Resolve Rmult_Ol : real v62. +Hint Resolve Rmult_0_l: real v62. (**********) -Lemma Rmult_ne:(r:R)``r*1==r``/\``1*r==r``. -Intro;Split;Ring. +Lemma Rmult_ne : forall r, r * 1 = r /\ 1 * r = r. +intro; split; ring. Qed. -Hints Resolve Rmult_ne : real v62. +Hint Resolve Rmult_ne: real v62. (**********) -Lemma Rmult_1r:(r:R)(``r*1==r``). -Intro; Ring. +Lemma Rmult_1_r : forall r, r * 1 = r. +intro; ring. Qed. -Hints Resolve Rmult_1r : real. +Hint Resolve Rmult_1_r: real. (**********) -Lemma Rmult_mult_r:(r,r1,r2:R)r1==r2->``r*r1==r*r2``. - Auto with real. +Lemma Rmult_eq_compat_l : forall r r1 r2, r1 = r2 -> r * r1 = r * r2. + auto with real. Qed. -(*i OLD i*)Hints Resolve Rmult_mult_r : v62. +(*i OLD i*)Hint Resolve Rmult_eq_compat_l: v62. (**********) -Lemma r_Rmult_mult:(r,r1,r2:R)(``r*r1==r*r2``)->``r<>0``->(r1==r2). - Intros; Transitivity ``(/r * r)*r1``. - Rewrite Rinv_l; Auto with real. - Transitivity ``(/r * r)*r2``. - Repeat Rewrite Rmult_assoc; Rewrite H; Trivial. - Rewrite Rinv_l; Auto with real. +Lemma Rmult_eq_reg_l : forall r r1 r2, r * r1 = r * r2 -> r <> 0 -> r1 = r2. + intros; transitivity (/ r * r * r1). + rewrite Rinv_l; auto with real. + transitivity (/ r * r * r2). + repeat rewrite Rmult_assoc; rewrite H; trivial. + rewrite Rinv_l; auto with real. Qed. (**********) -Lemma without_div_Od:(r1,r2:R)``r1*r2==0`` -> ``r1==0`` \/ ``r2==0``. - Intros; Case (Req_EM r1 ``0``); [Intro Hz | Intro Hnotz]. - Auto. - Right; Apply r_Rmult_mult with r1; Trivial. - Rewrite H; Auto with real. +Lemma Rmult_integral : forall r1 r2, r1 * r2 = 0 -> r1 = 0 \/ r2 = 0. + intros; case (Req_dec r1 0); [ intro Hz | intro Hnotz ]. + auto. + right; apply Rmult_eq_reg_l with r1; trivial. + rewrite H; auto with real. Qed. (**********) -Lemma without_div_Oi:(r1,r2:R) ``r1==0``\/``r2==0`` -> ``r1*r2==0``. - Intros r1 r2 [H | H]; Rewrite H; Auto with real. +Lemma Rmult_eq_0_compat : forall r1 r2, r1 = 0 \/ r2 = 0 -> r1 * r2 = 0. + intros r1 r2 [H| H]; rewrite H; auto with real. Qed. -Hints Resolve without_div_Oi : real. +Hint Resolve Rmult_eq_0_compat: real. (**********) -Lemma without_div_Oi1:(r1,r2:R) ``r1==0`` -> ``r1*r2==0``. - Auto with real. +Lemma Rmult_eq_0_compat_r : forall r1 r2, r1 = 0 -> r1 * r2 = 0. + auto with real. Qed. (**********) -Lemma without_div_Oi2:(r1,r2:R) ``r2==0`` -> ``r1*r2==0``. - Auto with real. +Lemma Rmult_eq_0_compat_l : forall r1 r2, r2 = 0 -> r1 * r2 = 0. + auto with real. Qed. (**********) -Lemma without_div_O_contr:(r1,r2:R)``r1*r2<>0`` -> ``r1<>0`` /\ ``r2<>0``. -Intros r1 r2 H; Split; Red; Intro; Apply H; Auto with real. +Lemma Rmult_neq_0_reg : forall r1 r2, r1 * r2 <> 0 -> r1 <> 0 /\ r2 <> 0. +intros r1 r2 H; split; red in |- *; intro; apply H; auto with real. Qed. (**********) -Lemma mult_non_zero :(r1,r2:R)``r1<>0`` /\ ``r2<>0`` -> ``r1*r2<>0``. -Red; Intros r1 r2 (H1,H2) H. -Case (without_div_Od r1 r2); Auto with real. +Lemma Rmult_integral_contrapositive : + forall r1 r2, r1 <> 0 /\ r2 <> 0 -> r1 * r2 <> 0. +red in |- *; intros r1 r2 [H1 H2] H. +case (Rmult_integral r1 r2); auto with real. Qed. -Hints Resolve mult_non_zero : real. +Hint Resolve Rmult_integral_contrapositive: real. (**********) -Lemma Rmult_Rplus_distrl: - (r1,r2,r3:R) ``(r1+r2)*r3 == (r1*r3)+(r2*r3)``. -Intros; Ring. +Lemma Rmult_plus_distr_r : + forall r1 r2 r3, (r1 + r2) * r3 = r1 * r3 + r2 * r3. +intros; ring. Qed. (** Square function *) (***********) -Definition Rsqr:R->R:=[r:R]``r*r``. -V7only[Notation "x ²" := (Rsqr x) (at level 2,left associativity).]. +Definition Rsqr r : R := r * r. (***********) -Lemma Rsqr_O:(Rsqr ``0``)==``0``. - Unfold Rsqr; Auto with real. +Lemma Rsqr_0 : Rsqr 0 = 0. + unfold Rsqr in |- *; auto with real. Qed. (***********) -Lemma Rsqr_r_R0:(r:R)(Rsqr r)==``0``->``r==0``. -Unfold Rsqr;Intros;Elim (without_div_Od r r H);Trivial. +Lemma Rsqr_0_uniq : forall r, Rsqr r = 0 -> r = 0. +unfold Rsqr in |- *; intros; elim (Rmult_integral r r H); trivial. Qed. (*********************************************************) @@ -429,736 +430,725 @@ Qed. (*********************************************************) (**********) -Lemma eq_Ropp:(r1,r2:R)(r1==r2)->``-r1 == -r2``. - Auto with real. +Lemma Ropp_eq_compat : forall r1 r2, r1 = r2 -> - r1 = - r2. + auto with real. Qed. -Hints Resolve eq_Ropp : real. +Hint Resolve Ropp_eq_compat: real. (**********) -Lemma Ropp_O:``-0==0``. - Ring. +Lemma Ropp_0 : -0 = 0. + ring. Qed. -Hints Resolve Ropp_O : real v62. +Hint Resolve Ropp_0: real v62. (**********) -Lemma eq_RoppO:(r:R)``r==0``-> ``-r==0``. - Intros; Rewrite -> H; Auto with real. +Lemma Ropp_eq_0_compat : forall r, r = 0 -> - r = 0. + intros; rewrite H; auto with real. Qed. -Hints Resolve eq_RoppO : real. +Hint Resolve Ropp_eq_0_compat: real. (**********) -Lemma Ropp_Ropp:(r:R)``-(-r)==r``. - Intro; Ring. +Lemma Ropp_involutive : forall r, - - r = r. + intro; ring. Qed. -Hints Resolve Ropp_Ropp : real. +Hint Resolve Ropp_involutive: real. (*********) -Lemma Ropp_neq:(r:R)``r<>0``->``-r<>0``. -Red;Intros r H H0. -Apply H. -Transitivity ``-(-r)``; Auto with real. +Lemma Ropp_neq_0_compat : forall r, r <> 0 -> - r <> 0. +red in |- *; intros r H H0. +apply H. +transitivity (- - r); auto with real. Qed. -Hints Resolve Ropp_neq : real. +Hint Resolve Ropp_neq_0_compat: real. (**********) -Lemma Ropp_distr1:(r1,r2:R)``-(r1+r2)==(-r1 + -r2)``. - Intros; Ring. +Lemma Ropp_plus_distr : forall r1 r2, - (r1 + r2) = - r1 + - r2. + intros; ring. Qed. -Hints Resolve Ropp_distr1 : real. +Hint Resolve Ropp_plus_distr: real. (** Opposite and multiplication *) -Lemma Ropp_mul1:(r1,r2:R)``(-r1)*r2 == -(r1*r2)``. - Intros; Ring. +Lemma Ropp_mult_distr_l_reverse : forall r1 r2, - r1 * r2 = - (r1 * r2). + intros; ring. Qed. -Hints Resolve Ropp_mul1 : real. +Hint Resolve Ropp_mult_distr_l_reverse: real. (**********) -Lemma Ropp_mul2:(r1,r2:R)``(-r1)*(-r2)==r1*r2``. - Intros; Ring. +Lemma Rmult_opp_opp : forall r1 r2, - r1 * - r2 = r1 * r2. + intros; ring. Qed. -Hints Resolve Ropp_mul2 : real. +Hint Resolve Rmult_opp_opp: real. -Lemma Ropp_mul3 : (r1,r2:R) ``r1*(-r2) == -(r1*r2)``. -Intros; Rewrite <- Ropp_mul1; Ring. +Lemma Ropp_mult_distr_r_reverse : forall r1 r2, r1 * - r2 = - (r1 * r2). +intros; rewrite <- Ropp_mult_distr_l_reverse; ring. Qed. (** Substraction *) -Lemma minus_R0:(r:R)``r-0==r``. -Intro;Ring. +Lemma Rminus_0_r : forall r, r - 0 = r. +intro; ring. Qed. -Hints Resolve minus_R0 : real. +Hint Resolve Rminus_0_r: real. -Lemma Rminus_Ropp:(r:R)``0-r==-r``. -Intro;Ring. +Lemma Rminus_0_l : forall r, 0 - r = - r. +intro; ring. Qed. -Hints Resolve Rminus_Ropp : real. +Hint Resolve Rminus_0_l: real. (**********) -Lemma Ropp_distr2:(r1,r2:R)``-(r1-r2)==r2-r1``. - Intros; Ring. +Lemma Ropp_minus_distr : forall r1 r2, - (r1 - r2) = r2 - r1. + intros; ring. Qed. -Hints Resolve Ropp_distr2 : real. +Hint Resolve Ropp_minus_distr: real. -Lemma Ropp_distr3:(r1,r2:R)``-(r2-r1)==r1-r2``. -Intros; Ring. +Lemma Ropp_minus_distr' : forall r1 r2, - (r2 - r1) = r1 - r2. +intros; ring. Qed. -Hints Resolve Ropp_distr3 : real. +Hint Resolve Ropp_minus_distr': real. (**********) -Lemma eq_Rminus:(r1,r2:R)(r1==r2)->``r1-r2==0``. - Intros; Rewrite H; Ring. +Lemma Rminus_diag_eq : forall r1 r2, r1 = r2 -> r1 - r2 = 0. + intros; rewrite H; ring. Qed. -Hints Resolve eq_Rminus : real. +Hint Resolve Rminus_diag_eq: real. (**********) -Lemma Rminus_eq:(r1,r2:R)``r1-r2==0`` -> r1==r2. - Intros r1 r2; Unfold Rminus; Rewrite -> Rplus_sym; Intro. - Rewrite <- (Ropp_Ropp r2); Apply (Rplus_Ropp (Ropp r2) r1 H). +Lemma Rminus_diag_uniq : forall r1 r2, r1 - r2 = 0 -> r1 = r2. + intros r1 r2; unfold Rminus in |- *; rewrite Rplus_comm; intro. + rewrite <- (Ropp_involutive r2); apply (Rplus_opp_r_uniq (- r2) r1 H). Qed. -Hints Immediate Rminus_eq : real. +Hint Immediate Rminus_diag_uniq: real. -Lemma Rminus_eq_right:(r1,r2:R)``r2-r1==0`` -> r1==r2. -Intros;Generalize (Rminus_eq r2 r1 H);Clear H;Intro H;Rewrite H;Ring. +Lemma Rminus_diag_uniq_sym : forall r1 r2, r2 - r1 = 0 -> r1 = r2. +intros; generalize (Rminus_diag_uniq r2 r1 H); clear H; intro H; rewrite H; + ring. Qed. -Hints Immediate Rminus_eq_right : real. +Hint Immediate Rminus_diag_uniq_sym: real. -Lemma Rplus_Rminus: (p,q:R)``p+(q-p)``==q. -Intros; Ring. +Lemma Rplus_minus : forall r1 r2, r1 + (r2 - r1) = r2. +intros; ring. Qed. -Hints Resolve Rplus_Rminus:real. +Hint Resolve Rplus_minus: real. (**********) -Lemma Rminus_eq_contra:(r1,r2:R)``r1<>r2``->``r1-r2<>0``. -Red; Intros r1 r2 H H0. -Apply H; Auto with real. +Lemma Rminus_eq_contra : forall r1 r2, r1 <> r2 -> r1 - r2 <> 0. +red in |- *; intros r1 r2 H H0. +apply H; auto with real. Qed. -Hints Resolve Rminus_eq_contra : real. +Hint Resolve Rminus_eq_contra: real. -Lemma Rminus_not_eq:(r1,r2:R)``r1-r2<>0``->``r1<>r2``. -Red; Intros; Elim H; Apply eq_Rminus; Auto. +Lemma Rminus_not_eq : forall r1 r2, r1 - r2 <> 0 -> r1 <> r2. +red in |- *; intros; elim H; apply Rminus_diag_eq; auto. Qed. -Hints Resolve Rminus_not_eq : real. +Hint Resolve Rminus_not_eq: real. -Lemma Rminus_not_eq_right:(r1,r2:R)``r2-r1<>0`` -> ``r1<>r2``. -Red; Intros;Elim H;Rewrite H0; Ring. +Lemma Rminus_not_eq_right : forall r1 r2, r2 - r1 <> 0 -> r1 <> r2. +red in |- *; intros; elim H; rewrite H0; ring. Qed. -Hints Resolve Rminus_not_eq_right : real. +Hint Resolve Rminus_not_eq_right: real. -V7only [Notation not_sym := (sym_not_eq R).]. (**********) -Lemma Rminus_distr: (x,y,z:R) ``x*(y-z)==(x*y) - (x*z)``. -Intros; Ring. +Lemma Rmult_minus_distr_l : + forall r1 r2 r3, r1 * (r2 - r3) = r1 * r2 - r1 * r3. +intros; ring. Qed. (** Inverse *) -Lemma Rinv_R1:``/1==1``. -Field;Auto with real. +Lemma Rinv_1 : / 1 = 1. +field; auto with real. Qed. -Hints Resolve Rinv_R1 : real. +Hint Resolve Rinv_1: real. (*********) -Lemma Rinv_neq_R0:(r:R)``r<>0``->``(/r)<>0``. -Red; Intros; Apply R1_neq_R0. -Replace ``1`` with ``(/r) * r``; Auto with real. +Lemma Rinv_neq_0_compat : forall r, r <> 0 -> / r <> 0. +red in |- *; intros; apply R1_neq_R0. +replace 1 with (/ r * r); auto with real. Qed. -Hints Resolve Rinv_neq_R0 : real. +Hint Resolve Rinv_neq_0_compat: real. (*********) -Lemma Rinv_Rinv:(r:R)``r<>0``->``/(/r)==r``. -Intros;Field;Auto with real. +Lemma Rinv_involutive : forall r, r <> 0 -> / / r = r. +intros; field; auto with real. Qed. -Hints Resolve Rinv_Rinv : real. +Hint Resolve Rinv_involutive: real. (*********) -Lemma Rinv_Rmult:(r1,r2:R)``r1<>0``->``r2<>0``->``/(r1*r2)==(/r1)*(/r2)``. -Intros;Field;Auto with real. +Lemma Rinv_mult_distr : + forall r1 r2, r1 <> 0 -> r2 <> 0 -> / (r1 * r2) = / r1 * / r2. +intros; field; auto with real. Qed. (*********) -Lemma Ropp_Rinv:(r:R)``r<>0``->``-(/r)==/(-r)``. -Intros;Field;Auto with real. +Lemma Ropp_inv_permute : forall r, r <> 0 -> - / r = / - r. +intros; field; auto with real. Qed. -Lemma Rinv_r_simpl_r : (r1,r2:R)``r1<>0``->``r1*(/r1)*r2==r2``. -Intros; Transitivity ``1*r2``; Auto with real. -Rewrite Rinv_r; Auto with real. +Lemma Rinv_r_simpl_r : forall r1 r2, r1 <> 0 -> r1 * / r1 * r2 = r2. +intros; transitivity (1 * r2); auto with real. +rewrite Rinv_r; auto with real. Qed. -Lemma Rinv_r_simpl_l : (r1,r2:R)``r1<>0``->``r2*r1*(/r1)==r2``. -Intros; Transitivity ``r2*1``; Auto with real. -Transitivity ``r2*(r1*/r1)``; Auto with real. +Lemma Rinv_r_simpl_l : forall r1 r2, r1 <> 0 -> r2 * r1 * / r1 = r2. +intros; transitivity (r2 * 1); auto with real. +transitivity (r2 * (r1 * / r1)); auto with real. Qed. -Lemma Rinv_r_simpl_m : (r1,r2:R)``r1<>0``->``r1*r2*(/r1)==r2``. -Intros; Transitivity ``r2*1``; Auto with real. -Transitivity ``r2*(r1*/r1)``; Auto with real. -Ring. +Lemma Rinv_r_simpl_m : forall r1 r2, r1 <> 0 -> r1 * r2 * / r1 = r2. +intros; transitivity (r2 * 1); auto with real. +transitivity (r2 * (r1 * / r1)); auto with real. +ring. Qed. -Hints Resolve Rinv_r_simpl_l Rinv_r_simpl_r Rinv_r_simpl_m : real. +Hint Resolve Rinv_r_simpl_l Rinv_r_simpl_r Rinv_r_simpl_m: real. (*********) -Lemma Rinv_Rmult_simpl:(a,b,c:R)``a<>0``->``(a*(/b))*(c*(/a))==c*(/b)``. -Intros a b c; Intros. -Transitivity ``(a*/a)*(c*(/b))``; Auto with real. -Ring. +Lemma Rinv_mult_simpl : + forall r1 r2 r3, r1 <> 0 -> r1 * / r2 * (r3 * / r1) = r3 * / r2. +intros a b c; intros. +transitivity (a * / a * (c * / b)); auto with real. +ring. Qed. (** Order and addition *) -Lemma Rlt_compatibility_r:(r,r1,r2:R)``r1<r2``->``r1+r<r2+r``. -Intros. -Rewrite (Rplus_sym r1 r); Rewrite (Rplus_sym r2 r); Auto with real. +Lemma Rplus_lt_compat_r : forall r r1 r2, r1 < r2 -> r1 + r < r2 + r. +intros. +rewrite (Rplus_comm r1 r); rewrite (Rplus_comm r2 r); auto with real. Qed. -Hints Resolve Rlt_compatibility_r : real. +Hint Resolve Rplus_lt_compat_r: real. (**********) -Lemma Rlt_anti_compatibility: (r,r1,r2:R)``r+r1 < r+r2`` -> ``r1<r2``. -Intros; Cut ``(-r+r)+r1 < (-r+r)+r2``. -Rewrite -> Rplus_Ropp_l. -Elim (Rplus_ne r1); Elim (Rplus_ne r2); Intros; Rewrite <- H3; - Rewrite <- H1; Auto with zarith real. -Rewrite -> Rplus_assoc; Rewrite -> Rplus_assoc; - Apply (Rlt_compatibility ``-r`` ``r+r1`` ``r+r2`` H). +Lemma Rplus_lt_reg_r : forall r r1 r2, r + r1 < r + r2 -> r1 < r2. +intros; cut (- r + r + r1 < - r + r + r2). +rewrite Rplus_opp_l. +elim (Rplus_ne r1); elim (Rplus_ne r2); intros; rewrite <- H3; rewrite <- H1; + auto with zarith real. +rewrite Rplus_assoc; rewrite Rplus_assoc; + apply (Rplus_lt_compat_l (- r) (r + r1) (r + r2) H). Qed. (**********) -Lemma Rle_compatibility:(r,r1,r2:R)``r1<=r2`` -> ``r+r1 <= r+r2 ``. -Unfold Rle; Intros; Elim H; Intro. -Left; Apply (Rlt_compatibility r r1 r2 H0). -Right; Rewrite <- H0; Auto with zarith real. +Lemma Rplus_le_compat_l : forall r r1 r2, r1 <= r2 -> r + r1 <= r + r2. +unfold Rle in |- *; intros; elim H; intro. +left; apply (Rplus_lt_compat_l r r1 r2 H0). +right; rewrite <- H0; auto with zarith real. Qed. (**********) -Lemma Rle_compatibility_r:(r,r1,r2:R)``r1<=r2`` -> ``r1+r<=r2+r``. -Unfold Rle; Intros; Elim H; Intro. -Left; Apply (Rlt_compatibility_r r r1 r2 H0). -Right; Rewrite <- H0; Auto with real. +Lemma Rplus_le_compat_r : forall r r1 r2, r1 <= r2 -> r1 + r <= r2 + r. +unfold Rle in |- *; intros; elim H; intro. +left; apply (Rplus_lt_compat_r r r1 r2 H0). +right; rewrite <- H0; auto with real. Qed. -Hints Resolve Rle_compatibility Rle_compatibility_r : real. +Hint Resolve Rplus_le_compat_l Rplus_le_compat_r: real. (**********) -Lemma Rle_anti_compatibility: (r,r1,r2:R)``r+r1<=r+r2`` -> ``r1<=r2``. -Unfold Rle; Intros; Elim H; Intro. -Left; Apply (Rlt_anti_compatibility r r1 r2 H0). -Right; Apply (r_Rplus_plus r r1 r2 H0). +Lemma Rplus_le_reg_l : forall r r1 r2, r + r1 <= r + r2 -> r1 <= r2. +unfold Rle in |- *; intros; elim H; intro. +left; apply (Rplus_lt_reg_r r r1 r2 H0). +right; apply (Rplus_eq_reg_l r r1 r2 H0). Qed. (**********) -Lemma sum_inequa_Rle_lt:(a,x,b,c,y,d:R)``a<=x`` -> ``x<b`` -> - ``c<y`` -> ``y<=d`` -> ``a+c < x+y < b+d``. -Intros;Split. -Apply Rlt_le_trans with ``a+y``; Auto with real. -Apply Rlt_le_trans with ``b+y``; Auto with real. +Lemma sum_inequa_Rle_lt : + forall a x b c y d:R, + a <= x -> x < b -> c < y -> y <= d -> a + c < x + y < b + d. +intros; split. +apply Rlt_le_trans with (a + y); auto with real. +apply Rlt_le_trans with (b + y); auto with real. Qed. (*********) -Lemma Rplus_lt:(r1,r2,r3,r4:R)``r1<r2`` -> ``r3<r4`` -> ``r1+r3 < r2+r4``. -Intros; Apply Rlt_trans with ``r2+r3``; Auto with real. +Lemma Rplus_lt_compat : + forall r1 r2 r3 r4, r1 < r2 -> r3 < r4 -> r1 + r3 < r2 + r4. +intros; apply Rlt_trans with (r2 + r3); auto with real. Qed. -Lemma Rplus_le:(r1,r2,r3,r4:R)``r1<=r2`` -> ``r3<=r4`` -> ``r1+r3 <= r2+r4``. -Intros; Apply Rle_trans with ``r2+r3``; Auto with real. +Lemma Rplus_le_compat : + forall r1 r2 r3 r4, r1 <= r2 -> r3 <= r4 -> r1 + r3 <= r2 + r4. +intros; apply Rle_trans with (r2 + r3); auto with real. Qed. (*********) -Lemma Rplus_lt_le_lt:(r1,r2,r3,r4:R)``r1<r2`` -> ``r3<=r4`` -> - ``r1+r3 < r2+r4``. -Intros; Apply Rlt_le_trans with ``r2+r3``; Auto with real. +Lemma Rplus_lt_le_compat : + forall r1 r2 r3 r4, r1 < r2 -> r3 <= r4 -> r1 + r3 < r2 + r4. +intros; apply Rlt_le_trans with (r2 + r3); auto with real. Qed. (*********) -Lemma Rplus_le_lt_lt:(r1,r2,r3,r4:R)``r1<=r2`` -> ``r3<r4`` -> - ``r1+r3 < r2+r4``. -Intros; Apply Rle_lt_trans with ``r2+r3``; Auto with real. +Lemma Rplus_le_lt_compat : + forall r1 r2 r3 r4, r1 <= r2 -> r3 < r4 -> r1 + r3 < r2 + r4. +intros; apply Rle_lt_trans with (r2 + r3); auto with real. Qed. -Hints Immediate Rplus_lt Rplus_le Rplus_lt_le_lt Rplus_le_lt_lt : real. +Hint Immediate Rplus_lt_compat Rplus_le_compat Rplus_lt_le_compat + Rplus_le_lt_compat: real. (** Order and Opposite *) (**********) -Lemma Rgt_Ropp:(r1,r2:R) ``r1 > r2`` -> ``-r1 < -r2``. -Unfold Rgt; Intros. -Apply (Rlt_anti_compatibility ``r2+r1``). -Replace ``r2+r1+(-r1)`` with r2. -Replace ``r2+r1+(-r2)`` with r1. -Trivial. -Ring. -Ring. +Lemma Ropp_gt_lt_contravar : forall r1 r2, r1 > r2 -> - r1 < - r2. +unfold Rgt in |- *; intros. +apply (Rplus_lt_reg_r (r2 + r1)). +replace (r2 + r1 + - r1) with r2. +replace (r2 + r1 + - r2) with r1. +trivial. +ring. +ring. Qed. -Hints Resolve Rgt_Ropp. +Hint Resolve Ropp_gt_lt_contravar. (**********) -Lemma Rlt_Ropp:(r1,r2:R) ``r1 < r2`` -> ``-r1 > -r2``. -Unfold Rgt; Auto with real. +Lemma Ropp_lt_gt_contravar : forall r1 r2, r1 < r2 -> - r1 > - r2. +unfold Rgt in |- *; auto with real. Qed. -Hints Resolve Rlt_Ropp : real. +Hint Resolve Ropp_lt_gt_contravar: real. -Lemma Ropp_Rlt: (x,y:R) ``-y < -x`` ->``x<y``. -Intros x y H'. -Rewrite <- (Ropp_Ropp x); Rewrite <- (Ropp_Ropp y); Auto with real. +Lemma Ropp_lt_cancel : forall r1 r2, - r2 < - r1 -> r1 < r2. +intros x y H'. +rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive y); + auto with real. Qed. -Hints Immediate Ropp_Rlt : real. +Hint Immediate Ropp_lt_cancel: real. -Lemma Rlt_Ropp1:(r1,r2:R) ``r2 < r1`` -> ``-r1 < -r2``. -Auto with real. +Lemma Ropp_lt_contravar : forall r1 r2, r2 < r1 -> - r1 < - r2. +auto with real. Qed. -Hints Resolve Rlt_Ropp1 : real. +Hint Resolve Ropp_lt_contravar: real. (**********) -Lemma Rle_Ropp:(r1,r2:R) ``r1 <= r2`` -> ``-r1 >= -r2``. -Unfold Rge; Intros r1 r2 [H|H]; Auto with real. +Lemma Ropp_le_ge_contravar : forall r1 r2, r1 <= r2 -> - r1 >= - r2. +unfold Rge in |- *; intros r1 r2 [H| H]; auto with real. Qed. -Hints Resolve Rle_Ropp : real. +Hint Resolve Ropp_le_ge_contravar: real. -Lemma Ropp_Rle: (x,y:R) ``-y <= -x`` ->``x <= y``. -Intros x y H. -Elim H;Auto with real. -Intro H1;Rewrite <-(Ropp_Ropp x);Rewrite <-(Ropp_Ropp y);Rewrite H1; - Auto with real. +Lemma Ropp_le_cancel : forall r1 r2, - r2 <= - r1 -> r1 <= r2. +intros x y H. +elim H; auto with real. +intro H1; rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive y); + rewrite H1; auto with real. Qed. -Hints Immediate Ropp_Rle : real. +Hint Immediate Ropp_le_cancel: real. -Lemma Rle_Ropp1:(r1,r2:R) ``r2 <= r1`` -> ``-r1 <= -r2``. -Intros r1 r2 H;Elim H;Auto with real. +Lemma Ropp_le_contravar : forall r1 r2, r2 <= r1 -> - r1 <= - r2. +intros r1 r2 H; elim H; auto with real. Qed. -Hints Resolve Rle_Ropp1 : real. +Hint Resolve Ropp_le_contravar: real. (**********) -Lemma Rge_Ropp:(r1,r2:R) ``r1 >= r2`` -> ``-r1 <= -r2``. -Unfold Rge; Intros r1 r2 [H|H]; Auto with real. +Lemma Ropp_ge_le_contravar : forall r1 r2, r1 >= r2 -> - r1 <= - r2. +unfold Rge in |- *; intros r1 r2 [H| H]; auto with real. Qed. -Hints Resolve Rge_Ropp : real. +Hint Resolve Ropp_ge_le_contravar: real. (**********) -Lemma Rlt_RO_Ropp:(r:R) ``0 < r`` -> ``0 > -r``. -Intros; Replace ``0`` with ``-0``; Auto with real. +Lemma Ropp_0_lt_gt_contravar : forall r, 0 < r -> 0 > - r. +intros; replace 0 with (-0); auto with real. Qed. -Hints Resolve Rlt_RO_Ropp : real. +Hint Resolve Ropp_0_lt_gt_contravar: real. (**********) -Lemma Rgt_RO_Ropp:(r:R) ``0 > r`` -> ``0 < -r``. -Intros; Replace ``0`` with ``-0``; Auto with real. +Lemma Ropp_0_gt_lt_contravar : forall r, 0 > r -> 0 < - r. +intros; replace 0 with (-0); auto with real. Qed. -Hints Resolve Rgt_RO_Ropp : real. +Hint Resolve Ropp_0_gt_lt_contravar: real. (**********) -Lemma Rgt_RoppO:(r:R)``r>0``->``(-r)<0``. -Intros; Rewrite <- Ropp_O; Auto with real. +Lemma Ropp_lt_gt_0_contravar : forall r, r > 0 -> - r < 0. +intros; rewrite <- Ropp_0; auto with real. Qed. (**********) -Lemma Rlt_RoppO:(r:R)``r<0``->``-r>0``. -Intros; Rewrite <- Ropp_O; Auto with real. +Lemma Ropp_gt_lt_0_contravar : forall r, r < 0 -> - r > 0. +intros; rewrite <- Ropp_0; auto with real. Qed. -Hints Resolve Rgt_RoppO Rlt_RoppO: real. +Hint Resolve Ropp_lt_gt_0_contravar Ropp_gt_lt_0_contravar: real. (**********) -Lemma Rle_RO_Ropp:(r:R) ``0 <= r`` -> ``0 >= -r``. -Intros; Replace ``0`` with ``-0``; Auto with real. +Lemma Ropp_0_le_ge_contravar : forall r, 0 <= r -> 0 >= - r. +intros; replace 0 with (-0); auto with real. Qed. -Hints Resolve Rle_RO_Ropp : real. +Hint Resolve Ropp_0_le_ge_contravar: real. (**********) -Lemma Rge_RO_Ropp:(r:R) ``0 >= r`` -> ``0 <= -r``. -Intros; Replace ``0`` with ``-0``; Auto with real. +Lemma Ropp_0_ge_le_contravar : forall r, 0 >= r -> 0 <= - r. +intros; replace 0 with (-0); auto with real. Qed. -Hints Resolve Rge_RO_Ropp : real. +Hint Resolve Ropp_0_ge_le_contravar: real. (** Order and multiplication *) -Lemma Rlt_monotony_r:(r,r1,r2:R)``0<r`` -> ``r1 < r2`` -> ``r1*r < r2*r``. -Intros; Rewrite (Rmult_sym r1 r); Rewrite (Rmult_sym r2 r); Auto with real. +Lemma Rmult_lt_compat_r : forall r r1 r2, 0 < r -> r1 < r2 -> r1 * r < r2 * r. +intros; rewrite (Rmult_comm r1 r); rewrite (Rmult_comm r2 r); auto with real. Qed. -Hints Resolve Rlt_monotony_r. +Hint Resolve Rmult_lt_compat_r. -Lemma Rlt_monotony_contra: (z, x, y:R) ``0<z`` ->``z*x<z*y`` ->``x<y``. -Intros z x y H H0. -Case (total_order x y); Intros Eq0; Auto; Elim Eq0; Clear Eq0; Intros Eq0. - Rewrite Eq0 in H0;ElimType False;Apply (Rlt_antirefl ``z*y``);Auto. -Generalize (Rlt_monotony z y x H Eq0);Intro;ElimType False; - Generalize (Rlt_trans ``z*x`` ``z*y`` ``z*x`` H0 H1);Intro; - Apply (Rlt_antirefl ``z*x``);Auto. +Lemma Rmult_lt_reg_l : forall r r1 r2, 0 < r -> r * r1 < r * r2 -> r1 < r2. +intros z x y H H0. +case (Rtotal_order x y); intros Eq0; auto; elim Eq0; clear Eq0; intros Eq0. + rewrite Eq0 in H0; elimtype False; apply (Rlt_irrefl (z * y)); auto. +generalize (Rmult_lt_compat_l z y x H Eq0); intro; elimtype False; + generalize (Rlt_trans (z * x) (z * y) (z * x) H0 H1); + intro; apply (Rlt_irrefl (z * x)); auto. Qed. -V7only [ -Notation Rlt_monotony_rev := Rlt_monotony_contra. -]. -Lemma Rlt_anti_monotony:(r,r1,r2:R)``r < 0`` -> ``r1 < r2`` -> ``r*r1 > r*r2``. -Intros; Replace r with ``-(-r)``; Auto with real. -Rewrite (Ropp_mul1 ``-r``); Rewrite (Ropp_mul1 ``-r``). -Apply Rlt_Ropp; Auto with real. +Lemma Rmult_lt_gt_compat_neg_l : + forall r r1 r2, r < 0 -> r1 < r2 -> r * r1 > r * r2. +intros; replace r with (- - r); auto with real. +rewrite (Ropp_mult_distr_l_reverse (- r)); + rewrite (Ropp_mult_distr_l_reverse (- r)). +apply Ropp_lt_gt_contravar; auto with real. Qed. (**********) -Lemma Rle_monotony: - (r,r1,r2:R)``0 <= r`` -> ``r1 <= r2`` -> ``r*r1 <= r*r2``. -Intros r r1 r2 H H0; NewDestruct H; NewDestruct H0; Unfold Rle; Auto with real. -Right; Rewrite <- H; Do 2 Rewrite Rmult_Ol; Reflexivity. +Lemma Rmult_le_compat_l : + forall r r1 r2, 0 <= r -> r1 <= r2 -> r * r1 <= r * r2. +intros r r1 r2 H H0; destruct H; destruct H0; unfold Rle in |- *; + auto with real. +right; rewrite <- H; do 2 rewrite Rmult_0_l; reflexivity. Qed. -Hints Resolve Rle_monotony : real. +Hint Resolve Rmult_le_compat_l: real. -Lemma Rle_monotony_r: - (r,r1,r2:R)``0 <= r`` -> ``r1 <= r2`` -> ``r1*r <= r2*r``. -Intros r r1 r2 H; -Rewrite (Rmult_sym r1 r); Rewrite (Rmult_sym r2 r); Auto with real. +Lemma Rmult_le_compat_r : + forall r r1 r2, 0 <= r -> r1 <= r2 -> r1 * r <= r2 * r. +intros r r1 r2 H; rewrite (Rmult_comm r1 r); rewrite (Rmult_comm r2 r); + auto with real. Qed. -Hints Resolve Rle_monotony_r : real. +Hint Resolve Rmult_le_compat_r: real. -Lemma Rle_monotony_contra: - (z, x, y:R) ``0<z`` ->``z*x<=z*y`` ->``x<=y``. -Intros z x y H H0;Case H0; Auto with real. -Intros H1; Apply Rlt_le. -Apply Rlt_monotony_contra with z := z;Auto. -Intros H1;Replace x with (Rmult (Rinv z) (Rmult z x)); Auto with real. -Replace y with (Rmult (Rinv z) (Rmult z y)). - Rewrite H1;Auto with real. -Rewrite <- Rmult_assoc; Rewrite Rinv_l; Auto with real. -Rewrite <- Rmult_assoc; Rewrite Rinv_l; Auto with real. +Lemma Rmult_le_reg_l : forall r r1 r2, 0 < r -> r * r1 <= r * r2 -> r1 <= r2. +intros z x y H H0; case H0; auto with real. +intros H1; apply Rlt_le. +apply Rmult_lt_reg_l with (r := z); auto. +intros H1; replace x with (/ z * (z * x)); auto with real. +replace y with (/ z * (z * y)). + rewrite H1; auto with real. +rewrite <- Rmult_assoc; rewrite Rinv_l; auto with real. +rewrite <- Rmult_assoc; rewrite Rinv_l; auto with real. Qed. -Lemma Rle_anti_monotony1 - :(r,r1,r2:R)``r <= 0`` -> ``r1 <= r2`` -> ``r*r2 <= r*r1``. -Intros; Replace r with ``-(-r)``; Auto with real. -Do 2 Rewrite (Ropp_mul1 ``-r``). -Apply Rle_Ropp1; Auto with real. +Lemma Rmult_le_compat_neg_l : + forall r r1 r2, r <= 0 -> r1 <= r2 -> r * r2 <= r * r1. +intros; replace r with (- - r); auto with real. +do 2 rewrite (Ropp_mult_distr_l_reverse (- r)). +apply Ropp_le_contravar; auto with real. Qed. -Hints Resolve Rle_anti_monotony1 : real. +Hint Resolve Rmult_le_compat_neg_l: real. -Lemma Rle_anti_monotony - :(r,r1,r2:R)``r <= 0`` -> ``r1 <= r2`` -> ``r*r1 >= r*r2``. -Intros; Apply Rle_ge; Auto with real. +Lemma Rmult_le_ge_compat_neg_l : + forall r r1 r2, r <= 0 -> r1 <= r2 -> r * r1 >= r * r2. +intros; apply Rle_ge; auto with real. Qed. -Hints Resolve Rle_anti_monotony : real. +Hint Resolve Rmult_le_ge_compat_neg_l: real. -Lemma Rle_Rmult_comp: - (x, y, z, t:R) ``0 <= x`` -> ``0 <= z`` -> ``x <= y`` -> ``z <= t`` -> - ``x*z <= y*t``. -Intros x y z t H' H'0 H'1 H'2. -Apply Rle_trans with r2 := ``x*t``; Auto with real. -Repeat Rewrite [x:?](Rmult_sym x t). -Apply Rle_monotony; Auto. -Apply Rle_trans with z; Auto. +Lemma Rmult_le_compat : + forall r1 r2 r3 r4, + 0 <= r1 -> 0 <= r3 -> r1 <= r2 -> r3 <= r4 -> r1 * r3 <= r2 * r4. +intros x y z t H' H'0 H'1 H'2. +apply Rle_trans with (r2 := x * t); auto with real. +repeat rewrite (fun x => Rmult_comm x t). +apply Rmult_le_compat_l; auto. +apply Rle_trans with z; auto. Qed. -Hints Resolve Rle_Rmult_comp :real. +Hint Resolve Rmult_le_compat: real. -Lemma Rmult_lt:(r1,r2,r3,r4:R)``r3>0`` -> ``r2>0`` -> - `` r1 < r2`` -> ``r3 < r4`` -> ``r1*r3 < r2*r4``. -Intros; Apply Rlt_trans with ``r2*r3``; Auto with real. +Lemma Rmult_gt_0_lt_compat : + forall r1 r2 r3 r4, + r3 > 0 -> r2 > 0 -> r1 < r2 -> r3 < r4 -> r1 * r3 < r2 * r4. +intros; apply Rlt_trans with (r2 * r3); auto with real. Qed. (*********) -Lemma Rmult_lt_0 - :(r1,r2,r3,r4:R)``r3>=0``->``r2>0``->``r1<r2``->``r3<r4``->``r1*r3<r2*r4``. -Intros; Apply Rle_lt_trans with ``r2*r3``; Auto with real. +Lemma Rmult_ge_0_gt_0_lt_compat : + forall r1 r2 r3 r4, + r3 >= 0 -> r2 > 0 -> r1 < r2 -> r3 < r4 -> r1 * r3 < r2 * r4. +intros; apply Rle_lt_trans with (r2 * r3); auto with real. Qed. (** Order and Substractions *) -Lemma Rlt_minus:(r1,r2:R)``r1 < r2`` -> ``r1-r2 < 0``. -Intros; Apply (Rlt_anti_compatibility ``r2``). -Replace ``r2+(r1-r2)`` with r1. -Replace ``r2+0`` with r2; Auto with real. -Ring. +Lemma Rlt_minus : forall r1 r2, r1 < r2 -> r1 - r2 < 0. +intros; apply (Rplus_lt_reg_r r2). +replace (r2 + (r1 - r2)) with r1. +replace (r2 + 0) with r2; auto with real. +ring. Qed. -Hints Resolve Rlt_minus : real. +Hint Resolve Rlt_minus: real. (**********) -Lemma Rle_minus:(r1,r2:R)``r1 <= r2`` -> ``r1-r2 <= 0``. -NewDestruct 1; Unfold Rle; Auto with real. +Lemma Rle_minus : forall r1 r2, r1 <= r2 -> r1 - r2 <= 0. +destruct 1; unfold Rle in |- *; auto with real. Qed. (**********) -Lemma Rminus_lt:(r1,r2:R)``r1-r2 < 0`` -> ``r1 < r2``. -Intros; Replace r1 with ``r1-r2+r2``. -Pattern 3 r2; Replace r2 with ``0+r2``; Auto with real. -Ring. +Lemma Rminus_lt : forall r1 r2, r1 - r2 < 0 -> r1 < r2. +intros; replace r1 with (r1 - r2 + r2). +pattern r2 at 3 in |- *; replace r2 with (0 + r2); auto with real. +ring. Qed. (**********) -Lemma Rminus_le:(r1,r2:R)``r1-r2 <= 0`` -> ``r1 <= r2``. -Intros; Replace r1 with ``r1-r2+r2``. -Pattern 3 r2; Replace r2 with ``0+r2``; Auto with real. -Ring. +Lemma Rminus_le : forall r1 r2, r1 - r2 <= 0 -> r1 <= r2. +intros; replace r1 with (r1 - r2 + r2). +pattern r2 at 3 in |- *; replace r2 with (0 + r2); auto with real. +ring. Qed. (**********) -Lemma tech_Rplus:(r,s:R)``0<=r`` -> ``0<s`` -> ``r+s<>0``. -Intros; Apply sym_not_eqT; Apply Rlt_not_eq. -Rewrite Rplus_sym; Replace ``0`` with ``0+0``; Auto with real. +Lemma tech_Rplus : forall r (s:R), 0 <= r -> 0 < s -> r + s <> 0. +intros; apply sym_not_eq; apply Rlt_not_eq. +rewrite Rplus_comm; replace 0 with (0 + 0); auto with real. Qed. -Hints Immediate tech_Rplus : real. +Hint Immediate tech_Rplus: real. (** Order and the square function *) -Lemma pos_Rsqr:(r:R)``0<=(Rsqr r)``. -Intro; Case (total_order_Rlt_Rle r ``0``); Unfold Rsqr; Intro. -Replace ``r*r`` with ``(-r)*(-r)``; Auto with real. -Replace ``0`` with ``-r*0``; Auto with real. -Replace ``0`` with ``0*r``; Auto with real. +Lemma Rle_0_sqr : forall r, 0 <= Rsqr r. +intro; case (Rlt_le_dec r 0); unfold Rsqr in |- *; intro. +replace (r * r) with (- r * - r); auto with real. +replace 0 with (- r * 0); auto with real. +replace 0 with (0 * r); auto with real. Qed. (***********) -Lemma pos_Rsqr1:(r:R)``r<>0``->``0<(Rsqr r)``. -Intros; Case (not_Req r ``0``); Trivial; Unfold Rsqr; Intro. -Replace ``r*r`` with ``(-r)*(-r)``; Auto with real. -Replace ``0`` with ``-r*0``; Auto with real. -Replace ``0`` with ``0*r``; Auto with real. +Lemma Rlt_0_sqr : forall r, r <> 0 -> 0 < Rsqr r. +intros; case (Rdichotomy r 0); trivial; unfold Rsqr in |- *; intro. +replace (r * r) with (- r * - r); auto with real. +replace 0 with (- r * 0); auto with real. +replace 0 with (0 * r); auto with real. Qed. -Hints Resolve pos_Rsqr pos_Rsqr1 : real. +Hint Resolve Rle_0_sqr Rlt_0_sqr: real. (** Zero is less than one *) -Lemma Rlt_R0_R1:``0<1``. -Replace ``1`` with ``(Rsqr 1)``; Auto with real. -Unfold Rsqr; Auto with real. +Lemma Rlt_0_1 : 0 < 1. +replace 1 with (Rsqr 1); auto with real. +unfold Rsqr in |- *; auto with real. Qed. -Hints Resolve Rlt_R0_R1 : real. +Hint Resolve Rlt_0_1: real. -Lemma Rle_R0_R1:``0<=1``. -Left. -Exact Rlt_R0_R1. +Lemma Rle_0_1 : 0 <= 1. +left. +exact Rlt_0_1. Qed. (** Order and inverse *) -Lemma Rlt_Rinv:(r:R)``0<r``->``0</r``. -Intros; Apply not_Rle; Red; Intros. -Absurd ``1<=0``; Auto with real. -Replace ``1`` with ``r*(/r)``; Auto with real. -Replace ``0`` with ``r*0``; Auto with real. +Lemma Rinv_0_lt_compat : forall r, 0 < r -> 0 < / r. +intros; apply Rnot_le_lt; red in |- *; intros. +absurd (1 <= 0); auto with real. +replace 1 with (r * / r); auto with real. +replace 0 with (r * 0); auto with real. Qed. -Hints Resolve Rlt_Rinv : real. +Hint Resolve Rinv_0_lt_compat: real. (*********) -Lemma Rlt_Rinv2:(r:R)``r < 0``->``/r < 0``. -Intros; Apply not_Rle; Red; Intros. -Absurd ``1<=0``; Auto with real. -Replace ``1`` with ``r*(/r)``; Auto with real. -Replace ``0`` with ``r*0``; Auto with real. +Lemma Rinv_lt_0_compat : forall r, r < 0 -> / r < 0. +intros; apply Rnot_le_lt; red in |- *; intros. +absurd (1 <= 0); auto with real. +replace 1 with (r * / r); auto with real. +replace 0 with (r * 0); auto with real. Qed. -Hints Resolve Rlt_Rinv2 : real. +Hint Resolve Rinv_lt_0_compat: real. (*********) -Lemma Rinv_lt:(r1,r2:R)``0 < r1*r2`` -> ``r1 < r2`` -> ``/r2 < /r1``. -Intros; Apply Rlt_monotony_rev with ``r1*r2``; Auto with real. -Case (without_div_O_contr r1 r2 ); Intros; Auto with real. -Replace ``r1*r2*/r2`` with r1. -Replace ``r1*r2*/r1`` with r2; Trivial. -Symmetry; Auto with real. -Symmetry; Auto with real. -Qed. - -Lemma Rlt_Rinv_R1: (x, y:R) ``1 <= x`` -> ``x<y`` ->``/y< /x``. -Intros x y H' H'0. -Cut (Rlt R0 x); [Intros Lt0 | Apply Rlt_le_trans with r2 := R1]; - Auto with real. -Apply Rlt_monotony_contra with z := x; Auto with real. -Rewrite (Rmult_sym x (Rinv x)); Rewrite Rinv_l; Auto with real. -Apply Rlt_monotony_contra with z := y; Auto with real. -Apply Rlt_trans with r2:=x;Auto. -Cut ``y*(x*/y)==x``. -Intro H1;Rewrite H1;Rewrite (Rmult_1r y);Auto. -Rewrite (Rmult_sym x); Rewrite <- Rmult_assoc; Rewrite (Rmult_sym y (Rinv y)); - Rewrite Rinv_l; Auto with real. -Apply imp_not_Req; Right. -Red; Apply Rlt_trans with r2 := x; Auto with real. -Qed. -Hints Resolve Rlt_Rinv_R1 :real. +Lemma Rinv_lt_contravar : forall r1 r2, 0 < r1 * r2 -> r1 < r2 -> / r2 < / r1. +intros; apply Rmult_lt_reg_l with (r1 * r2); auto with real. +case (Rmult_neq_0_reg r1 r2); intros; auto with real. +replace (r1 * r2 * / r2) with r1. +replace (r1 * r2 * / r1) with r2; trivial. +symmetry in |- *; auto with real. +symmetry in |- *; auto with real. +Qed. + +Lemma Rinv_1_lt_contravar : forall r1 r2, 1 <= r1 -> r1 < r2 -> / r2 < / r1. +intros x y H' H'0. +cut (0 < x); [ intros Lt0 | apply Rlt_le_trans with (r2 := 1) ]; + auto with real. +apply Rmult_lt_reg_l with (r := x); auto with real. +rewrite (Rmult_comm x (/ x)); rewrite Rinv_l; auto with real. +apply Rmult_lt_reg_l with (r := y); auto with real. +apply Rlt_trans with (r2 := x); auto. +cut (y * (x * / y) = x). +intro H1; rewrite H1; rewrite (Rmult_1_r y); auto. +rewrite (Rmult_comm x); rewrite <- Rmult_assoc; rewrite (Rmult_comm y (/ y)); + rewrite Rinv_l; auto with real. +apply Rlt_dichotomy_converse; right. +red in |- *; apply Rlt_trans with (r2 := x); auto with real. +Qed. +Hint Resolve Rinv_1_lt_contravar: real. (*********************************************************) (** Greater *) (*********************************************************) (**********) -Lemma Rge_ge_eq:(r1,r2:R)``r1 >= r2`` -> ``r2 >= r1`` -> r1==r2. -Intros; Apply Rle_antisym; Auto with real. +Lemma Rge_antisym : forall r1 r2, r1 >= r2 -> r2 >= r1 -> r1 = r2. +intros; apply Rle_antisym; auto with real. Qed. (**********) -Lemma Rlt_not_ge:(r1,r2:R)~(``r1<r2``)->``r1>=r2``. -Intros; Unfold Rge; Elim (total_order r1 r2); Intro. -Absurd ``r1<r2``; Trivial. -Case H0; Auto. +Lemma Rnot_lt_ge : forall r1 r2, ~ r1 < r2 -> r1 >= r2. +intros; unfold Rge in |- *; elim (Rtotal_order r1 r2); intro. +absurd (r1 < r2); trivial. +case H0; auto. Qed. (**********) -Lemma Rnot_lt_le:(r1,r2:R)~(``r1<r2``)->``r2<=r1``. -Intros; Apply Rge_le; Apply Rlt_not_ge; Assumption. +Lemma Rnot_lt_le : forall r1 r2, ~ r1 < r2 -> r2 <= r1. +intros; apply Rge_le; apply Rnot_lt_ge; assumption. Qed. (**********) -Lemma Rgt_not_le:(r1,r2:R)~(``r1>r2``)->``r1<=r2``. -Intros r1 r2 H; Apply Rge_le. -Exact (Rlt_not_ge r2 r1 H). +Lemma Rnot_gt_le : forall r1 r2, ~ r1 > r2 -> r1 <= r2. +intros r1 r2 H; apply Rge_le. +exact (Rnot_lt_ge r2 r1 H). Qed. (**********) -Lemma Rgt_ge:(r1,r2:R)``r1>r2`` -> ``r1 >= r2``. -Red; Auto with real. +Lemma Rgt_ge : forall r1 r2, r1 > r2 -> r1 >= r2. +red in |- *; auto with real. Qed. -V7only [ -(**********) -Lemma Rlt_sym:(r1,r2:R)``r1<r2`` <-> ``r2>r1``. -Split; Unfold Rgt; Auto with real. -Qed. - -(**********) -Lemma Rle_sym1:(r1,r2:R)``r1<=r2``->``r2>=r1``. -Proof Rle_ge. - -Notation "'Rle_sym2' a b c" := (Rge_le b a c) - (at level 10, a,b,c at level 9, only parsing). -Notation Rle_sym2 := Rge_le (only parsing). -(* -(**********) -Lemma Rle_sym2:(r1,r2:R)``r2>=r1`` -> ``r1<=r2``. -Proof [r1,r2](Rge_le r2 r1). -*) - -(**********) -Lemma Rle_sym:(r1,r2:R)``r1<=r2``<->``r2>=r1``. -Split; Auto with real. -Qed. -]. (**********) -Lemma Rge_gt_trans:(r1,r2,r3:R)``r1>=r2``->``r2>r3``->``r1>r3``. -Unfold Rgt; Intros; Apply Rlt_le_trans with r2; Auto with real. +Lemma Rge_gt_trans : forall r1 r2 r3, r1 >= r2 -> r2 > r3 -> r1 > r3. +unfold Rgt in |- *; intros; apply Rlt_le_trans with r2; auto with real. Qed. (**********) -Lemma Rgt_ge_trans:(r1,r2,r3:R)``r1>r2`` -> ``r2>=r3`` -> ``r1>r3``. -Unfold Rgt; Intros; Apply Rle_lt_trans with r2; Auto with real. +Lemma Rgt_ge_trans : forall r1 r2 r3, r1 > r2 -> r2 >= r3 -> r1 > r3. +unfold Rgt in |- *; intros; apply Rle_lt_trans with r2; auto with real. Qed. (**********) -Lemma Rgt_trans:(r1,r2,r3:R)``r1>r2`` -> ``r2>r3`` -> ``r1>r3``. -Unfold Rgt; Intros; Apply Rlt_trans with r2; Auto with real. +Lemma Rgt_trans : forall r1 r2 r3, r1 > r2 -> r2 > r3 -> r1 > r3. +unfold Rgt in |- *; intros; apply Rlt_trans with r2; auto with real. Qed. (**********) -Lemma Rge_trans:(r1,r2,r3:R)``r1>=r2`` -> ``r2>=r3`` -> ``r1>=r3``. -Intros; Apply Rle_ge. -Apply Rle_trans with r2; Auto with real. +Lemma Rge_trans : forall r1 r2 r3, r1 >= r2 -> r2 >= r3 -> r1 >= r3. +intros; apply Rle_ge. +apply Rle_trans with r2; auto with real. Qed. (**********) -Lemma Rlt_r_plus_R1:(r:R)``0<=r`` -> ``0<r+1``. -Intros. -Apply Rlt_le_trans with ``1``; Auto with real. -Pattern 1 ``1``; Replace ``1`` with ``0+1``; Auto with real. +Lemma Rle_lt_0_plus_1 : forall r, 0 <= r -> 0 < r + 1. +intros. +apply Rlt_le_trans with 1; auto with real. +pattern 1 at 1 in |- *; replace 1 with (0 + 1); auto with real. Qed. -Hints Resolve Rlt_r_plus_R1: real. +Hint Resolve Rle_lt_0_plus_1: real. (**********) -Lemma Rlt_r_r_plus_R1:(r:R)``r<r+1``. -Intros. -Pattern 1 r; Replace r with ``r+0``; Auto with real. +Lemma Rlt_plus_1 : forall r, r < r + 1. +intros. +pattern r at 1 in |- *; replace r with (r + 0); auto with real. Qed. -Hints Resolve Rlt_r_r_plus_R1: real. +Hint Resolve Rlt_plus_1: real. (**********) -Lemma tech_Rgt_minus:(r1,r2:R)``0<r2``->``r1>r1-r2``. -Red; Unfold Rminus; Intros. -Pattern 2 r1; Replace r1 with ``r1+0``; Auto with real. +Lemma tech_Rgt_minus : forall r1 r2, 0 < r2 -> r1 > r1 - r2. +red in |- *; unfold Rminus in |- *; intros. +pattern r1 at 2 in |- *; replace r1 with (r1 + 0); auto with real. Qed. (***********) -Lemma Rgt_plus_plus_r:(r,r1,r2:R)``r1>r2``->``r+r1 > r+r2``. -Unfold Rgt; Auto with real. +Lemma Rplus_gt_compat_l : forall r r1 r2, r1 > r2 -> r + r1 > r + r2. +unfold Rgt in |- *; auto with real. Qed. -Hints Resolve Rgt_plus_plus_r : real. +Hint Resolve Rplus_gt_compat_l: real. (***********) -Lemma Rgt_r_plus_plus:(r,r1,r2:R)``r+r1 > r+r2`` -> ``r1 > r2``. -Unfold Rgt; Intros; Apply (Rlt_anti_compatibility r r2 r1 H). +Lemma Rplus_gt_reg_l : forall r r1 r2, r + r1 > r + r2 -> r1 > r2. +unfold Rgt in |- *; intros; apply (Rplus_lt_reg_r r r2 r1 H). Qed. (***********) -Lemma Rge_plus_plus_r:(r,r1,r2:R)``r1>=r2`` -> ``r+r1 >= r+r2``. -Intros; Apply Rle_ge; Auto with real. +Lemma Rplus_ge_compat_l : forall r r1 r2, r1 >= r2 -> r + r1 >= r + r2. +intros; apply Rle_ge; auto with real. Qed. -Hints Resolve Rge_plus_plus_r : real. +Hint Resolve Rplus_ge_compat_l: real. (***********) -Lemma Rge_r_plus_plus:(r,r1,r2:R)``r+r1 >= r+r2`` -> ``r1>=r2``. -Intros; Apply Rle_ge; Apply Rle_anti_compatibility with r; Auto with real. +Lemma Rplus_ge_reg_l : forall r r1 r2, r + r1 >= r + r2 -> r1 >= r2. +intros; apply Rle_ge; apply Rplus_le_reg_l with r; auto with real. Qed. (***********) -Lemma Rge_monotony: - (x,y,z:R) ``z>=0`` -> ``x>=y`` -> ``x*z >= y*z``. -Intros x y z; Intros; Apply Rle_ge; Apply Rle_monotony_r; Apply Rge_le; Assumption. +Lemma Rmult_ge_compat_r : + forall r r1 r2, r2 >= 0 -> r >= r1 -> r * r2 >= r1 * r2. +intros x y z; intros; apply Rle_ge; apply Rmult_le_compat_r; apply Rge_le; + assumption. Qed. (***********) -Lemma Rgt_minus:(r1,r2:R)``r1>r2`` -> ``r1-r2 > 0``. -Intros; Replace ``0`` with ``r2-r2``; Auto with real. -Unfold Rgt Rminus; Auto with real. +Lemma Rgt_minus : forall r1 r2, r1 > r2 -> r1 - r2 > 0. +intros; replace 0 with (r2 - r2); auto with real. +unfold Rgt, Rminus in |- *; auto with real. Qed. (*********) -Lemma minus_Rgt:(r1,r2:R)``r1-r2 > 0`` -> ``r1>r2``. -Intros; Replace r2 with ``r2+0``; Auto with real. -Intros; Replace r1 with ``r2+(r1-r2)``; Auto with real. +Lemma minus_Rgt : forall r1 r2, r1 - r2 > 0 -> r1 > r2. +intros; replace r2 with (r2 + 0); auto with real. +intros; replace r1 with (r2 + (r1 - r2)); auto with real. Qed. (**********) -Lemma Rge_minus:(r1,r2:R)``r1>=r2`` -> ``r1-r2 >= 0``. -Unfold Rge; Intros; Elim H; Intro. -Left; Apply (Rgt_minus r1 r2 H0). -Right; Apply (eq_Rminus r1 r2 H0). +Lemma Rge_minus : forall r1 r2, r1 >= r2 -> r1 - r2 >= 0. +unfold Rge in |- *; intros; elim H; intro. +left; apply (Rgt_minus r1 r2 H0). +right; apply (Rminus_diag_eq r1 r2 H0). Qed. (*********) -Lemma minus_Rge:(r1,r2:R)``r1-r2 >= 0`` -> ``r1>=r2``. -Intros; Replace r2 with ``r2+0``; Auto with real. -Intros; Replace r1 with ``r2+(r1-r2)``; Auto with real. +Lemma minus_Rge : forall r1 r2, r1 - r2 >= 0 -> r1 >= r2. +intros; replace r2 with (r2 + 0); auto with real. +intros; replace r1 with (r2 + (r1 - r2)); auto with real. Qed. (*********) -Lemma Rmult_gt:(r1,r2:R)``r1>0`` -> ``r2>0`` -> ``r1*r2>0``. -Unfold Rgt;Intros. -Replace ``0`` with ``0*r2``; Auto with real. +Lemma Rmult_gt_0_compat : forall r1 r2, r1 > 0 -> r2 > 0 -> r1 * r2 > 0. +unfold Rgt in |- *; intros. +replace 0 with (0 * r2); auto with real. Qed. (*********) -Lemma Rmult_lt_pos:(x,y:R)``0<x`` -> ``0<y`` -> ``0<x*y``. -Proof Rmult_gt. +Lemma Rmult_lt_0_compat : forall r1 r2, 0 < r1 -> 0 < r2 -> 0 < r1 * r2. +Proof Rmult_gt_0_compat. (***********) -Lemma Rplus_eq_R0_l:(a,b:R)``0<=a`` -> ``0<=b`` -> ``a+b==0`` -> ``a==0``. -Intros a b [H|H] H0 H1; Auto with real. -Absurd ``0<a+b``. -Rewrite H1; Auto with real. -Replace ``0`` with ``0+0``; Auto with real. +Lemma Rplus_eq_0_l : + forall r1 r2, 0 <= r1 -> 0 <= r2 -> r1 + r2 = 0 -> r1 = 0. +intros a b [H| H] H0 H1; auto with real. +absurd (0 < a + b). +rewrite H1; auto with real. +replace 0 with (0 + 0); auto with real. Qed. -Lemma Rplus_eq_R0 - :(a,b:R)``0<=a`` -> ``0<=b`` -> ``a+b==0`` -> ``a==0``/\``b==0``. -Intros a b; Split. -Apply Rplus_eq_R0_l with b; Auto with real. -Apply Rplus_eq_R0_l with a; Auto with real. -Rewrite Rplus_sym; Auto with real. +Lemma Rplus_eq_R0 : + forall r1 r2, 0 <= r1 -> 0 <= r2 -> r1 + r2 = 0 -> r1 = 0 /\ r2 = 0. +intros a b; split. +apply Rplus_eq_0_l with b; auto with real. +apply Rplus_eq_0_l with a; auto with real. +rewrite Rplus_comm; auto with real. Qed. (***********) -Lemma Rplus_Rsr_eq_R0_l:(a,b:R)``(Rsqr a)+(Rsqr b)==0``->``a==0``. -Intros a b; Intros; Apply Rsqr_r_R0; Apply Rplus_eq_R0_l with (Rsqr b); Auto with real. +Lemma Rplus_sqr_eq_0_l : forall r1 r2, Rsqr r1 + Rsqr r2 = 0 -> r1 = 0. +intros a b; intros; apply Rsqr_0_uniq; apply Rplus_eq_0_l with (Rsqr b); + auto with real. Qed. -Lemma Rplus_Rsr_eq_R0:(a,b:R)``(Rsqr a)+(Rsqr b)==0``->``a==0``/\``b==0``. -Intros a b; Split. -Apply Rplus_Rsr_eq_R0_l with b; Auto with real. -Apply Rplus_Rsr_eq_R0_l with a; Auto with real. -Rewrite Rplus_sym; Auto with real. +Lemma Rplus_sqr_eq_0 : + forall r1 r2, Rsqr r1 + Rsqr r2 = 0 -> r1 = 0 /\ r2 = 0. +intros a b; split. +apply Rplus_sqr_eq_0_l with b; auto with real. +apply Rplus_sqr_eq_0_l with a; auto with real. +rewrite Rplus_comm; auto with real. Qed. @@ -1167,448 +1157,476 @@ Qed. (**********************************************************) (**********) -Lemma S_INR:(n:nat)(INR (S n))==``(INR n)+1``. -Intro; Case n; Auto with real. +Lemma S_INR : forall n:nat, INR (S n) = INR n + 1. +intro; case n; auto with real. Qed. (**********) -Lemma S_O_plus_INR:(n:nat) - (INR (plus (S O) n))==``(INR (S O))+(INR n)``. -Intro; Simpl; Case n; Intros; Auto with real. +Lemma S_O_plus_INR : forall n:nat, INR (1 + n) = INR 1 + INR n. +intro; simpl in |- *; case n; intros; auto with real. Qed. (**********) -Lemma plus_INR:(n,m:nat)(INR (plus n m))==``(INR n)+(INR m)``. -Intros n m; Induction n. -Simpl; Auto with real. -Replace (plus (S n) m) with (S (plus n m)); Auto with arith. -Repeat Rewrite S_INR. -Rewrite Hrecn; Ring. +Lemma plus_INR : forall n m:nat, INR (n + m) = INR n + INR m. +intros n m; induction n as [| n Hrecn]. +simpl in |- *; auto with real. +replace (S n + m)%nat with (S (n + m)); auto with arith. +repeat rewrite S_INR. +rewrite Hrecn; ring. Qed. (**********) -Lemma minus_INR:(n,m:nat)(le m n)->(INR (minus n m))==``(INR n)-(INR m)``. -Intros n m le; Pattern m n; Apply le_elim_rel; Auto with real. -Intros; Rewrite <- minus_n_O; Auto with real. -Intros; Repeat Rewrite S_INR; Simpl. -Rewrite H0; Ring. +Lemma minus_INR : forall n m:nat, (m <= n)%nat -> INR (n - m) = INR n - INR m. +intros n m le; pattern m, n in |- *; apply le_elim_rel; auto with real. +intros; rewrite <- minus_n_O; auto with real. +intros; repeat rewrite S_INR; simpl in |- *. +rewrite H0; ring. Qed. (*********) -Lemma mult_INR:(n,m:nat)(INR (mult n m))==(Rmult (INR n) (INR m)). -Intros n m; Induction n. -Simpl; Auto with real. -Intros; Repeat Rewrite S_INR; Simpl. -Rewrite plus_INR; Rewrite Hrecn; Ring. +Lemma mult_INR : forall n m:nat, INR (n * m) = INR n * INR m. +intros n m; induction n as [| n Hrecn]. +simpl in |- *; auto with real. +intros; repeat rewrite S_INR; simpl in |- *. +rewrite plus_INR; rewrite Hrecn; ring. Qed. -Hints Resolve plus_INR minus_INR mult_INR : real. +Hint Resolve plus_INR minus_INR mult_INR: real. (*********) -Lemma lt_INR_0:(n:nat)(lt O n)->``0 < (INR n)``. -Induction 1; Intros; Auto with real. -Rewrite S_INR; Auto with real. +Lemma lt_INR_0 : forall n:nat, (0 < n)%nat -> 0 < INR n. +simple induction 1; intros; auto with real. +rewrite S_INR; auto with real. Qed. -Hints Resolve lt_INR_0: real. +Hint Resolve lt_INR_0: real. -Lemma lt_INR:(n,m:nat)(lt n m)->``(INR n) < (INR m)``. -Induction 1; Intros; Auto with real. -Rewrite S_INR; Auto with real. -Rewrite S_INR; Apply Rlt_trans with (INR m0); Auto with real. +Lemma lt_INR : forall n m:nat, (n < m)%nat -> INR n < INR m. +simple induction 1; intros; auto with real. +rewrite S_INR; auto with real. +rewrite S_INR; apply Rlt_trans with (INR m0); auto with real. Qed. -Hints Resolve lt_INR: real. +Hint Resolve lt_INR: real. -Lemma INR_lt_1:(n:nat)(lt (S O) n)->``1 < (INR n)``. -Intros;Replace ``1`` with (INR (S O));Auto with real. +Lemma INR_lt_1 : forall n:nat, (1 < n)%nat -> 1 < INR n. +intros; replace 1 with (INR 1); auto with real. Qed. -Hints Resolve INR_lt_1: real. +Hint Resolve INR_lt_1: real. (**********) -Lemma INR_pos : (p:positive)``0<(INR (convert p))``. -Intro; Apply lt_INR_0. -Simpl; Auto with real. -Apply compare_convert_O. +Lemma INR_pos : forall p:positive, 0 < INR (nat_of_P p). +intro; apply lt_INR_0. +simpl in |- *; auto with real. +apply lt_O_nat_of_P. Qed. -Hints Resolve INR_pos : real. +Hint Resolve INR_pos: real. (**********) -Lemma pos_INR:(n:nat)``0 <= (INR n)``. -Intro n; Case n. -Simpl; Auto with real. -Auto with arith real. +Lemma pos_INR : forall n:nat, 0 <= INR n. +intro n; case n. +simpl in |- *; auto with real. +auto with arith real. Qed. -Hints Resolve pos_INR: real. +Hint Resolve pos_INR: real. -Lemma INR_lt:(n,m:nat)``(INR n) < (INR m)``->(lt n m). -Double Induction n m;Intros. -Simpl;ElimType False;Apply (Rlt_antirefl R0);Auto. -Auto with arith. -Generalize (pos_INR (S n0));Intro;Cut (INR O)==R0; - [Intro H2;Rewrite H2 in H0;Idtac|Simpl;Trivial]. -Generalize (Rle_lt_trans ``0`` (INR (S n0)) ``0`` H1 H0);Intro; - ElimType False;Apply (Rlt_antirefl R0);Auto. -Do 2 Rewrite S_INR in H1;Cut ``(INR n1) < (INR n0)``. -Intro H2;Generalize (H0 n0 H2);Intro;Auto with arith. -Apply (Rlt_anti_compatibility ``1`` (INR n1) (INR n0)). -Rewrite Rplus_sym;Rewrite (Rplus_sym ``1`` (INR n0));Trivial. +Lemma INR_lt : forall n m:nat, INR n < INR m -> (n < m)%nat. +double induction n m; intros. +simpl in |- *; elimtype False; apply (Rlt_irrefl 0); auto. +auto with arith. +generalize (pos_INR (S n0)); intro; cut (INR 0 = 0); + [ intro H2; rewrite H2 in H0; idtac | simpl in |- *; trivial ]. +generalize (Rle_lt_trans 0 (INR (S n0)) 0 H1 H0); intro; elimtype False; + apply (Rlt_irrefl 0); auto. +do 2 rewrite S_INR in H1; cut (INR n1 < INR n0). +intro H2; generalize (H0 n0 H2); intro; auto with arith. +apply (Rplus_lt_reg_r 1 (INR n1) (INR n0)). +rewrite Rplus_comm; rewrite (Rplus_comm 1 (INR n0)); trivial. Qed. -Hints Resolve INR_lt: real. +Hint Resolve INR_lt: real. (*********) -Lemma le_INR:(n,m:nat)(le n m)->``(INR n)<=(INR m)``. -Induction 1; Intros; Auto with real. -Rewrite S_INR. -Apply Rle_trans with (INR m0); Auto with real. +Lemma le_INR : forall n m:nat, (n <= m)%nat -> INR n <= INR m. +simple induction 1; intros; auto with real. +rewrite S_INR. +apply Rle_trans with (INR m0); auto with real. Qed. -Hints Resolve le_INR: real. +Hint Resolve le_INR: real. (**********) -Lemma not_INR_O:(n:nat)``(INR n)<>0``->~n=O. -Red; Intros n H H1. -Apply H. -Rewrite H1; Trivial. +Lemma not_INR_O : forall n:nat, INR n <> 0 -> n <> 0%nat. +red in |- *; intros n H H1. +apply H. +rewrite H1; trivial. Qed. -Hints Immediate not_INR_O : real. +Hint Immediate not_INR_O: real. (**********) -Lemma not_O_INR:(n:nat)~n=O->``(INR n)<>0``. -Intro n; Case n. -Intro; Absurd (0)=(0); Trivial. -Intros; Rewrite S_INR. -Apply Rgt_not_eq; Red; Auto with real. +Lemma not_O_INR : forall n:nat, n <> 0%nat -> INR n <> 0. +intro n; case n. +intro; absurd (0%nat = 0%nat); trivial. +intros; rewrite S_INR. +apply Rgt_not_eq; red in |- *; auto with real. Qed. -Hints Resolve not_O_INR : real. +Hint Resolve not_O_INR: real. -Lemma not_nm_INR:(n,m:nat)~n=m->``(INR n)<>(INR m)``. -Intros n m H; Case (le_or_lt n m); Intros H1. -Case (le_lt_or_eq ? ? H1); Intros H2. -Apply imp_not_Req; Auto with real. -ElimType False;Auto. -Apply sym_not_eqT; Apply imp_not_Req; Auto with real. +Lemma not_nm_INR : forall n m:nat, n <> m -> INR n <> INR m. +intros n m H; case (le_or_lt n m); intros H1. +case (le_lt_or_eq _ _ H1); intros H2. +apply Rlt_dichotomy_converse; auto with real. +elimtype False; auto. +apply sym_not_eq; apply Rlt_dichotomy_converse; auto with real. Qed. -Hints Resolve not_nm_INR : real. +Hint Resolve not_nm_INR: real. -Lemma INR_eq: (n,m:nat)(INR n)==(INR m)->n=m. -Intros;Case (le_or_lt n m); Intros H1. -Case (le_lt_or_eq ? ? H1); Intros H2;Auto. -Cut ~n=m. -Intro H3;Generalize (not_nm_INR n m H3);Intro H4; - ElimType False;Auto. -Omega. -Symmetry;Cut ~m=n. -Intro H3;Generalize (not_nm_INR m n H3);Intro H4; - ElimType False;Auto. -Omega. +Lemma INR_eq : forall n m:nat, INR n = INR m -> n = m. +intros; case (le_or_lt n m); intros H1. +case (le_lt_or_eq _ _ H1); intros H2; auto. +cut (n <> m). +intro H3; generalize (not_nm_INR n m H3); intro H4; elimtype False; auto. +omega. +symmetry in |- *; cut (m <> n). +intro H3; generalize (not_nm_INR m n H3); intro H4; elimtype False; auto. +omega. Qed. -Hints Resolve INR_eq : real. +Hint Resolve INR_eq: real. -Lemma INR_le: (n, m : nat) (Rle (INR n) (INR m)) -> (le n m). -Intros;Elim H;Intro. -Generalize (INR_lt n m H0);Intro;Auto with arith. -Generalize (INR_eq n m H0);Intro;Rewrite H1;Auto. +Lemma INR_le : forall n m:nat, INR n <= INR m -> (n <= m)%nat. +intros; elim H; intro. +generalize (INR_lt n m H0); intro; auto with arith. +generalize (INR_eq n m H0); intro; rewrite H1; auto. Qed. -Hints Resolve INR_le : real. +Hint Resolve INR_le: real. -Lemma not_1_INR:(n:nat)~n=(S O)->``(INR n)<>1``. -Replace ``1`` with (INR (S O)); Auto with real. +Lemma not_1_INR : forall n:nat, n <> 1%nat -> INR n <> 1. +replace 1 with (INR 1); auto with real. Qed. -Hints Resolve not_1_INR : real. +Hint Resolve not_1_INR: real. (**********************************************************) (** Injection from [Z] to [R] *) (**********************************************************) -V7only [ -(**********) -Definition Z_of_nat := inject_nat. -Notation INZ:=Z_of_nat. -]. (**********) -Lemma IZN:(z:Z)(`0<=z`)->(Ex [m:nat] z=(INZ m)). -Intros z; Unfold INZ; Apply inject_nat_complete; Assumption. +Lemma IZN : forall n:Z, (0 <= n)%Z -> exists m : nat | n = Z_of_nat m. +intros z; idtac; apply Z_of_nat_complete; assumption. Qed. (**********) -Lemma INR_IZR_INZ:(n:nat)(INR n)==(IZR (INZ n)). -Induction n; Auto with real. -Intros; Simpl; Rewrite bij1; Auto with real. +Lemma INR_IZR_INZ : forall n:nat, INR n = IZR (Z_of_nat n). +simple induction n; auto with real. +intros; simpl in |- *; rewrite nat_of_P_o_P_of_succ_nat_eq_succ; + auto with real. Qed. -Lemma plus_IZR_NEG_POS : - (p,q:positive)(IZR `(POS p)+(NEG q)`)==``(IZR (POS p))+(IZR (NEG q))``. -Intros. -Case (lt_eq_lt_dec (convert p) (convert q)). -Intros [H | H]; Simpl. -Rewrite convert_compare_INFERIEUR; Simpl; Trivial. -Rewrite (true_sub_convert q p). -Rewrite minus_INR; Auto with arith; Ring. -Apply ZC2; Apply convert_compare_INFERIEUR; Trivial. -Rewrite (convert_intro p q); Trivial. -Rewrite convert_compare_EGAL; Simpl; Auto with real. -Intro H; Simpl. -Rewrite convert_compare_SUPERIEUR; Simpl; Auto with arith. -Rewrite (true_sub_convert p q). -Rewrite minus_INR; Auto with arith; Ring. -Apply ZC2; Apply convert_compare_INFERIEUR; Trivial. +Lemma plus_IZR_NEG_POS : + forall p q:positive, IZR (Zpos p + Zneg q) = IZR (Zpos p) + IZR (Zneg q). +intros. +case (lt_eq_lt_dec (nat_of_P p) (nat_of_P q)). +intros [H| H]; simpl in |- *. +rewrite nat_of_P_lt_Lt_compare_complement_morphism; simpl in |- *; trivial. +rewrite (nat_of_P_minus_morphism q p). +rewrite minus_INR; auto with arith; ring. +apply ZC2; apply nat_of_P_lt_Lt_compare_complement_morphism; trivial. +rewrite (nat_of_P_inj p q); trivial. +rewrite Pcompare_refl; simpl in |- *; auto with real. +intro H; simpl in |- *. +rewrite nat_of_P_gt_Gt_compare_complement_morphism; simpl in |- *; + auto with arith. +rewrite (nat_of_P_minus_morphism p q). +rewrite minus_INR; auto with arith; ring. +apply ZC2; apply nat_of_P_lt_Lt_compare_complement_morphism; trivial. Qed. (**********) -Lemma plus_IZR:(z,t:Z)(IZR `z+t`)==``(IZR z)+(IZR t)``. -Intro z; NewDestruct z; Intro t; NewDestruct t; Intros; Auto with real. -Simpl; Intros; Rewrite convert_add; Auto with real. -Apply plus_IZR_NEG_POS. -Rewrite Zplus_sym; Rewrite Rplus_sym; Apply plus_IZR_NEG_POS. -Simpl; Intros; Rewrite convert_add; Rewrite plus_INR; Auto with real. +Lemma plus_IZR : forall n m:Z, IZR (n + m) = IZR n + IZR m. +intro z; destruct z; intro t; destruct t; intros; auto with real. +simpl in |- *; intros; rewrite nat_of_P_plus_morphism; auto with real. +apply plus_IZR_NEG_POS. +rewrite Zplus_comm; rewrite Rplus_comm; apply plus_IZR_NEG_POS. +simpl in |- *; intros; rewrite nat_of_P_plus_morphism; rewrite plus_INR; + auto with real. Qed. (**********) -Lemma mult_IZR:(z,t:Z)(IZR `z*t`)==``(IZR z)*(IZR t)``. -Intros z t; Case z; Case t; Simpl; Auto with real. -Intros t1 z1; Rewrite times_convert; Auto with real. -Intros t1 z1; Rewrite times_convert; Auto with real. -Rewrite Rmult_sym. -Rewrite Ropp_mul1; Auto with real. -Apply eq_Ropp; Rewrite mult_sym; Auto with real. -Intros t1 z1; Rewrite times_convert; Auto with real. -Rewrite Ropp_mul1; Auto with real. -Intros t1 z1; Rewrite times_convert; Auto with real. -Rewrite Ropp_mul2; Auto with real. +Lemma mult_IZR : forall n m:Z, IZR (n * m) = IZR n * IZR m. +intros z t; case z; case t; simpl in |- *; auto with real. +intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real. +intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real. +rewrite Rmult_comm. +rewrite Ropp_mult_distr_l_reverse; auto with real. +apply Ropp_eq_compat; rewrite mult_comm; auto with real. +intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real. +rewrite Ropp_mult_distr_l_reverse; auto with real. +intros t1 z1; rewrite nat_of_P_mult_morphism; auto with real. +rewrite Rmult_opp_opp; auto with real. Qed. (**********) -Lemma Ropp_Ropp_IZR:(z:Z)(IZR (`-z`))==``-(IZR z)``. -Intro z; Case z; Simpl; Auto with real. +Lemma Ropp_Ropp_IZR : forall n:Z, IZR (- n) = - IZR n. +intro z; case z; simpl in |- *; auto with real. Qed. (**********) -Lemma Z_R_minus:(z1,z2:Z)``(IZR z1)-(IZR z2)``==(IZR `z1-z2`). -Intros z1 z2; Unfold Rminus; Unfold Zminus. -Rewrite <-(Ropp_Ropp_IZR z2); Symmetry; Apply plus_IZR. +Lemma Z_R_minus : forall n m:Z, IZR n - IZR m = IZR (n - m). +intros z1 z2; unfold Rminus in |- *; unfold Zminus in |- *. +rewrite <- (Ropp_Ropp_IZR z2); symmetry in |- *; apply plus_IZR. Qed. (**********) -Lemma lt_O_IZR:(z:Z)``0 < (IZR z)``->`0<z`. -Intro z; Case z; Simpl; Intros. -Absurd ``0<0``; Auto with real. -Unfold Zlt; Simpl; Trivial. -Case Rlt_le_not with 1:=H. -Replace ``0`` with ``-0``; Auto with real. +Lemma lt_O_IZR : forall n:Z, 0 < IZR n -> (0 < n)%Z. +intro z; case z; simpl in |- *; intros. +absurd (0 < 0); auto with real. +unfold Zlt in |- *; simpl in |- *; trivial. +case Rlt_not_le with (1 := H). +replace 0 with (-0); auto with real. Qed. (**********) -Lemma lt_IZR:(z1,z2:Z)``(IZR z1)<(IZR z2)``->`z1<z2`. -Intros z1 z2 H; Apply Zlt_O_minus_lt. -Apply lt_O_IZR. -Rewrite <- Z_R_minus. -Exact (Rgt_minus (IZR z2) (IZR z1) H). +Lemma lt_IZR : forall n m:Z, IZR n < IZR m -> (n < m)%Z. +intros z1 z2 H; apply Zlt_O_minus_lt. +apply lt_O_IZR. +rewrite <- Z_R_minus. +exact (Rgt_minus (IZR z2) (IZR z1) H). Qed. (**********) -Lemma eq_IZR_R0:(z:Z)``(IZR z)==0``->`z=0`. -Intro z; NewDestruct z; Simpl; Intros; Auto with zarith. -Case (Rlt_not_eq ``0`` (INR (convert p))); Auto with real. -Case (Rlt_not_eq ``-(INR (convert p))`` ``0`` ); Auto with real. -Apply Rgt_RoppO. Unfold Rgt; Apply INR_pos. +Lemma eq_IZR_R0 : forall n:Z, IZR n = 0 -> n = 0%Z. +intro z; destruct z; simpl in |- *; intros; auto with zarith. +case (Rlt_not_eq 0 (INR (nat_of_P p))); auto with real. +case (Rlt_not_eq (- INR (nat_of_P p)) 0); auto with real. +apply Ropp_lt_gt_0_contravar. unfold Rgt in |- *; apply INR_pos. Qed. (**********) -Lemma eq_IZR:(z1,z2:Z)(IZR z1)==(IZR z2)->z1=z2. -Intros z1 z2 H;Generalize (eq_Rminus (IZR z1) (IZR z2) H); - Rewrite (Z_R_minus z1 z2);Intro;Generalize (eq_IZR_R0 `z1-z2` H0); - Intro;Omega. +Lemma eq_IZR : forall n m:Z, IZR n = IZR m -> n = m. +intros z1 z2 H; generalize (Rminus_diag_eq (IZR z1) (IZR z2) H); + rewrite (Z_R_minus z1 z2); intro; generalize (eq_IZR_R0 (z1 - z2) H0); + intro; omega. Qed. (**********) -Lemma not_O_IZR:(z:Z)`z<>0`->``(IZR z)<>0``. -Intros z H; Red; Intros H0; Case H. -Apply eq_IZR; Auto. +Lemma not_O_IZR : forall n:Z, n <> 0%Z -> IZR n <> 0. +intros z H; red in |- *; intros H0; case H. +apply eq_IZR; auto. Qed. (*********) -Lemma le_O_IZR:(z:Z)``0<= (IZR z)``->`0<=z`. -Unfold Rle; Intros z [H|H]. -Red;Intro;Apply (Zlt_le_weak `0` z (lt_O_IZR z H)); Assumption. -Rewrite (eq_IZR_R0 z); Auto with zarith real. +Lemma le_O_IZR : forall n:Z, 0 <= IZR n -> (0 <= n)%Z. +unfold Rle in |- *; intros z [H| H]. +red in |- *; intro; apply (Zlt_le_weak 0 z (lt_O_IZR z H)); assumption. +rewrite (eq_IZR_R0 z); auto with zarith real. Qed. (**********) -Lemma le_IZR:(z1,z2:Z)``(IZR z1)<=(IZR z2)``->`z1<=z2`. -Unfold Rle; Intros z1 z2 [H|H]. -Apply (Zlt_le_weak z1 z2); Auto with real. -Apply lt_IZR; Trivial. -Rewrite (eq_IZR z1 z2); Auto with zarith real. +Lemma le_IZR : forall n m:Z, IZR n <= IZR m -> (n <= m)%Z. +unfold Rle in |- *; intros z1 z2 [H| H]. +apply (Zlt_le_weak z1 z2); auto with real. +apply lt_IZR; trivial. +rewrite (eq_IZR z1 z2); auto with zarith real. Qed. (**********) -Lemma le_IZR_R1:(z:Z)``(IZR z)<=1``-> `z<=1`. -Pattern 1 ``1``; Replace ``1`` with (IZR `1`); Intros; Auto. -Apply le_IZR; Trivial. +Lemma le_IZR_R1 : forall n:Z, IZR n <= 1 -> (n <= 1)%Z. +pattern 1 at 1 in |- *; replace 1 with (IZR 1); intros; auto. +apply le_IZR; trivial. Qed. (**********) -Lemma IZR_ge: (m,n:Z) `m>= n` -> ``(IZR m)>=(IZR n)``. -Intros m n H; Apply Rlt_not_ge;Red;Intro. -Generalize (lt_IZR m n H0); Intro; Omega. +Lemma IZR_ge : forall n m:Z, (n >= m)%Z -> IZR n >= IZR m. +intros m n H; apply Rnot_lt_ge; red in |- *; intro. +generalize (lt_IZR m n H0); intro; omega. Qed. -Lemma IZR_le: (m,n:Z) `m<= n` -> ``(IZR m)<=(IZR n)``. -Intros m n H;Apply Rgt_not_le;Red;Intro. -Unfold Rgt in H0;Generalize (lt_IZR n m H0); Intro; Omega. +Lemma IZR_le : forall n m:Z, (n <= m)%Z -> IZR n <= IZR m. +intros m n H; apply Rnot_gt_le; red in |- *; intro. +unfold Rgt in H0; generalize (lt_IZR n m H0); intro; omega. Qed. -Lemma IZR_lt: (m,n:Z) `m< n` -> ``(IZR m)<(IZR n)``. -Intros m n H;Cut `m<=n`. -Intro H0;Elim (IZR_le m n H0);Intro;Auto. -Generalize (eq_IZR m n H1);Intro;ElimType False;Omega. -Omega. +Lemma IZR_lt : forall n m:Z, (n < m)%Z -> IZR n < IZR m. +intros m n H; cut (m <= n)%Z. +intro H0; elim (IZR_le m n H0); intro; auto. +generalize (eq_IZR m n H1); intro; elimtype False; omega. +omega. Qed. -Lemma one_IZR_lt1 : (z:Z)``-1<(IZR z)<1``->`z=0`. -Intros z (H1,H2). -Apply Zle_antisym. -Apply Zlt_n_Sm_le; Apply lt_IZR; Trivial. -Replace `0` with (Zs `-1`); Trivial. -Apply Zlt_le_S; Apply lt_IZR; Trivial. +Lemma one_IZR_lt1 : forall n:Z, -1 < IZR n < 1 -> n = 0%Z. +intros z [H1 H2]. +apply Zle_antisym. +apply Zlt_succ_le; apply lt_IZR; trivial. +replace 0%Z with (Zsucc (-1)); trivial. +apply Zlt_le_succ; apply lt_IZR; trivial. Qed. -Lemma one_IZR_r_R1 - : (r:R)(z,x:Z)``r<(IZR z)<=r+1``->``r<(IZR x)<=r+1``->z=x. -Intros r z x (H1,H2) (H3,H4). -Cut `z-x=0`; Auto with zarith. -Apply one_IZR_lt1. -Rewrite <- Z_R_minus; Split. -Replace ``-1`` with ``r-(r+1)``. -Unfold Rminus; Apply Rplus_lt_le_lt; Auto with real. -Ring. -Replace ``1`` with ``(r+1)-r``. -Unfold Rminus; Apply Rplus_le_lt_lt; Auto with real. -Ring. +Lemma one_IZR_r_R1 : + forall r (n m:Z), r < IZR n <= r + 1 -> r < IZR m <= r + 1 -> n = m. +intros r z x [H1 H2] [H3 H4]. +cut ((z - x)%Z = 0%Z); auto with zarith. +apply one_IZR_lt1. +rewrite <- Z_R_minus; split. +replace (-1) with (r - (r + 1)). +unfold Rminus in |- *; apply Rplus_lt_le_compat; auto with real. +ring. +replace 1 with (r + 1 - r). +unfold Rminus in |- *; apply Rplus_le_lt_compat; auto with real. +ring. Qed. (**********) -Lemma single_z_r_R1: - (r:R)(z,x:Z)``r<(IZR z)``->``(IZR z)<=r+1``->``r<(IZR x)``-> - ``(IZR x)<=r+1``->z=x. -Intros; Apply one_IZR_r_R1 with r; Auto. +Lemma single_z_r_R1 : + forall r (n m:Z), + r < IZR n -> IZR n <= r + 1 -> r < IZR m -> IZR m <= r + 1 -> n = m. +intros; apply one_IZR_r_R1 with r; auto. Qed. (**********) -Lemma tech_single_z_r_R1 - :(r:R)(z:Z)``r<(IZR z)``->``(IZR z)<=r+1`` - -> (Ex [s:Z] (~s=z/\``r<(IZR s)``/\``(IZR s)<=r+1``))->False. -Intros r z H1 H2 (s, (H3,(H4,H5))). -Apply H3; Apply single_z_r_R1 with r; Trivial. +Lemma tech_single_z_r_R1 : + forall r (n:Z), + r < IZR n -> + IZR n <= r + 1 -> + ( exists s : Z | s <> n /\ r < IZR s /\ IZR s <= r + 1) -> False. +intros r z H1 H2 [s [H3 [H4 H5]]]. +apply H3; apply single_z_r_R1 with r; trivial. Qed. (*****************************************************************) (** Definitions of new types *) (*****************************************************************) -Record nonnegreal : Type := mknonnegreal { -nonneg :> R; -cond_nonneg : ``0<=nonneg`` }. +Record nonnegreal : Type := mknonnegreal + {nonneg :> R; cond_nonneg : 0 <= nonneg}. -Record posreal : Type := mkposreal { -pos :> R; -cond_pos : ``0<pos`` }. +Record posreal : Type := mkposreal {pos :> R; cond_pos : 0 < pos}. -Record nonposreal : Type := mknonposreal { -nonpos :> R; -cond_nonpos : ``nonpos<=0`` }. +Record nonposreal : Type := mknonposreal + {nonpos :> R; cond_nonpos : nonpos <= 0}. -Record negreal : Type := mknegreal { -neg :> R; -cond_neg : ``neg<0`` }. +Record negreal : Type := mknegreal {neg :> R; cond_neg : neg < 0}. -Record nonzeroreal : Type := mknonzeroreal { -nonzero :> R; -cond_nonzero : ~``nonzero==0`` }. +Record nonzeroreal : Type := mknonzeroreal + {nonzero :> R; cond_nonzero : nonzero <> 0}. (**********) -Lemma prod_neq_R0 : (x,y:R) ~``x==0``->~``y==0``->~``x*y==0``. -Intros x y; Intros; Red; Intro; Generalize (without_div_Od x y H1); Intro; Elim H2; Intro; [Rewrite H3 in H; Elim H | Rewrite H3 in H0; Elim H0]; Reflexivity. +Lemma prod_neq_R0 : forall r1 r2, r1 <> 0 -> r2 <> 0 -> r1 * r2 <> 0. +intros x y; intros; red in |- *; intro; generalize (Rmult_integral x y H1); + intro; elim H2; intro; + [ rewrite H3 in H; elim H | rewrite H3 in H0; elim H0 ]; + reflexivity. Qed. (*********) -Lemma Rmult_le_pos : (x,y:R) ``0<=x`` -> ``0<=y`` -> ``0<=x*y``. -Intros x y H H0; Rewrite <- (Rmult_Ol x); Rewrite <- (Rmult_sym x); Apply (Rle_monotony x R0 y H H0). +Lemma Rmult_le_pos : forall r1 r2, 0 <= r1 -> 0 <= r2 -> 0 <= r1 * r2. +intros x y H H0; rewrite <- (Rmult_0_l x); rewrite <- (Rmult_comm x); + apply (Rmult_le_compat_l x 0 y H H0). Qed. -Lemma double : (x:R) ``2*x==x+x``. -Intro; Ring. +Lemma double : forall r1, 2 * r1 = r1 + r1. +intro; ring. Qed. -Lemma double_var : (x:R) ``x == x/2 + x/2``. -Intro; Rewrite <- double; Unfold Rdiv; Rewrite <- Rmult_assoc; Symmetry; Apply Rinv_r_simpl_m. -Replace ``2`` with (INR (2)); [Apply not_O_INR; Discriminate | Unfold INR; Ring]. +Lemma double_var : forall r1, r1 = r1 / 2 + r1 / 2. +intro; rewrite <- double; unfold Rdiv in |- *; rewrite <- Rmult_assoc; + symmetry in |- *; apply Rinv_r_simpl_m. +replace 2 with (INR 2); + [ apply not_O_INR; discriminate | unfold INR in |- *; ring ]. Qed. (**********************************************************) (** Other rules about < and <= *) (**********************************************************) -Lemma gt0_plus_gt0_is_gt0 : (x,y:R) ``0<x`` -> ``0<y`` -> ``0<x+y``. -Intros x y; Intros; Apply Rlt_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rlt_compatibility; Assumption]. -Qed. - -Lemma ge0_plus_gt0_is_gt0 : (x,y:R) ``0<=x`` -> ``0<y`` -> ``0<x+y``. -Intros x y; Intros; Apply Rle_lt_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rlt_compatibility; Assumption]. -Qed. - -Lemma gt0_plus_ge0_is_gt0 : (x,y:R) ``0<x`` -> ``0<=y`` -> ``0<x+y``. -Intros x y; Intros; Rewrite <- Rplus_sym; Apply ge0_plus_gt0_is_gt0; Assumption. -Qed. - -Lemma ge0_plus_ge0_is_ge0 : (x,y:R) ``0<=x`` -> ``0<=y`` -> ``0<=x+y``. -Intros x y; Intros; Apply Rle_trans with x; [Assumption | Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption]. -Qed. - -Lemma plus_le_is_le : (x,y,z:R) ``0<=y`` -> ``x+y<=z`` -> ``x<=z``. -Intros x y z; Intros; Apply Rle_trans with ``x+y``; [Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption | Assumption]. -Qed. - -Lemma plus_lt_is_lt : (x,y,z:R) ``0<=y`` -> ``x+y<z`` -> ``x<z``. -Intros x y z; Intros; Apply Rle_lt_trans with ``x+y``; [Pattern 1 x; Rewrite <- (Rplus_Or x); Apply Rle_compatibility; Assumption | Assumption]. -Qed. - -Lemma Rmult_lt2 : (r1,r2,r3,r4:R) ``0<=r1`` -> ``0<=r3`` -> ``r1<r2`` -> ``r3<r4`` -> ``r1*r3<r2*r4``. -Intros; Apply Rle_lt_trans with ``r2*r3``; [Apply Rle_monotony_r; [Assumption | Left; Assumption] | Apply Rlt_monotony; [Apply Rle_lt_trans with r1; Assumption | Assumption]]. -Qed. - -Lemma le_epsilon : (x,y:R) ((eps : R) ``0<eps``->``x<=y+eps``) -> ``x<=y``. -Intros x y; Intros; Elim (total_order x y); Intro. -Left; Assumption. -Elim H0; Intro. -Right; Assumption. -Clear H0; Generalize (Rgt_minus x y H1); Intro H2; Change ``0<x-y`` in H2. -Cut ``0<2``. -Intro. -Generalize (Rmult_lt_pos ``x-y`` ``/2`` H2 (Rlt_Rinv ``2`` H0)); Intro H3; Generalize (H ``(x-y)*/2`` H3); Replace ``y+(x-y)*/2`` with ``(y+x)*/2``. -Intro H4; Generalize (Rle_monotony ``2`` x ``(y+x)*/2`` (Rlt_le ``0`` ``2`` H0) H4); Rewrite <- (Rmult_sym ``((y+x)*/2)``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Replace ``2*x`` with ``x+x``. -Rewrite (Rplus_sym y); Intro H5; Apply Rle_anti_compatibility with x; Assumption. -Ring. -Replace ``2`` with (INR (S (S O))); [Apply not_O_INR; Discriminate | Ring]. -Pattern 2 y; Replace y with ``y/2+y/2``. -Unfold Rminus Rdiv. -Repeat Rewrite Rmult_Rplus_distrl. -Ring. -Cut (z:R) ``2*z == z + z``. -Intro. -Rewrite <- (H4 ``y/2``). -Unfold Rdiv. -Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m. -Replace ``2`` with (INR (2)). -Apply not_O_INR. -Discriminate. -Unfold INR; Reflexivity. -Intro; Ring. -Cut ~(O=(2)); [Intro H0; Generalize (lt_INR_0 (2) (neq_O_lt (2) H0)); Unfold INR; Intro; Assumption | Discriminate]. -Qed. - -(**********) -Lemma complet_weak : (E:R->Prop) (bound E) -> (ExT [x:R] (E x)) -> (ExT [m:R] (is_lub E m)). -Intros; Elim (complet E H H0); Intros; Split with x; Assumption. -Qed. +Lemma Rplus_lt_0_compat : forall r1 r2, 0 < r1 -> 0 < r2 -> 0 < r1 + r2. +intros x y; intros; apply Rlt_trans with x; + [ assumption + | pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_lt_compat_l; + assumption ]. +Qed. + +Lemma Rplus_le_lt_0_compat : forall r1 r2, 0 <= r1 -> 0 < r2 -> 0 < r1 + r2. +intros x y; intros; apply Rle_lt_trans with x; + [ assumption + | pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_lt_compat_l; + assumption ]. +Qed. + +Lemma Rplus_lt_le_0_compat : forall r1 r2, 0 < r1 -> 0 <= r2 -> 0 < r1 + r2. +intros x y; intros; rewrite <- Rplus_comm; apply Rplus_le_lt_0_compat; + assumption. +Qed. + +Lemma Rplus_le_le_0_compat : forall r1 r2, 0 <= r1 -> 0 <= r2 -> 0 <= r1 + r2. +intros x y; intros; apply Rle_trans with x; + [ assumption + | pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + assumption ]. +Qed. + +Lemma plus_le_is_le : forall r1 r2 r3, 0 <= r2 -> r1 + r2 <= r3 -> r1 <= r3. +intros x y z; intros; apply Rle_trans with (x + y); + [ pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + assumption + | assumption ]. +Qed. + +Lemma plus_lt_is_lt : forall r1 r2 r3, 0 <= r2 -> r1 + r2 < r3 -> r1 < r3. +intros x y z; intros; apply Rle_lt_trans with (x + y); + [ pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + assumption + | assumption ]. +Qed. + +Lemma Rmult_le_0_lt_compat : + forall r1 r2 r3 r4, + 0 <= r1 -> 0 <= r3 -> r1 < r2 -> r3 < r4 -> r1 * r3 < r2 * r4. +intros; apply Rle_lt_trans with (r2 * r3); + [ apply Rmult_le_compat_r; [ assumption | left; assumption ] + | apply Rmult_lt_compat_l; + [ apply Rle_lt_trans with r1; assumption | assumption ] ]. +Qed. + +Lemma le_epsilon : + forall r1 r2, (forall eps:R, 0 < eps -> r1 <= r2 + eps) -> r1 <= r2. +intros x y; intros; elim (Rtotal_order x y); intro. +left; assumption. +elim H0; intro. +right; assumption. +clear H0; generalize (Rgt_minus x y H1); intro H2; change (0 < x - y) in H2. +cut (0 < 2). +intro. +generalize (Rmult_lt_0_compat (x - y) (/ 2) H2 (Rinv_0_lt_compat 2 H0)); + intro H3; generalize (H ((x - y) * / 2) H3); + replace (y + (x - y) * / 2) with ((y + x) * / 2). +intro H4; + generalize (Rmult_le_compat_l 2 x ((y + x) * / 2) (Rlt_le 0 2 H0) H4); + rewrite <- (Rmult_comm ((y + x) * / 2)); rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; replace (2 * x) with (x + x). +rewrite (Rplus_comm y); intro H5; apply Rplus_le_reg_l with x; assumption. +ring. +replace 2 with (INR 2); [ apply not_O_INR; discriminate | ring ]. +pattern y at 2 in |- *; replace y with (y / 2 + y / 2). +unfold Rminus, Rdiv in |- *. +repeat rewrite Rmult_plus_distr_r. +ring. +cut (forall z:R, 2 * z = z + z). +intro. +rewrite <- (H4 (y / 2)). +unfold Rdiv in |- *. +rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. +replace 2 with (INR 2). +apply not_O_INR. +discriminate. +unfold INR in |- *; reflexivity. +intro; ring. +cut (0%nat <> 2%nat); + [ intro H0; generalize (lt_INR_0 2 (neq_O_lt 2 H0)); unfold INR in |- *; + intro; assumption + | discriminate ]. +Qed. + +(**********) +Lemma completeness_weak : + forall E:R -> Prop, + bound E -> ( exists x : R | E x) -> exists m : R | is_lub E m. +intros; elim (completeness E H H0); intros; split with x; assumption. +Qed.
\ No newline at end of file diff --git a/theories/Reals/RList.v b/theories/Reals/RList.v index 6e6f2716b..40848009a 100644 --- a/theories/Reals/RList.v +++ b/theories/Reals/RList.v @@ -8,420 +8,737 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. Open Local Scope R_scope. Inductive Rlist : Type := -| nil : Rlist -| cons : R -> Rlist -> Rlist. - -Fixpoint In [x:R;l:Rlist] : Prop := -Cases l of -| nil => False -| (cons a l') => ``x==a``\/(In x l') end. - -Fixpoint Rlength [l:Rlist] : nat := -Cases l of -| nil => O -| (cons a l') => (S (Rlength l')) end. - -Fixpoint MaxRlist [l:Rlist] : R := - Cases l of - | nil => R0 - | (cons a l1) => - Cases l1 of - | nil => a - | (cons a' l2) => (Rmax a (MaxRlist l1)) - end -end. - -Fixpoint MinRlist [l:Rlist] : R := -Cases l of - | nil => R1 - | (cons a l1) => - Cases l1 of - | nil => a - | (cons a' l2) => (Rmin a (MinRlist l1)) - end -end. - -Lemma MaxRlist_P1 : (l:Rlist;x:R) (In x l)->``x<=(MaxRlist l)``. -Intros; Induction l. -Simpl in H; Elim H. -Induction l. -Simpl in H; Elim H; Intro. -Simpl; Right; Assumption. -Elim H0. -Replace (MaxRlist (cons r (cons r0 l))) with (Rmax r (MaxRlist (cons r0 l))). -Simpl in H; Decompose [or] H. -Rewrite H0; Apply RmaxLess1. -Unfold Rmax; Case (total_order_Rle r (MaxRlist (cons r0 l))); Intro. -Apply Hrecl; Simpl; Tauto. -Apply Rle_trans with (MaxRlist (cons r0 l)); [Apply Hrecl; Simpl; Tauto | Left; Auto with real]. -Unfold Rmax; Case (total_order_Rle r (MaxRlist (cons r0 l))); Intro. -Apply Hrecl; Simpl; Tauto. -Apply Rle_trans with (MaxRlist (cons r0 l)); [Apply Hrecl; Simpl; Tauto | Left; Auto with real]. -Reflexivity. -Qed. - -Fixpoint AbsList [l:Rlist] : R->Rlist := -[x:R] Cases l of -| nil => nil -| (cons a l') => (cons ``(Rabsolu (a-x))/2`` (AbsList l' x)) -end. - -Lemma MinRlist_P1 : (l:Rlist;x:R) (In x l)->``(MinRlist l)<=x``. -Intros; Induction l. -Simpl in H; Elim H. -Induction l. -Simpl in H; Elim H; Intro. -Simpl; Right; Symmetry; Assumption. -Elim H0. -Replace (MinRlist (cons r (cons r0 l))) with (Rmin r (MinRlist (cons r0 l))). -Simpl in H; Decompose [or] H. -Rewrite H0; Apply Rmin_l. -Unfold Rmin; Case (total_order_Rle r (MinRlist (cons r0 l))); Intro. -Apply Rle_trans with (MinRlist (cons r0 l)). -Assumption. -Apply Hrecl; Simpl; Tauto. -Apply Hrecl; Simpl; Tauto. -Apply Rle_trans with (MinRlist (cons r0 l)). -Apply Rmin_r. -Apply Hrecl; Simpl; Tauto. -Reflexivity. -Qed. - -Lemma AbsList_P1 : (l:Rlist;x,y:R) (In y l) -> (In ``(Rabsolu (y-x))/2`` (AbsList l x)). -Intros; Induction l. -Elim H. -Simpl; Simpl in H; Elim H; Intro. -Left; Rewrite H0; Reflexivity. -Right; Apply Hrecl; Assumption. -Qed. - -Lemma MinRlist_P2 : (l:Rlist) ((y:R)(In y l)->``0<y``)->``0<(MinRlist l)``. -Intros; Induction l. -Apply Rlt_R0_R1. -Induction l. -Simpl; Apply H; Simpl; Tauto. -Replace (MinRlist (cons r (cons r0 l))) with (Rmin r (MinRlist (cons r0 l))). -Unfold Rmin; Case (total_order_Rle r (MinRlist (cons r0 l))); Intro. -Apply H; Simpl; Tauto. -Apply Hrecl; Intros; Apply H; Simpl; Simpl in H0; Tauto. -Reflexivity. -Qed. - -Lemma AbsList_P2 : (l:Rlist;x,y:R) (In y (AbsList l x)) -> (EXT z : R | (In z l)/\``y==(Rabsolu (z-x))/2``). -Intros; Induction l. -Elim H. -Elim H; Intro. -Exists r; Split. -Simpl; Tauto. -Assumption. -Assert H1 := (Hrecl H0); Elim H1; Intros; Elim H2; Clear H2; Intros; Exists x0; Simpl; Simpl in H2; Tauto. -Qed. - -Lemma MaxRlist_P2 : (l:Rlist) (EXT y:R | (In y l)) -> (In (MaxRlist l) l). -Intros; Induction l. -Simpl in H; Elim H; Trivial. -Induction l. -Simpl; Left; Reflexivity. -Change (In (Rmax r (MaxRlist (cons r0 l))) (cons r (cons r0 l))); Unfold Rmax; Case (total_order_Rle r (MaxRlist (cons r0 l))); Intro. -Right; Apply Hrecl; Exists r0; Left; Reflexivity. -Left; Reflexivity. -Qed. - -Fixpoint pos_Rl [l:Rlist] : nat->R := -[i:nat] Cases l of -| nil => R0 -| (cons a l') => - Cases i of - | O => a - | (S i') => (pos_Rl l' i') - end -end. - -Lemma pos_Rl_P1 : (l:Rlist;a:R) (lt O (Rlength l)) -> (pos_Rl (cons a l) (Rlength l))==(pos_Rl l (pred (Rlength l))). -Intros; Induction l; [Elim (lt_n_O ? H) | Simpl; Case (Rlength l); [Reflexivity | Intro; Reflexivity]]. -Qed. - -Lemma pos_Rl_P2 : (l:Rlist;x:R) (In x l)<->(EX i:nat | (lt i (Rlength l))/\x==(pos_Rl l i)). -Intros; Induction l. -Split; Intro; [Elim H | Elim H; Intros; Elim H0; Intros; Elim (lt_n_O ? H1)]. -Split; Intro. -Elim H; Intro. -Exists O; Split; [Simpl; Apply lt_O_Sn | Simpl; Apply H0]. -Elim Hrecl; Intros; Assert H3 := (H1 H0); Elim H3; Intros; Elim H4; Intros; Exists (S x0); Split; [Simpl; Apply lt_n_S; Assumption | Simpl; Assumption]. -Elim H; Intros; Elim H0; Intros; Elim (zerop x0); Intro. -Rewrite a in H2; Simpl in H2; Left; Assumption. -Right; Elim Hrecl; Intros; Apply H4; Assert H5 : (S (pred x0))=x0. -Symmetry; Apply S_pred with O; Assumption. -Exists (pred x0); Split; [Simpl in H1; Apply lt_S_n; Rewrite H5; Assumption | Rewrite <- H5 in H2; Simpl in H2; Assumption]. -Qed. - -Lemma Rlist_P1 : (l:Rlist;P:R->R->Prop) ((x:R)(In x l)->(EXT y:R | (P x y))) -> (EXT l':Rlist | (Rlength l)=(Rlength l')/\(i:nat) (lt i (Rlength l))->(P (pos_Rl l i) (pos_Rl l' i))). -Intros; Induction l. -Exists nil; Intros; Split; [Reflexivity | Intros; Simpl in H0; Elim (lt_n_O ? H0)]. -Assert H0 : (In r (cons r l)). -Simpl; Left; Reflexivity. -Assert H1 := (H ? H0); Assert H2 : (x:R)(In x l)->(EXT y:R | (P x y)). -Intros; Apply H; Simpl; Right; Assumption. -Assert H3 := (Hrecl H2); Elim H1; Intros; Elim H3; Intros; Exists (cons x x0); Intros; Elim H5; Clear H5; Intros; Split. -Simpl; Rewrite H5; Reflexivity. -Intros; Elim (zerop i); Intro. -Rewrite a; Simpl; Assumption. -Assert H8 : i=(S (pred i)). -Apply S_pred with O; Assumption. -Rewrite H8; Simpl; Apply H6; Simpl in H7; Apply lt_S_n; Rewrite <- H8; Assumption. -Qed. - -Definition ordered_Rlist [l:Rlist] : Prop := (i:nat) (lt i (pred (Rlength l))) -> (Rle (pos_Rl l i) (pos_Rl l (S i))). - -Fixpoint insert [l:Rlist] : R->Rlist := -[x:R] Cases l of -| nil => (cons x nil) -| (cons a l') => - Cases (total_order_Rle a x) of - | (leftT _) => (cons a (insert l' x)) - | (rightT _) => (cons x l) - end -end. - -Fixpoint cons_Rlist [l:Rlist] : Rlist->Rlist := -[k:Rlist] Cases l of -| nil => k -| (cons a l') => (cons a (cons_Rlist l' k)) end. - -Fixpoint cons_ORlist [k:Rlist] : Rlist->Rlist := -[l:Rlist] Cases k of -| nil => l -| (cons a k') => (cons_ORlist k' (insert l a)) -end. - -Fixpoint app_Rlist [l:Rlist] : (R->R)->Rlist := -[f:R->R] Cases l of -| nil => nil -| (cons a l') => (cons (f a) (app_Rlist l' f)) -end. - -Fixpoint mid_Rlist [l:Rlist] : R->Rlist := -[x:R] Cases l of -| nil => nil -| (cons a l') => (cons ``(x+a)/2`` (mid_Rlist l' a)) -end. - -Definition Rtail [l:Rlist] : Rlist := -Cases l of -| nil => nil -| (cons a l') => l' -end. - -Definition FF [l:Rlist;f:R->R] : Rlist := -Cases l of -| nil => nil -| (cons a l') => (app_Rlist (mid_Rlist l' a) f) -end. - -Lemma RList_P0 : (l:Rlist;a:R) ``(pos_Rl (insert l a) O) == a`` \/ ``(pos_Rl (insert l a) O) == (pos_Rl l O)``. -Intros; Induction l; [Left; Reflexivity | Simpl; Case (total_order_Rle r a); Intro; [Right; Reflexivity | Left; Reflexivity]]. -Qed. - -Lemma RList_P1 : (l:Rlist;a:R) (ordered_Rlist l) -> (ordered_Rlist (insert l a)). -Intros; Induction l. -Simpl; Unfold ordered_Rlist; Intros; Simpl in H0; Elim (lt_n_O ? H0). -Simpl; Case (total_order_Rle r a); Intro. -Assert H1 : (ordered_Rlist l). -Unfold ordered_Rlist; Unfold ordered_Rlist in H; Intros; Assert H1 : (lt (S i) (pred (Rlength (cons r l)))); [Simpl; Replace (Rlength l) with (S (pred (Rlength l))); [Apply lt_n_S; Assumption | Symmetry; Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H1 in H0; Simpl in H0; Elim (lt_n_O ? H0)] | Apply (H ? H1)]. -Assert H2 := (Hrecl H1); Unfold ordered_Rlist; Intros; Induction i. -Simpl; Assert H3 := (RList_P0 l a); Elim H3; Intro. -Rewrite H4; Assumption. -Induction l; [Simpl; Assumption | Rewrite H4; Apply (H O); Simpl; Apply lt_O_Sn]. -Simpl; Apply H2; Simpl in H0; Apply lt_S_n; Replace (S (pred (Rlength (insert l a)))) with (Rlength (insert l a)); [Assumption | Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H3 in H0; Elim (lt_n_O ? H0)]. -Unfold ordered_Rlist; Intros; Induction i; [Simpl; Auto with real | Change ``(pos_Rl (cons r l) i)<=(pos_Rl (cons r l) (S i))``; Apply H; Simpl in H0; Simpl; Apply (lt_S_n ? ? H0)]. -Qed. - -Lemma RList_P2 : (l1,l2:Rlist) (ordered_Rlist l2) ->(ordered_Rlist (cons_ORlist l1 l2)). -Induction l1; [Intros; Simpl; Apply H | Intros; Simpl; Apply H; Apply RList_P1; Assumption]. -Qed. - -Lemma RList_P3 : (l:Rlist;x:R) (In x l) <-> (EX i:nat | x==(pos_Rl l i)/\(lt i (Rlength l))). -Intros; Split; Intro; Induction l. -Elim H. -Elim H; Intro; [Exists O; Split; [Apply H0 | Simpl; Apply lt_O_Sn] | Elim (Hrecl H0); Intros; Elim H1; Clear H1; Intros; Exists (S x0); Split; [Apply H1 | Simpl; Apply lt_n_S; Assumption]]. -Elim H; Intros; Elim H0; Intros; Elim (lt_n_O ? H2). -Simpl; Elim H; Intros; Elim H0; Clear H0; Intros; Induction x0; [Left; Apply H0 | Right; Apply Hrecl; Exists x0; Split; [Apply H0 | Simpl in H1; Apply lt_S_n; Assumption]]. -Qed. - -Lemma RList_P4 : (l1:Rlist;a:R) (ordered_Rlist (cons a l1)) -> (ordered_Rlist l1). -Intros; Unfold ordered_Rlist; Intros; Apply (H (S i)); Simpl; Replace (Rlength l1) with (S (pred (Rlength l1))); [Apply lt_n_S; Assumption | Symmetry; Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H1 in H0; Elim (lt_n_O ? H0)]. -Qed. - -Lemma RList_P5 : (l:Rlist;x:R) (ordered_Rlist l) -> (In x l) -> ``(pos_Rl l O)<=x``. -Intros; Induction l; [Elim H0 | Simpl; Elim H0; Intro; [Rewrite H1; Right; Reflexivity | Apply Rle_trans with (pos_Rl l O); [Apply (H O); Simpl; Induction l; [Elim H1 | Simpl; Apply lt_O_Sn] | Apply Hrecl; [EApply RList_P4; Apply H | Assumption]]]]. -Qed. - -Lemma RList_P6 : (l:Rlist) (ordered_Rlist l)<->((i,j:nat)(le i j)->(lt j (Rlength l))->``(pos_Rl l i)<=(pos_Rl l j)``). -Induction l; Split; Intro. -Intros; Right; Reflexivity. -Unfold ordered_Rlist; Intros; Simpl in H0; Elim (lt_n_O ? H0). -Intros; Induction i; [Induction j; [Right; Reflexivity | Simpl; Apply Rle_trans with (pos_Rl r0 O); [Apply (H0 O); Simpl; Simpl in H2; Apply neq_O_lt; Red; Intro; Rewrite <- H3 in H2; Assert H4 := (lt_S_n ? ? H2); Elim (lt_n_O ? H4) | Elim H; Intros; Apply H3; [Apply RList_P4 with r; Assumption | Apply le_O_n | Simpl in H2; Apply lt_S_n; Assumption]]] | Induction j; [Elim (le_Sn_O ? H1) | Simpl; Elim H; Intros; Apply H3; [Apply RList_P4 with r; Assumption | Apply le_S_n; Assumption | Simpl in H2; Apply lt_S_n; Assumption]]]. -Unfold ordered_Rlist; Intros; Apply H0; [Apply le_n_Sn | Simpl; Simpl in H1; Apply lt_n_S; Assumption]. -Qed. - -Lemma RList_P7 : (l:Rlist;x:R) (ordered_Rlist l) -> (In x l) -> ``x<=(pos_Rl l (pred (Rlength l)))``. -Intros; Assert H1 := (RList_P6 l); Elim H1; Intros H2 _; Assert H3 := (H2 H); Clear H1 H2; Assert H1 := (RList_P3 l x); Elim H1; Clear H1; Intros; Assert H4 := (H1 H0); Elim H4; Clear H4; Intros; Elim H4; Clear H4; Intros; Rewrite H4; Assert H6 : (Rlength l)=(S (pred (Rlength l))). -Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H6 in H5; Elim (lt_n_O ? H5). -Apply H3; [Rewrite H6 in H5; Apply lt_n_Sm_le; Assumption | Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H7 in H5; Elim (lt_n_O ? H5)]. -Qed. - -Lemma RList_P8 : (l:Rlist;a,x:R) (In x (insert l a)) <-> x==a\/(In x l). -Induction l. -Intros; Split; Intro; Simpl in H; Apply H. -Intros; Split; Intro; [Simpl in H0; Generalize H0; Case (total_order_Rle r a); Intros; [Simpl in H1; Elim H1; Intro; [Right; Left; Assumption |Elim (H a x); Intros; Elim (H3 H2); Intro; [Left; Assumption | Right; Right; Assumption]] | Simpl in H1; Decompose [or] H1; [Left; Assumption | Right; Left; Assumption | Right; Right; Assumption]] | Simpl; Case (total_order_Rle r a); Intro; [Simpl in H0; Decompose [or] H0; [Right; Elim (H a x); Intros; Apply H3; Left | Left | Right; Elim (H a x); Intros; Apply H3; Right] | Simpl in H0; Decompose [or] H0; [Left | Right; Left | Right; Right]]; Assumption]. -Qed. - -Lemma RList_P9 : (l1,l2:Rlist;x:R) (In x (cons_ORlist l1 l2)) <-> (In x l1)\/(In x l2). -Induction l1. -Intros; Split; Intro; [Simpl in H; Right; Assumption | Simpl; Elim H; Intro; [Elim H0 | Assumption]]. -Intros; Split. -Simpl; Intros; Elim (H (insert l2 r) x); Intros; Assert H3 := (H1 H0); Elim H3; Intro; [Left; Right; Assumption | Elim (RList_P8 l2 r x); Intros H5 _; Assert H6 := (H5 H4); Elim H6; Intro; [Left; Left; Assumption | Right; Assumption]]. -Intro; Simpl; Elim (H (insert l2 r) x); Intros _ H1; Apply H1; Elim H0; Intro; [Elim H2; Intro; [Right; Elim (RList_P8 l2 r x); Intros _ H4; Apply H4; Left; Assumption | Left; Assumption] | Right; Elim (RList_P8 l2 r x); Intros _ H3; Apply H3; Right; Assumption]. -Qed. - -Lemma RList_P10 : (l:Rlist;a:R) (Rlength (insert l a))==(S (Rlength l)). -Intros; Induction l; [Reflexivity | Simpl; Case (total_order_Rle r a); Intro; [Simpl; Rewrite Hrecl; Reflexivity | Reflexivity]]. -Qed. - -Lemma RList_P11 : (l1,l2:Rlist) (Rlength (cons_ORlist l1 l2))=(plus (Rlength l1) (Rlength l2)). -Induction l1; [Intro; Reflexivity | Intros; Simpl; Rewrite (H (insert l2 r)); Rewrite RList_P10; Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite S_INR; Ring]. -Qed. - -Lemma RList_P12 : (l:Rlist;i:nat;f:R->R) (lt i (Rlength l)) -> (pos_Rl (app_Rlist l f) i)==(f (pos_Rl l i)). -Induction l; [Intros; Elim (lt_n_O ? H) | Intros; Induction i; [Reflexivity | Simpl; Apply H; Apply lt_S_n; Apply H0]]. -Qed. - -Lemma RList_P13 : (l:Rlist;i:nat;a:R) (lt i (pred (Rlength l))) -> ``(pos_Rl (mid_Rlist l a) (S i)) == ((pos_Rl l i)+(pos_Rl l (S i)))/2``. -Induction l. -Intros; Simpl in H; Elim (lt_n_O ? H). -Induction r0. -Intros; Simpl in H0; Elim (lt_n_O ? H0). -Intros; Simpl in H1; Induction i. -Reflexivity. -Change ``(pos_Rl (mid_Rlist (cons r1 r2) r) (S i)) == ((pos_Rl (cons r1 r2) i)+(pos_Rl (cons r1 r2) (S i)))/2``; Apply H0; Simpl; Apply lt_S_n; Assumption. -Qed. - -Lemma RList_P14 : (l:Rlist;a:R) (Rlength (mid_Rlist l a))=(Rlength l). -Induction l; Intros; [Reflexivity | Simpl; Rewrite (H r); Reflexivity]. -Qed. - -Lemma RList_P15 : (l1,l2:Rlist) (ordered_Rlist l1) -> (ordered_Rlist l2) -> (pos_Rl l1 O)==(pos_Rl l2 O) -> (pos_Rl (cons_ORlist l1 l2) O)==(pos_Rl l1 O). -Intros; Apply Rle_antisym. -Induction l1; [Simpl; Simpl in H1; Right; Symmetry; Assumption | Elim (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) (0))); Intros; Assert H4 : (In (pos_Rl (cons r l1) (0)) (cons r l1))\/(In (pos_Rl (cons r l1) (0)) l2); [Left; Left; Reflexivity | Assert H5 := (H3 H4); Apply RList_P5; [Apply RList_P2; Assumption | Assumption]]]. -Induction l1; [Simpl; Simpl in H1; Right; Assumption | Assert H2 : (In (pos_Rl (cons_ORlist (cons r l1) l2) (0)) (cons_ORlist (cons r l1) l2)); [Elim (RList_P3 (cons_ORlist (cons r l1) l2) (pos_Rl (cons_ORlist (cons r l1) l2) (0))); Intros; Apply H3; Exists O; Split; [Reflexivity | Rewrite RList_P11; Simpl; Apply lt_O_Sn] | Elim (RList_P9 (cons r l1) l2 (pos_Rl (cons_ORlist (cons r l1) l2) (0))); Intros; Assert H5 := (H3 H2); Elim H5; Intro; [Apply RList_P5; Assumption | Rewrite H1; Apply RList_P5; Assumption]]]. -Qed. - -Lemma RList_P16 : (l1,l2:Rlist) (ordered_Rlist l1) -> (ordered_Rlist l2) -> (pos_Rl l1 (pred (Rlength l1)))==(pos_Rl l2 (pred (Rlength l2))) -> (pos_Rl (cons_ORlist l1 l2) (pred (Rlength (cons_ORlist l1 l2))))==(pos_Rl l1 (pred (Rlength l1))). -Intros; Apply Rle_antisym. -Induction l1. -Simpl; Simpl in H1; Right; Symmetry; Assumption. -Assert H2 : (In (pos_Rl (cons_ORlist (cons r l1) l2) (pred (Rlength (cons_ORlist (cons r l1) l2)))) (cons_ORlist (cons r l1) l2)); [Elim (RList_P3 (cons_ORlist (cons r l1) l2) (pos_Rl (cons_ORlist (cons r l1) l2) (pred (Rlength (cons_ORlist (cons r l1) l2))))); Intros; Apply H3; Exists (pred (Rlength (cons_ORlist (cons r l1) l2))); Split; [Reflexivity | Rewrite RList_P11; Simpl; Apply lt_n_Sn] | Elim (RList_P9 (cons r l1) l2 (pos_Rl (cons_ORlist (cons r l1) l2) (pred (Rlength (cons_ORlist (cons r l1) l2))))); Intros; Assert H5 := (H3 H2); Elim H5; Intro; [Apply RList_P7; Assumption | Rewrite H1; Apply RList_P7; Assumption]]. -Induction l1. -Simpl; Simpl in H1; Right; Assumption. -Elim (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); Intros; Assert H4 : (In (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))) (cons r l1))\/(In (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))) l2); [Left; Change (In (pos_Rl (cons r l1) (Rlength l1)) (cons r l1)); Elim (RList_P3 (cons r l1) (pos_Rl (cons r l1) (Rlength l1))); Intros; Apply H5; Exists (Rlength l1); Split; [Reflexivity | Simpl; Apply lt_n_Sn] | Assert H5 := (H3 H4); Apply RList_P7; [Apply RList_P2; Assumption | Elim (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); Intros; Apply H7; Left; Elim (RList_P3 (cons r l1) (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); Intros; Apply H9; Exists (pred (Rlength (cons r l1))); Split; [Reflexivity | Simpl; Apply lt_n_Sn]]]. -Qed. - -Lemma RList_P17 : (l1:Rlist;x:R;i:nat) (ordered_Rlist l1) -> (In x l1) -> ``(pos_Rl l1 i)<x`` -> (lt i (pred (Rlength l1))) -> ``(pos_Rl l1 (S i))<=x``. -Induction l1. -Intros; Elim H0. -Intros; Induction i. -Simpl; Elim H1; Intro; [Simpl in H2; Rewrite H4 in H2; Elim (Rlt_antirefl ? H2) | Apply RList_P5; [Apply RList_P4 with r; Assumption | Assumption]]. -Simpl; Simpl in H2; Elim H1; Intro. -Rewrite H4 in H2; Assert H5 : ``r<=(pos_Rl r0 i)``; [Apply Rle_trans with (pos_Rl r0 O); [Apply (H0 O); Simpl; Simpl in H3; Apply neq_O_lt; Red; Intro; Rewrite <- H5 in H3; Elim (lt_n_O ? H3) | Elim (RList_P6 r0); Intros; Apply H5; [Apply RList_P4 with r; Assumption | Apply le_O_n | Simpl in H3; Apply lt_S_n; Apply lt_trans with (Rlength r0); [Apply H3 | Apply lt_n_Sn]]] | Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H5 H2))]. -Apply H; Try Assumption; [Apply RList_P4 with r; Assumption | Simpl in H3; Apply lt_S_n; Replace (S (pred (Rlength r0))) with (Rlength r0); [Apply H3 | Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H5 in H3; Elim (lt_n_O ? H3)]]. -Qed. - -Lemma RList_P18 : (l:Rlist;f:R->R) (Rlength (app_Rlist l f))=(Rlength l). -Induction l; Intros; [Reflexivity | Simpl; Rewrite H; Reflexivity]. -Qed. - -Lemma RList_P19 : (l:Rlist) ~l==nil -> (EXT r:R | (EXT r0:Rlist | l==(cons r r0))). -Intros; Induction l; [Elim H; Reflexivity | Exists r; Exists l; Reflexivity]. -Qed. - -Lemma RList_P20 : (l:Rlist) (le (2) (Rlength l)) -> (EXT r:R | (EXT r1:R | (EXT l':Rlist | l==(cons r (cons r1 l'))))). -Intros; Induction l; [Simpl in H; Elim (le_Sn_O ? H) | Induction l; [Simpl in H; Elim (le_Sn_O ? (le_S_n ? ? H)) | Exists r; Exists r0; Exists l; Reflexivity]]. -Qed. - -Lemma RList_P21 : (l,l':Rlist) l==l' -> (Rtail l)==(Rtail l'). -Intros; Rewrite H; Reflexivity. -Qed. - -Lemma RList_P22 : (l1,l2:Rlist) ~l1==nil -> (pos_Rl (cons_Rlist l1 l2) O)==(pos_Rl l1 O). -Induction l1; [Intros; Elim H; Reflexivity | Intros; Reflexivity]. -Qed. - -Lemma RList_P23 : (l1,l2:Rlist) (Rlength (cons_Rlist l1 l2))==(plus (Rlength l1) (Rlength l2)). -Induction l1; [Intro; Reflexivity | Intros; Simpl; Rewrite H; Reflexivity]. -Qed. - -Lemma RList_P24 : (l1,l2:Rlist) ~l2==nil -> (pos_Rl (cons_Rlist l1 l2) (pred (Rlength (cons_Rlist l1 l2)))) == (pos_Rl l2 (pred (Rlength l2))). -Induction l1. -Intros; Reflexivity. -Intros; Rewrite <- (H l2 H0); Induction l2. -Elim H0; Reflexivity. -Do 2 Rewrite RList_P23; Replace (plus (Rlength (cons r r0)) (Rlength (cons r1 l2))) with (S (S (plus (Rlength r0) (Rlength l2)))); [Replace (plus (Rlength r0) (Rlength (cons r1 l2))) with (S (plus (Rlength r0) (Rlength l2))); [Reflexivity | Simpl; Apply INR_eq; Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite S_INR; Ring] | Simpl; Apply INR_eq; Do 3 Rewrite S_INR; Do 2 Rewrite plus_INR; Rewrite S_INR; Ring]. -Qed. - -Lemma RList_P25 : (l1,l2:Rlist) (ordered_Rlist l1) -> (ordered_Rlist l2) -> ``(pos_Rl l1 (pred (Rlength l1)))<=(pos_Rl l2 O)`` -> (ordered_Rlist (cons_Rlist l1 l2)). -Induction l1. -Intros; Simpl; Assumption. -Induction r0. -Intros; Simpl; Simpl in H2; Unfold ordered_Rlist; Intros; Simpl in H3. -Induction i. -Simpl; Assumption. -Change ``(pos_Rl l2 i)<=(pos_Rl l2 (S i))``; Apply (H1 i); Apply lt_S_n; Replace (S (pred (Rlength l2))) with (Rlength l2); [Assumption | Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H4 in H3; Elim (lt_n_O ? H3)]. -Intros; Clear H; Assert H : (ordered_Rlist (cons_Rlist (cons r1 r2) l2)). -Apply H0; Try Assumption. -Apply RList_P4 with r; Assumption. -Unfold ordered_Rlist; Intros; Simpl in H4; Induction i. -Simpl; Apply (H1 O); Simpl; Apply lt_O_Sn. -Change ``(pos_Rl (cons_Rlist (cons r1 r2) l2) i)<=(pos_Rl (cons_Rlist (cons r1 r2) l2) (S i))``; Apply (H i); Simpl; Apply lt_S_n; Assumption. -Qed. - -Lemma RList_P26 : (l1,l2:Rlist;i:nat) (lt i (Rlength l1)) -> (pos_Rl (cons_Rlist l1 l2) i)==(pos_Rl l1 i). -Induction l1. -Intros; Elim (lt_n_O ? H). -Intros; Induction i. -Apply RList_P22; Discriminate. -Apply (H l2 i); Simpl in H0; Apply lt_S_n; Assumption. -Qed. - -Lemma RList_P27 : (l1,l2,l3:Rlist) (cons_Rlist l1 (cons_Rlist l2 l3))==(cons_Rlist (cons_Rlist l1 l2) l3). -Induction l1; Intros; [Reflexivity | Simpl; Rewrite (H l2 l3); Reflexivity]. -Qed. - -Lemma RList_P28 : (l:Rlist) (cons_Rlist l nil)==l. -Induction l; [Reflexivity | Intros; Simpl; Rewrite H; Reflexivity]. -Qed. - -Lemma RList_P29 : (l2,l1:Rlist;i:nat) (le (Rlength l1) i) -> (lt i (Rlength (cons_Rlist l1 l2))) -> (pos_Rl (cons_Rlist l1 l2) i)==(pos_Rl l2 (minus i (Rlength l1))). -Induction l2. -Intros; Rewrite RList_P28 in H0; Elim (lt_n_n ? (le_lt_trans ? ? ? H H0)). -Intros; Replace (cons_Rlist l1 (cons r r0)) with (cons_Rlist (cons_Rlist l1 (cons r nil)) r0). -Inversion H0. -Rewrite <- minus_n_n; Simpl; Rewrite RList_P26. -Clear l2 r0 H i H0 H1 H2; Induction l1. -Reflexivity. -Simpl; Assumption. -Rewrite RList_P23; Rewrite plus_sym; Simpl; Apply lt_n_Sn. -Replace (minus (S m) (Rlength l1)) with (S (minus (S m) (S (Rlength l1)))). -Rewrite H3; Simpl; Replace (S (Rlength l1)) with (Rlength (cons_Rlist l1 (cons r nil))). -Apply (H (cons_Rlist l1 (cons r nil)) i). -Rewrite RList_P23; Rewrite plus_sym; Simpl; Rewrite <- H3; Apply le_n_S; Assumption. -Repeat Rewrite RList_P23; Simpl; Rewrite RList_P23 in H1; Rewrite plus_sym in H1; Simpl in H1; Rewrite (plus_sym (Rlength l1)); Simpl; Rewrite plus_sym; Apply H1. -Rewrite RList_P23; Rewrite plus_sym; Reflexivity. -Change (S (minus m (Rlength l1)))=(minus (S m) (Rlength l1)); Apply minus_Sn_m; Assumption. -Replace (cons r r0) with (cons_Rlist (cons r nil) r0); [Symmetry; Apply RList_P27 | Reflexivity]. -Qed. + | nil : Rlist + | cons : R -> Rlist -> Rlist. + +Fixpoint In (x:R) (l:Rlist) {struct l} : Prop := + match l with + | nil => False + | cons a l' => x = a \/ In x l' + end. + +Fixpoint Rlength (l:Rlist) : nat := + match l with + | nil => 0%nat + | cons a l' => S (Rlength l') + end. + +Fixpoint MaxRlist (l:Rlist) : R := + match l with + | nil => 0 + | cons a l1 => + match l1 with + | nil => a + | cons a' l2 => Rmax a (MaxRlist l1) + end + end. + +Fixpoint MinRlist (l:Rlist) : R := + match l with + | nil => 1 + | cons a l1 => + match l1 with + | nil => a + | cons a' l2 => Rmin a (MinRlist l1) + end + end. + +Lemma MaxRlist_P1 : forall (l:Rlist) (x:R), In x l -> x <= MaxRlist l. +intros; induction l as [| r l Hrecl]. +simpl in H; elim H. +induction l as [| r0 l Hrecl0]. +simpl in H; elim H; intro. +simpl in |- *; right; assumption. +elim H0. +replace (MaxRlist (cons r (cons r0 l))) with (Rmax r (MaxRlist (cons r0 l))). +simpl in H; decompose [or] H. +rewrite H0; apply RmaxLess1. +unfold Rmax in |- *; case (Rle_dec r (MaxRlist (cons r0 l))); intro. +apply Hrecl; simpl in |- *; tauto. +apply Rle_trans with (MaxRlist (cons r0 l)); + [ apply Hrecl; simpl in |- *; tauto | left; auto with real ]. +unfold Rmax in |- *; case (Rle_dec r (MaxRlist (cons r0 l))); intro. +apply Hrecl; simpl in |- *; tauto. +apply Rle_trans with (MaxRlist (cons r0 l)); + [ apply Hrecl; simpl in |- *; tauto | left; auto with real ]. +reflexivity. +Qed. + +Fixpoint AbsList (l:Rlist) (x:R) {struct l} : Rlist := + match l with + | nil => nil + | cons a l' => cons (Rabs (a - x) / 2) (AbsList l' x) + end. + +Lemma MinRlist_P1 : forall (l:Rlist) (x:R), In x l -> MinRlist l <= x. +intros; induction l as [| r l Hrecl]. +simpl in H; elim H. +induction l as [| r0 l Hrecl0]. +simpl in H; elim H; intro. +simpl in |- *; right; symmetry in |- *; assumption. +elim H0. +replace (MinRlist (cons r (cons r0 l))) with (Rmin r (MinRlist (cons r0 l))). +simpl in H; decompose [or] H. +rewrite H0; apply Rmin_l. +unfold Rmin in |- *; case (Rle_dec r (MinRlist (cons r0 l))); intro. +apply Rle_trans with (MinRlist (cons r0 l)). +assumption. +apply Hrecl; simpl in |- *; tauto. +apply Hrecl; simpl in |- *; tauto. +apply Rle_trans with (MinRlist (cons r0 l)). +apply Rmin_r. +apply Hrecl; simpl in |- *; tauto. +reflexivity. +Qed. + +Lemma AbsList_P1 : + forall (l:Rlist) (x y:R), In y l -> In (Rabs (y - x) / 2) (AbsList l x). +intros; induction l as [| r l Hrecl]. +elim H. +simpl in |- *; simpl in H; elim H; intro. +left; rewrite H0; reflexivity. +right; apply Hrecl; assumption. +Qed. + +Lemma MinRlist_P2 : + forall l:Rlist, (forall y:R, In y l -> 0 < y) -> 0 < MinRlist l. +intros; induction l as [| r l Hrecl]. +apply Rlt_0_1. +induction l as [| r0 l Hrecl0]. +simpl in |- *; apply H; simpl in |- *; tauto. +replace (MinRlist (cons r (cons r0 l))) with (Rmin r (MinRlist (cons r0 l))). +unfold Rmin in |- *; case (Rle_dec r (MinRlist (cons r0 l))); intro. +apply H; simpl in |- *; tauto. +apply Hrecl; intros; apply H; simpl in |- *; simpl in H0; tauto. +reflexivity. +Qed. + +Lemma AbsList_P2 : + forall (l:Rlist) (x y:R), + In y (AbsList l x) -> exists z : R | In z l /\ y = Rabs (z - x) / 2. +intros; induction l as [| r l Hrecl]. +elim H. +elim H; intro. +exists r; split. +simpl in |- *; tauto. +assumption. +assert (H1 := Hrecl H0); elim H1; intros; elim H2; clear H2; intros; + exists x0; simpl in |- *; simpl in H2; tauto. +Qed. + +Lemma MaxRlist_P2 : + forall l:Rlist, ( exists y : R | In y l) -> In (MaxRlist l) l. +intros; induction l as [| r l Hrecl]. +simpl in H; elim H; trivial. +induction l as [| r0 l Hrecl0]. +simpl in |- *; left; reflexivity. +change (In (Rmax r (MaxRlist (cons r0 l))) (cons r (cons r0 l))) in |- *; + unfold Rmax in |- *; case (Rle_dec r (MaxRlist (cons r0 l))); + intro. +right; apply Hrecl; exists r0; left; reflexivity. +left; reflexivity. +Qed. + +Fixpoint pos_Rl (l:Rlist) (i:nat) {struct l} : R := + match l with + | nil => 0 + | cons a l' => match i with + | O => a + | S i' => pos_Rl l' i' + end + end. + +Lemma pos_Rl_P1 : + forall (l:Rlist) (a:R), + (0 < Rlength l)%nat -> + pos_Rl (cons a l) (Rlength l) = pos_Rl l (pred (Rlength l)). +intros; induction l as [| r l Hrecl]; + [ elim (lt_n_O _ H) + | simpl in |- *; case (Rlength l); [ reflexivity | intro; reflexivity ] ]. +Qed. + +Lemma pos_Rl_P2 : + forall (l:Rlist) (x:R), + In x l <-> ( exists i : nat | (i < Rlength l)%nat /\ x = pos_Rl l i). +intros; induction l as [| r l Hrecl]. +split; intro; + [ elim H | elim H; intros; elim H0; intros; elim (lt_n_O _ H1) ]. +split; intro. +elim H; intro. +exists 0%nat; split; + [ simpl in |- *; apply lt_O_Sn | simpl in |- *; apply H0 ]. +elim Hrecl; intros; assert (H3 := H1 H0); elim H3; intros; elim H4; intros; + exists (S x0); split; + [ simpl in |- *; apply lt_n_S; assumption | simpl in |- *; assumption ]. +elim H; intros; elim H0; intros; elim (zerop x0); intro. +rewrite a in H2; simpl in H2; left; assumption. +right; elim Hrecl; intros; apply H4; assert (H5 : S (pred x0) = x0). +symmetry in |- *; apply S_pred with 0%nat; assumption. +exists (pred x0); split; + [ simpl in H1; apply lt_S_n; rewrite H5; assumption + | rewrite <- H5 in H2; simpl in H2; assumption ]. +Qed. + +Lemma Rlist_P1 : + forall (l:Rlist) (P:R -> R -> Prop), + (forall x:R, In x l -> exists y : R | P x y) -> + exists l' : Rlist + | Rlength l = Rlength l' /\ + (forall i:nat, (i < Rlength l)%nat -> P (pos_Rl l i) (pos_Rl l' i)). +intros; induction l as [| r l Hrecl]. +exists nil; intros; split; + [ reflexivity | intros; simpl in H0; elim (lt_n_O _ H0) ]. +assert (H0 : In r (cons r l)). +simpl in |- *; left; reflexivity. +assert (H1 := H _ H0); + assert (H2 : forall x:R, In x l -> exists y : R | P x y). +intros; apply H; simpl in |- *; right; assumption. +assert (H3 := Hrecl H2); elim H1; intros; elim H3; intros; exists (cons x x0); + intros; elim H5; clear H5; intros; split. +simpl in |- *; rewrite H5; reflexivity. +intros; elim (zerop i); intro. +rewrite a; simpl in |- *; assumption. +assert (H8 : i = S (pred i)). +apply S_pred with 0%nat; assumption. +rewrite H8; simpl in |- *; apply H6; simpl in H7; apply lt_S_n; rewrite <- H8; + assumption. +Qed. + +Definition ordered_Rlist (l:Rlist) : Prop := + forall i:nat, (i < pred (Rlength l))%nat -> pos_Rl l i <= pos_Rl l (S i). + +Fixpoint insert (l:Rlist) (x:R) {struct l} : Rlist := + match l with + | nil => cons x nil + | cons a l' => + match Rle_dec a x with + | left _ => cons a (insert l' x) + | right _ => cons x l + end + end. + +Fixpoint cons_Rlist (l k:Rlist) {struct l} : Rlist := + match l with + | nil => k + | cons a l' => cons a (cons_Rlist l' k) + end. + +Fixpoint cons_ORlist (k l:Rlist) {struct k} : Rlist := + match k with + | nil => l + | cons a k' => cons_ORlist k' (insert l a) + end. + +Fixpoint app_Rlist (l:Rlist) (f:R -> R) {struct l} : Rlist := + match l with + | nil => nil + | cons a l' => cons (f a) (app_Rlist l' f) + end. + +Fixpoint mid_Rlist (l:Rlist) (x:R) {struct l} : Rlist := + match l with + | nil => nil + | cons a l' => cons ((x + a) / 2) (mid_Rlist l' a) + end. + +Definition Rtail (l:Rlist) : Rlist := + match l with + | nil => nil + | cons a l' => l' + end. + +Definition FF (l:Rlist) (f:R -> R) : Rlist := + match l with + | nil => nil + | cons a l' => app_Rlist (mid_Rlist l' a) f + end. + +Lemma RList_P0 : + forall (l:Rlist) (a:R), + pos_Rl (insert l a) 0 = a \/ pos_Rl (insert l a) 0 = pos_Rl l 0. +intros; induction l as [| r l Hrecl]; + [ left; reflexivity + | simpl in |- *; case (Rle_dec r a); intro; + [ right; reflexivity | left; reflexivity ] ]. +Qed. + +Lemma RList_P1 : + forall (l:Rlist) (a:R), ordered_Rlist l -> ordered_Rlist (insert l a). +intros; induction l as [| r l Hrecl]. +simpl in |- *; unfold ordered_Rlist in |- *; intros; simpl in H0; + elim (lt_n_O _ H0). +simpl in |- *; case (Rle_dec r a); intro. +assert (H1 : ordered_Rlist l). +unfold ordered_Rlist in |- *; unfold ordered_Rlist in H; intros; + assert (H1 : (S i < pred (Rlength (cons r l)))%nat); + [ simpl in |- *; replace (Rlength l) with (S (pred (Rlength l))); + [ apply lt_n_S; assumption + | symmetry in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *; + intro; rewrite <- H1 in H0; simpl in H0; elim (lt_n_O _ H0) ] + | apply (H _ H1) ]. +assert (H2 := Hrecl H1); unfold ordered_Rlist in |- *; intros; + induction i as [| i Hreci]. +simpl in |- *; assert (H3 := RList_P0 l a); elim H3; intro. +rewrite H4; assumption. +induction l as [| r1 l Hrecl0]; + [ simpl in |- *; assumption + | rewrite H4; apply (H 0%nat); simpl in |- *; apply lt_O_Sn ]. +simpl in |- *; apply H2; simpl in H0; apply lt_S_n; + replace (S (pred (Rlength (insert l a)))) with (Rlength (insert l a)); + [ assumption + | apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro; + rewrite <- H3 in H0; elim (lt_n_O _ H0) ]. +unfold ordered_Rlist in |- *; intros; induction i as [| i Hreci]; + [ simpl in |- *; auto with real + | change (pos_Rl (cons r l) i <= pos_Rl (cons r l) (S i)) in |- *; apply H; + simpl in H0; simpl in |- *; apply (lt_S_n _ _ H0) ]. +Qed. + +Lemma RList_P2 : + forall l1 l2:Rlist, ordered_Rlist l2 -> ordered_Rlist (cons_ORlist l1 l2). +simple induction l1; + [ intros; simpl in |- *; apply H + | intros; simpl in |- *; apply H; apply RList_P1; assumption ]. +Qed. + +Lemma RList_P3 : + forall (l:Rlist) (x:R), + In x l <-> ( exists i : nat | x = pos_Rl l i /\ (i < Rlength l)%nat). +intros; split; intro; + [ induction l as [| r l Hrecl] | induction l as [| r l Hrecl] ]. +elim H. +elim H; intro; + [ exists 0%nat; split; [ apply H0 | simpl in |- *; apply lt_O_Sn ] + | elim (Hrecl H0); intros; elim H1; clear H1; intros; exists (S x0); split; + [ apply H1 | simpl in |- *; apply lt_n_S; assumption ] ]. +elim H; intros; elim H0; intros; elim (lt_n_O _ H2). +simpl in |- *; elim H; intros; elim H0; clear H0; intros; + induction x0 as [| x0 Hrecx0]; + [ left; apply H0 + | right; apply Hrecl; exists x0; split; + [ apply H0 | simpl in H1; apply lt_S_n; assumption ] ]. +Qed. + +Lemma RList_P4 : + forall (l1:Rlist) (a:R), ordered_Rlist (cons a l1) -> ordered_Rlist l1. +intros; unfold ordered_Rlist in |- *; intros; apply (H (S i)); simpl in |- *; + replace (Rlength l1) with (S (pred (Rlength l1))); + [ apply lt_n_S; assumption + | symmetry in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *; + intro; rewrite <- H1 in H0; elim (lt_n_O _ H0) ]. +Qed. + +Lemma RList_P5 : + forall (l:Rlist) (x:R), ordered_Rlist l -> In x l -> pos_Rl l 0 <= x. +intros; induction l as [| r l Hrecl]; + [ elim H0 + | simpl in |- *; elim H0; intro; + [ rewrite H1; right; reflexivity + | apply Rle_trans with (pos_Rl l 0); + [ apply (H 0%nat); simpl in |- *; induction l as [| r0 l Hrecl0]; + [ elim H1 | simpl in |- *; apply lt_O_Sn ] + | apply Hrecl; [ eapply RList_P4; apply H | assumption ] ] ] ]. +Qed. + +Lemma RList_P6 : + forall l:Rlist, + ordered_Rlist l <-> + (forall i j:nat, + (i <= j)%nat -> (j < Rlength l)%nat -> pos_Rl l i <= pos_Rl l j). +simple induction l; split; intro. +intros; right; reflexivity. +unfold ordered_Rlist in |- *; intros; simpl in H0; elim (lt_n_O _ H0). +intros; induction i as [| i Hreci]; + [ induction j as [| j Hrecj]; + [ right; reflexivity + | simpl in |- *; apply Rle_trans with (pos_Rl r0 0); + [ apply (H0 0%nat); simpl in |- *; simpl in H2; apply neq_O_lt; + red in |- *; intro; rewrite <- H3 in H2; + assert (H4 := lt_S_n _ _ H2); elim (lt_n_O _ H4) + | elim H; intros; apply H3; + [ apply RList_P4 with r; assumption + | apply le_O_n + | simpl in H2; apply lt_S_n; assumption ] ] ] + | induction j as [| j Hrecj]; + [ elim (le_Sn_O _ H1) + | simpl in |- *; elim H; intros; apply H3; + [ apply RList_P4 with r; assumption + | apply le_S_n; assumption + | simpl in H2; apply lt_S_n; assumption ] ] ]. +unfold ordered_Rlist in |- *; intros; apply H0; + [ apply le_n_Sn | simpl in |- *; simpl in H1; apply lt_n_S; assumption ]. +Qed. + +Lemma RList_P7 : + forall (l:Rlist) (x:R), + ordered_Rlist l -> In x l -> x <= pos_Rl l (pred (Rlength l)). +intros; assert (H1 := RList_P6 l); elim H1; intros H2 _; assert (H3 := H2 H); + clear H1 H2; assert (H1 := RList_P3 l x); elim H1; + clear H1; intros; assert (H4 := H1 H0); elim H4; clear H4; + intros; elim H4; clear H4; intros; rewrite H4; + assert (H6 : Rlength l = S (pred (Rlength l))). +apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro; + rewrite <- H6 in H5; elim (lt_n_O _ H5). +apply H3; + [ rewrite H6 in H5; apply lt_n_Sm_le; assumption + | apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H7 in H5; + elim (lt_n_O _ H5) ]. +Qed. + +Lemma RList_P8 : + forall (l:Rlist) (a x:R), In x (insert l a) <-> x = a \/ In x l. +simple induction l. +intros; split; intro; simpl in H; apply H. +intros; split; intro; + [ simpl in H0; generalize H0; case (Rle_dec r a); intros; + [ simpl in H1; elim H1; intro; + [ right; left; assumption + | elim (H a x); intros; elim (H3 H2); intro; + [ left; assumption | right; right; assumption ] ] + | simpl in H1; decompose [or] H1; + [ left; assumption + | right; left; assumption + | right; right; assumption ] ] + | simpl in |- *; case (Rle_dec r a); intro; + [ simpl in H0; decompose [or] H0; + [ right; elim (H a x); intros; apply H3; left + | left + | right; elim (H a x); intros; apply H3; right ] + | simpl in H0; decompose [or] H0; [ left | right; left | right; right ] ]; + assumption ]. +Qed. + +Lemma RList_P9 : + forall (l1 l2:Rlist) (x:R), In x (cons_ORlist l1 l2) <-> In x l1 \/ In x l2. +simple induction l1. +intros; split; intro; + [ simpl in H; right; assumption + | simpl in |- *; elim H; intro; [ elim H0 | assumption ] ]. +intros; split. +simpl in |- *; intros; elim (H (insert l2 r) x); intros; assert (H3 := H1 H0); + elim H3; intro; + [ left; right; assumption + | elim (RList_P8 l2 r x); intros H5 _; assert (H6 := H5 H4); elim H6; intro; + [ left; left; assumption | right; assumption ] ]. +intro; simpl in |- *; elim (H (insert l2 r) x); intros _ H1; apply H1; + elim H0; intro; + [ elim H2; intro; + [ right; elim (RList_P8 l2 r x); intros _ H4; apply H4; left; assumption + | left; assumption ] + | right; elim (RList_P8 l2 r x); intros _ H3; apply H3; right; assumption ]. +Qed. + +Lemma RList_P10 : + forall (l:Rlist) (a:R), Rlength (insert l a) = S (Rlength l). +intros; induction l as [| r l Hrecl]; + [ reflexivity + | simpl in |- *; case (Rle_dec r a); intro; + [ simpl in |- *; rewrite Hrecl; reflexivity | reflexivity ] ]. +Qed. + +Lemma RList_P11 : + forall l1 l2:Rlist, + Rlength (cons_ORlist l1 l2) = (Rlength l1 + Rlength l2)%nat. +simple induction l1; + [ intro; reflexivity + | intros; simpl in |- *; rewrite (H (insert l2 r)); rewrite RList_P10; + apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; + rewrite S_INR; ring ]. +Qed. + +Lemma RList_P12 : + forall (l:Rlist) (i:nat) (f:R -> R), + (i < Rlength l)%nat -> pos_Rl (app_Rlist l f) i = f (pos_Rl l i). +simple induction l; + [ intros; elim (lt_n_O _ H) + | intros; induction i as [| i Hreci]; + [ reflexivity | simpl in |- *; apply H; apply lt_S_n; apply H0 ] ]. +Qed. + +Lemma RList_P13 : + forall (l:Rlist) (i:nat) (a:R), + (i < pred (Rlength l))%nat -> + pos_Rl (mid_Rlist l a) (S i) = (pos_Rl l i + pos_Rl l (S i)) / 2. +simple induction l. +intros; simpl in H; elim (lt_n_O _ H). +simple induction r0. +intros; simpl in H0; elim (lt_n_O _ H0). +intros; simpl in H1; induction i as [| i Hreci]. +reflexivity. +change + (pos_Rl (mid_Rlist (cons r1 r2) r) (S i) = + (pos_Rl (cons r1 r2) i + pos_Rl (cons r1 r2) (S i)) / 2) + in |- *; apply H0; simpl in |- *; apply lt_S_n; assumption. +Qed. + +Lemma RList_P14 : forall (l:Rlist) (a:R), Rlength (mid_Rlist l a) = Rlength l. +simple induction l; intros; + [ reflexivity | simpl in |- *; rewrite (H r); reflexivity ]. +Qed. + +Lemma RList_P15 : + forall l1 l2:Rlist, + ordered_Rlist l1 -> + ordered_Rlist l2 -> + pos_Rl l1 0 = pos_Rl l2 0 -> pos_Rl (cons_ORlist l1 l2) 0 = pos_Rl l1 0. +intros; apply Rle_antisym. +induction l1 as [| r l1 Hrecl1]; + [ simpl in |- *; simpl in H1; right; symmetry in |- *; assumption + | elim (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) 0)); intros; + assert + (H4 : + In (pos_Rl (cons r l1) 0) (cons r l1) \/ In (pos_Rl (cons r l1) 0) l2); + [ left; left; reflexivity + | assert (H5 := H3 H4); apply RList_P5; + [ apply RList_P2; assumption | assumption ] ] ]. +induction l1 as [| r l1 Hrecl1]; + [ simpl in |- *; simpl in H1; right; assumption + | assert + (H2 : + In (pos_Rl (cons_ORlist (cons r l1) l2) 0) (cons_ORlist (cons r l1) l2)); + [ elim + (RList_P3 (cons_ORlist (cons r l1) l2) + (pos_Rl (cons_ORlist (cons r l1) l2) 0)); + intros; apply H3; exists 0%nat; split; + [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_O_Sn ] + | elim (RList_P9 (cons r l1) l2 (pos_Rl (cons_ORlist (cons r l1) l2) 0)); + intros; assert (H5 := H3 H2); elim H5; intro; + [ apply RList_P5; assumption + | rewrite H1; apply RList_P5; assumption ] ] ]. +Qed. + +Lemma RList_P16 : + forall l1 l2:Rlist, + ordered_Rlist l1 -> + ordered_Rlist l2 -> + pos_Rl l1 (pred (Rlength l1)) = pos_Rl l2 (pred (Rlength l2)) -> + pos_Rl (cons_ORlist l1 l2) (pred (Rlength (cons_ORlist l1 l2))) = + pos_Rl l1 (pred (Rlength l1)). +intros; apply Rle_antisym. +induction l1 as [| r l1 Hrecl1]. +simpl in |- *; simpl in H1; right; symmetry in |- *; assumption. +assert + (H2 : + In + (pos_Rl (cons_ORlist (cons r l1) l2) + (pred (Rlength (cons_ORlist (cons r l1) l2)))) + (cons_ORlist (cons r l1) l2)); + [ elim + (RList_P3 (cons_ORlist (cons r l1) l2) + (pos_Rl (cons_ORlist (cons r l1) l2) + (pred (Rlength (cons_ORlist (cons r l1) l2))))); + intros; apply H3; exists (pred (Rlength (cons_ORlist (cons r l1) l2))); + split; [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_n_Sn ] + | elim + (RList_P9 (cons r l1) l2 + (pos_Rl (cons_ORlist (cons r l1) l2) + (pred (Rlength (cons_ORlist (cons r l1) l2))))); + intros; assert (H5 := H3 H2); elim H5; intro; + [ apply RList_P7; assumption | rewrite H1; apply RList_P7; assumption ] ]. +induction l1 as [| r l1 Hrecl1]. +simpl in |- *; simpl in H1; right; assumption. +elim + (RList_P9 (cons r l1) l2 (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); + intros; + assert + (H4 : + In (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))) (cons r l1) \/ + In (pos_Rl (cons r l1) (pred (Rlength (cons r l1)))) l2); + [ left; change (In (pos_Rl (cons r l1) (Rlength l1)) (cons r l1)) in |- *; + elim (RList_P3 (cons r l1) (pos_Rl (cons r l1) (Rlength l1))); + intros; apply H5; exists (Rlength l1); split; + [ reflexivity | simpl in |- *; apply lt_n_Sn ] + | assert (H5 := H3 H4); apply RList_P7; + [ apply RList_P2; assumption + | elim + (RList_P9 (cons r l1) l2 + (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); + intros; apply H7; left; + elim + (RList_P3 (cons r l1) + (pos_Rl (cons r l1) (pred (Rlength (cons r l1))))); + intros; apply H9; exists (pred (Rlength (cons r l1))); + split; [ reflexivity | simpl in |- *; apply lt_n_Sn ] ] ]. +Qed. + +Lemma RList_P17 : + forall (l1:Rlist) (x:R) (i:nat), + ordered_Rlist l1 -> + In x l1 -> + pos_Rl l1 i < x -> (i < pred (Rlength l1))%nat -> pos_Rl l1 (S i) <= x. +simple induction l1. +intros; elim H0. +intros; induction i as [| i Hreci]. +simpl in |- *; elim H1; intro; + [ simpl in H2; rewrite H4 in H2; elim (Rlt_irrefl _ H2) + | apply RList_P5; [ apply RList_P4 with r; assumption | assumption ] ]. +simpl in |- *; simpl in H2; elim H1; intro. +rewrite H4 in H2; assert (H5 : r <= pos_Rl r0 i); + [ apply Rle_trans with (pos_Rl r0 0); + [ apply (H0 0%nat); simpl in |- *; simpl in H3; apply neq_O_lt; + red in |- *; intro; rewrite <- H5 in H3; elim (lt_n_O _ H3) + | elim (RList_P6 r0); intros; apply H5; + [ apply RList_P4 with r; assumption + | apply le_O_n + | simpl in H3; apply lt_S_n; apply lt_trans with (Rlength r0); + [ apply H3 | apply lt_n_Sn ] ] ] + | elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H5 H2)) ]. +apply H; try assumption; + [ apply RList_P4 with r; assumption + | simpl in H3; apply lt_S_n; + replace (S (pred (Rlength r0))) with (Rlength r0); + [ apply H3 + | apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro; + rewrite <- H5 in H3; elim (lt_n_O _ H3) ] ]. +Qed. + +Lemma RList_P18 : + forall (l:Rlist) (f:R -> R), Rlength (app_Rlist l f) = Rlength l. +simple induction l; intros; + [ reflexivity | simpl in |- *; rewrite H; reflexivity ]. +Qed. + +Lemma RList_P19 : + forall l:Rlist, + l <> nil -> exists r : R | ( exists r0 : Rlist | l = cons r r0). +intros; induction l as [| r l Hrecl]; + [ elim H; reflexivity | exists r; exists l; reflexivity ]. +Qed. + +Lemma RList_P20 : + forall l:Rlist, + (2 <= Rlength l)%nat -> + exists r : R + | ( exists r1 : R | ( exists l' : Rlist | l = cons r (cons r1 l'))). +intros; induction l as [| r l Hrecl]; + [ simpl in H; elim (le_Sn_O _ H) + | induction l as [| r0 l Hrecl0]; + [ simpl in H; elim (le_Sn_O _ (le_S_n _ _ H)) + | exists r; exists r0; exists l; reflexivity ] ]. +Qed. + +Lemma RList_P21 : forall l l':Rlist, l = l' -> Rtail l = Rtail l'. +intros; rewrite H; reflexivity. +Qed. + +Lemma RList_P22 : + forall l1 l2:Rlist, l1 <> nil -> pos_Rl (cons_Rlist l1 l2) 0 = pos_Rl l1 0. +simple induction l1; [ intros; elim H; reflexivity | intros; reflexivity ]. +Qed. + +Lemma RList_P23 : + forall l1 l2:Rlist, + Rlength (cons_Rlist l1 l2) = (Rlength l1 + Rlength l2)%nat. +simple induction l1; + [ intro; reflexivity | intros; simpl in |- *; rewrite H; reflexivity ]. +Qed. + +Lemma RList_P24 : + forall l1 l2:Rlist, + l2 <> nil -> + pos_Rl (cons_Rlist l1 l2) (pred (Rlength (cons_Rlist l1 l2))) = + pos_Rl l2 (pred (Rlength l2)). +simple induction l1. +intros; reflexivity. +intros; rewrite <- (H l2 H0); induction l2 as [| r1 l2 Hrecl2]. +elim H0; reflexivity. +do 2 rewrite RList_P23; + replace (Rlength (cons r r0) + Rlength (cons r1 l2))%nat with + (S (S (Rlength r0 + Rlength l2))); + [ replace (Rlength r0 + Rlength (cons r1 l2))%nat with + (S (Rlength r0 + Rlength l2)); + [ reflexivity + | simpl in |- *; apply INR_eq; rewrite S_INR; do 2 rewrite plus_INR; + rewrite S_INR; ring ] + | simpl in |- *; apply INR_eq; do 3 rewrite S_INR; do 2 rewrite plus_INR; + rewrite S_INR; ring ]. +Qed. + +Lemma RList_P25 : + forall l1 l2:Rlist, + ordered_Rlist l1 -> + ordered_Rlist l2 -> + pos_Rl l1 (pred (Rlength l1)) <= pos_Rl l2 0 -> + ordered_Rlist (cons_Rlist l1 l2). +simple induction l1. +intros; simpl in |- *; assumption. +simple induction r0. +intros; simpl in |- *; simpl in H2; unfold ordered_Rlist in |- *; intros; + simpl in H3. +induction i as [| i Hreci]. +simpl in |- *; assumption. +change (pos_Rl l2 i <= pos_Rl l2 (S i)) in |- *; apply (H1 i); apply lt_S_n; + replace (S (pred (Rlength l2))) with (Rlength l2); + [ assumption + | apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro; + rewrite <- H4 in H3; elim (lt_n_O _ H3) ]. +intros; clear H; assert (H : ordered_Rlist (cons_Rlist (cons r1 r2) l2)). +apply H0; try assumption. +apply RList_P4 with r; assumption. +unfold ordered_Rlist in |- *; intros; simpl in H4; + induction i as [| i Hreci]. +simpl in |- *; apply (H1 0%nat); simpl in |- *; apply lt_O_Sn. +change + (pos_Rl (cons_Rlist (cons r1 r2) l2) i <= + pos_Rl (cons_Rlist (cons r1 r2) l2) (S i)) in |- *; + apply (H i); simpl in |- *; apply lt_S_n; assumption. +Qed. + +Lemma RList_P26 : + forall (l1 l2:Rlist) (i:nat), + (i < Rlength l1)%nat -> pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i. +simple induction l1. +intros; elim (lt_n_O _ H). +intros; induction i as [| i Hreci]. +apply RList_P22; discriminate. +apply (H l2 i); simpl in H0; apply lt_S_n; assumption. +Qed. + +Lemma RList_P27 : + forall l1 l2 l3:Rlist, + cons_Rlist l1 (cons_Rlist l2 l3) = cons_Rlist (cons_Rlist l1 l2) l3. +simple induction l1; intros; + [ reflexivity | simpl in |- *; rewrite (H l2 l3); reflexivity ]. +Qed. + +Lemma RList_P28 : forall l:Rlist, cons_Rlist l nil = l. +simple induction l; + [ reflexivity | intros; simpl in |- *; rewrite H; reflexivity ]. +Qed. + +Lemma RList_P29 : + forall (l2 l1:Rlist) (i:nat), + (Rlength l1 <= i)%nat -> + (i < Rlength (cons_Rlist l1 l2))%nat -> + pos_Rl (cons_Rlist l1 l2) i = pos_Rl l2 (i - Rlength l1). +simple induction l2. +intros; rewrite RList_P28 in H0; elim (lt_irrefl _ (le_lt_trans _ _ _ H H0)). +intros; + replace (cons_Rlist l1 (cons r r0)) with + (cons_Rlist (cons_Rlist l1 (cons r nil)) r0). +inversion H0. +rewrite <- minus_n_n; simpl in |- *; rewrite RList_P26. +clear l2 r0 H i H0 H1 H2; induction l1 as [| r0 l1 Hrecl1]. +reflexivity. +simpl in |- *; assumption. +rewrite RList_P23; rewrite plus_comm; simpl in |- *; apply lt_n_Sn. +replace (S m - Rlength l1)%nat with (S (S m - S (Rlength l1))). +rewrite H3; simpl in |- *; + replace (S (Rlength l1)) with (Rlength (cons_Rlist l1 (cons r nil))). +apply (H (cons_Rlist l1 (cons r nil)) i). +rewrite RList_P23; rewrite plus_comm; simpl in |- *; rewrite <- H3; + apply le_n_S; assumption. +repeat rewrite RList_P23; simpl in |- *; rewrite RList_P23 in H1; + rewrite plus_comm in H1; simpl in H1; rewrite (plus_comm (Rlength l1)); + simpl in |- *; rewrite plus_comm; apply H1. +rewrite RList_P23; rewrite plus_comm; reflexivity. +change (S (m - Rlength l1) = (S m - Rlength l1)%nat) in |- *; + apply minus_Sn_m; assumption. +replace (cons r r0) with (cons_Rlist (cons r nil) r0); + [ symmetry in |- *; apply RList_P27 | reflexivity ]. +Qed.
\ No newline at end of file diff --git a/theories/Reals/R_Ifp.v b/theories/Reals/R_Ifp.v index b167b6ef9..37d987855 100644 --- a/theories/Reals/R_Ifp.v +++ b/theories/Reals/R_Ifp.v @@ -13,9 +13,8 @@ (* *) (**********************************************************) -Require Rbase. -Require Omega. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Omega. Open Local Scope R_scope. (*********************************************************) @@ -23,83 +22,81 @@ Open Local Scope R_scope. (*********************************************************) (**********) -Definition Int_part:R->Z:=[r:R](`(up r)-1`). +Definition Int_part (r:R) : Z := (up r - 1)%Z. (**********) -Definition frac_part:R->R:=[r:R](Rminus r (IZR (Int_part r))). +Definition frac_part (r:R) : R := r - IZR (Int_part r). (**********) -Lemma tech_up:(r:R)(z:Z)(Rlt r (IZR z))->(Rle (IZR z) (Rplus r R1))-> - z=(up r). -Intros;Generalize (archimed r);Intro;Elim H1;Intros;Clear H1; - Unfold Rgt in H2;Unfold Rminus in H3; -Generalize (Rle_compatibility r (Rplus (IZR (up r)) - (Ropp r)) R1 H3);Intro;Clear H3; - Rewrite (Rplus_sym (IZR (up r)) (Ropp r)) in H1; - Rewrite <-(Rplus_assoc r (Ropp r) (IZR (up r))) in H1; - Rewrite (Rplus_Ropp_r r) in H1;Elim (Rplus_ne (IZR (up r)));Intros a b; - Rewrite b in H1;Clear a b;Apply (single_z_r_R1 r z (up r));Auto with zarith real. +Lemma tech_up : forall (r:R) (z:Z), r < IZR z -> IZR z <= r + 1 -> z = up r. +intros; generalize (archimed r); intro; elim H1; intros; clear H1; + unfold Rgt in H2; unfold Rminus in H3; + generalize (Rplus_le_compat_l r (IZR (up r) + - r) 1 H3); + intro; clear H3; rewrite (Rplus_comm (IZR (up r)) (- r)) in H1; + rewrite <- (Rplus_assoc r (- r) (IZR (up r))) in H1; + rewrite (Rplus_opp_r r) in H1; elim (Rplus_ne (IZR (up r))); + intros a b; rewrite b in H1; clear a b; apply (single_z_r_R1 r z (up r)); + auto with zarith real. Qed. (**********) -Lemma up_tech:(r:R)(z:Z)(Rle (IZR z) r)->(Rlt r (IZR `z+1`))-> - `z+1`=(up r). -Intros;Generalize (Rle_compatibility R1 (IZR z) r H);Intro;Clear H; - Rewrite (Rplus_sym R1 (IZR z)) in H1;Rewrite (Rplus_sym R1 r) in H1; - Cut (R1==(IZR `1`));Auto with zarith real. -Intro;Generalize H1;Pattern 1 R1;Rewrite H;Intro;Clear H H1; - Rewrite <-(plus_IZR z `1`) in H2;Apply (tech_up r `z+1`);Auto with zarith real. +Lemma up_tech : + forall (r:R) (z:Z), IZR z <= r -> r < IZR (z + 1) -> (z + 1)%Z = up r. +intros; generalize (Rplus_le_compat_l 1 (IZR z) r H); intro; clear H; + rewrite (Rplus_comm 1 (IZR z)) in H1; rewrite (Rplus_comm 1 r) in H1; + cut (1 = IZR 1); auto with zarith real. +intro; generalize H1; pattern 1 at 1 in |- *; rewrite H; intro; clear H H1; + rewrite <- (plus_IZR z 1) in H2; apply (tech_up r (z + 1)); + auto with zarith real. Qed. (**********) -Lemma fp_R0:(frac_part R0)==R0. -Unfold frac_part; Unfold Int_part; Elim (archimed R0); - Intros; Unfold Rminus; - Elim (Rplus_ne (Ropp (IZR `(up R0)-1`))); Intros a b; - Rewrite b;Clear a b;Rewrite <- Z_R_minus;Cut (up R0)=`1`. -Intro;Rewrite H1; - Rewrite (eq_Rminus (IZR `1`) (IZR `1`) (refl_eqT R (IZR `1`))); - Apply Ropp_O. -Elim (archimed R0);Intros;Clear H2;Unfold Rgt in H1; - Rewrite (minus_R0 (IZR (up R0))) in H0; - Generalize (lt_O_IZR (up R0) H1);Intro;Clear H1; - Generalize (le_IZR_R1 (up R0) H0);Intro;Clear H H0;Omega. +Lemma fp_R0 : frac_part 0 = 0. +unfold frac_part in |- *; unfold Int_part in |- *; elim (archimed 0); intros; + unfold Rminus in |- *; elim (Rplus_ne (- IZR (up 0 - 1))); + intros a b; rewrite b; clear a b; rewrite <- Z_R_minus; + cut (up 0 = 1%Z). +intro; rewrite H1; + rewrite (Rminus_diag_eq (IZR 1) (IZR 1) (refl_equal (IZR 1))); + apply Ropp_0. +elim (archimed 0); intros; clear H2; unfold Rgt in H1; + rewrite (Rminus_0_r (IZR (up 0))) in H0; generalize (lt_O_IZR (up 0) H1); + intro; clear H1; generalize (le_IZR_R1 (up 0) H0); + intro; clear H H0; omega. Qed. (**********) -Lemma for_base_fp:(r:R)(Rgt (Rminus (IZR (up r)) r) R0)/\ - (Rle (Rminus (IZR (up r)) r) R1). -Intro; Split; - Cut (Rgt (IZR (up r)) r)/\(Rle (Rminus (IZR (up r)) r) R1). -Intro; Elim H; Intros. -Apply (Rgt_minus (IZR (up r)) r H0). -Apply archimed. -Intro; Elim H; Intros. -Exact H1. -Apply archimed. +Lemma for_base_fp : forall r:R, IZR (up r) - r > 0 /\ IZR (up r) - r <= 1. +intro; split; cut (IZR (up r) > r /\ IZR (up r) - r <= 1). +intro; elim H; intros. +apply (Rgt_minus (IZR (up r)) r H0). +apply archimed. +intro; elim H; intros. +exact H1. +apply archimed. Qed. (**********) -Lemma base_fp:(r:R)(Rge (frac_part r) R0)/\(Rlt (frac_part r) R1). -Intro; Unfold frac_part; Unfold Int_part; Split. +Lemma base_fp : forall r:R, frac_part r >= 0 /\ frac_part r < 1. +intro; unfold frac_part in |- *; unfold Int_part in |- *; split. (*sup a O*) -Cut (Rge (Rminus r (IZR (up r))) (Ropp R1)). -Rewrite <- Z_R_minus;Simpl;Intro; Unfold Rminus; - Rewrite Ropp_distr1;Rewrite <-Rplus_assoc; - Fold (Rminus r (IZR (up r))); - Fold (Rminus (Rminus r (IZR (up r))) (Ropp R1)); - Apply Rge_minus;Auto with zarith real. -Rewrite <- Ropp_distr2;Apply Rle_Ropp;Elim (for_base_fp r); Auto with zarith real. +cut (r - IZR (up r) >= -1). +rewrite <- Z_R_minus; simpl in |- *; intro; unfold Rminus in |- *; + rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; + fold (r - IZR (up r)) in |- *; fold (r - IZR (up r) - -1) in |- *; + apply Rge_minus; auto with zarith real. +rewrite <- Ropp_minus_distr; apply Ropp_le_ge_contravar; elim (for_base_fp r); + auto with zarith real. (*inf a 1*) -Cut (Rlt (Rminus r (IZR (up r))) R0). -Rewrite <- Z_R_minus; Simpl;Intro; Unfold Rminus; - Rewrite Ropp_distr1;Rewrite <-Rplus_assoc; - Fold (Rminus r (IZR (up r)));Rewrite Ropp_Ropp; - Elim (Rplus_ne R1);Intros a b;Pattern 2 R1;Rewrite <-a;Clear a b; - Rewrite (Rplus_sym (Rminus r (IZR (up r))) R1); - Apply Rlt_compatibility;Auto with zarith real. -Elim (for_base_fp r);Intros;Rewrite <-Ropp_O; - Rewrite<-Ropp_distr2;Apply Rgt_Ropp;Auto with zarith real. +cut (r - IZR (up r) < 0). +rewrite <- Z_R_minus; simpl in |- *; intro; unfold Rminus in |- *; + rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; + fold (r - IZR (up r)) in |- *; rewrite Ropp_involutive; + elim (Rplus_ne 1); intros a b; pattern 1 at 2 in |- *; + rewrite <- a; clear a b; rewrite (Rplus_comm (r - IZR (up r)) 1); + apply Rplus_lt_compat_l; auto with zarith real. +elim (for_base_fp r); intros; rewrite <- Ropp_0; rewrite <- Ropp_minus_distr; + apply Ropp_gt_lt_contravar; auto with zarith real. Qed. (*********************************************************) @@ -107,446 +104,442 @@ Qed. (*********************************************************) (**********) -Lemma base_Int_part:(r:R)(Rle (IZR (Int_part r)) r)/\ - (Rgt (Rminus (IZR (Int_part r)) r) (Ropp R1)). -Intro;Unfold Int_part;Elim (archimed r);Intros. -Split;Rewrite <- (Z_R_minus (up r) `1`);Simpl. -Generalize (Rle_minus (Rminus (IZR (up r)) r) R1 H0);Intro; - Unfold Rminus in H1; - Rewrite (Rplus_assoc (IZR (up r)) (Ropp r) (Ropp R1)) in - H1;Rewrite (Rplus_sym (Ropp r) (Ropp R1)) in H1; - Rewrite <-(Rplus_assoc (IZR (up r)) (Ropp R1) (Ropp r)) in - H1;Fold (Rminus (IZR (up r)) R1) in H1; - Fold (Rminus (Rminus (IZR (up r)) R1) r) in H1; - Apply Rminus_le;Auto with zarith real. -Generalize (Rgt_plus_plus_r (Ropp R1) (IZR (up r)) r H);Intro; - Rewrite (Rplus_sym (Ropp R1) (IZR (up r))) in H1; - Generalize (Rgt_plus_plus_r (Ropp r) - (Rplus (IZR (up r)) (Ropp R1)) (Rplus (Ropp R1) r) H1); - Intro;Clear H H0 H1; - Rewrite (Rplus_sym (Ropp r) (Rplus (IZR (up r)) (Ropp R1))) - in H2;Fold (Rminus (IZR (up r)) R1) in H2; - Fold (Rminus (Rminus (IZR (up r)) R1) r) in H2; - Rewrite (Rplus_sym (Ropp r) (Rplus (Ropp R1) r)) in H2; - Rewrite (Rplus_assoc (Ropp R1) r (Ropp r)) in H2; - Rewrite (Rplus_Ropp_r r) in H2;Elim (Rplus_ne (Ropp R1));Intros a b; - Rewrite a in H2;Clear a b;Auto with zarith real. +Lemma base_Int_part : + forall r:R, IZR (Int_part r) <= r /\ IZR (Int_part r) - r > -1. +intro; unfold Int_part in |- *; elim (archimed r); intros. +split; rewrite <- (Z_R_minus (up r) 1); simpl in |- *. +generalize (Rle_minus (IZR (up r) - r) 1 H0); intro; unfold Rminus in H1; + rewrite (Rplus_assoc (IZR (up r)) (- r) (-1)) in H1; + rewrite (Rplus_comm (- r) (-1)) in H1; + rewrite <- (Rplus_assoc (IZR (up r)) (-1) (- r)) in H1; + fold (IZR (up r) - 1) in H1; fold (IZR (up r) - 1 - r) in H1; + apply Rminus_le; auto with zarith real. +generalize (Rplus_gt_compat_l (-1) (IZR (up r)) r H); intro; + rewrite (Rplus_comm (-1) (IZR (up r))) in H1; + generalize (Rplus_gt_compat_l (- r) (IZR (up r) + -1) (-1 + r) H1); + intro; clear H H0 H1; rewrite (Rplus_comm (- r) (IZR (up r) + -1)) in H2; + fold (IZR (up r) - 1) in H2; fold (IZR (up r) - 1 - r) in H2; + rewrite (Rplus_comm (- r) (-1 + r)) in H2; + rewrite (Rplus_assoc (-1) r (- r)) in H2; rewrite (Rplus_opp_r r) in H2; + elim (Rplus_ne (-1)); intros a b; rewrite a in H2; + clear a b; auto with zarith real. Qed. (**********) -Lemma Int_part_INR:(n : nat) (Int_part (INR n)) = (inject_nat n). -Intros n; Unfold Int_part. -Cut (up (INR n)) = (Zplus (inject_nat n) (inject_nat (1))). -Intros H'; Rewrite H'; Simpl; Ring. -Apply sym_equal; Apply tech_up; Auto. -Replace (Zplus (inject_nat n) (inject_nat (1))) with (INZ (S n)). -Repeat Rewrite <- INR_IZR_INZ. -Apply lt_INR; Auto. -Rewrite Zplus_sym; Rewrite <- inj_plus; Simpl; Auto. -Rewrite plus_IZR; Simpl; Auto with real. -Repeat Rewrite <- INR_IZR_INZ; Auto with real. +Lemma Int_part_INR : forall n:nat, Int_part (INR n) = Z_of_nat n. +intros n; unfold Int_part in |- *. +cut (up (INR n) = (Z_of_nat n + Z_of_nat 1)%Z). +intros H'; rewrite H'; simpl in |- *; ring. +apply sym_equal; apply tech_up; auto. +replace (Z_of_nat n + Z_of_nat 1)%Z with (Z_of_nat (S n)). +repeat rewrite <- INR_IZR_INZ. +apply lt_INR; auto. +rewrite Zplus_comm; rewrite <- Znat.inj_plus; simpl in |- *; auto. +rewrite plus_IZR; simpl in |- *; auto with real. +repeat rewrite <- INR_IZR_INZ; auto with real. Qed. (**********) -Lemma fp_nat:(r:R)(frac_part r)==R0->(Ex [c:Z](r==(IZR c))). -Unfold frac_part;Intros;Split with (Int_part r);Apply Rminus_eq; Auto with zarith real. +Lemma fp_nat : forall r:R, frac_part r = 0 -> exists c : Z | r = IZR c. +unfold frac_part in |- *; intros; split with (Int_part r); + apply Rminus_diag_uniq; auto with zarith real. Qed. (**********) -Lemma R0_fp_O:(r:R)~R0==(frac_part r)->~R0==r. -Red;Intros;Rewrite <- H0 in H;Generalize fp_R0;Intro;Auto with zarith real. +Lemma R0_fp_O : forall r:R, 0 <> frac_part r -> 0 <> r. +red in |- *; intros; rewrite <- H0 in H; generalize fp_R0; intro; + auto with zarith real. Qed. (**********) -Lemma Rminus_Int_part1:(r1,r2:R)(Rge (frac_part r1) (frac_part r2))-> - (Int_part (Rminus r1 r2))=(Zminus (Int_part r1) (Int_part r2)). -Intros;Elim (base_fp r1);Elim (base_fp r2);Intros; - Generalize (Rle_sym2 R0 (frac_part r2) H0);Intro;Clear H0; - Generalize (Rle_Ropp R0 (frac_part r2) H4);Intro;Clear H4; - Rewrite (Ropp_O) in H0; - Generalize (Rle_sym2 (Ropp (frac_part r2)) R0 H0);Intro;Clear H0; - Generalize (Rle_sym2 R0 (frac_part r1) H2);Intro;Clear H2; - Generalize (Rlt_Ropp (frac_part r2) R1 H1);Intro;Clear H1; - Unfold Rgt in H2; - Generalize (sum_inequa_Rle_lt R0 (frac_part r1) R1 (Ropp R1) - (Ropp (frac_part r2)) R0 H0 H3 H2 H4);Intro;Elim H1;Intros; - Clear H1;Elim (Rplus_ne R1);Intros a b;Rewrite a in H6;Clear a b H5; - Generalize (Rge_minus (frac_part r1) (frac_part r2) H);Intro;Clear H; - Fold (Rminus (frac_part r1) (frac_part r2)) in H6; - Generalize (Rle_sym2 R0 (Rminus (frac_part r1) (frac_part r2)) H1); - Intro;Clear H1 H3 H4 H0 H2;Unfold frac_part in H6 H; - Unfold Rminus in H6 H; - Rewrite (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))) in H; - Rewrite (Ropp_Ropp (IZR (Int_part r2))) in H; - Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) - (Rplus (Ropp r2) (IZR (Int_part r2)))) in H; - Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2) - (IZR (Int_part r2))) in H; - Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (Ropp r2)) in H; - Rewrite (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1))) - (IZR (Int_part r2))) in H; - Rewrite <-(Rplus_assoc r1 (Ropp r2) - (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))) in H; - Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) in H; - Fold (Rminus r1 r2) in H;Fold (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) - in H;Generalize (Rle_compatibility - (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) R0 - (Rplus (Rminus r1 r2) (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) H);Intro; - Clear H;Rewrite (Rplus_sym (Rminus r1 r2) - (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) in H0; - Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) (Rminus r1 r2)) in H0; - Unfold Rminus in H0;Fold (Rminus r1 r2) in H0; - Rewrite (Rplus_assoc (IZR (Int_part r1)) (Ropp (IZR (Int_part r2))) - (Rplus (IZR (Int_part r2)) (Ropp (IZR (Int_part r1))))) in H0; - Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r2))) (IZR (Int_part r2)) - (Ropp (IZR (Int_part r1)))) in H0;Rewrite (Rplus_Ropp_l (IZR (Int_part r2))) in - H0;Elim (Rplus_ne (Ropp (IZR (Int_part r1))));Intros a b;Rewrite b in H0; - Clear a b; - Elim (Rplus_ne (Rplus (IZR (Int_part r1)) (Ropp (IZR (Int_part r2))))); - Intros a b;Rewrite a in H0;Clear a b;Rewrite (Rplus_Ropp_r (IZR (Int_part r1))) - in H0;Elim (Rplus_ne (Rminus r1 r2));Intros a b;Rewrite b in H0; - Clear a b;Fold (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) in H0; - Rewrite (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))) in H6; - Rewrite (Ropp_Ropp (IZR (Int_part r2))) in H6; - Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) - (Rplus (Ropp r2) (IZR (Int_part r2)))) in H6; - Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2) - (IZR (Int_part r2))) in H6; - Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (Ropp r2)) in H6; - Rewrite (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1))) - (IZR (Int_part r2))) in H6; - Rewrite <-(Rplus_assoc r1 (Ropp r2) - (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))) in H6; - Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) in H6; - Fold (Rminus r1 r2) in H6;Fold (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) - in H6;Generalize (Rlt_compatibility - (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Rplus (Rminus r1 r2) (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) R1 H6); - Intro;Clear H6; - Rewrite (Rplus_sym (Rminus r1 r2) - (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) in H; - Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) (Rminus r1 r2)) in H; - Rewrite <-(Ropp_distr2 (IZR (Int_part r1)) (IZR (Int_part r2))) in H; - Rewrite (Rplus_Ropp_r (Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H; - Elim (Rplus_ne (Rminus r1 r2));Intros a b;Rewrite b in H;Clear a b; - Rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0; - Rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H; - Cut R1==(IZR `1`);Auto with zarith real. -Intro;Rewrite H1 in H;Clear H1; - Rewrite <-(plus_IZR `(Int_part r1)-(Int_part r2)` `1`) in H; - Generalize (up_tech (Rminus r1 r2) `(Int_part r1)-(Int_part r2)` - H0 H);Intros;Clear H H0;Unfold 1 Int_part;Omega. +Lemma Rminus_Int_part1 : + forall r1 r2:R, + frac_part r1 >= frac_part r2 -> + Int_part (r1 - r2) = (Int_part r1 - Int_part r2)%Z. +intros; elim (base_fp r1); elim (base_fp r2); intros; + generalize (Rge_le (frac_part r2) 0 H0); intro; clear H0; + generalize (Ropp_le_ge_contravar 0 (frac_part r2) H4); + intro; clear H4; rewrite Ropp_0 in H0; + generalize (Rge_le 0 (- frac_part r2) H0); intro; + clear H0; generalize (Rge_le (frac_part r1) 0 H2); + intro; clear H2; generalize (Ropp_lt_gt_contravar (frac_part r2) 1 H1); + intro; clear H1; unfold Rgt in H2; + generalize + (sum_inequa_Rle_lt 0 (frac_part r1) 1 (-1) (- frac_part r2) 0 H0 H3 H2 H4); + intro; elim H1; intros; clear H1; elim (Rplus_ne 1); + intros a b; rewrite a in H6; clear a b H5; + generalize (Rge_minus (frac_part r1) (frac_part r2) H); + intro; clear H; fold (frac_part r1 - frac_part r2) in H6; + generalize (Rge_le (frac_part r1 - frac_part r2) 0 H1); + intro; clear H1 H3 H4 H0 H2; unfold frac_part in H6, H; + unfold Rminus in H6, H; + rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))) in H; + rewrite (Ropp_involutive (IZR (Int_part r2))) in H; + rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2))) + in H; + rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2))) + in H; rewrite (Rplus_comm (- IZR (Int_part r1)) (- r2)) in H; + rewrite (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))) in H; + rewrite <- (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2))) + in H; rewrite (Rplus_comm (- IZR (Int_part r1)) (IZR (Int_part r2))) in H; + fold (r1 - r2) in H; fold (IZR (Int_part r2) - IZR (Int_part r1)) in H; + generalize + (Rplus_le_compat_l (IZR (Int_part r1) - IZR (Int_part r2)) 0 + (r1 - r2 + (IZR (Int_part r2) - IZR (Int_part r1))) H); + intro; clear H; + rewrite (Rplus_comm (r1 - r2) (IZR (Int_part r2) - IZR (Int_part r1))) in H0; + rewrite <- + (Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2)) + (IZR (Int_part r2) - IZR (Int_part r1)) (r1 - r2)) + in H0; unfold Rminus in H0; fold (r1 - r2) in H0; + rewrite + (Rplus_assoc (IZR (Int_part r1)) (- IZR (Int_part r2)) + (IZR (Int_part r2) + - IZR (Int_part r1))) in H0; + rewrite <- + (Rplus_assoc (- IZR (Int_part r2)) (IZR (Int_part r2)) + (- IZR (Int_part r1))) in H0; + rewrite (Rplus_opp_l (IZR (Int_part r2))) in H0; + elim (Rplus_ne (- IZR (Int_part r1))); intros a b; + rewrite b in H0; clear a b; + elim (Rplus_ne (IZR (Int_part r1) + - IZR (Int_part r2))); + intros a b; rewrite a in H0; clear a b; + rewrite (Rplus_opp_r (IZR (Int_part r1))) in H0; elim (Rplus_ne (r1 - r2)); + intros a b; rewrite b in H0; clear a b; + fold (IZR (Int_part r1) - IZR (Int_part r2)) in H0; + rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))) in H6; + rewrite (Ropp_involutive (IZR (Int_part r2))) in H6; + rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2))) + in H6; + rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2))) + in H6; rewrite (Rplus_comm (- IZR (Int_part r1)) (- r2)) in H6; + rewrite (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))) in H6; + rewrite <- (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2))) + in H6; + rewrite (Rplus_comm (- IZR (Int_part r1)) (IZR (Int_part r2))) in H6; + fold (r1 - r2) in H6; fold (IZR (Int_part r2) - IZR (Int_part r1)) in H6; + generalize + (Rplus_lt_compat_l (IZR (Int_part r1) - IZR (Int_part r2)) + (r1 - r2 + (IZR (Int_part r2) - IZR (Int_part r1))) 1 H6); + intro; clear H6; + rewrite (Rplus_comm (r1 - r2) (IZR (Int_part r2) - IZR (Int_part r1))) in H; + rewrite <- + (Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2)) + (IZR (Int_part r2) - IZR (Int_part r1)) (r1 - r2)) + in H; + rewrite <- (Ropp_minus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H; + rewrite (Rplus_opp_r (IZR (Int_part r1) - IZR (Int_part r2))) in H; + elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H; + clear a b; rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0; + rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H; + cut (1 = IZR 1); auto with zarith real. +intro; rewrite H1 in H; clear H1; + rewrite <- (plus_IZR (Int_part r1 - Int_part r2) 1) in H; + generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2) H0 H); + intros; clear H H0; unfold Int_part at 1 in |- *; + omega. Qed. (**********) -Lemma Rminus_Int_part2:(r1,r2:R)(Rlt (frac_part r1) (frac_part r2))-> - (Int_part (Rminus r1 r2))=(Zminus (Zminus (Int_part r1) (Int_part r2)) `1`). -Intros;Elim (base_fp r1);Elim (base_fp r2);Intros; - Generalize (Rle_sym2 R0 (frac_part r2) H0);Intro;Clear H0; - Generalize (Rle_Ropp R0 (frac_part r2) H4);Intro;Clear H4; - Rewrite (Ropp_O) in H0; - Generalize (Rle_sym2 (Ropp (frac_part r2)) R0 H0);Intro;Clear H0; - Generalize (Rle_sym2 R0 (frac_part r1) H2);Intro;Clear H2; - Generalize (Rlt_Ropp (frac_part r2) R1 H1);Intro;Clear H1; - Unfold Rgt in H2; - Generalize (sum_inequa_Rle_lt R0 (frac_part r1) R1 (Ropp R1) - (Ropp (frac_part r2)) R0 H0 H3 H2 H4);Intro;Elim H1;Intros; - Clear H1;Elim (Rplus_ne (Ropp R1));Intros a b;Rewrite b in H5; - Clear a b H6;Generalize (Rlt_minus (frac_part r1) (frac_part r2) H); - Intro;Clear H;Fold (Rminus (frac_part r1) (frac_part r2)) in H5; - Clear H3 H4 H0 H2;Unfold frac_part in H5 H1; - Unfold Rminus in H5 H1; - Rewrite (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))) in H5; - Rewrite (Ropp_Ropp (IZR (Int_part r2))) in H5; - Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) - (Rplus (Ropp r2) (IZR (Int_part r2)))) in H5; - Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2) - (IZR (Int_part r2))) in H5; - Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (Ropp r2)) in H5; - Rewrite (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1))) - (IZR (Int_part r2))) in H5; - Rewrite <-(Rplus_assoc r1 (Ropp r2) - (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))) in H5; - Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) in H5; - Fold (Rminus r1 r2) in H5;Fold (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) - in H5;Generalize (Rlt_compatibility - (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) (Ropp R1) - (Rplus (Rminus r1 r2) (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) H5); - Intro;Clear H5;Rewrite (Rplus_sym (Rminus r1 r2) - (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) in H; - Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) (Rminus r1 r2)) in H; - Unfold Rminus in H;Fold (Rminus r1 r2) in H; - Rewrite (Rplus_assoc (IZR (Int_part r1)) (Ropp (IZR (Int_part r2))) - (Rplus (IZR (Int_part r2)) (Ropp (IZR (Int_part r1))))) in H; - Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r2))) (IZR (Int_part r2)) - (Ropp (IZR (Int_part r1)))) in H;Rewrite (Rplus_Ropp_l (IZR (Int_part r2))) in - H;Elim (Rplus_ne (Ropp (IZR (Int_part r1))));Intros a b;Rewrite b in H; - Clear a b;Rewrite (Rplus_Ropp_r (IZR (Int_part r1))) in H; - Elim (Rplus_ne (Rminus r1 r2));Intros a b;Rewrite b in H; - Clear a b;Fold (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) in H; - Fold (Rminus (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) R1) in H; - Rewrite (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))) in H1; - Rewrite (Ropp_Ropp (IZR (Int_part r2))) in H1; - Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) - (Rplus (Ropp r2) (IZR (Int_part r2)))) in H1; - Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2) - (IZR (Int_part r2))) in H1; - Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (Ropp r2)) in H1; - Rewrite (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1))) - (IZR (Int_part r2))) in H1; - Rewrite <-(Rplus_assoc r1 (Ropp r2) - (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))) in H1; - Rewrite (Rplus_sym (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) in H1; - Fold (Rminus r1 r2) in H1;Fold (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) - in H1;Generalize (Rlt_compatibility - (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Rplus (Rminus r1 r2) (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) R0 H1); - Intro;Clear H1; - Rewrite (Rplus_sym (Rminus r1 r2) - (Rminus (IZR (Int_part r2)) (IZR (Int_part r1)))) in H0; - Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Rminus (IZR (Int_part r2)) (IZR (Int_part r1))) (Rminus r1 r2)) in H0; - Rewrite <-(Ropp_distr2 (IZR (Int_part r1)) (IZR (Int_part r2))) in H0; - Rewrite (Rplus_Ropp_r (Rminus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H0; - Elim (Rplus_ne (Rminus r1 r2));Intros a b;Rewrite b in H0;Clear a b; - Rewrite <-(Rplus_Ropp_l R1) in H0; - Rewrite <-(Rplus_assoc (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Ropp R1) R1) in H0; - Fold (Rminus (Rminus (IZR (Int_part r1)) (IZR (Int_part r2))) R1) in H0; - Rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0; - Rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H; - Cut R1==(IZR `1`);Auto with zarith real. -Intro;Rewrite H1 in H;Rewrite H1 in H0;Clear H1; - Rewrite (Z_R_minus `(Int_part r1)-(Int_part r2)` `1`) in H; - Rewrite (Z_R_minus `(Int_part r1)-(Int_part r2)` `1`) in H0; - Rewrite <-(plus_IZR `(Int_part r1)-(Int_part r2)-1` `1`) in H0; - Generalize (Rlt_le (IZR `(Int_part r1)-(Int_part r2)-1`) (Rminus r1 r2) H); - Intro;Clear H; - Generalize (up_tech (Rminus r1 r2) `(Int_part r1)-(Int_part r2)-1` - H1 H0);Intros;Clear H0 H1;Unfold 1 Int_part;Omega. +Lemma Rminus_Int_part2 : + forall r1 r2:R, + frac_part r1 < frac_part r2 -> + Int_part (r1 - r2) = (Int_part r1 - Int_part r2 - 1)%Z. +intros; elim (base_fp r1); elim (base_fp r2); intros; + generalize (Rge_le (frac_part r2) 0 H0); intro; clear H0; + generalize (Ropp_le_ge_contravar 0 (frac_part r2) H4); + intro; clear H4; rewrite Ropp_0 in H0; + generalize (Rge_le 0 (- frac_part r2) H0); intro; + clear H0; generalize (Rge_le (frac_part r1) 0 H2); + intro; clear H2; generalize (Ropp_lt_gt_contravar (frac_part r2) 1 H1); + intro; clear H1; unfold Rgt in H2; + generalize + (sum_inequa_Rle_lt 0 (frac_part r1) 1 (-1) (- frac_part r2) 0 H0 H3 H2 H4); + intro; elim H1; intros; clear H1; elim (Rplus_ne (-1)); + intros a b; rewrite b in H5; clear a b H6; + generalize (Rlt_minus (frac_part r1) (frac_part r2) H); + intro; clear H; fold (frac_part r1 - frac_part r2) in H5; + clear H3 H4 H0 H2; unfold frac_part in H5, H1; unfold Rminus in H5, H1; + rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))) in H5; + rewrite (Ropp_involutive (IZR (Int_part r2))) in H5; + rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2))) + in H5; + rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2))) + in H5; rewrite (Rplus_comm (- IZR (Int_part r1)) (- r2)) in H5; + rewrite (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))) in H5; + rewrite <- (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2))) + in H5; + rewrite (Rplus_comm (- IZR (Int_part r1)) (IZR (Int_part r2))) in H5; + fold (r1 - r2) in H5; fold (IZR (Int_part r2) - IZR (Int_part r1)) in H5; + generalize + (Rplus_lt_compat_l (IZR (Int_part r1) - IZR (Int_part r2)) (-1) + (r1 - r2 + (IZR (Int_part r2) - IZR (Int_part r1))) H5); + intro; clear H5; + rewrite (Rplus_comm (r1 - r2) (IZR (Int_part r2) - IZR (Int_part r1))) in H; + rewrite <- + (Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2)) + (IZR (Int_part r2) - IZR (Int_part r1)) (r1 - r2)) + in H; unfold Rminus in H; fold (r1 - r2) in H; + rewrite + (Rplus_assoc (IZR (Int_part r1)) (- IZR (Int_part r2)) + (IZR (Int_part r2) + - IZR (Int_part r1))) in H; + rewrite <- + (Rplus_assoc (- IZR (Int_part r2)) (IZR (Int_part r2)) + (- IZR (Int_part r1))) in H; + rewrite (Rplus_opp_l (IZR (Int_part r2))) in H; + elim (Rplus_ne (- IZR (Int_part r1))); intros a b; + rewrite b in H; clear a b; rewrite (Rplus_opp_r (IZR (Int_part r1))) in H; + elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H; + clear a b; fold (IZR (Int_part r1) - IZR (Int_part r2)) in H; + fold (IZR (Int_part r1) - IZR (Int_part r2) - 1) in H; + rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))) in H1; + rewrite (Ropp_involutive (IZR (Int_part r2))) in H1; + rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2))) + in H1; + rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2))) + in H1; rewrite (Rplus_comm (- IZR (Int_part r1)) (- r2)) in H1; + rewrite (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))) in H1; + rewrite <- (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2))) + in H1; + rewrite (Rplus_comm (- IZR (Int_part r1)) (IZR (Int_part r2))) in H1; + fold (r1 - r2) in H1; fold (IZR (Int_part r2) - IZR (Int_part r1)) in H1; + generalize + (Rplus_lt_compat_l (IZR (Int_part r1) - IZR (Int_part r2)) + (r1 - r2 + (IZR (Int_part r2) - IZR (Int_part r1))) 0 H1); + intro; clear H1; + rewrite (Rplus_comm (r1 - r2) (IZR (Int_part r2) - IZR (Int_part r1))) in H0; + rewrite <- + (Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2)) + (IZR (Int_part r2) - IZR (Int_part r1)) (r1 - r2)) + in H0; + rewrite <- (Ropp_minus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H0; + rewrite (Rplus_opp_r (IZR (Int_part r1) - IZR (Int_part r2))) in H0; + elim (Rplus_ne (r1 - r2)); intros a b; rewrite b in H0; + clear a b; rewrite <- (Rplus_opp_l 1) in H0; + rewrite <- (Rplus_assoc (IZR (Int_part r1) - IZR (Int_part r2)) (-1) 1) + in H0; fold (IZR (Int_part r1) - IZR (Int_part r2) - 1) in H0; + rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H0; + rewrite (Z_R_minus (Int_part r1) (Int_part r2)) in H; + cut (1 = IZR 1); auto with zarith real. +intro; rewrite H1 in H; rewrite H1 in H0; clear H1; + rewrite (Z_R_minus (Int_part r1 - Int_part r2) 1) in H; + rewrite (Z_R_minus (Int_part r1 - Int_part r2) 1) in H0; + rewrite <- (plus_IZR (Int_part r1 - Int_part r2 - 1) 1) in H0; + generalize (Rlt_le (IZR (Int_part r1 - Int_part r2 - 1)) (r1 - r2) H); + intro; clear H; + generalize (up_tech (r1 - r2) (Int_part r1 - Int_part r2 - 1) H1 H0); + intros; clear H0 H1; unfold Int_part at 1 in |- *; + omega. Qed. (**********) -Lemma Rminus_fp1:(r1,r2:R)(Rge (frac_part r1) (frac_part r2))-> - (frac_part (Rminus r1 r2))==(Rminus (frac_part r1) (frac_part r2)). -Intros;Unfold frac_part; - Generalize (Rminus_Int_part1 r1 r2 H);Intro;Rewrite -> H0; - Rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));Unfold Rminus; - Rewrite -> (Ropp_distr1 (IZR (Int_part r1)) (Ropp (IZR (Int_part r2)))); - Rewrite -> (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))); - Rewrite -> (Ropp_Ropp (IZR (Int_part r2))); - Rewrite -> (Rplus_assoc r1 (Ropp r2) - (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))); - Rewrite -> (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) - (Rplus (Ropp r2) (IZR (Int_part r2)))); - Rewrite <- (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1))) - (IZR (Int_part r2))); - Rewrite <- (Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2) - (IZR (Int_part r2))); - Rewrite -> (Rplus_sym (Ropp r2) (Ropp (IZR (Int_part r1))));Auto with zarith real. +Lemma Rminus_fp1 : + forall r1 r2:R, + frac_part r1 >= frac_part r2 -> + frac_part (r1 - r2) = frac_part r1 - frac_part r2. +intros; unfold frac_part in |- *; generalize (Rminus_Int_part1 r1 r2 H); + intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1) (Int_part r2)); + unfold Rminus in |- *; + rewrite (Ropp_plus_distr (IZR (Int_part r1)) (- IZR (Int_part r2))); + rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))); + rewrite (Ropp_involutive (IZR (Int_part r2))); + rewrite (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2))); + rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2))); + rewrite <- (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))); + rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2))); + rewrite (Rplus_comm (- r2) (- IZR (Int_part r1))); + auto with zarith real. Qed. (**********) -Lemma Rminus_fp2:(r1,r2:R)(Rlt (frac_part r1) (frac_part r2))-> - (frac_part (Rminus r1 r2))== - (Rplus (Rminus (frac_part r1) (frac_part r2)) R1). -Intros;Unfold frac_part;Generalize (Rminus_Int_part2 r1 r2 H);Intro; - Rewrite -> H0; - Rewrite <- (Z_R_minus (Zminus (Int_part r1) (Int_part r2)) `1`); - Rewrite <- (Z_R_minus (Int_part r1) (Int_part r2));Unfold Rminus; - Rewrite -> (Ropp_distr1 (Rplus (IZR (Int_part r1)) (Ropp (IZR (Int_part r2)))) - (Ropp (IZR `1`))); - Rewrite -> (Ropp_distr1 r2 (Ropp (IZR (Int_part r2)))); - Rewrite -> (Ropp_Ropp (IZR `1`)); - Rewrite -> (Ropp_Ropp (IZR (Int_part r2))); - Rewrite -> (Ropp_distr1 (IZR (Int_part r1))); - Rewrite -> (Ropp_Ropp (IZR (Int_part r2)));Simpl; - Rewrite <- (Rplus_assoc (Rplus r1 (Ropp r2)) - (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2))) R1); - Rewrite -> (Rplus_assoc r1 (Ropp r2) - (Rplus (Ropp (IZR (Int_part r1))) (IZR (Int_part r2)))); - Rewrite -> (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) - (Rplus (Ropp r2) (IZR (Int_part r2)))); - Rewrite <- (Rplus_assoc (Ropp r2) (Ropp (IZR (Int_part r1))) - (IZR (Int_part r2))); - Rewrite <- (Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp r2) - (IZR (Int_part r2))); - Rewrite -> (Rplus_sym (Ropp r2) (Ropp (IZR (Int_part r1))));Auto with zarith real. +Lemma Rminus_fp2 : + forall r1 r2:R, + frac_part r1 < frac_part r2 -> + frac_part (r1 - r2) = frac_part r1 - frac_part r2 + 1. +intros; unfold frac_part in |- *; generalize (Rminus_Int_part2 r1 r2 H); + intro; rewrite H0; rewrite <- (Z_R_minus (Int_part r1 - Int_part r2) 1); + rewrite <- (Z_R_minus (Int_part r1) (Int_part r2)); + unfold Rminus in |- *; + rewrite + (Ropp_plus_distr (IZR (Int_part r1) + - IZR (Int_part r2)) (- IZR 1)) + ; rewrite (Ropp_plus_distr r2 (- IZR (Int_part r2))); + rewrite (Ropp_involutive (IZR 1)); + rewrite (Ropp_involutive (IZR (Int_part r2))); + rewrite (Ropp_plus_distr (IZR (Int_part r1))); + rewrite (Ropp_involutive (IZR (Int_part r2))); simpl in |- *; + rewrite <- + (Rplus_assoc (r1 + - r2) (- IZR (Int_part r1) + IZR (Int_part r2)) 1) + ; rewrite (Rplus_assoc r1 (- r2) (- IZR (Int_part r1) + IZR (Int_part r2))); + rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (- r2 + IZR (Int_part r2))); + rewrite <- (Rplus_assoc (- r2) (- IZR (Int_part r1)) (IZR (Int_part r2))); + rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- r2) (IZR (Int_part r2))); + rewrite (Rplus_comm (- r2) (- IZR (Int_part r1))); + auto with zarith real. Qed. (**********) -Lemma plus_Int_part1:(r1,r2:R)(Rge (Rplus (frac_part r1) (frac_part r2)) R1)-> - (Int_part (Rplus r1 r2))=(Zplus (Zplus (Int_part r1) (Int_part r2)) `1`). -Intros; - Generalize (Rle_sym2 R1 (Rplus (frac_part r1) (frac_part r2)) H); - Intro;Clear H;Elim (base_fp r1);Elim (base_fp r2);Intros;Clear H H2; - Generalize (Rlt_compatibility (frac_part r2) (frac_part r1) R1 H3); - Intro;Clear H3; - Generalize (Rlt_compatibility R1 (frac_part r2) R1 H1);Intro;Clear H1; - Rewrite (Rplus_sym R1 (frac_part r2)) in H2; - Generalize (Rlt_trans (Rplus (frac_part r2) (frac_part r1)) - (Rplus (frac_part r2) R1) (Rplus R1 R1) H H2);Intro;Clear H H2; - Rewrite (Rplus_sym (frac_part r2) (frac_part r1)) in H1; - Unfold frac_part in H0 H1;Unfold Rminus in H0 H1; - Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) - (Rplus r2 (Ropp (IZR (Int_part r2))))) in H1; - Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))) in H1; - Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))) - r2) in H1; - Rewrite (Rplus_sym - (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2) in H1; - Rewrite <-(Rplus_assoc r1 r2 - (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))) in H1; - Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))) in H1; - Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) - (Rplus r2 (Ropp (IZR (Int_part r2))))) in H0; - Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))) in H0; - Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))) - r2) in H0; - Rewrite (Rplus_sym - (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2) in H0; - Rewrite <-(Rplus_assoc r1 r2 - (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))) in H0; - Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))) in H0; - Generalize (Rle_compatibility (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) - R1 (Rplus (Rplus r1 r2) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) H0);Intro; - Clear H0; - Generalize (Rlt_compatibility (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Rplus (Rplus r1 r2) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) (Rplus R1 R1) H1); - Intro;Clear H1; - Rewrite (Rplus_sym (Rplus r1 r2) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) in H; - Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) (Rplus r1 r2)) in H; - Rewrite (Rplus_Ropp_r (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H; - Elim (Rplus_ne (Rplus r1 r2));Intros a b;Rewrite b in H;Clear a b; - Rewrite (Rplus_sym (Rplus r1 r2) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) in H0; - Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) (Rplus r1 r2)) in H0; - Rewrite (Rplus_Ropp_r (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H0; - Elim (Rplus_ne (Rplus r1 r2));Intros a b;Rewrite b in H0;Clear a b; - Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) R1 R1) in - H0;Cut R1==(IZR `1`);Auto with zarith real. -Intro;Rewrite H1 in H0;Rewrite H1 in H;Clear H1; - Rewrite <-(plus_IZR (Int_part r1) (Int_part r2)) in H; - Rewrite <-(plus_IZR (Int_part r1) (Int_part r2)) in H0; - Rewrite <-(plus_IZR `(Int_part r1)+(Int_part r2)` `1`) in H; - Rewrite <-(plus_IZR `(Int_part r1)+(Int_part r2)` `1`) in H0; - Rewrite <-(plus_IZR `(Int_part r1)+(Int_part r2)+1` `1`) in H0; - Generalize (up_tech (Rplus r1 r2) `(Int_part r1)+(Int_part r2)+1` H H0);Intro; - Clear H H0;Unfold 1 Int_part;Omega. +Lemma plus_Int_part1 : + forall r1 r2:R, + frac_part r1 + frac_part r2 >= 1 -> + Int_part (r1 + r2) = (Int_part r1 + Int_part r2 + 1)%Z. +intros; generalize (Rge_le (frac_part r1 + frac_part r2) 1 H); intro; clear H; + elim (base_fp r1); elim (base_fp r2); intros; clear H H2; + generalize (Rplus_lt_compat_l (frac_part r2) (frac_part r1) 1 H3); + intro; clear H3; generalize (Rplus_lt_compat_l 1 (frac_part r2) 1 H1); + intro; clear H1; rewrite (Rplus_comm 1 (frac_part r2)) in H2; + generalize + (Rlt_trans (frac_part r2 + frac_part r1) (frac_part r2 + 1) 2 H H2); + intro; clear H H2; rewrite (Rplus_comm (frac_part r2) (frac_part r1)) in H1; + unfold frac_part in H0, H1; unfold Rminus in H0, H1; + rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2))) + in H1; rewrite (Rplus_comm r2 (- IZR (Int_part r2))) in H1; + rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2) + in H1; + rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2) in H1; + rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2))) + in H1; + rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H1; + rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2))) + in H0; rewrite (Rplus_comm r2 (- IZR (Int_part r2))) in H0; + rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2) + in H0; + rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2) in H0; + rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2))) + in H0; + rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H0; + generalize + (Rplus_le_compat_l (IZR (Int_part r1) + IZR (Int_part r2)) 1 + (r1 + r2 + - (IZR (Int_part r1) + IZR (Int_part r2))) H0); + intro; clear H0; + generalize + (Rplus_lt_compat_l (IZR (Int_part r1) + IZR (Int_part r2)) + (r1 + r2 + - (IZR (Int_part r1) + IZR (Int_part r2))) 2 H1); + intro; clear H1; + rewrite (Rplus_comm (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2)))) + in H; + rewrite <- + (Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2)) + (- (IZR (Int_part r1) + IZR (Int_part r2))) (r1 + r2)) + in H; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H; + elim (Rplus_ne (r1 + r2)); intros a b; rewrite b in H; + clear a b; + rewrite (Rplus_comm (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2)))) + in H0; + rewrite <- + (Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2)) + (- (IZR (Int_part r1) + IZR (Int_part r2))) (r1 + r2)) + in H0; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H0; + elim (Rplus_ne (r1 + r2)); intros a b; rewrite b in H0; + clear a b; + rewrite <- (Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2)) 1 1) in H0; + cut (1 = IZR 1); auto with zarith real. +intro; rewrite H1 in H0; rewrite H1 in H; clear H1; + rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H; + rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H0; + rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H; + rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H0; + rewrite <- (plus_IZR (Int_part r1 + Int_part r2 + 1) 1) in H0; + generalize (up_tech (r1 + r2) (Int_part r1 + Int_part r2 + 1) H H0); + intro; clear H H0; unfold Int_part at 1 in |- *; omega. Qed. (**********) -Lemma plus_Int_part2:(r1,r2:R)(Rlt (Rplus (frac_part r1) (frac_part r2)) R1)-> - (Int_part (Rplus r1 r2))=(Zplus (Int_part r1) (Int_part r2)). -Intros;Elim (base_fp r1);Elim (base_fp r2);Intros;Clear H1 H3; - Generalize (Rle_sym2 R0 (frac_part r2) H0);Intro;Clear H0; - Generalize (Rle_sym2 R0 (frac_part r1) H2);Intro;Clear H2; - Generalize (Rle_compatibility (frac_part r1) R0 (frac_part r2) H1); - Intro;Clear H1;Elim (Rplus_ne (frac_part r1));Intros a b; - Rewrite a in H2;Clear a b;Generalize (Rle_trans R0 (frac_part r1) - (Rplus (frac_part r1) (frac_part r2)) H0 H2);Intro;Clear H0 H2; - Unfold frac_part in H H1;Unfold Rminus in H H1; - Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) - (Rplus r2 (Ropp (IZR (Int_part r2))))) in H1; - Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))) in H1; - Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))) - r2) in H1; - Rewrite (Rplus_sym - (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2) in H1; - Rewrite <-(Rplus_assoc r1 r2 - (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))) in H1; - Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))) in H1; - Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) - (Rplus r2 (Ropp (IZR (Int_part r2))))) in H; - Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))) in H; - Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))) - r2) in H; - Rewrite (Rplus_sym - (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2) in H; - Rewrite <-(Rplus_assoc r1 r2 - (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))) in H; - Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))) in H; - Generalize (Rle_compatibility (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) - R0 (Rplus (Rplus r1 r2) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) H1);Intro; - Clear H1; - Generalize (Rlt_compatibility (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Rplus (Rplus r1 r2) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) R1 H); - Intro;Clear H; - Rewrite (Rplus_sym (Rplus r1 r2) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) in H1; - Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) (Rplus r1 r2)) in H1; - Rewrite (Rplus_Ropp_r (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H1; - Elim (Rplus_ne (Rplus r1 r2));Intros a b;Rewrite b in H1;Clear a b; - Rewrite (Rplus_sym (Rplus r1 r2) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))))) in H0; - Rewrite <-(Rplus_assoc (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) (Rplus r1 r2)) in H0; - Rewrite (Rplus_Ropp_r (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) in H0; - Elim (Rplus_ne (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))));Intros a b; - Rewrite a in H0;Clear a b;Elim (Rplus_ne (Rplus r1 r2));Intros a b; - Rewrite b in H0;Clear a b;Cut R1==(IZR `1`);Auto with zarith real. -Intro;Rewrite H in H1;Clear H; - Rewrite <-(plus_IZR (Int_part r1) (Int_part r2)) in H0; - Rewrite <-(plus_IZR (Int_part r1) (Int_part r2)) in H1; - Rewrite <-(plus_IZR `(Int_part r1)+(Int_part r2)` `1`) in H1; - Generalize (up_tech (Rplus r1 r2) `(Int_part r1)+(Int_part r2)` H0 H1);Intro; - Clear H0 H1;Unfold 1 Int_part;Omega. +Lemma plus_Int_part2 : + forall r1 r2:R, + frac_part r1 + frac_part r2 < 1 -> + Int_part (r1 + r2) = (Int_part r1 + Int_part r2)%Z. +intros; elim (base_fp r1); elim (base_fp r2); intros; clear H1 H3; + generalize (Rge_le (frac_part r2) 0 H0); intro; clear H0; + generalize (Rge_le (frac_part r1) 0 H2); intro; clear H2; + generalize (Rplus_le_compat_l (frac_part r1) 0 (frac_part r2) H1); + intro; clear H1; elim (Rplus_ne (frac_part r1)); intros a b; + rewrite a in H2; clear a b; + generalize (Rle_trans 0 (frac_part r1) (frac_part r1 + frac_part r2) H0 H2); + intro; clear H0 H2; unfold frac_part in H, H1; unfold Rminus in H, H1; + rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2))) + in H1; rewrite (Rplus_comm r2 (- IZR (Int_part r2))) in H1; + rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2) + in H1; + rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2) in H1; + rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2))) + in H1; + rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H1; + rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2))) + in H; rewrite (Rplus_comm r2 (- IZR (Int_part r2))) in H; + rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2) in H; + rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2) in H; + rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2))) + in H; + rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))) in H; + generalize + (Rplus_le_compat_l (IZR (Int_part r1) + IZR (Int_part r2)) 0 + (r1 + r2 + - (IZR (Int_part r1) + IZR (Int_part r2))) H1); + intro; clear H1; + generalize + (Rplus_lt_compat_l (IZR (Int_part r1) + IZR (Int_part r2)) + (r1 + r2 + - (IZR (Int_part r1) + IZR (Int_part r2))) 1 H); + intro; clear H; + rewrite (Rplus_comm (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2)))) + in H1; + rewrite <- + (Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2)) + (- (IZR (Int_part r1) + IZR (Int_part r2))) (r1 + r2)) + in H1; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H1; + elim (Rplus_ne (r1 + r2)); intros a b; rewrite b in H1; + clear a b; + rewrite (Rplus_comm (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2)))) + in H0; + rewrite <- + (Rplus_assoc (IZR (Int_part r1) + IZR (Int_part r2)) + (- (IZR (Int_part r1) + IZR (Int_part r2))) (r1 + r2)) + in H0; rewrite (Rplus_opp_r (IZR (Int_part r1) + IZR (Int_part r2))) in H0; + elim (Rplus_ne (IZR (Int_part r1) + IZR (Int_part r2))); + intros a b; rewrite a in H0; clear a b; elim (Rplus_ne (r1 + r2)); + intros a b; rewrite b in H0; clear a b; cut (1 = IZR 1); + auto with zarith real. +intro; rewrite H in H1; clear H; + rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H0; + rewrite <- (plus_IZR (Int_part r1) (Int_part r2)) in H1; + rewrite <- (plus_IZR (Int_part r1 + Int_part r2) 1) in H1; + generalize (up_tech (r1 + r2) (Int_part r1 + Int_part r2) H0 H1); + intro; clear H0 H1; unfold Int_part at 1 in |- *; + omega. Qed. (**********) -Lemma plus_frac_part1:(r1,r2:R) - (Rge (Rplus (frac_part r1) (frac_part r2)) R1)-> - (frac_part (Rplus r1 r2))== - (Rminus (Rplus (frac_part r1) (frac_part r2)) R1). -Intros;Unfold frac_part; - Generalize (plus_Int_part1 r1 r2 H);Intro;Rewrite H0; - Rewrite (plus_IZR `(Int_part r1)+(Int_part r2)` `1`); - Rewrite (plus_IZR (Int_part r1) (Int_part r2));Simpl;Unfold 3 4 Rminus; - Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) - (Rplus r2 (Ropp (IZR (Int_part r2))))); - Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))); - Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))) - r2); - Rewrite (Rplus_sym - (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2); - Rewrite <-(Rplus_assoc r1 r2 - (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))); - Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2))); - Unfold Rminus; - Rewrite (Rplus_assoc (Rplus r1 r2) - (Ropp (Rplus (IZR (Int_part r1)) (IZR (Int_part r2)))) - (Ropp R1)); - Rewrite <-(Ropp_distr1 (Rplus (IZR (Int_part r1)) (IZR (Int_part r2))) R1); - Trivial with zarith real. +Lemma plus_frac_part1 : + forall r1 r2:R, + frac_part r1 + frac_part r2 >= 1 -> + frac_part (r1 + r2) = frac_part r1 + frac_part r2 - 1. +intros; unfold frac_part in |- *; generalize (plus_Int_part1 r1 r2 H); intro; + rewrite H0; rewrite (plus_IZR (Int_part r1 + Int_part r2) 1); + rewrite (plus_IZR (Int_part r1) (Int_part r2)); simpl in |- *; + unfold Rminus at 3 4 in |- *; + rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2))); + rewrite (Rplus_comm r2 (- IZR (Int_part r2))); + rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2); + rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2); + rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2))); + rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))); + unfold Rminus in |- *; + rewrite + (Rplus_assoc (r1 + r2) (- (IZR (Int_part r1) + IZR (Int_part r2))) (-1)) + ; rewrite <- (Ropp_plus_distr (IZR (Int_part r1) + IZR (Int_part r2)) 1); + trivial with zarith real. Qed. (**********) -Lemma plus_frac_part2:(r1,r2:R) - (Rlt (Rplus (frac_part r1) (frac_part r2)) R1)-> -(frac_part (Rplus r1 r2))==(Rplus (frac_part r1) (frac_part r2)). -Intros;Unfold frac_part; - Generalize (plus_Int_part2 r1 r2 H);Intro;Rewrite H0; - Rewrite (plus_IZR (Int_part r1) (Int_part r2));Unfold 2 3 Rminus; - Rewrite (Rplus_assoc r1 (Ropp (IZR (Int_part r1))) - (Rplus r2 (Ropp (IZR (Int_part r2))))); - Rewrite (Rplus_sym r2 (Ropp (IZR (Int_part r2)))); - Rewrite <-(Rplus_assoc (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))) - r2); - Rewrite (Rplus_sym - (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2)))) r2); - Rewrite <-(Rplus_assoc r1 r2 - (Rplus (Ropp (IZR (Int_part r1))) (Ropp (IZR (Int_part r2))))); - Rewrite <-(Ropp_distr1 (IZR (Int_part r1)) (IZR (Int_part r2)));Unfold Rminus; - Trivial with zarith real. -Qed. +Lemma plus_frac_part2 : + forall r1 r2:R, + frac_part r1 + frac_part r2 < 1 -> + frac_part (r1 + r2) = frac_part r1 + frac_part r2. +intros; unfold frac_part in |- *; generalize (plus_Int_part2 r1 r2 H); intro; + rewrite H0; rewrite (plus_IZR (Int_part r1) (Int_part r2)); + unfold Rminus at 2 3 in |- *; + rewrite (Rplus_assoc r1 (- IZR (Int_part r1)) (r2 + - IZR (Int_part r2))); + rewrite (Rplus_comm r2 (- IZR (Int_part r2))); + rewrite <- (Rplus_assoc (- IZR (Int_part r1)) (- IZR (Int_part r2)) r2); + rewrite (Rplus_comm (- IZR (Int_part r1) + - IZR (Int_part r2)) r2); + rewrite <- (Rplus_assoc r1 r2 (- IZR (Int_part r1) + - IZR (Int_part r2))); + rewrite <- (Ropp_plus_distr (IZR (Int_part r1)) (IZR (Int_part r2))); + unfold Rminus in |- *; trivial with zarith real. +Qed.
\ No newline at end of file diff --git a/theories/Reals/R_sqr.v b/theories/Reals/R_sqr.v index 0610db3be..1abe6d925 100644 --- a/theories/Reals/R_sqr.v +++ b/theories/Reals/R_sqr.v @@ -8,225 +8,323 @@ (*i $Id$ i*) -Require Rbase. -Require Rbasic_fun. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rbasic_fun. Open Local Scope R_scope. (****************************************************) (* Rsqr : some results *) (****************************************************) -Tactic Definition SqRing := Unfold Rsqr; Ring. +Ltac ring_Rsqr := unfold Rsqr in |- *; ring. -Lemma Rsqr_neg : (x:R) ``(Rsqr x)==(Rsqr (-x))``. -Intros; SqRing. +Lemma Rsqr_neg : forall x:R, Rsqr x = Rsqr (- x). +intros; ring_Rsqr. Qed. -Lemma Rsqr_times : (x,y:R) ``(Rsqr (x*y))==(Rsqr x)*(Rsqr y)``. -Intros; SqRing. +Lemma Rsqr_mult : forall x y:R, Rsqr (x * y) = Rsqr x * Rsqr y. +intros; ring_Rsqr. Qed. -Lemma Rsqr_plus : (x,y:R) ``(Rsqr (x+y))==(Rsqr x)+(Rsqr y)+2*x*y``. -Intros; SqRing. +Lemma Rsqr_plus : forall x y:R, Rsqr (x + y) = Rsqr x + Rsqr y + 2 * x * y. +intros; ring_Rsqr. Qed. -Lemma Rsqr_minus : (x,y:R) ``(Rsqr (x-y))==(Rsqr x)+(Rsqr y)-2*x*y``. -Intros; SqRing. +Lemma Rsqr_minus : forall x y:R, Rsqr (x - y) = Rsqr x + Rsqr y - 2 * x * y. +intros; ring_Rsqr. Qed. -Lemma Rsqr_neg_minus : (x,y:R) ``(Rsqr (x-y))==(Rsqr (y-x))``. -Intros; SqRing. +Lemma Rsqr_neg_minus : forall x y:R, Rsqr (x - y) = Rsqr (y - x). +intros; ring_Rsqr. Qed. -Lemma Rsqr_1 : ``(Rsqr 1)==1``. -SqRing. +Lemma Rsqr_1 : Rsqr 1 = 1. +ring_Rsqr. Qed. -Lemma Rsqr_gt_0_0 : (x:R) ``0<(Rsqr x)`` -> ~``x==0``. -Intros; Red; Intro; Rewrite H0 in H; Rewrite Rsqr_O in H; Elim (Rlt_antirefl ``0`` H). +Lemma Rsqr_gt_0_0 : forall x:R, 0 < Rsqr x -> x <> 0. +intros; red in |- *; intro; rewrite H0 in H; rewrite Rsqr_0 in H; + elim (Rlt_irrefl 0 H). Qed. -Lemma Rsqr_pos_lt : (x:R) ~(x==R0)->``0<(Rsqr x)``. -Intros; Case (total_order R0 x); Intro; [Unfold Rsqr; Apply Rmult_lt_pos; Assumption | Elim H0; Intro; [Elim H; Symmetry; Exact H1 | Rewrite Rsqr_neg; Generalize (Rlt_Ropp x ``0`` H1); Rewrite Ropp_O; Intro; Unfold Rsqr; Apply Rmult_lt_pos; Assumption]]. +Lemma Rsqr_pos_lt : forall x:R, x <> 0 -> 0 < Rsqr x. +intros; case (Rtotal_order 0 x); intro; + [ unfold Rsqr in |- *; apply Rmult_lt_0_compat; assumption + | elim H0; intro; + [ elim H; symmetry in |- *; exact H1 + | rewrite Rsqr_neg; generalize (Ropp_lt_gt_contravar x 0 H1); + rewrite Ropp_0; intro; unfold Rsqr in |- *; + apply Rmult_lt_0_compat; assumption ] ]. Qed. -Lemma Rsqr_div : (x,y:R) ~``y==0`` -> ``(Rsqr (x/y))==(Rsqr x)/(Rsqr y)``. -Intros; Unfold Rsqr. -Unfold Rdiv. -Rewrite Rinv_Rmult. -Repeat Rewrite Rmult_assoc. -Apply Rmult_mult_r. -Pattern 2 x; Rewrite Rmult_sym. -Repeat Rewrite Rmult_assoc. -Apply Rmult_mult_r. -Reflexivity. -Assumption. -Assumption. +Lemma Rsqr_div : forall x y:R, y <> 0 -> Rsqr (x / y) = Rsqr x / Rsqr y. +intros; unfold Rsqr in |- *. +unfold Rdiv in |- *. +rewrite Rinv_mult_distr. +repeat rewrite Rmult_assoc. +apply Rmult_eq_compat_l. +pattern x at 2 in |- *; rewrite Rmult_comm. +repeat rewrite Rmult_assoc. +apply Rmult_eq_compat_l. +reflexivity. +assumption. +assumption. Qed. -Lemma Rsqr_eq_0 : (x:R) ``(Rsqr x)==0`` -> ``x==0``. -Unfold Rsqr; Intros; Generalize (without_div_Od x x H); Intro; Elim H0; Intro ; Assumption. +Lemma Rsqr_eq_0 : forall x:R, Rsqr x = 0 -> x = 0. +unfold Rsqr in |- *; intros; generalize (Rmult_integral x x H); intro; + elim H0; intro; assumption. Qed. -Lemma Rsqr_minus_plus : (a,b:R) ``(a-b)*(a+b)==(Rsqr a)-(Rsqr b)``. -Intros; SqRing. +Lemma Rsqr_minus_plus : forall a b:R, (a - b) * (a + b) = Rsqr a - Rsqr b. +intros; ring_Rsqr. Qed. -Lemma Rsqr_plus_minus : (a,b:R) ``(a+b)*(a-b)==(Rsqr a)-(Rsqr b)``. -Intros; SqRing. +Lemma Rsqr_plus_minus : forall a b:R, (a + b) * (a - b) = Rsqr a - Rsqr b. +intros; ring_Rsqr. Qed. -Lemma Rsqr_incr_0 : (x,y:R) ``(Rsqr x)<=(Rsqr y)`` -> ``0<=x`` -> ``0<=y`` -> ``x<=y``. -Intros; Case (total_order_Rle x y); Intro; [Assumption | Cut ``y<x``; [Intro; Unfold Rsqr in H; Generalize (Rmult_lt2 y x y x H1 H1 H2 H2); Intro; Generalize (Rle_lt_trans ``x*x`` ``y*y`` ``x*x`` H H3); Intro; Elim (Rlt_antirefl ``x*x`` H4) | Auto with real]]. +Lemma Rsqr_incr_0 : + forall x y:R, Rsqr x <= Rsqr y -> 0 <= x -> 0 <= y -> x <= y. +intros; case (Rle_dec x y); intro; + [ assumption + | cut (y < x); + [ intro; unfold Rsqr in H; + generalize (Rmult_le_0_lt_compat y x y x H1 H1 H2 H2); + intro; generalize (Rle_lt_trans (x * x) (y * y) (x * x) H H3); + intro; elim (Rlt_irrefl (x * x) H4) + | auto with real ] ]. Qed. -Lemma Rsqr_incr_0_var : (x,y:R) ``(Rsqr x)<=(Rsqr y)`` -> ``0<=y`` -> ``x<=y``. -Intros; Case (total_order_Rle x y); Intro; [Assumption | Cut ``y<x``; [Intro; Unfold Rsqr in H; Generalize (Rmult_lt2 y x y x H0 H0 H1 H1); Intro; Generalize (Rle_lt_trans ``x*x`` ``y*y`` ``x*x`` H H2); Intro; Elim (Rlt_antirefl ``x*x`` H3) | Auto with real]]. +Lemma Rsqr_incr_0_var : forall x y:R, Rsqr x <= Rsqr y -> 0 <= y -> x <= y. +intros; case (Rle_dec x y); intro; + [ assumption + | cut (y < x); + [ intro; unfold Rsqr in H; + generalize (Rmult_le_0_lt_compat y x y x H0 H0 H1 H1); + intro; generalize (Rle_lt_trans (x * x) (y * y) (x * x) H H2); + intro; elim (Rlt_irrefl (x * x) H3) + | auto with real ] ]. Qed. -Lemma Rsqr_incr_1 : (x,y:R) ``x<=y``->``0<=x``->``0<= y``->``(Rsqr x)<=(Rsqr y)``. -Intros; Unfold Rsqr; Apply Rle_Rmult_comp; Assumption. +Lemma Rsqr_incr_1 : + forall x y:R, x <= y -> 0 <= x -> 0 <= y -> Rsqr x <= Rsqr y. +intros; unfold Rsqr in |- *; apply Rmult_le_compat; assumption. Qed. -Lemma Rsqr_incrst_0 : (x,y:R) ``(Rsqr x)<(Rsqr y)``->``0<=x``->``0<=y``-> ``x<y``. -Intros; Case (total_order x y); Intro; [Assumption | Elim H2; Intro; [Rewrite H3 in H; Elim (Rlt_antirefl (Rsqr y) H) | Generalize (Rmult_lt2 y x y x H1 H1 H3 H3); Intro; Unfold Rsqr in H; Generalize (Rlt_trans ``x*x`` ``y*y`` ``x*x`` H H4); Intro; Elim (Rlt_antirefl ``x*x`` H5)]]. +Lemma Rsqr_incrst_0 : + forall x y:R, Rsqr x < Rsqr y -> 0 <= x -> 0 <= y -> x < y. +intros; case (Rtotal_order x y); intro; + [ assumption + | elim H2; intro; + [ rewrite H3 in H; elim (Rlt_irrefl (Rsqr y) H) + | generalize (Rmult_le_0_lt_compat y x y x H1 H1 H3 H3); intro; + unfold Rsqr in H; generalize (Rlt_trans (x * x) (y * y) (x * x) H H4); + intro; elim (Rlt_irrefl (x * x) H5) ] ]. Qed. -Lemma Rsqr_incrst_1 : (x,y:R) ``x<y``->``0<=x``->``0<=y``->``(Rsqr x)<(Rsqr y)``. -Intros; Unfold Rsqr; Apply Rmult_lt2; Assumption. +Lemma Rsqr_incrst_1 : + forall x y:R, x < y -> 0 <= x -> 0 <= y -> Rsqr x < Rsqr y. +intros; unfold Rsqr in |- *; apply Rmult_le_0_lt_compat; assumption. Qed. -Lemma Rsqr_neg_pos_le_0 : (x,y:R) ``(Rsqr x)<=(Rsqr y)``->``0<=y``->``-y<=x``. -Intros; Case (case_Rabsolu x); Intro. -Generalize (Rlt_Ropp x ``0`` r); Rewrite Ropp_O; Intro; Generalize (Rlt_le ``0`` ``-x`` H1); Intro; Rewrite (Rsqr_neg x) in H; Generalize (Rsqr_incr_0 (Ropp x) y H H2 H0); Intro; Rewrite <- (Ropp_Ropp x); Apply Rge_Ropp; Apply Rle_sym1; Assumption. -Apply Rle_trans with ``0``; [Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Assumption | Apply Rle_sym2; Assumption]. -Qed. - -Lemma Rsqr_neg_pos_le_1 : (x,y:R) ``(-y)<=x`` -> ``x<=y`` -> ``0<=y`` -> ``(Rsqr x)<=(Rsqr y)``. -Intros; Case (case_Rabsolu x); Intro. -Generalize (Rlt_Ropp x ``0`` r); Rewrite Ropp_O; Intro; Generalize (Rlt_le ``0`` ``-x`` H2); Intro; Generalize (Rle_Ropp ``-y`` x H); Rewrite Ropp_Ropp; Intro; Generalize (Rle_sym2 ``-x`` y H4); Intro; Rewrite (Rsqr_neg x); Apply Rsqr_incr_1; Assumption. -Generalize (Rle_sym2 ``0`` x r); Intro; Apply Rsqr_incr_1; Assumption. -Qed. - -Lemma neg_pos_Rsqr_le : (x,y:R) ``(-y)<=x``->``x<=y``->``(Rsqr x)<=(Rsqr y)``. -Intros; Case (case_Rabsolu x); Intro. -Generalize (Rlt_Ropp x ``0`` r); Rewrite Ropp_O; Intro; Generalize (Rle_Ropp ``-y`` x H); Rewrite Ropp_Ropp; Intro; Generalize (Rle_sym2 ``-x`` y H2); Intro; Generalize (Rlt_le ``0`` ``-x`` H1); Intro; Generalize (Rle_trans ``0`` ``-x`` y H4 H3); Intro; Rewrite (Rsqr_neg x); Apply Rsqr_incr_1; Assumption. -Generalize (Rle_sym2 ``0`` x r); Intro; Generalize (Rle_trans ``0`` x y H1 H0); Intro; Apply Rsqr_incr_1; Assumption. -Qed. - -Lemma Rsqr_abs : (x:R) ``(Rsqr x)==(Rsqr (Rabsolu x))``. -Intro; Unfold Rabsolu; Case (case_Rabsolu x); Intro; [Apply Rsqr_neg | Reflexivity]. -Qed. - -Lemma Rsqr_le_abs_0 : (x,y:R) ``(Rsqr x)<=(Rsqr y)`` -> ``(Rabsolu x)<=(Rabsolu y)``. -Intros; Apply Rsqr_incr_0; Repeat Rewrite <- Rsqr_abs; [Assumption | Apply Rabsolu_pos | Apply Rabsolu_pos]. -Qed. - -Lemma Rsqr_le_abs_1 : (x,y:R) ``(Rabsolu x)<=(Rabsolu y)`` -> ``(Rsqr x)<=(Rsqr y)``. -Intros; Rewrite (Rsqr_abs x); Rewrite (Rsqr_abs y); Apply (Rsqr_incr_1 (Rabsolu x) (Rabsolu y) H (Rabsolu_pos x) (Rabsolu_pos y)). -Qed. - -Lemma Rsqr_lt_abs_0 : (x,y:R) ``(Rsqr x)<(Rsqr y)`` -> ``(Rabsolu x)<(Rabsolu y)``. -Intros; Apply Rsqr_incrst_0; Repeat Rewrite <- Rsqr_abs; [Assumption | Apply Rabsolu_pos | Apply Rabsolu_pos]. -Qed. - -Lemma Rsqr_lt_abs_1 : (x,y:R) ``(Rabsolu x)<(Rabsolu y)`` -> ``(Rsqr x)<(Rsqr y)``. -Intros; Rewrite (Rsqr_abs x); Rewrite (Rsqr_abs y); Apply (Rsqr_incrst_1 (Rabsolu x) (Rabsolu y) H (Rabsolu_pos x) (Rabsolu_pos y)). -Qed. - -Lemma Rsqr_inj : (x,y:R) ``0<=x`` -> ``0<=y`` -> (Rsqr x)==(Rsqr y) -> x==y. -Intros; Generalize (Rle_le_eq (Rsqr x) (Rsqr y)); Intro; Elim H2; Intros _ H3; Generalize (H3 H1); Intro; Elim H4; Intros; Apply Rle_antisym; Apply Rsqr_incr_0; Assumption. -Qed. - -Lemma Rsqr_eq_abs_0 : (x,y:R) (Rsqr x)==(Rsqr y) -> (Rabsolu x)==(Rabsolu y). -Intros; Unfold Rabsolu; Case (case_Rabsolu x); Case (case_Rabsolu y); Intros. -Rewrite -> (Rsqr_neg x) in H; Rewrite -> (Rsqr_neg y) in H; Generalize (Rlt_Ropp y ``0`` r); Generalize (Rlt_Ropp x ``0`` r0); Rewrite Ropp_O; Intros; Generalize (Rlt_le ``0`` ``-x`` H0); Generalize (Rlt_le ``0`` ``-y`` H1); Intros; Apply Rsqr_inj; Assumption. -Rewrite -> (Rsqr_neg x) in H; Generalize (Rle_sym2 ``0`` y r); Intro; Generalize (Rlt_Ropp x ``0`` r0); Rewrite Ropp_O; Intro; Generalize (Rlt_le ``0`` ``-x`` H1); Intro; Apply Rsqr_inj; Assumption. -Rewrite -> (Rsqr_neg y) in H; Generalize (Rle_sym2 ``0`` x r0); Intro; Generalize (Rlt_Ropp y ``0`` r); Rewrite Ropp_O; Intro; Generalize (Rlt_le ``0`` ``-y`` H1); Intro; Apply Rsqr_inj; Assumption. -Generalize (Rle_sym2 ``0`` x r0); Generalize (Rle_sym2 ``0`` y r); Intros; Apply Rsqr_inj; Assumption. -Qed. - -Lemma Rsqr_eq_asb_1 : (x,y:R) (Rabsolu x)==(Rabsolu y) -> (Rsqr x)==(Rsqr y). -Intros; Cut ``(Rsqr (Rabsolu x))==(Rsqr (Rabsolu y))``. -Intro; Repeat Rewrite <- Rsqr_abs in H0; Assumption. -Rewrite H; Reflexivity. -Qed. - -Lemma triangle_rectangle : (x,y,z:R) ``0<=z``->``(Rsqr x)+(Rsqr y)<=(Rsqr z)``->``-z<=x<=z`` /\``-z<=y<=z``. -Intros; Generalize (plus_le_is_le (Rsqr x) (Rsqr y) (Rsqr z) (pos_Rsqr y) H0); Rewrite Rplus_sym in H0; Generalize (plus_le_is_le (Rsqr y) (Rsqr x) (Rsqr z) (pos_Rsqr x) H0); Intros; Split; [Split; [Apply Rsqr_neg_pos_le_0; Assumption | Apply Rsqr_incr_0_var; Assumption] | Split; [Apply Rsqr_neg_pos_le_0; Assumption | Apply Rsqr_incr_0_var; Assumption]]. -Qed. - -Lemma triangle_rectangle_lt : (x,y,z:R) ``(Rsqr x)+(Rsqr y)<(Rsqr z)`` -> ``(Rabsolu x)<(Rabsolu z)``/\``(Rabsolu y)<(Rabsolu z)``. -Intros; Split; [Generalize (plus_lt_is_lt (Rsqr x) (Rsqr y) (Rsqr z) (pos_Rsqr y) H); Intro; Apply Rsqr_lt_abs_0; Assumption | Rewrite Rplus_sym in H; Generalize (plus_lt_is_lt (Rsqr y) (Rsqr x) (Rsqr z) (pos_Rsqr x) H); Intro; Apply Rsqr_lt_abs_0; Assumption]. -Qed. - -Lemma triangle_rectangle_le : (x,y,z:R) ``(Rsqr x)+(Rsqr y)<=(Rsqr z)`` -> ``(Rabsolu x)<=(Rabsolu z)``/\``(Rabsolu y)<=(Rabsolu z)``. -Intros; Split; [Generalize (plus_le_is_le (Rsqr x) (Rsqr y) (Rsqr z) (pos_Rsqr y) H); Intro; Apply Rsqr_le_abs_0; Assumption | Rewrite Rplus_sym in H; Generalize (plus_le_is_le (Rsqr y) (Rsqr x) (Rsqr z) (pos_Rsqr x) H); Intro; Apply Rsqr_le_abs_0; Assumption]. -Qed. - -Lemma Rsqr_inv : (x:R) ~``x==0`` -> ``(Rsqr (/x))==/(Rsqr x)``. -Intros; Unfold Rsqr. -Rewrite Rinv_Rmult; Try Reflexivity Orelse Assumption. -Qed. - -Lemma canonical_Rsqr : (a:nonzeroreal;b,c,x:R) ``a*(Rsqr x)+b*x+c == a* (Rsqr (x+b/(2*a))) + (4*a*c - (Rsqr b))/(4*a)``. -Intros. -Rewrite Rsqr_plus. -Repeat Rewrite Rmult_Rplus_distr. -Repeat Rewrite Rplus_assoc. -Apply Rplus_plus_r. -Unfold Rdiv Rminus. -Replace ``2*1+2*1`` with ``4``; [Idtac | Ring]. -Rewrite (Rmult_Rplus_distrl ``4*a*c`` ``-(Rsqr b)`` ``/(4*a)``). -Rewrite Rsqr_times. -Repeat Rewrite Rinv_Rmult. -Repeat Rewrite (Rmult_sym a). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Rewrite (Rmult_sym ``2``). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Rewrite (Rmult_sym ``/2``). -Rewrite (Rmult_sym ``2``). -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Rewrite (Rmult_sym a). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Rewrite (Rmult_sym ``2``). -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Repeat Rewrite Rplus_assoc. -Rewrite (Rplus_sym ``(Rsqr b)*((Rsqr (/a*/2))*a)``). -Repeat Rewrite Rplus_assoc. -Rewrite (Rmult_sym x). -Apply Rplus_plus_r. -Rewrite (Rmult_sym ``/a``). -Unfold Rsqr; Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Ring. -Apply (cond_nonzero a). -DiscrR. -Apply (cond_nonzero a). -DiscrR. -DiscrR. -Apply (cond_nonzero a). -DiscrR. -DiscrR. -DiscrR. -Apply (cond_nonzero a). -DiscrR. -Apply (cond_nonzero a). -Qed. - -Lemma Rsqr_eq : (x,y:R) (Rsqr x)==(Rsqr y) -> x==y \/ x==``-y``. -Intros; Unfold Rsqr in H; Generalize (Rplus_plus_r ``-(y*y)`` ``x*x`` ``y*y`` H); Rewrite Rplus_Ropp_l; Replace ``-(y*y)+x*x`` with ``(x-y)*(x+y)``. -Intro; Generalize (without_div_Od ``x-y`` ``x+y`` H0); Intro; Elim H1; Intros. -Left; Apply Rminus_eq; Assumption. -Right; Apply Rminus_eq; Unfold Rminus; Rewrite Ropp_Ropp; Assumption. -Ring. -Qed. +Lemma Rsqr_neg_pos_le_0 : + forall x y:R, Rsqr x <= Rsqr y -> 0 <= y -> - y <= x. +intros; case (Rcase_abs x); intro. +generalize (Ropp_lt_gt_contravar x 0 r); rewrite Ropp_0; intro; + generalize (Rlt_le 0 (- x) H1); intro; rewrite (Rsqr_neg x) in H; + generalize (Rsqr_incr_0 (- x) y H H2 H0); intro; + rewrite <- (Ropp_involutive x); apply Ropp_ge_le_contravar; + apply Rle_ge; assumption. +apply Rle_trans with 0; + [ rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; assumption + | apply Rge_le; assumption ]. +Qed. + +Lemma Rsqr_neg_pos_le_1 : + forall x y:R, - y <= x -> x <= y -> 0 <= y -> Rsqr x <= Rsqr y. +intros; case (Rcase_abs x); intro. +generalize (Ropp_lt_gt_contravar x 0 r); rewrite Ropp_0; intro; + generalize (Rlt_le 0 (- x) H2); intro; + generalize (Ropp_le_ge_contravar (- y) x H); rewrite Ropp_involutive; + intro; generalize (Rge_le y (- x) H4); intro; rewrite (Rsqr_neg x); + apply Rsqr_incr_1; assumption. +generalize (Rge_le x 0 r); intro; apply Rsqr_incr_1; assumption. +Qed. + +Lemma neg_pos_Rsqr_le : forall x y:R, - y <= x -> x <= y -> Rsqr x <= Rsqr y. +intros; case (Rcase_abs x); intro. +generalize (Ropp_lt_gt_contravar x 0 r); rewrite Ropp_0; intro; + generalize (Ropp_le_ge_contravar (- y) x H); rewrite Ropp_involutive; + intro; generalize (Rge_le y (- x) H2); intro; generalize (Rlt_le 0 (- x) H1); + intro; generalize (Rle_trans 0 (- x) y H4 H3); intro; + rewrite (Rsqr_neg x); apply Rsqr_incr_1; assumption. +generalize (Rge_le x 0 r); intro; generalize (Rle_trans 0 x y H1 H0); intro; + apply Rsqr_incr_1; assumption. +Qed. + +Lemma Rsqr_abs : forall x:R, Rsqr x = Rsqr (Rabs x). +intro; unfold Rabs in |- *; case (Rcase_abs x); intro; + [ apply Rsqr_neg | reflexivity ]. +Qed. + +Lemma Rsqr_le_abs_0 : forall x y:R, Rsqr x <= Rsqr y -> Rabs x <= Rabs y. +intros; apply Rsqr_incr_0; repeat rewrite <- Rsqr_abs; + [ assumption | apply Rabs_pos | apply Rabs_pos ]. +Qed. + +Lemma Rsqr_le_abs_1 : forall x y:R, Rabs x <= Rabs y -> Rsqr x <= Rsqr y. +intros; rewrite (Rsqr_abs x); rewrite (Rsqr_abs y); + apply (Rsqr_incr_1 (Rabs x) (Rabs y) H (Rabs_pos x) (Rabs_pos y)). +Qed. + +Lemma Rsqr_lt_abs_0 : forall x y:R, Rsqr x < Rsqr y -> Rabs x < Rabs y. +intros; apply Rsqr_incrst_0; repeat rewrite <- Rsqr_abs; + [ assumption | apply Rabs_pos | apply Rabs_pos ]. +Qed. + +Lemma Rsqr_lt_abs_1 : forall x y:R, Rabs x < Rabs y -> Rsqr x < Rsqr y. +intros; rewrite (Rsqr_abs x); rewrite (Rsqr_abs y); + apply (Rsqr_incrst_1 (Rabs x) (Rabs y) H (Rabs_pos x) (Rabs_pos y)). +Qed. + +Lemma Rsqr_inj : forall x y:R, 0 <= x -> 0 <= y -> Rsqr x = Rsqr y -> x = y. +intros; generalize (Rle_le_eq (Rsqr x) (Rsqr y)); intro; elim H2; intros _ H3; + generalize (H3 H1); intro; elim H4; intros; apply Rle_antisym; + apply Rsqr_incr_0; assumption. +Qed. + +Lemma Rsqr_eq_abs_0 : forall x y:R, Rsqr x = Rsqr y -> Rabs x = Rabs y. +intros; unfold Rabs in |- *; case (Rcase_abs x); case (Rcase_abs y); intros. +rewrite (Rsqr_neg x) in H; rewrite (Rsqr_neg y) in H; + generalize (Ropp_lt_gt_contravar y 0 r); + generalize (Ropp_lt_gt_contravar x 0 r0); rewrite Ropp_0; + intros; generalize (Rlt_le 0 (- x) H0); generalize (Rlt_le 0 (- y) H1); + intros; apply Rsqr_inj; assumption. +rewrite (Rsqr_neg x) in H; generalize (Rge_le y 0 r); intro; + generalize (Ropp_lt_gt_contravar x 0 r0); rewrite Ropp_0; + intro; generalize (Rlt_le 0 (- x) H1); intro; apply Rsqr_inj; + assumption. +rewrite (Rsqr_neg y) in H; generalize (Rge_le x 0 r0); intro; + generalize (Ropp_lt_gt_contravar y 0 r); rewrite Ropp_0; + intro; generalize (Rlt_le 0 (- y) H1); intro; apply Rsqr_inj; + assumption. +generalize (Rge_le x 0 r0); generalize (Rge_le y 0 r); intros; apply Rsqr_inj; + assumption. +Qed. + +Lemma Rsqr_eq_asb_1 : forall x y:R, Rabs x = Rabs y -> Rsqr x = Rsqr y. +intros; cut (Rsqr (Rabs x) = Rsqr (Rabs y)). +intro; repeat rewrite <- Rsqr_abs in H0; assumption. +rewrite H; reflexivity. +Qed. + +Lemma triangle_rectangle : + forall x y z:R, + 0 <= z -> Rsqr x + Rsqr y <= Rsqr z -> - z <= x <= z /\ - z <= y <= z. +intros; + generalize (plus_le_is_le (Rsqr x) (Rsqr y) (Rsqr z) (Rle_0_sqr y) H0); + rewrite Rplus_comm in H0; + generalize (plus_le_is_le (Rsqr y) (Rsqr x) (Rsqr z) (Rle_0_sqr x) H0); + intros; split; + [ split; + [ apply Rsqr_neg_pos_le_0; assumption + | apply Rsqr_incr_0_var; assumption ] + | split; + [ apply Rsqr_neg_pos_le_0; assumption + | apply Rsqr_incr_0_var; assumption ] ]. +Qed. + +Lemma triangle_rectangle_lt : + forall x y z:R, + Rsqr x + Rsqr y < Rsqr z -> Rabs x < Rabs z /\ Rabs y < Rabs z. +intros; split; + [ generalize (plus_lt_is_lt (Rsqr x) (Rsqr y) (Rsqr z) (Rle_0_sqr y) H); + intro; apply Rsqr_lt_abs_0; assumption + | rewrite Rplus_comm in H; + generalize (plus_lt_is_lt (Rsqr y) (Rsqr x) (Rsqr z) (Rle_0_sqr x) H); + intro; apply Rsqr_lt_abs_0; assumption ]. +Qed. + +Lemma triangle_rectangle_le : + forall x y z:R, + Rsqr x + Rsqr y <= Rsqr z -> Rabs x <= Rabs z /\ Rabs y <= Rabs z. +intros; split; + [ generalize (plus_le_is_le (Rsqr x) (Rsqr y) (Rsqr z) (Rle_0_sqr y) H); + intro; apply Rsqr_le_abs_0; assumption + | rewrite Rplus_comm in H; + generalize (plus_le_is_le (Rsqr y) (Rsqr x) (Rsqr z) (Rle_0_sqr x) H); + intro; apply Rsqr_le_abs_0; assumption ]. +Qed. + +Lemma Rsqr_inv : forall x:R, x <> 0 -> Rsqr (/ x) = / Rsqr x. +intros; unfold Rsqr in |- *. +rewrite Rinv_mult_distr; try reflexivity || assumption. +Qed. + +Lemma canonical_Rsqr : + forall (a:nonzeroreal) (b c x:R), + a * Rsqr x + b * x + c = + a * Rsqr (x + b / (2 * a)) + (4 * a * c - Rsqr b) / (4 * a). +intros. +rewrite Rsqr_plus. +repeat rewrite Rmult_plus_distr_l. +repeat rewrite Rplus_assoc. +apply Rplus_eq_compat_l. +unfold Rdiv, Rminus in |- *. +replace (2 * 1 + 2 * 1) with 4; [ idtac | ring ]. +rewrite (Rmult_plus_distr_r (4 * a * c) (- Rsqr b) (/ (4 * a))). +rewrite Rsqr_mult. +repeat rewrite Rinv_mult_distr. +repeat rewrite (Rmult_comm a). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite (Rmult_comm 2). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite (Rmult_comm (/ 2)). +rewrite (Rmult_comm 2). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite (Rmult_comm a). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite (Rmult_comm 2). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +repeat rewrite Rplus_assoc. +rewrite (Rplus_comm (Rsqr b * (Rsqr (/ a * / 2) * a))). +repeat rewrite Rplus_assoc. +rewrite (Rmult_comm x). +apply Rplus_eq_compat_l. +rewrite (Rmult_comm (/ a)). +unfold Rsqr in |- *; repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +ring. +apply (cond_nonzero a). +discrR. +apply (cond_nonzero a). +discrR. +discrR. +apply (cond_nonzero a). +discrR. +discrR. +discrR. +apply (cond_nonzero a). +discrR. +apply (cond_nonzero a). +Qed. + +Lemma Rsqr_eq : forall x y:R, Rsqr x = Rsqr y -> x = y \/ x = - y. +intros; unfold Rsqr in H; + generalize (Rplus_eq_compat_l (- (y * y)) (x * x) (y * y) H); + rewrite Rplus_opp_l; replace (- (y * y) + x * x) with ((x - y) * (x + y)). +intro; generalize (Rmult_integral (x - y) (x + y) H0); intro; elim H1; intros. +left; apply Rminus_diag_uniq; assumption. +right; apply Rminus_diag_uniq; unfold Rminus in |- *; rewrite Ropp_involutive; + assumption. +ring. +Qed.
\ No newline at end of file diff --git a/theories/Reals/R_sqrt.v b/theories/Reals/R_sqrt.v index 759e4b164..f4d5ccf1a 100644 --- a/theories/Reals/R_sqrt.v +++ b/theories/Reals/R_sqrt.v @@ -8,244 +8,392 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Rsqrt_def. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rfunctions. +Require Import Rsqrt_def. Open Local Scope R_scope. (* Here is a continuous extension of Rsqrt on R *) -Definition sqrt : R->R := [x:R](Cases (case_Rabsolu x) of - (leftT _) => R0 - | (rightT a) => (Rsqrt (mknonnegreal x (Rle_sym2 ? ? a))) end). - -Lemma sqrt_positivity : (x:R) ``0<=x`` -> ``0<=(sqrt x)``. -Intros. -Unfold sqrt. -Case (case_Rabsolu x); Intro. -Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? r H)). -Apply Rsqrt_positivity. +Definition sqrt (x:R) : R := + match Rcase_abs x with + | left _ => 0 + | right a => Rsqrt (mknonnegreal x (Rge_le _ _ a)) + end. + +Lemma sqrt_positivity : forall x:R, 0 <= x -> 0 <= sqrt x. +intros. +unfold sqrt in |- *. +case (Rcase_abs x); intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ r H)). +apply Rsqrt_positivity. Qed. -Lemma sqrt_sqrt : (x:R) ``0<=x`` -> ``(sqrt x)*(sqrt x)==x``. -Intros. -Unfold sqrt. -Case (case_Rabsolu x); Intro. -Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? r H)). -Rewrite Rsqrt_Rsqrt; Reflexivity. +Lemma sqrt_sqrt : forall x:R, 0 <= x -> sqrt x * sqrt x = x. +intros. +unfold sqrt in |- *. +case (Rcase_abs x); intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ r H)). +rewrite Rsqrt_Rsqrt; reflexivity. Qed. -Lemma sqrt_0 : ``(sqrt 0)==0``. -Apply Rsqr_eq_0; Unfold Rsqr; Apply sqrt_sqrt; Right; Reflexivity. +Lemma sqrt_0 : sqrt 0 = 0. +apply Rsqr_eq_0; unfold Rsqr in |- *; apply sqrt_sqrt; right; reflexivity. Qed. -Lemma sqrt_1 : ``(sqrt 1)==1``. -Apply (Rsqr_inj (sqrt R1) R1); [Apply sqrt_positivity; Left | Left | Unfold Rsqr; Rewrite -> sqrt_sqrt; [Ring | Left]]; Apply Rlt_R0_R1. +Lemma sqrt_1 : sqrt 1 = 1. +apply (Rsqr_inj (sqrt 1) 1); + [ apply sqrt_positivity; left + | left + | unfold Rsqr in |- *; rewrite sqrt_sqrt; [ ring | left ] ]; + apply Rlt_0_1. Qed. -Lemma sqrt_eq_0 : (x:R) ``0<=x``->``(sqrt x)==0``->``x==0``. -Intros; Cut ``(Rsqr (sqrt x))==0``. -Intro; Unfold Rsqr in H1; Rewrite -> sqrt_sqrt in H1; Assumption. -Rewrite H0; Apply Rsqr_O. +Lemma sqrt_eq_0 : forall x:R, 0 <= x -> sqrt x = 0 -> x = 0. +intros; cut (Rsqr (sqrt x) = 0). +intro; unfold Rsqr in H1; rewrite sqrt_sqrt in H1; assumption. +rewrite H0; apply Rsqr_0. Qed. -Lemma sqrt_lem_0 : (x,y:R) ``0<=x``->``0<=y``->(sqrt x)==y->``y*y==x``. -Intros; Rewrite <- H1; Apply (sqrt_sqrt x H). +Lemma sqrt_lem_0 : forall x y:R, 0 <= x -> 0 <= y -> sqrt x = y -> y * y = x. +intros; rewrite <- H1; apply (sqrt_sqrt x H). Qed. -Lemma sqtr_lem_1 : (x,y:R) ``0<=x``->``0<=y``->``y*y==x``->(sqrt x)==y. -Intros; Apply Rsqr_inj; [Apply (sqrt_positivity x H) | Assumption | Unfold Rsqr; Rewrite -> H1; Apply (sqrt_sqrt x H)]. +Lemma sqtr_lem_1 : forall x y:R, 0 <= x -> 0 <= y -> y * y = x -> sqrt x = y. +intros; apply Rsqr_inj; + [ apply (sqrt_positivity x H) + | assumption + | unfold Rsqr in |- *; rewrite H1; apply (sqrt_sqrt x H) ]. Qed. -Lemma sqrt_def : (x:R) ``0<=x``->``(sqrt x)*(sqrt x)==x``. -Intros; Apply (sqrt_sqrt x H). +Lemma sqrt_def : forall x:R, 0 <= x -> sqrt x * sqrt x = x. +intros; apply (sqrt_sqrt x H). Qed. -Lemma sqrt_square : (x:R) ``0<=x``->``(sqrt (x*x))==x``. -Intros; Apply (Rsqr_inj (sqrt (Rsqr x)) x (sqrt_positivity (Rsqr x) (pos_Rsqr x)) H); Unfold Rsqr; Apply (sqrt_sqrt (Rsqr x) (pos_Rsqr x)). +Lemma sqrt_square : forall x:R, 0 <= x -> sqrt (x * x) = x. +intros; + apply + (Rsqr_inj (sqrt (Rsqr x)) x (sqrt_positivity (Rsqr x) (Rle_0_sqr x)) H); + unfold Rsqr in |- *; apply (sqrt_sqrt (Rsqr x) (Rle_0_sqr x)). Qed. -Lemma sqrt_Rsqr : (x:R) ``0<=x``->``(sqrt (Rsqr x))==x``. -Intros; Unfold Rsqr; Apply sqrt_square; Assumption. +Lemma sqrt_Rsqr : forall x:R, 0 <= x -> sqrt (Rsqr x) = x. +intros; unfold Rsqr in |- *; apply sqrt_square; assumption. Qed. -Lemma sqrt_Rsqr_abs : (x:R) (sqrt (Rsqr x))==(Rabsolu x). -Intro x; Rewrite -> Rsqr_abs; Apply sqrt_Rsqr; Apply Rabsolu_pos. +Lemma sqrt_Rsqr_abs : forall x:R, sqrt (Rsqr x) = Rabs x. +intro x; rewrite Rsqr_abs; apply sqrt_Rsqr; apply Rabs_pos. Qed. -Lemma Rsqr_sqrt : (x:R) ``0<=x``->(Rsqr (sqrt x))==x. -Intros x H1; Unfold Rsqr; Apply (sqrt_sqrt x H1). +Lemma Rsqr_sqrt : forall x:R, 0 <= x -> Rsqr (sqrt x) = x. +intros x H1; unfold Rsqr in |- *; apply (sqrt_sqrt x H1). Qed. -Lemma sqrt_times : (x,y:R) ``0<=x``->``0<=y``->``(sqrt (x*y))==(sqrt x)*(sqrt y)``. -Intros x y H1 H2; Apply (Rsqr_inj (sqrt (Rmult x y)) (Rmult (sqrt x) (sqrt y)) (sqrt_positivity (Rmult x y) (Rmult_le_pos x y H1 H2)) (Rmult_le_pos (sqrt x) (sqrt y) (sqrt_positivity x H1) (sqrt_positivity y H2))); Rewrite Rsqr_times; Repeat Rewrite Rsqr_sqrt; [Ring | Assumption |Assumption | Apply (Rmult_le_pos x y H1 H2)]. +Lemma sqrt_mult : + forall x y:R, 0 <= x -> 0 <= y -> sqrt (x * y) = sqrt x * sqrt y. +intros x y H1 H2; + apply + (Rsqr_inj (sqrt (x * y)) (sqrt x * sqrt y) + (sqrt_positivity (x * y) (Rmult_le_pos x y H1 H2)) + (Rmult_le_pos (sqrt x) (sqrt y) (sqrt_positivity x H1) + (sqrt_positivity y H2))); rewrite Rsqr_mult; + repeat rewrite Rsqr_sqrt; + [ ring | assumption | assumption | apply (Rmult_le_pos x y H1 H2) ]. Qed. -Lemma sqrt_lt_R0 : (x:R) ``0<x`` -> ``0<(sqrt x)``. -Intros x H1; Apply Rsqr_incrst_0; [Rewrite Rsqr_O; Rewrite Rsqr_sqrt ; [Assumption | Left; Assumption] | Right; Reflexivity | Apply (sqrt_positivity x (Rlt_le R0 x H1))]. +Lemma sqrt_lt_R0 : forall x:R, 0 < x -> 0 < sqrt x. +intros x H1; apply Rsqr_incrst_0; + [ rewrite Rsqr_0; rewrite Rsqr_sqrt; [ assumption | left; assumption ] + | right; reflexivity + | apply (sqrt_positivity x (Rlt_le 0 x H1)) ]. Qed. -Lemma sqrt_div : (x,y:R) ``0<=x``->``0<y``->``(sqrt (x/y))==(sqrt x)/(sqrt y)``. -Intros x y H1 H2; Apply Rsqr_inj; [ Apply sqrt_positivity; Apply (Rmult_le_pos x (Rinv y)); [ Assumption | Generalize (Rlt_Rinv y H2); Clear H2; Intro H2; Left; Assumption] | Apply (Rmult_le_pos (sqrt x) (Rinv (sqrt y))) ; [ Apply (sqrt_positivity x H1) | Generalize (sqrt_lt_R0 y H2); Clear H2; Intro H2; Generalize (Rlt_Rinv (sqrt y) H2); Clear H2; Intro H2; Left; Assumption] | Rewrite Rsqr_div; Repeat Rewrite Rsqr_sqrt; [ Reflexivity | Left; Assumption | Assumption | Generalize (Rlt_Rinv y H2); Intro H3; Generalize (Rlt_le R0 (Rinv y) H3); Intro H4; Apply (Rmult_le_pos x (Rinv y) H1 H4) |Red; Intro H3; Generalize (Rlt_le R0 y H2); Intro H4; Generalize (sqrt_eq_0 y H4 H3); Intro H5; Rewrite H5 in H2; Elim (Rlt_antirefl R0 H2)]]. +Lemma sqrt_div : + forall x y:R, 0 <= x -> 0 < y -> sqrt (x / y) = sqrt x / sqrt y. +intros x y H1 H2; apply Rsqr_inj; + [ apply sqrt_positivity; apply (Rmult_le_pos x (/ y)); + [ assumption + | generalize (Rinv_0_lt_compat y H2); clear H2; intro H2; left; + assumption ] + | apply (Rmult_le_pos (sqrt x) (/ sqrt y)); + [ apply (sqrt_positivity x H1) + | generalize (sqrt_lt_R0 y H2); clear H2; intro H2; + generalize (Rinv_0_lt_compat (sqrt y) H2); clear H2; + intro H2; left; assumption ] + | rewrite Rsqr_div; repeat rewrite Rsqr_sqrt; + [ reflexivity + | left; assumption + | assumption + | generalize (Rinv_0_lt_compat y H2); intro H3; + generalize (Rlt_le 0 (/ y) H3); intro H4; + apply (Rmult_le_pos x (/ y) H1 H4) + | red in |- *; intro H3; generalize (Rlt_le 0 y H2); intro H4; + generalize (sqrt_eq_0 y H4 H3); intro H5; rewrite H5 in H2; + elim (Rlt_irrefl 0 H2) ] ]. Qed. -Lemma sqrt_lt_0 : (x,y:R) ``0<=x``->``0<=y``->``(sqrt x)<(sqrt y)``->``x<y``. -Intros x y H1 H2 H3; Generalize (Rsqr_incrst_1 (sqrt x) (sqrt y) H3 (sqrt_positivity x H1) (sqrt_positivity y H2)); Intro H4; Rewrite (Rsqr_sqrt x H1) in H4; Rewrite (Rsqr_sqrt y H2) in H4; Assumption. +Lemma sqrt_lt_0 : forall x y:R, 0 <= x -> 0 <= y -> sqrt x < sqrt y -> x < y. +intros x y H1 H2 H3; + generalize + (Rsqr_incrst_1 (sqrt x) (sqrt y) H3 (sqrt_positivity x H1) + (sqrt_positivity y H2)); intro H4; rewrite (Rsqr_sqrt x H1) in H4; + rewrite (Rsqr_sqrt y H2) in H4; assumption. Qed. -Lemma sqrt_lt_1 : (x,y:R) ``0<=x``->``0<=y``->``x<y``->``(sqrt x)<(sqrt y)``. -Intros x y H1 H2 H3; Apply Rsqr_incrst_0; [Rewrite (Rsqr_sqrt x H1); Rewrite (Rsqr_sqrt y H2); Assumption | Apply (sqrt_positivity x H1) | Apply (sqrt_positivity y H2)]. +Lemma sqrt_lt_1 : forall x y:R, 0 <= x -> 0 <= y -> x < y -> sqrt x < sqrt y. +intros x y H1 H2 H3; apply Rsqr_incrst_0; + [ rewrite (Rsqr_sqrt x H1); rewrite (Rsqr_sqrt y H2); assumption + | apply (sqrt_positivity x H1) + | apply (sqrt_positivity y H2) ]. Qed. -Lemma sqrt_le_0 : (x,y:R) ``0<=x``->``0<=y``->``(sqrt x)<=(sqrt y)``->``x<=y``. -Intros x y H1 H2 H3; Generalize (Rsqr_incr_1 (sqrt x) (sqrt y) H3 (sqrt_positivity x H1) (sqrt_positivity y H2)); Intro H4; Rewrite (Rsqr_sqrt x H1) in H4; Rewrite (Rsqr_sqrt y H2) in H4; Assumption. +Lemma sqrt_le_0 : + forall x y:R, 0 <= x -> 0 <= y -> sqrt x <= sqrt y -> x <= y. +intros x y H1 H2 H3; + generalize + (Rsqr_incr_1 (sqrt x) (sqrt y) H3 (sqrt_positivity x H1) + (sqrt_positivity y H2)); intro H4; rewrite (Rsqr_sqrt x H1) in H4; + rewrite (Rsqr_sqrt y H2) in H4; assumption. Qed. -Lemma sqrt_le_1 : (x,y:R) ``0<=x``->``0<=y``->``x<=y``->``(sqrt x)<=(sqrt y)``. -Intros x y H1 H2 H3; Apply Rsqr_incr_0; [ Rewrite (Rsqr_sqrt x H1); Rewrite (Rsqr_sqrt y H2); Assumption | Apply (sqrt_positivity x H1) | Apply (sqrt_positivity y H2)]. +Lemma sqrt_le_1 : + forall x y:R, 0 <= x -> 0 <= y -> x <= y -> sqrt x <= sqrt y. +intros x y H1 H2 H3; apply Rsqr_incr_0; + [ rewrite (Rsqr_sqrt x H1); rewrite (Rsqr_sqrt y H2); assumption + | apply (sqrt_positivity x H1) + | apply (sqrt_positivity y H2) ]. Qed. -Lemma sqrt_inj : (x,y:R) ``0<=x``->``0<=y``->(sqrt x)==(sqrt y)->x==y. -Intros; Cut ``(Rsqr (sqrt x))==(Rsqr (sqrt y))``. -Intro; Rewrite (Rsqr_sqrt x H) in H2; Rewrite (Rsqr_sqrt y H0) in H2; Assumption. -Rewrite H1; Reflexivity. +Lemma sqrt_inj : forall x y:R, 0 <= x -> 0 <= y -> sqrt x = sqrt y -> x = y. +intros; cut (Rsqr (sqrt x) = Rsqr (sqrt y)). +intro; rewrite (Rsqr_sqrt x H) in H2; rewrite (Rsqr_sqrt y H0) in H2; + assumption. +rewrite H1; reflexivity. Qed. -Lemma sqrt_less : (x:R) ``0<=x``->``1<x``->``(sqrt x)<x``. -Intros x H1 H2; Generalize (sqrt_lt_1 R1 x (Rlt_le R0 R1 (Rlt_R0_R1)) H1 H2); Intro H3; Rewrite sqrt_1 in H3; Generalize (Rmult_ne (sqrt x)); Intro H4; Elim H4; Intros H5 H6; Rewrite <- H5; Pattern 2 x; Rewrite <- (sqrt_def x H1); Apply (Rlt_monotony (sqrt x) R1 (sqrt x) (sqrt_lt_R0 x (Rlt_trans R0 R1 x Rlt_R0_R1 H2)) H3). +Lemma sqrt_less : forall x:R, 0 <= x -> 1 < x -> sqrt x < x. +intros x H1 H2; generalize (sqrt_lt_1 1 x (Rlt_le 0 1 Rlt_0_1) H1 H2); + intro H3; rewrite sqrt_1 in H3; generalize (Rmult_ne (sqrt x)); + intro H4; elim H4; intros H5 H6; rewrite <- H5; pattern x at 2 in |- *; + rewrite <- (sqrt_def x H1); + apply + (Rmult_lt_compat_l (sqrt x) 1 (sqrt x) + (sqrt_lt_R0 x (Rlt_trans 0 1 x Rlt_0_1 H2)) H3). Qed. -Lemma sqrt_more : (x:R) ``0<x``->``x<1``->``x<(sqrt x)``. -Intros x H1 H2; Generalize (sqrt_lt_1 x R1 (Rlt_le R0 x H1) (Rlt_le R0 R1 (Rlt_R0_R1)) H2); Intro H3; Rewrite sqrt_1 in H3; Generalize (Rmult_ne (sqrt x)); Intro H4; Elim H4; Intros H5 H6; Rewrite <- H5; Pattern 1 x; Rewrite <- (sqrt_def x (Rlt_le R0 x H1)); Apply (Rlt_monotony (sqrt x) (sqrt x) R1 (sqrt_lt_R0 x H1) H3). +Lemma sqrt_more : forall x:R, 0 < x -> x < 1 -> x < sqrt x. +intros x H1 H2; + generalize (sqrt_lt_1 x 1 (Rlt_le 0 x H1) (Rlt_le 0 1 Rlt_0_1) H2); + intro H3; rewrite sqrt_1 in H3; generalize (Rmult_ne (sqrt x)); + intro H4; elim H4; intros H5 H6; rewrite <- H5; pattern x at 1 in |- *; + rewrite <- (sqrt_def x (Rlt_le 0 x H1)); + apply (Rmult_lt_compat_l (sqrt x) (sqrt x) 1 (sqrt_lt_R0 x H1) H3). Qed. -Lemma sqrt_cauchy : (a,b,c,d:R) ``a*c+b*d<=(sqrt ((Rsqr a)+(Rsqr b)))*(sqrt ((Rsqr c)+(Rsqr d)))``. -Intros a b c d; Apply Rsqr_incr_0_var; [Rewrite Rsqr_times; Repeat Rewrite Rsqr_sqrt; Unfold Rsqr; [Replace ``(a*c+b*d)*(a*c+b*d)`` with ``(a*a*c*c+b*b*d*d)+(2*a*b*c*d)``; [Replace ``(a*a+b*b)*(c*c+d*d)`` with ``(a*a*c*c+b*b*d*d)+(a*a*d*d+b*b*c*c)``; [Apply Rle_compatibility; Replace ``a*a*d*d+b*b*c*c`` with ``(2*a*b*c*d)+(a*a*d*d+b*b*c*c-2*a*b*c*d)``; [Pattern 1 ``2*a*b*c*d``; Rewrite <- Rplus_Or; Apply Rle_compatibility; Replace ``a*a*d*d+b*b*c*c-2*a*b*c*d`` with (Rsqr (Rminus (Rmult a d) (Rmult b c))); [Apply pos_Rsqr | Unfold Rsqr; Ring] | Ring] | Ring] | Ring] | Apply (ge0_plus_ge0_is_ge0 (Rsqr c) (Rsqr d) (pos_Rsqr c) (pos_Rsqr d)) | Apply (ge0_plus_ge0_is_ge0 (Rsqr a) (Rsqr b) (pos_Rsqr a) (pos_Rsqr b))] | Apply Rmult_le_pos; Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0; Apply pos_Rsqr]. +Lemma sqrt_cauchy : + forall a b c d:R, + a * c + b * d <= sqrt (Rsqr a + Rsqr b) * sqrt (Rsqr c + Rsqr d). +intros a b c d; apply Rsqr_incr_0_var; + [ rewrite Rsqr_mult; repeat rewrite Rsqr_sqrt; unfold Rsqr in |- *; + [ replace ((a * c + b * d) * (a * c + b * d)) with + (a * a * c * c + b * b * d * d + 2 * a * b * c * d); + [ replace ((a * a + b * b) * (c * c + d * d)) with + (a * a * c * c + b * b * d * d + (a * a * d * d + b * b * c * c)); + [ apply Rplus_le_compat_l; + replace (a * a * d * d + b * b * c * c) with + (2 * a * b * c * d + + (a * a * d * d + b * b * c * c - 2 * a * b * c * d)); + [ pattern (2 * a * b * c * d) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l; + replace (a * a * d * d + b * b * c * c - 2 * a * b * c * d) + with (Rsqr (a * d - b * c)); + [ apply Rle_0_sqr | unfold Rsqr in |- *; ring ] + | ring ] + | ring ] + | ring ] + | apply + (Rplus_le_le_0_compat (Rsqr c) (Rsqr d) (Rle_0_sqr c) (Rle_0_sqr d)) + | apply + (Rplus_le_le_0_compat (Rsqr a) (Rsqr b) (Rle_0_sqr a) (Rle_0_sqr b)) ] + | apply Rmult_le_pos; apply sqrt_positivity; apply Rplus_le_le_0_compat; + apply Rle_0_sqr ]. Qed. (************************************************************) (* Resolution of [a*X^2+b*X+c=0] *) (************************************************************) -Definition Delta [a:nonzeroreal;b,c:R] : R := ``(Rsqr b)-4*a*c``. - -Definition Delta_is_pos [a:nonzeroreal;b,c:R] : Prop := ``0<=(Delta a b c)``. - -Definition sol_x1 [a:nonzeroreal;b,c:R] : R := ``(-b+(sqrt (Delta a b c)))/(2*a)``. - -Definition sol_x2 [a:nonzeroreal;b,c:R] : R := ``(-b-(sqrt (Delta a b c)))/(2*a)``. - -Lemma Rsqr_sol_eq_0_1 : (a:nonzeroreal;b,c,x:R) (Delta_is_pos a b c) -> (x==(sol_x1 a b c))\/(x==(sol_x2 a b c)) -> ``a*(Rsqr x)+b*x+c==0``. -Intros; Elim H0; Intro. -Unfold sol_x1 in H1; Unfold Delta in H1; Rewrite H1; Unfold Rdiv; Repeat Rewrite Rsqr_times; Rewrite Rsqr_plus; Rewrite <- Rsqr_neg; Rewrite Rsqr_sqrt. -Rewrite Rsqr_inv. -Unfold Rsqr; Repeat Rewrite Rinv_Rmult. -Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym a). -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite Rmult_Rplus_distrl. -Repeat Rewrite Rmult_assoc. -Pattern 2 ``2``; Rewrite (Rmult_sym ``2``). -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Rewrite (Rmult_Rplus_distrl ``-b`` ``(sqrt (b*b-(2*(2*(a*c)))))`` ``(/2*/a)``). -Rewrite Rmult_Rplus_distr; Repeat Rewrite Rplus_assoc. -Replace ``( -b*((sqrt (b*b-(2*(2*(a*c)))))*(/2*/a))+(b*( -b*(/2*/a))+(b*((sqrt (b*b-(2*(2*(a*c)))))*(/2*/a))+c)))`` with ``(b*( -b*(/2*/a)))+c``. -Unfold Rminus; Repeat Rewrite <- Rplus_assoc. -Replace ``b*b+b*b`` with ``2*(b*b)``. -Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Rewrite Ropp_mul1; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym ``2``). -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite (Rmult_sym ``/2``); Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym ``2``). -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym a); Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite <- Ropp_mul2. -Ring. -Apply (cond_nonzero a). -DiscrR. -DiscrR. -DiscrR. -Ring. -Ring. -DiscrR. -Apply (cond_nonzero a). -DiscrR. -Apply (cond_nonzero a). -Apply prod_neq_R0; [DiscrR | Apply (cond_nonzero a)]. -Apply prod_neq_R0; [DiscrR | Apply (cond_nonzero a)]. -Apply prod_neq_R0; [DiscrR | Apply (cond_nonzero a)]. -Assumption. -Unfold sol_x2 in H1; Unfold Delta in H1; Rewrite H1; Unfold Rdiv; Repeat Rewrite Rsqr_times; Rewrite Rsqr_minus; Rewrite <- Rsqr_neg; Rewrite Rsqr_sqrt. -Rewrite Rsqr_inv. -Unfold Rsqr; Repeat Rewrite Rinv_Rmult; Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym a); Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Unfold Rminus; Rewrite Rmult_Rplus_distrl. -Rewrite Ropp_mul1; Repeat Rewrite Rmult_assoc; Pattern 2 ``2``; Rewrite (Rmult_sym ``2``). -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite (Rmult_Rplus_distrl ``-b`` ``-(sqrt (b*b+ -(2*(2*(a*c))))) `` ``(/2*/a)``). -Rewrite Rmult_Rplus_distr; Repeat Rewrite Rplus_assoc. -Rewrite Ropp_mul1; Rewrite Ropp_Ropp. -Replace ``(b*((sqrt (b*b+ -(2*(2*(a*c)))))*(/2*/a))+(b*( -b*(/2*/a))+(b*( -(sqrt (b*b+ -(2*(2*(a*c)))))*(/2*/a))+c)))`` with ``(b*( -b*(/2*/a)))+c``. -Repeat Rewrite <- Rplus_assoc; Replace ``b*b+b*b`` with ``2*(b*b)``. -Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Ropp_mul1; Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite (Rmult_sym ``/2``); Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym a); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite <- Ropp_mul2; Ring. -Apply (cond_nonzero a). -DiscrR. -DiscrR. -DiscrR. -Ring. -Ring. -DiscrR. -Apply (cond_nonzero a). -DiscrR. -DiscrR. -Apply (cond_nonzero a). -Apply prod_neq_R0; DiscrR Orelse Apply (cond_nonzero a). -Apply prod_neq_R0; DiscrR Orelse Apply (cond_nonzero a). -Apply prod_neq_R0; DiscrR Orelse Apply (cond_nonzero a). -Assumption. -Qed. - -Lemma Rsqr_sol_eq_0_0 : (a:nonzeroreal;b,c,x:R) (Delta_is_pos a b c) -> ``a*(Rsqr x)+b*x+c==0`` -> (x==(sol_x1 a b c))\/(x==(sol_x2 a b c)). -Intros; Rewrite (canonical_Rsqr a b c x) in H0; Rewrite Rplus_sym in H0; Generalize (Rplus_Ropp ``(4*a*c-(Rsqr b))/(4*a)`` ``a*(Rsqr (x+b/(2*a)))`` H0); Cut ``(Rsqr b)-4*a*c==(Delta a b c)``. -Intro; Replace ``-((4*a*c-(Rsqr b))/(4*a))`` with ``((Rsqr b)-4*a*c)/(4*a)``. -Rewrite H1; Intro; Generalize (Rmult_mult_r ``/a`` ``a*(Rsqr (x+b/(2*a)))`` ``(Delta a b c)/(4*a)`` H2); Replace ``/a*(a*(Rsqr (x+b/(2*a))))`` with ``(Rsqr (x+b/(2*a)))``. -Replace ``/a*(Delta a b c)/(4*a)`` with ``(Rsqr ((sqrt (Delta a b c))/(2*a)))``. -Intro; Generalize (Rsqr_eq ``(x+b/(2*a))`` ``((sqrt (Delta a b c))/(2*a))`` H3); Intro; Elim H4; Intro. -Left; Unfold sol_x1; Generalize (Rplus_plus_r ``-(b/(2*a))`` ``x+b/(2*a)`` ``(sqrt (Delta a b c))/(2*a)`` H5); Replace `` -(b/(2*a))+(x+b/(2*a))`` with x. -Intro; Rewrite H6; Unfold Rdiv; Ring. -Ring. -Right; Unfold sol_x2; Generalize (Rplus_plus_r ``-(b/(2*a))`` ``x+b/(2*a)`` ``-((sqrt (Delta a b c))/(2*a))`` H5); Replace `` -(b/(2*a))+(x+b/(2*a))`` with x. -Intro; Rewrite H6; Unfold Rdiv; Ring. -Ring. -Rewrite Rsqr_div. -Rewrite Rsqr_sqrt. -Unfold Rdiv. -Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym ``/a``). -Rewrite Rmult_assoc. -Rewrite <- Rinv_Rmult. -Replace ``(2*(2*a))*a`` with ``(Rsqr (2*a))``. -Reflexivity. -SqRing. -Rewrite <- Rmult_assoc; Apply prod_neq_R0; [DiscrR | Apply (cond_nonzero a)]. -Apply (cond_nonzero a). -Assumption. -Apply prod_neq_R0; [DiscrR | Apply (cond_nonzero a)]. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Symmetry; Apply Rmult_1l. -Apply (cond_nonzero a). -Unfold Rdiv; Rewrite <- Ropp_mul1. -Rewrite Ropp_distr2. -Reflexivity. -Reflexivity. +Definition Delta (a:nonzeroreal) (b c:R) : R := Rsqr b - 4 * a * c. + +Definition Delta_is_pos (a:nonzeroreal) (b c:R) : Prop := 0 <= Delta a b c. + +Definition sol_x1 (a:nonzeroreal) (b c:R) : R := + (- b + sqrt (Delta a b c)) / (2 * a). + +Definition sol_x2 (a:nonzeroreal) (b c:R) : R := + (- b - sqrt (Delta a b c)) / (2 * a). + +Lemma Rsqr_sol_eq_0_1 : + forall (a:nonzeroreal) (b c x:R), + Delta_is_pos a b c -> + x = sol_x1 a b c \/ x = sol_x2 a b c -> a * Rsqr x + b * x + c = 0. +intros; elim H0; intro. +unfold sol_x1 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv in |- *; + repeat rewrite Rsqr_mult; rewrite Rsqr_plus; rewrite <- Rsqr_neg; + rewrite Rsqr_sqrt. +rewrite Rsqr_inv. +unfold Rsqr in |- *; repeat rewrite Rinv_mult_distr. +repeat rewrite Rmult_assoc; rewrite (Rmult_comm a). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite Rmult_plus_distr_r. +repeat rewrite Rmult_assoc. +pattern 2 at 2 in |- *; rewrite (Rmult_comm 2). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite + (Rmult_plus_distr_r (- b) (sqrt (b * b - 2 * (2 * (a * c)))) (/ 2 * / a)) + . +rewrite Rmult_plus_distr_l; repeat rewrite Rplus_assoc. +replace + (- b * (sqrt (b * b - 2 * (2 * (a * c))) * (/ 2 * / a)) + + (b * (- b * (/ 2 * / a)) + + (b * (sqrt (b * b - 2 * (2 * (a * c))) * (/ 2 * / a)) + c))) with + (b * (- b * (/ 2 * / a)) + c). +unfold Rminus in |- *; repeat rewrite <- Rplus_assoc. +replace (b * b + b * b) with (2 * (b * b)). +rewrite Rmult_plus_distr_r; repeat rewrite Rmult_assoc. +rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc; + rewrite (Rmult_comm 2). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite (Rmult_comm (/ 2)); repeat rewrite Rmult_assoc; + rewrite (Rmult_comm 2). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; repeat rewrite Rmult_assoc. +rewrite (Rmult_comm a); rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite <- Rmult_opp_opp. +ring. +apply (cond_nonzero a). +discrR. +discrR. +discrR. +ring. +ring. +discrR. +apply (cond_nonzero a). +discrR. +apply (cond_nonzero a). +apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. +apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. +apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. +assumption. +unfold sol_x2 in H1; unfold Delta in H1; rewrite H1; unfold Rdiv in |- *; + repeat rewrite Rsqr_mult; rewrite Rsqr_minus; rewrite <- Rsqr_neg; + rewrite Rsqr_sqrt. +rewrite Rsqr_inv. +unfold Rsqr in |- *; repeat rewrite Rinv_mult_distr; + repeat rewrite Rmult_assoc. +rewrite (Rmult_comm a); repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; unfold Rminus in |- *; rewrite Rmult_plus_distr_r. +rewrite Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc; + pattern 2 at 2 in |- *; rewrite (Rmult_comm 2). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; + rewrite + (Rmult_plus_distr_r (- b) (- sqrt (b * b + - (2 * (2 * (a * c))))) + (/ 2 * / a)). +rewrite Rmult_plus_distr_l; repeat rewrite Rplus_assoc. +rewrite Ropp_mult_distr_l_reverse; rewrite Ropp_involutive. +replace + (b * (sqrt (b * b + - (2 * (2 * (a * c)))) * (/ 2 * / a)) + + (b * (- b * (/ 2 * / a)) + + (b * (- sqrt (b * b + - (2 * (2 * (a * c)))) * (/ 2 * / a)) + c))) with + (b * (- b * (/ 2 * / a)) + c). +repeat rewrite <- Rplus_assoc; replace (b * b + b * b) with (2 * (b * b)). +rewrite Rmult_plus_distr_r; repeat rewrite Rmult_assoc; + rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc. +rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite (Rmult_comm (/ 2)); repeat rewrite Rmult_assoc. +rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; repeat rewrite Rmult_assoc; rewrite (Rmult_comm a); + rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite <- Rmult_opp_opp; ring. +apply (cond_nonzero a). +discrR. +discrR. +discrR. +ring. +ring. +discrR. +apply (cond_nonzero a). +discrR. +discrR. +apply (cond_nonzero a). +apply prod_neq_R0; discrR || apply (cond_nonzero a). +apply prod_neq_R0; discrR || apply (cond_nonzero a). +apply prod_neq_R0; discrR || apply (cond_nonzero a). +assumption. Qed. + +Lemma Rsqr_sol_eq_0_0 : + forall (a:nonzeroreal) (b c x:R), + Delta_is_pos a b c -> + a * Rsqr x + b * x + c = 0 -> x = sol_x1 a b c \/ x = sol_x2 a b c. +intros; rewrite (canonical_Rsqr a b c x) in H0; rewrite Rplus_comm in H0; + generalize + (Rplus_opp_r_uniq ((4 * a * c - Rsqr b) / (4 * a)) + (a * Rsqr (x + b / (2 * a))) H0); cut (Rsqr b - 4 * a * c = Delta a b c). +intro; + replace (- ((4 * a * c - Rsqr b) / (4 * a))) with + ((Rsqr b - 4 * a * c) / (4 * a)). +rewrite H1; intro; + generalize + (Rmult_eq_compat_l (/ a) (a * Rsqr (x + b / (2 * a))) + (Delta a b c / (4 * a)) H2); + replace (/ a * (a * Rsqr (x + b / (2 * a)))) with (Rsqr (x + b / (2 * a))). +replace (/ a * (Delta a b c / (4 * a))) with + (Rsqr (sqrt (Delta a b c) / (2 * a))). +intro; + generalize (Rsqr_eq (x + b / (2 * a)) (sqrt (Delta a b c) / (2 * a)) H3); + intro; elim H4; intro. +left; unfold sol_x1 in |- *; + generalize + (Rplus_eq_compat_l (- (b / (2 * a))) (x + b / (2 * a)) + (sqrt (Delta a b c) / (2 * a)) H5); + replace (- (b / (2 * a)) + (x + b / (2 * a))) with x. +intro; rewrite H6; unfold Rdiv in |- *; ring. +ring. +right; unfold sol_x2 in |- *; + generalize + (Rplus_eq_compat_l (- (b / (2 * a))) (x + b / (2 * a)) + (- (sqrt (Delta a b c) / (2 * a))) H5); + replace (- (b / (2 * a)) + (x + b / (2 * a))) with x. +intro; rewrite H6; unfold Rdiv in |- *; ring. +ring. +rewrite Rsqr_div. +rewrite Rsqr_sqrt. +unfold Rdiv in |- *. +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (/ a)). +rewrite Rmult_assoc. +rewrite <- Rinv_mult_distr. +replace (2 * (2 * a) * a) with (Rsqr (2 * a)). +reflexivity. +ring_Rsqr. +rewrite <- Rmult_assoc; apply prod_neq_R0; + [ discrR | apply (cond_nonzero a) ]. +apply (cond_nonzero a). +assumption. +apply prod_neq_R0; [ discrR | apply (cond_nonzero a) ]. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +symmetry in |- *; apply Rmult_1_l. +apply (cond_nonzero a). +unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse. +rewrite Ropp_minus_distr. +reflexivity. +reflexivity. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Ranalysis.v b/theories/Reals/Ranalysis.v index 4f944995c..eee3f2daf 100644 --- a/theories/Reals/Ranalysis.v +++ b/theories/Reals/Ranalysis.v @@ -8,10 +8,10 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Rtrigo. -Require SeqSeries. +Require Import Rbase. +Require Import Rfunctions. +Require Import Rtrigo. +Require Import SeqSeries. Require Export Ranalysis1. Require Export Ranalysis2. Require Export Ranalysis3. @@ -27,451 +27,776 @@ Require Export Rgeom. Require Export RList. Require Export Sqrt_reg. Require Export Ranalysis4. -Require Export Rpower. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Export Rpower. Open Local Scope R_scope. Axiom AppVar : R. (**********) -Recursive Tactic Definition IntroHypG trm := -Match trm With -|[(plus_fct ?1 ?2)] -> - (Match Context With - |[|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 - |[|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 - | _ -> Idtac) -|[(minus_fct ?1 ?2)] -> - (Match Context With - |[|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 - |[|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 - | _ -> Idtac) -|[(mult_fct ?1 ?2)] -> - (Match Context With - |[|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 - |[|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 - | _ -> Idtac) -|[(div_fct ?1 ?2)] -> Let aux = ?2 In - (Match Context With - |[_:(x0:R)``(aux x0)<>0``|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 - |[_:(x0:R)``(aux x0)<>0``|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 - |[|-(derivable ?)] -> Cut ((x0:R)``(aux x0)<>0``); [Intro; IntroHypG ?1; IntroHypG ?2 | Try Assumption] - |[|-(continuity ?)] -> Cut ((x0:R)``(aux x0)<>0``); [Intro; IntroHypG ?1; IntroHypG ?2 | Try Assumption] - | _ -> Idtac) -|[(comp ?1 ?2)] -> - (Match Context With - |[|-(derivable ?)] -> IntroHypG ?1; IntroHypG ?2 - |[|-(continuity ?)] -> IntroHypG ?1; IntroHypG ?2 - | _ -> Idtac) -|[(opp_fct ?1)] -> - (Match Context With - |[|-(derivable ?)] -> IntroHypG ?1 - |[|-(continuity ?)] -> IntroHypG ?1 - | _ -> Idtac) -|[(inv_fct ?1)] -> Let aux = ?1 In - (Match Context With - |[_:(x0:R)``(aux x0)<>0``|-(derivable ?)] -> IntroHypG ?1 - |[_:(x0:R)``(aux x0)<>0``|-(continuity ?)] -> IntroHypG ?1 - |[|-(derivable ?)] -> Cut ((x0:R)``(aux x0)<>0``); [Intro; IntroHypG ?1 | Try Assumption] - |[|-(continuity ?)] -> Cut ((x0:R)``(aux x0)<>0``); [Intro; IntroHypG ?1| Try Assumption] - | _ -> Idtac) -|[cos] -> Idtac -|[sin] -> Idtac -|[cosh] -> Idtac -|[sinh] -> Idtac -|[exp] -> Idtac -|[Rsqr] -> Idtac -|[sqrt] -> Idtac -|[id] -> Idtac -|[(fct_cte ?)] -> Idtac -|[(pow_fct ?)] -> Idtac -|[Rabsolu] -> Idtac -|[?1] -> Let p = ?1 In - (Match Context With - |[_:(derivable p)|- ?] -> Idtac - |[|-(derivable p)] -> Idtac - |[|-(derivable ?)] -> Cut True -> (derivable p); [Intro HYPPD; Cut (derivable p); [Intro; Clear HYPPD | Apply HYPPD; Clear HYPPD; Trivial] | Idtac] - | [_:(continuity p)|- ?] -> Idtac - |[|-(continuity p)] -> Idtac - |[|-(continuity ?)] -> Cut True -> (continuity p); [Intro HYPPD; Cut (continuity p); [Intro; Clear HYPPD | Apply HYPPD; Clear HYPPD; Trivial] | Idtac] - | _ -> Idtac). +Ltac intro_hyp_glob trm := + match constr:trm with + | (?X1 + ?X2)%F => + match goal with + | |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2 + | |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2 + | _ => idtac + end + | (?X1 - ?X2)%F => + match goal with + | |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2 + | |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2 + | _ => idtac + end + | (?X1 * ?X2)%F => + match goal with + | |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2 + | |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2 + | _ => idtac + end + | (?X1 / ?X2)%F => + let aux := constr:X2 in + match goal with + | _:(forall x0:R, aux x0 <> 0) |- (derivable _) => + intro_hyp_glob X1; intro_hyp_glob X2 + | _:(forall x0:R, aux x0 <> 0) |- (continuity _) => + intro_hyp_glob X1; intro_hyp_glob X2 + | |- (derivable _) => + cut (forall x0:R, aux x0 <> 0); + [ intro; intro_hyp_glob X1; intro_hyp_glob X2 | try assumption ] + | |- (continuity _) => + cut (forall x0:R, aux x0 <> 0); + [ intro; intro_hyp_glob X1; intro_hyp_glob X2 | try assumption ] + | _ => idtac + end + | (comp ?X1 ?X2) => + match goal with + | |- (derivable _) => intro_hyp_glob X1; intro_hyp_glob X2 + | |- (continuity _) => intro_hyp_glob X1; intro_hyp_glob X2 + | _ => idtac + end + | (- ?X1)%F => + match goal with + | |- (derivable _) => intro_hyp_glob X1 + | |- (continuity _) => intro_hyp_glob X1 + | _ => idtac + end + | (/ ?X1)%F => + let aux := constr:X1 in + match goal with + | _:(forall x0:R, aux x0 <> 0) |- (derivable _) => + intro_hyp_glob X1 + | _:(forall x0:R, aux x0 <> 0) |- (continuity _) => + intro_hyp_glob X1 + | |- (derivable _) => + cut (forall x0:R, aux x0 <> 0); + [ intro; intro_hyp_glob X1 | try assumption ] + | |- (continuity _) => + cut (forall x0:R, aux x0 <> 0); + [ intro; intro_hyp_glob X1 | try assumption ] + | _ => idtac + end + | cos => idtac + | sin => idtac + | cosh => idtac + | sinh => idtac + | exp => idtac + | Rsqr => idtac + | sqrt => idtac + | id => idtac + | (fct_cte _) => idtac + | (pow_fct _) => idtac + | Rabs => idtac + | ?X1 => + let p := constr:X1 in + match goal with + | _:(derivable p) |- _ => idtac + | |- (derivable p) => idtac + | |- (derivable _) => + cut (True -> derivable p); + [ intro HYPPD; cut (derivable p); + [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ] + | idtac ] + | _:(continuity p) |- _ => idtac + | |- (continuity p) => idtac + | |- (continuity _) => + cut (True -> continuity p); + [ intro HYPPD; cut (continuity p); + [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ] + | idtac ] + | _ => idtac + end + end. (**********) -Recursive Tactic Definition IntroHypL trm pt := -Match trm With -|[(plus_fct ?1 ?2)] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(continuity_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - | _ -> Idtac) -|[(minus_fct ?1 ?2)] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(continuity_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - | _ -> Idtac) -|[(mult_fct ?1 ?2)] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(continuity_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - | _ -> Idtac) -|[(div_fct ?1 ?2)] -> Let aux = ?2 In - (Match Context With - |[_:``(aux pt)<>0``|-(derivable_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[_:``(aux pt)<>0``|-(continuity_pt ? ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[_:``(aux pt)<>0``|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt; IntroHypL ?2 pt - |[id:(x0:R)``(aux x0)<>0``|-(derivable_pt ? ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt; IntroHypL ?2 pt - |[id:(x0:R)``(aux x0)<>0``|-(continuity_pt ? ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt; IntroHypL ?2 pt - |[id:(x0:R)``(aux x0)<>0``|-(eqT ? (derive_pt ? ? ?) ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt; IntroHypL ?2 pt - |[|-(derivable_pt ? ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt; IntroHypL ?2 pt | Try Assumption] - |[|-(continuity_pt ? ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt; IntroHypL ?2 pt | Try Assumption] - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt; IntroHypL ?2 pt | Try Assumption] - | _ -> Idtac) -|[(comp ?1 ?2)] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In IntroHypL ?1 pt_f1; IntroHypL ?2 pt - |[|-(continuity_pt ? ?)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In IntroHypL ?1 pt_f1; IntroHypL ?2 pt - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In IntroHypL ?1 pt_f1; IntroHypL ?2 pt - | _ -> Idtac) -|[(opp_fct ?1)] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> IntroHypL ?1 pt - |[|-(continuity_pt ? ?)] -> IntroHypL ?1 pt - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt - | _ -> Idtac) -|[(inv_fct ?1)] -> Let aux = ?1 In - (Match Context With - |[_:``(aux pt)<>0``|-(derivable_pt ? ?)] -> IntroHypL ?1 pt - |[_:``(aux pt)<>0``|-(continuity_pt ? ?)] -> IntroHypL ?1 pt - |[_:``(aux pt)<>0``|-(eqT ? (derive_pt ? ? ?) ?)] -> IntroHypL ?1 pt - |[id:(x0:R)``(aux x0)<>0``|-(derivable_pt ? ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt - |[id:(x0:R)``(aux x0)<>0``|-(continuity_pt ? ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt - |[id:(x0:R)``(aux x0)<>0``|-(eqT ? (derive_pt ? ? ?) ?)] -> Generalize (id pt); Intro; IntroHypL ?1 pt - |[|-(derivable_pt ? ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt | Try Assumption] - |[|-(continuity_pt ? ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt| Try Assumption] - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Cut ``(aux pt)<>0``; [Intro; IntroHypL ?1 pt | Try Assumption] - | _ -> Idtac) -|[cos] -> Idtac -|[sin] -> Idtac -|[cosh] -> Idtac -|[sinh] -> Idtac -|[exp] -> Idtac -|[Rsqr] -> Idtac -|[id] -> Idtac -|[(fct_cte ?)] -> Idtac -|[(pow_fct ?)] -> Idtac -|[sqrt] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> Cut ``0<pt``; [Intro | Try Assumption] - |[|-(continuity_pt ? ?)] -> Cut ``0<=pt``; [Intro | Try Assumption] - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Cut ``0<pt``; [Intro | Try Assumption] - | _ -> Idtac) -|[Rabsolu] -> - (Match Context With - |[|-(derivable_pt ? ?)] -> Cut ``pt<>0``; [Intro | Try Assumption] - | _ -> Idtac) -|[?1] -> Let p = ?1 In - (Match Context With - |[_:(derivable_pt p pt)|- ?] -> Idtac - |[|-(derivable_pt p pt)] -> Idtac - |[|-(derivable_pt ? ?)] -> Cut True -> (derivable_pt p pt); [Intro HYPPD; Cut (derivable_pt p pt); [Intro; Clear HYPPD | Apply HYPPD; Clear HYPPD; Trivial] | Idtac] - |[_:(continuity_pt p pt)|- ?] -> Idtac - |[|-(continuity_pt p pt)] -> Idtac - |[|-(continuity_pt ? ?)] -> Cut True -> (continuity_pt p pt); [Intro HYPPD; Cut (continuity_pt p pt); [Intro; Clear HYPPD | Apply HYPPD; Clear HYPPD; Trivial] | Idtac] - |[|-(eqT ? (derive_pt ? ? ?) ?)] -> Cut True -> (derivable_pt p pt); [Intro HYPPD; Cut (derivable_pt p pt); [Intro; Clear HYPPD | Apply HYPPD; Clear HYPPD; Trivial] | Idtac] - | _ -> Idtac). +Ltac intro_hyp_pt trm pt := + match constr:trm with + | (?X1 + ?X2)%F => + match goal with + | |- (derivable_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | |- (continuity_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | |- (derive_pt _ _ _ = _) => + intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | _ => idtac + end + | (?X1 - ?X2)%F => + match goal with + | |- (derivable_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | |- (continuity_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | |- (derive_pt _ _ _ = _) => + intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | _ => idtac + end + | (?X1 * ?X2)%F => + match goal with + | |- (derivable_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | |- (continuity_pt _ _) => intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | |- (derive_pt _ _ _ = _) => + intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | _ => idtac + end + | (?X1 / ?X2)%F => + let aux := constr:X2 in + match goal with + | _:(aux pt <> 0) |- (derivable_pt _ _) => + intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | _:(aux pt <> 0) |- (continuity_pt _ _) => + intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | _:(aux pt <> 0) |- (derive_pt _ _ _ = _) => + intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | id:(forall x0:R, aux x0 <> 0) |- (derivable_pt _ _) => + generalize (id pt); intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | id:(forall x0:R, aux x0 <> 0) |- (continuity_pt _ _) => + generalize (id pt); intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | id:(forall x0:R, aux x0 <> 0) |- (derive_pt _ _ _ = _) => + generalize (id pt); intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt + | |- (derivable_pt _ _) => + cut (aux pt <> 0); + [ intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt | try assumption ] + | |- (continuity_pt _ _) => + cut (aux pt <> 0); + [ intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt | try assumption ] + | |- (derive_pt _ _ _ = _) => + cut (aux pt <> 0); + [ intro; intro_hyp_pt X1 pt; intro_hyp_pt X2 pt | try assumption ] + | _ => idtac + end + | (comp ?X1 ?X2) => + match goal with + | |- (derivable_pt _ _) => + let pt_f1 := eval cbv beta in (X2 pt) in + (intro_hyp_pt X1 pt_f1; intro_hyp_pt X2 pt) + | |- (continuity_pt _ _) => + let pt_f1 := eval cbv beta in (X2 pt) in + (intro_hyp_pt X1 pt_f1; intro_hyp_pt X2 pt) + | |- (derive_pt _ _ _ = _) => + let pt_f1 := eval cbv beta in (X2 pt) in + (intro_hyp_pt X1 pt_f1; intro_hyp_pt X2 pt) + | _ => idtac + end + | (- ?X1)%F => + match goal with + | |- (derivable_pt _ _) => intro_hyp_pt X1 pt + | |- (continuity_pt _ _) => intro_hyp_pt X1 pt + | |- (derive_pt _ _ _ = _) => intro_hyp_pt X1 pt + | _ => idtac + end + | (/ ?X1)%F => + let aux := constr:X1 in + match goal with + | _:(aux pt <> 0) |- (derivable_pt _ _) => + intro_hyp_pt X1 pt + | _:(aux pt <> 0) |- (continuity_pt _ _) => + intro_hyp_pt X1 pt + | _:(aux pt <> 0) |- (derive_pt _ _ _ = _) => + intro_hyp_pt X1 pt + | id:(forall x0:R, aux x0 <> 0) |- (derivable_pt _ _) => + generalize (id pt); intro; intro_hyp_pt X1 pt + | id:(forall x0:R, aux x0 <> 0) |- (continuity_pt _ _) => + generalize (id pt); intro; intro_hyp_pt X1 pt + | id:(forall x0:R, aux x0 <> 0) |- (derive_pt _ _ _ = _) => + generalize (id pt); intro; intro_hyp_pt X1 pt + | |- (derivable_pt _ _) => + cut (aux pt <> 0); [ intro; intro_hyp_pt X1 pt | try assumption ] + | |- (continuity_pt _ _) => + cut (aux pt <> 0); [ intro; intro_hyp_pt X1 pt | try assumption ] + | |- (derive_pt _ _ _ = _) => + cut (aux pt <> 0); [ intro; intro_hyp_pt X1 pt | try assumption ] + | _ => idtac + end + | cos => idtac + | sin => idtac + | cosh => idtac + | sinh => idtac + | exp => idtac + | Rsqr => idtac + | id => idtac + | (fct_cte _) => idtac + | (pow_fct _) => idtac + | sqrt => + match goal with + | |- (derivable_pt _ _) => cut (0 < pt); [ intro | try assumption ] + | |- (continuity_pt _ _) => + cut (0 <= pt); [ intro | try assumption ] + | |- (derive_pt _ _ _ = _) => + cut (0 < pt); [ intro | try assumption ] + | _ => idtac + end + | Rabs => + match goal with + | |- (derivable_pt _ _) => + cut (pt <> 0); [ intro | try assumption ] + | _ => idtac + end + | ?X1 => + let p := constr:X1 in + match goal with + | _:(derivable_pt p pt) |- _ => idtac + | |- (derivable_pt p pt) => idtac + | |- (derivable_pt _ _) => + cut (True -> derivable_pt p pt); + [ intro HYPPD; cut (derivable_pt p pt); + [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ] + | idtac ] + | _:(continuity_pt p pt) |- _ => idtac + | |- (continuity_pt p pt) => idtac + | |- (continuity_pt _ _) => + cut (True -> continuity_pt p pt); + [ intro HYPPD; cut (continuity_pt p pt); + [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ] + | idtac ] + | |- (derive_pt _ _ _ = _) => + cut (True -> derivable_pt p pt); + [ intro HYPPD; cut (derivable_pt p pt); + [ intro; clear HYPPD | apply HYPPD; clear HYPPD; trivial ] + | idtac ] + | _ => idtac + end + end. (**********) -Recursive Tactic Definition IsDiff_pt := -Match Context With - (* fonctions de base *) - [|-(derivable_pt Rsqr ?)] -> Apply derivable_pt_Rsqr -|[|-(derivable_pt id ?1)] -> Apply (derivable_pt_id ?1) -|[|-(derivable_pt (fct_cte ?) ?)] -> Apply derivable_pt_const -|[|-(derivable_pt sin ?)] -> Apply derivable_pt_sin -|[|-(derivable_pt cos ?)] -> Apply derivable_pt_cos -|[|-(derivable_pt sinh ?)] -> Apply derivable_pt_sinh -|[|-(derivable_pt cosh ?)] -> Apply derivable_pt_cosh -|[|-(derivable_pt exp ?)] -> Apply derivable_pt_exp -|[|-(derivable_pt (pow_fct ?) ?)] -> Unfold pow_fct; Apply derivable_pt_pow -|[|-(derivable_pt sqrt ?1)] -> Apply (derivable_pt_sqrt ?1); Assumption Orelse Unfold plus_fct minus_fct opp_fct mult_fct div_fct inv_fct comp id fct_cte pow_fct -|[|-(derivable_pt Rabsolu ?1)] -> Apply (derivable_pt_Rabsolu ?1); Assumption Orelse Unfold plus_fct minus_fct opp_fct mult_fct div_fct inv_fct comp id fct_cte pow_fct - (* regles de differentiabilite *) - (* PLUS *) -|[|-(derivable_pt (plus_fct ?1 ?2) ?3)] -> Apply (derivable_pt_plus ?1 ?2 ?3); IsDiff_pt - (* MOINS *) -|[|-(derivable_pt (minus_fct ?1 ?2) ?3)] -> Apply (derivable_pt_minus ?1 ?2 ?3); IsDiff_pt - (* OPPOSE *) -|[|-(derivable_pt (opp_fct ?1) ?2)] -> Apply (derivable_pt_opp ?1 ?2); IsDiff_pt - (* MULTIPLICATION PAR UN SCALAIRE *) -|[|-(derivable_pt (mult_real_fct ?1 ?2) ?3)] -> Apply (derivable_pt_scal ?2 ?1 ?3); IsDiff_pt - (* MULTIPLICATION *) -|[|-(derivable_pt (mult_fct ?1 ?2) ?3)] -> Apply (derivable_pt_mult ?1 ?2 ?3); IsDiff_pt - (* DIVISION *) - |[|-(derivable_pt (div_fct ?1 ?2) ?3)] -> Apply (derivable_pt_div ?1 ?2 ?3); [IsDiff_pt | IsDiff_pt | Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct comp pow_fct id fct_cte] - (* INVERSION *) - |[|-(derivable_pt (inv_fct ?1) ?2)] -> Apply (derivable_pt_inv ?1 ?2); [Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct comp pow_fct id fct_cte | IsDiff_pt] - (* COMPOSITION *) -|[|-(derivable_pt (comp ?1 ?2) ?3)] -> Apply (derivable_pt_comp ?2 ?1 ?3); IsDiff_pt -|[_:(derivable_pt ?1 ?2)|-(derivable_pt ?1 ?2)] -> Assumption -|[_:(derivable ?1) |- (derivable_pt ?1 ?2)] -> Cut (derivable ?1); [Intro HypDDPT; Apply HypDDPT | Assumption] -|[|-True->(derivable_pt ? ?)] -> Intro HypTruE; Clear HypTruE; IsDiff_pt -| _ -> Try Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct. +Ltac is_diff_pt := + match goal with + | |- (derivable_pt Rsqr _) => + + (* fonctions de base *) + apply derivable_pt_Rsqr + | |- (derivable_pt id ?X1) => apply (derivable_pt_id X1) + | |- (derivable_pt (fct_cte _) _) => apply derivable_pt_const + | |- (derivable_pt sin _) => apply derivable_pt_sin + | |- (derivable_pt cos _) => apply derivable_pt_cos + | |- (derivable_pt sinh _) => apply derivable_pt_sinh + | |- (derivable_pt cosh _) => apply derivable_pt_cosh + | |- (derivable_pt exp _) => apply derivable_pt_exp + | |- (derivable_pt (pow_fct _) _) => + unfold pow_fct in |- *; apply derivable_pt_pow + | |- (derivable_pt sqrt ?X1) => + apply (derivable_pt_sqrt X1); + assumption || + unfold plus_fct, minus_fct, opp_fct, mult_fct, div_fct, inv_fct, + comp, id, fct_cte, pow_fct in |- * + | |- (derivable_pt Rabs ?X1) => + apply (Rderivable_pt_abs X1); + assumption || + unfold plus_fct, minus_fct, opp_fct, mult_fct, div_fct, inv_fct, + comp, id, fct_cte, pow_fct in |- * + (* regles de differentiabilite *) + (* PLUS *) + | |- (derivable_pt (?X1 + ?X2) ?X3) => + apply (derivable_pt_plus X1 X2 X3); is_diff_pt + (* MOINS *) + | |- (derivable_pt (?X1 - ?X2) ?X3) => + apply (derivable_pt_minus X1 X2 X3); is_diff_pt + (* OPPOSE *) + | |- (derivable_pt (- ?X1) ?X2) => + apply (derivable_pt_opp X1 X2); + is_diff_pt + (* MULTIPLICATION PAR UN SCALAIRE *) + | |- (derivable_pt (mult_real_fct ?X1 ?X2) ?X3) => + apply (derivable_pt_scal X2 X1 X3); is_diff_pt + (* MULTIPLICATION *) + | |- (derivable_pt (?X1 * ?X2) ?X3) => + apply (derivable_pt_mult X1 X2 X3); is_diff_pt + (* DIVISION *) + | |- (derivable_pt (?X1 / ?X2) ?X3) => + apply (derivable_pt_div X1 X2 X3); + [ is_diff_pt + | is_diff_pt + | try + assumption || + unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, + comp, pow_fct, id, fct_cte in |- * ] + | |- (derivable_pt (/ ?X1) ?X2) => + + (* INVERSION *) + apply (derivable_pt_inv X1 X2); + [ assumption || + unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, + comp, pow_fct, id, fct_cte in |- * + | is_diff_pt ] + | |- (derivable_pt (comp ?X1 ?X2) ?X3) => + + (* COMPOSITION *) + apply (derivable_pt_comp X2 X1 X3); is_diff_pt + | _:(derivable_pt ?X1 ?X2) |- (derivable_pt ?X1 ?X2) => + assumption + | _:(derivable ?X1) |- (derivable_pt ?X1 ?X2) => + cut (derivable X1); [ intro HypDDPT; apply HypDDPT | assumption ] + | |- (True -> derivable_pt _ _) => + intro HypTruE; clear HypTruE; is_diff_pt + | _ => + try + unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id, + fct_cte, comp, pow_fct in |- * + end. (**********) -Recursive Tactic Definition IsDiff_glob := -Match Context With - (* fonctions de base *) - [|-(derivable Rsqr)] -> Apply derivable_Rsqr - |[|-(derivable id)] -> Apply derivable_id - |[|-(derivable (fct_cte ?))] -> Apply derivable_const - |[|-(derivable sin)] -> Apply derivable_sin - |[|-(derivable cos)] -> Apply derivable_cos - |[|-(derivable cosh)] -> Apply derivable_cosh - |[|-(derivable sinh)] -> Apply derivable_sinh - |[|-(derivable exp)] -> Apply derivable_exp - |[|-(derivable (pow_fct ?))] -> Unfold pow_fct; Apply derivable_pow - (* regles de differentiabilite *) - (* PLUS *) - |[|-(derivable (plus_fct ?1 ?2))] -> Apply (derivable_plus ?1 ?2); IsDiff_glob - (* MOINS *) - |[|-(derivable (minus_fct ?1 ?2))] -> Apply (derivable_minus ?1 ?2); IsDiff_glob - (* OPPOSE *) - |[|-(derivable (opp_fct ?1))] -> Apply (derivable_opp ?1); IsDiff_glob - (* MULTIPLICATION PAR UN SCALAIRE *) - |[|-(derivable (mult_real_fct ?1 ?2))] -> Apply (derivable_scal ?2 ?1); IsDiff_glob - (* MULTIPLICATION *) - |[|-(derivable (mult_fct ?1 ?2))] -> Apply (derivable_mult ?1 ?2); IsDiff_glob - (* DIVISION *) - |[|-(derivable (div_fct ?1 ?2))] -> Apply (derivable_div ?1 ?2); [IsDiff_glob | IsDiff_glob | Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct] - (* INVERSION *) - |[|-(derivable (inv_fct ?1))] -> Apply (derivable_inv ?1); [Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct | IsDiff_glob] - (* COMPOSITION *) - |[|-(derivable (comp sqrt ?))] -> Unfold derivable; Intro; Try IsDiff_pt - |[|-(derivable (comp Rabsolu ?))] -> Unfold derivable; Intro; Try IsDiff_pt - |[|-(derivable (comp ?1 ?2))] -> Apply (derivable_comp ?2 ?1); IsDiff_glob - |[_:(derivable ?1)|-(derivable ?1)] -> Assumption - |[|-True->(derivable ?)] -> Intro HypTruE; Clear HypTruE; IsDiff_glob - | _ -> Try Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct. +Ltac is_diff_glob := + match goal with + | |- (derivable Rsqr) => + (* fonctions de base *) + apply derivable_Rsqr + | |- (derivable id) => apply derivable_id + | |- (derivable (fct_cte _)) => apply derivable_const + | |- (derivable sin) => apply derivable_sin + | |- (derivable cos) => apply derivable_cos + | |- (derivable cosh) => apply derivable_cosh + | |- (derivable sinh) => apply derivable_sinh + | |- (derivable exp) => apply derivable_exp + | |- (derivable (pow_fct _)) => + unfold pow_fct in |- *; + apply derivable_pow + (* regles de differentiabilite *) + (* PLUS *) + | |- (derivable (?X1 + ?X2)) => + apply (derivable_plus X1 X2); is_diff_glob + (* MOINS *) + | |- (derivable (?X1 - ?X2)) => + apply (derivable_minus X1 X2); is_diff_glob + (* OPPOSE *) + | |- (derivable (- ?X1)) => + apply (derivable_opp X1); + is_diff_glob + (* MULTIPLICATION PAR UN SCALAIRE *) + | |- (derivable (mult_real_fct ?X1 ?X2)) => + apply (derivable_scal X2 X1); is_diff_glob + (* MULTIPLICATION *) + | |- (derivable (?X1 * ?X2)) => + apply (derivable_mult X1 X2); is_diff_glob + (* DIVISION *) + | |- (derivable (?X1 / ?X2)) => + apply (derivable_div X1 X2); + [ is_diff_glob + | is_diff_glob + | try + assumption || + unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, + id, fct_cte, comp, pow_fct in |- * ] + | |- (derivable (/ ?X1)) => + + (* INVERSION *) + apply (derivable_inv X1); + [ try + assumption || + unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, + id, fct_cte, comp, pow_fct in |- * + | is_diff_glob ] + | |- (derivable (comp sqrt _)) => + + (* COMPOSITION *) + unfold derivable in |- *; intro; try is_diff_pt + | |- (derivable (comp Rabs _)) => + unfold derivable in |- *; intro; try is_diff_pt + | |- (derivable (comp ?X1 ?X2)) => + apply (derivable_comp X2 X1); is_diff_glob + | _:(derivable ?X1) |- (derivable ?X1) => assumption + | |- (True -> derivable _) => + intro HypTruE; clear HypTruE; is_diff_glob + | _ => + try + unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id, + fct_cte, comp, pow_fct in |- * + end. (**********) -Recursive Tactic Definition IsCont_pt := -Match Context With - (* fonctions de base *) - [|-(continuity_pt Rsqr ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_Rsqr -|[|-(continuity_pt id ?1)] -> Apply derivable_continuous_pt; Apply (derivable_pt_id ?1) -|[|-(continuity_pt (fct_cte ?) ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_const -|[|-(continuity_pt sin ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_sin -|[|-(continuity_pt cos ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_cos -|[|-(continuity_pt sinh ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_sinh -|[|-(continuity_pt cosh ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_cosh -|[|-(continuity_pt exp ?)] -> Apply derivable_continuous_pt; Apply derivable_pt_exp -|[|-(continuity_pt (pow_fct ?) ?)] -> Unfold pow_fct; Apply derivable_continuous_pt; Apply derivable_pt_pow -|[|-(continuity_pt sqrt ?1)] -> Apply continuity_pt_sqrt; Assumption Orelse Unfold plus_fct minus_fct opp_fct mult_fct div_fct inv_fct comp id fct_cte pow_fct -|[|-(continuity_pt Rabsolu ?1)] -> Apply (continuity_Rabsolu ?1) - (* regles de differentiabilite *) - (* PLUS *) -|[|-(continuity_pt (plus_fct ?1 ?2) ?3)] -> Apply (continuity_pt_plus ?1 ?2 ?3); IsCont_pt - (* MOINS *) -|[|-(continuity_pt (minus_fct ?1 ?2) ?3)] -> Apply (continuity_pt_minus ?1 ?2 ?3); IsCont_pt - (* OPPOSE *) -|[|-(continuity_pt (opp_fct ?1) ?2)] -> Apply (continuity_pt_opp ?1 ?2); IsCont_pt - (* MULTIPLICATION PAR UN SCALAIRE *) -|[|-(continuity_pt (mult_real_fct ?1 ?2) ?3)] -> Apply (continuity_pt_scal ?2 ?1 ?3); IsCont_pt - (* MULTIPLICATION *) -|[|-(continuity_pt (mult_fct ?1 ?2) ?3)] -> Apply (continuity_pt_mult ?1 ?2 ?3); IsCont_pt - (* DIVISION *) - |[|-(continuity_pt (div_fct ?1 ?2) ?3)] -> Apply (continuity_pt_div ?1 ?2 ?3); [IsCont_pt | IsCont_pt | Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct comp id fct_cte pow_fct] - (* INVERSION *) - |[|-(continuity_pt (inv_fct ?1) ?2)] -> Apply (continuity_pt_inv ?1 ?2); [IsCont_pt | Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct comp id fct_cte pow_fct] - (* COMPOSITION *) -|[|-(continuity_pt (comp ?1 ?2) ?3)] -> Apply (continuity_pt_comp ?2 ?1 ?3); IsCont_pt -|[_:(continuity_pt ?1 ?2)|-(continuity_pt ?1 ?2)] -> Assumption -|[_:(continuity ?1) |- (continuity_pt ?1 ?2)] -> Cut (continuity ?1); [Intro HypDDPT; Apply HypDDPT | Assumption] -|[_:(derivable_pt ?1 ?2)|-(continuity_pt ?1 ?2)] -> Apply derivable_continuous_pt; Assumption -|[_:(derivable ?1)|-(continuity_pt ?1 ?2)] -> Cut (continuity ?1); [Intro HypDDPT; Apply HypDDPT | Apply derivable_continuous; Assumption] -|[|-True->(continuity_pt ? ?)] -> Intro HypTruE; Clear HypTruE; IsCont_pt -| _ -> Try Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct. +Ltac is_cont_pt := + match goal with + | |- (continuity_pt Rsqr _) => + + (* fonctions de base *) + apply derivable_continuous_pt; apply derivable_pt_Rsqr + | |- (continuity_pt id ?X1) => + apply derivable_continuous_pt; apply (derivable_pt_id X1) + | |- (continuity_pt (fct_cte _) _) => + apply derivable_continuous_pt; apply derivable_pt_const + | |- (continuity_pt sin _) => + apply derivable_continuous_pt; apply derivable_pt_sin + | |- (continuity_pt cos _) => + apply derivable_continuous_pt; apply derivable_pt_cos + | |- (continuity_pt sinh _) => + apply derivable_continuous_pt; apply derivable_pt_sinh + | |- (continuity_pt cosh _) => + apply derivable_continuous_pt; apply derivable_pt_cosh + | |- (continuity_pt exp _) => + apply derivable_continuous_pt; apply derivable_pt_exp + | |- (continuity_pt (pow_fct _) _) => + unfold pow_fct in |- *; apply derivable_continuous_pt; + apply derivable_pt_pow + | |- (continuity_pt sqrt ?X1) => + apply continuity_pt_sqrt; + assumption || + unfold plus_fct, minus_fct, opp_fct, mult_fct, div_fct, inv_fct, + comp, id, fct_cte, pow_fct in |- * + | |- (continuity_pt Rabs ?X1) => + apply (Rcontinuity_abs X1) + (* regles de differentiabilite *) + (* PLUS *) + | |- (continuity_pt (?X1 + ?X2) ?X3) => + apply (continuity_pt_plus X1 X2 X3); is_cont_pt + (* MOINS *) + | |- (continuity_pt (?X1 - ?X2) ?X3) => + apply (continuity_pt_minus X1 X2 X3); is_cont_pt + (* OPPOSE *) + | |- (continuity_pt (- ?X1) ?X2) => + apply (continuity_pt_opp X1 X2); + is_cont_pt + (* MULTIPLICATION PAR UN SCALAIRE *) + | |- (continuity_pt (mult_real_fct ?X1 ?X2) ?X3) => + apply (continuity_pt_scal X2 X1 X3); is_cont_pt + (* MULTIPLICATION *) + | |- (continuity_pt (?X1 * ?X2) ?X3) => + apply (continuity_pt_mult X1 X2 X3); is_cont_pt + (* DIVISION *) + | |- (continuity_pt (?X1 / ?X2) ?X3) => + apply (continuity_pt_div X1 X2 X3); + [ is_cont_pt + | is_cont_pt + | try + assumption || + unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, + comp, id, fct_cte, pow_fct in |- * ] + | |- (continuity_pt (/ ?X1) ?X2) => + + (* INVERSION *) + apply (continuity_pt_inv X1 X2); + [ is_cont_pt + | assumption || + unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, + comp, id, fct_cte, pow_fct in |- * ] + | |- (continuity_pt (comp ?X1 ?X2) ?X3) => + + (* COMPOSITION *) + apply (continuity_pt_comp X2 X1 X3); is_cont_pt + | _:(continuity_pt ?X1 ?X2) |- (continuity_pt ?X1 ?X2) => + assumption + | _:(continuity ?X1) |- (continuity_pt ?X1 ?X2) => + cut (continuity X1); [ intro HypDDPT; apply HypDDPT | assumption ] + | _:(derivable_pt ?X1 ?X2) |- (continuity_pt ?X1 ?X2) => + apply derivable_continuous_pt; assumption + | _:(derivable ?X1) |- (continuity_pt ?X1 ?X2) => + cut (continuity X1); + [ intro HypDDPT; apply HypDDPT + | apply derivable_continuous; assumption ] + | |- (True -> continuity_pt _ _) => + intro HypTruE; clear HypTruE; is_cont_pt + | _ => + try + unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id, + fct_cte, comp, pow_fct in |- * + end. (**********) -Recursive Tactic Definition IsCont_glob := -Match Context With - (* fonctions de base *) - [|-(continuity Rsqr)] -> Apply derivable_continuous; Apply derivable_Rsqr - |[|-(continuity id)] -> Apply derivable_continuous; Apply derivable_id - |[|-(continuity (fct_cte ?))] -> Apply derivable_continuous; Apply derivable_const - |[|-(continuity sin)] -> Apply derivable_continuous; Apply derivable_sin - |[|-(continuity cos)] -> Apply derivable_continuous; Apply derivable_cos - |[|-(continuity exp)] -> Apply derivable_continuous; Apply derivable_exp - |[|-(continuity (pow_fct ?))] -> Unfold pow_fct; Apply derivable_continuous; Apply derivable_pow - |[|-(continuity sinh)] -> Apply derivable_continuous; Apply derivable_sinh - |[|-(continuity cosh)] -> Apply derivable_continuous; Apply derivable_cosh - |[|-(continuity Rabsolu)] -> Apply continuity_Rabsolu - (* regles de continuite *) - (* PLUS *) -|[|-(continuity (plus_fct ?1 ?2))] -> Apply (continuity_plus ?1 ?2); Try IsCont_glob Orelse Assumption - (* MOINS *) -|[|-(continuity (minus_fct ?1 ?2))] -> Apply (continuity_minus ?1 ?2); Try IsCont_glob Orelse Assumption - (* OPPOSE *) -|[|-(continuity (opp_fct ?1))] -> Apply (continuity_opp ?1); Try IsCont_glob Orelse Assumption - (* INVERSE *) -|[|-(continuity (inv_fct ?1))] -> Apply (continuity_inv ?1); Try IsCont_glob Orelse Assumption - (* MULTIPLICATION PAR UN SCALAIRE *) -|[|-(continuity (mult_real_fct ?1 ?2))] -> Apply (continuity_scal ?2 ?1); Try IsCont_glob Orelse Assumption - (* MULTIPLICATION *) -|[|-(continuity (mult_fct ?1 ?2))] -> Apply (continuity_mult ?1 ?2); Try IsCont_glob Orelse Assumption - (* DIVISION *) - |[|-(continuity (div_fct ?1 ?2))] -> Apply (continuity_div ?1 ?2); [Try IsCont_glob Orelse Assumption | Try IsCont_glob Orelse Assumption | Try Assumption Orelse Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte pow_fct] - (* COMPOSITION *) - |[|-(continuity (comp sqrt ?))] -> Unfold continuity_pt; Intro; Try IsCont_pt - |[|-(continuity (comp ?1 ?2))] -> Apply (continuity_comp ?2 ?1); Try IsCont_glob Orelse Assumption - |[_:(continuity ?1)|-(continuity ?1)] -> Assumption - |[|-True->(continuity ?)] -> Intro HypTruE; Clear HypTruE; IsCont_glob - |[_:(derivable ?1)|-(continuity ?1)] -> Apply derivable_continuous; Assumption - | _ -> Try Unfold plus_fct mult_fct div_fct minus_fct opp_fct inv_fct id fct_cte comp pow_fct. +Ltac is_cont_glob := + match goal with + | |- (continuity Rsqr) => + + (* fonctions de base *) + apply derivable_continuous; apply derivable_Rsqr + | |- (continuity id) => apply derivable_continuous; apply derivable_id + | |- (continuity (fct_cte _)) => + apply derivable_continuous; apply derivable_const + | |- (continuity sin) => apply derivable_continuous; apply derivable_sin + | |- (continuity cos) => apply derivable_continuous; apply derivable_cos + | |- (continuity exp) => apply derivable_continuous; apply derivable_exp + | |- (continuity (pow_fct _)) => + unfold pow_fct in |- *; apply derivable_continuous; apply derivable_pow + | |- (continuity sinh) => + apply derivable_continuous; apply derivable_sinh + | |- (continuity cosh) => + apply derivable_continuous; apply derivable_cosh + | |- (continuity Rabs) => + apply Rcontinuity_abs + (* regles de continuite *) + (* PLUS *) + | |- (continuity (?X1 + ?X2)) => + apply (continuity_plus X1 X2); + try is_cont_glob || assumption + (* MOINS *) + | |- (continuity (?X1 - ?X2)) => + apply (continuity_minus X1 X2); + try is_cont_glob || assumption + (* OPPOSE *) + | |- (continuity (- ?X1)) => + apply (continuity_opp X1); try is_cont_glob || assumption + (* INVERSE *) + | |- (continuity (/ ?X1)) => + apply (continuity_inv X1); + try is_cont_glob || assumption + (* MULTIPLICATION PAR UN SCALAIRE *) + | |- (continuity (mult_real_fct ?X1 ?X2)) => + apply (continuity_scal X2 X1); + try is_cont_glob || assumption + (* MULTIPLICATION *) + | |- (continuity (?X1 * ?X2)) => + apply (continuity_mult X1 X2); + try is_cont_glob || assumption + (* DIVISION *) + | |- (continuity (?X1 / ?X2)) => + apply (continuity_div X1 X2); + [ try is_cont_glob || assumption + | try is_cont_glob || assumption + | try + assumption || + unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, + id, fct_cte, pow_fct in |- * ] + | |- (continuity (comp sqrt _)) => + + (* COMPOSITION *) + unfold continuity_pt in |- *; intro; try is_cont_pt + | |- (continuity (comp ?X1 ?X2)) => + apply (continuity_comp X2 X1); try is_cont_glob || assumption + | _:(continuity ?X1) |- (continuity ?X1) => assumption + | |- (True -> continuity _) => + intro HypTruE; clear HypTruE; is_cont_glob + | _:(derivable ?X1) |- (continuity ?X1) => + apply derivable_continuous; assumption + | _ => + try + unfold plus_fct, mult_fct, div_fct, minus_fct, opp_fct, inv_fct, id, + fct_cte, comp, pow_fct in |- * + end. (**********) -Recursive Tactic Definition RewTerm trm := -Match trm With -| [(Rplus ?1 ?2)] -> Let p1= (RewTerm ?1) And p2 = (RewTerm ?2) In - (Match p1 With - [(fct_cte ?3)] -> - (Match p2 With - | [(fct_cte ?4)] -> '(fct_cte (Rplus ?3 ?4)) - | _ -> '(plus_fct p1 p2)) - | _ -> '(plus_fct p1 p2)) -| [(Rminus ?1 ?2)] -> Let p1 = (RewTerm ?1) And p2 = (RewTerm ?2) In - (Match p1 With - [(fct_cte ?3)] -> - (Match p2 With - | [(fct_cte ?4)] -> '(fct_cte (Rminus ?3 ?4)) - | _ -> '(minus_fct p1 p2)) - | _ -> '(minus_fct p1 p2)) -| [(Rdiv ?1 ?2)] -> Let p1 = (RewTerm ?1) And p2 = (RewTerm ?2) In - (Match p1 With - [(fct_cte ?3)] -> - (Match p2 With - | [(fct_cte ?4)] -> '(fct_cte (Rdiv ?3 ?4)) - | _ -> '(div_fct p1 p2)) - | _ -> - (Match p2 With - | [(fct_cte ?4)] -> '(mult_fct p1 (fct_cte (Rinv ?4))) - | _ -> '(div_fct p1 p2))) -| [(Rmult ?1 (Rinv ?2))] -> Let p1 = (RewTerm ?1) And p2 = (RewTerm ?2) In - (Match p1 With - [(fct_cte ?3)] -> - (Match p2 With - | [(fct_cte ?4)] -> '(fct_cte (Rdiv ?3 ?4)) - | _ -> '(div_fct p1 p2)) - | _ -> - (Match p2 With - | [(fct_cte ?4)] -> '(mult_fct p1 (fct_cte (Rinv ?4))) - | _ -> '(div_fct p1 p2))) -| [(Rmult ?1 ?2)] -> Let p1 = (RewTerm ?1) And p2 = (RewTerm ?2) In - (Match p1 With - [(fct_cte ?3)] -> - (Match p2 With - | [(fct_cte ?4)] -> '(fct_cte (Rmult ?3 ?4)) - | _ -> '(mult_fct p1 p2)) - | _ -> '(mult_fct p1 p2)) -| [(Ropp ?1)] -> Let p = (RewTerm ?1) In - (Match p With - [(fct_cte ?2)] -> '(fct_cte (Ropp ?2)) - | _ -> '(opp_fct p)) -| [(Rinv ?1)] -> Let p = (RewTerm ?1) In - (Match p With - [(fct_cte ?2)] -> '(fct_cte (Rinv ?2)) - | _ -> '(inv_fct p)) -| [(?1 AppVar)] -> '?1 -| [(?1 ?2)] -> Let p = (RewTerm ?2) In - (Match p With - | [(fct_cte ?3)] -> '(fct_cte (?1 ?3)) - | _ -> '(comp ?1 p)) -| [AppVar] -> 'id -| [(pow AppVar ?1)] -> '(pow_fct ?1) -| [(pow ?1 ?2)] -> Let p = (RewTerm ?1) In - (Match p With - | [(fct_cte ?3)] -> '(fct_cte (pow_fct ?2 ?3)) - | _ -> '(comp (pow_fct ?2) p)) -| [?1]-> '(fct_cte ?1). +Ltac rew_term trm := + match constr:trm with + | (?X1 + ?X2) => + let p1 := rew_term X1 with p2 := rew_term X2 in + match constr:p1 with + | (fct_cte ?X3) => + match constr:p2 with + | (fct_cte ?X4) => constr:(fct_cte (X3 + X4)) + | _ => constr:(p1 + p2)%F + end + | _ => constr:(p1 + p2)%F + end + | (?X1 - ?X2) => + let p1 := rew_term X1 with p2 := rew_term X2 in + match constr:p1 with + | (fct_cte ?X3) => + match constr:p2 with + | (fct_cte ?X4) => constr:(fct_cte (X3 - X4)) + | _ => constr:(p1 - p2)%F + end + | _ => constr:(p1 - p2)%F + end + | (?X1 / ?X2) => + let p1 := rew_term X1 with p2 := rew_term X2 in + match constr:p1 with + | (fct_cte ?X3) => + match constr:p2 with + | (fct_cte ?X4) => constr:(fct_cte (X3 / X4)) + | _ => constr:(p1 / p2)%F + end + | _ => + match constr:p2 with + | (fct_cte ?X4) => constr:(p1 * fct_cte (/ X4))%F + | _ => constr:(p1 / p2)%F + end + end + | (?X1 * / ?X2) => + let p1 := rew_term X1 with p2 := rew_term X2 in + match constr:p1 with + | (fct_cte ?X3) => + match constr:p2 with + | (fct_cte ?X4) => constr:(fct_cte (X3 / X4)) + | _ => constr:(p1 / p2)%F + end + | _ => + match constr:p2 with + | (fct_cte ?X4) => constr:(p1 * fct_cte (/ X4))%F + | _ => constr:(p1 / p2)%F + end + end + | (?X1 * ?X2) => + let p1 := rew_term X1 with p2 := rew_term X2 in + match constr:p1 with + | (fct_cte ?X3) => + match constr:p2 with + | (fct_cte ?X4) => constr:(fct_cte (X3 * X4)) + | _ => constr:(p1 * p2)%F + end + | _ => constr:(p1 * p2)%F + end + | (- ?X1) => + let p := rew_term X1 in + match constr:p with + | (fct_cte ?X2) => constr:(fct_cte (- X2)) + | _ => constr:(- p)%F + end + | (/ ?X1) => + let p := rew_term X1 in + match constr:p with + | (fct_cte ?X2) => constr:(fct_cte (/ X2)) + | _ => constr:(/ p)%F + end + | (?X1 AppVar) => constr:X1 + | (?X1 ?X2) => + let p := rew_term X2 in + match constr:p with + | (fct_cte ?X3) => constr:(fct_cte (X1 X3)) + | _ => constr:(comp X1 p) + end + | AppVar => constr:id + | (AppVar ^ ?X1) => constr:(pow_fct X1) + | (?X1 ^ ?X2) => + let p := rew_term X1 in + match constr:p with + | (fct_cte ?X3) => constr:(fct_cte (pow_fct X2 X3)) + | _ => constr:(comp (pow_fct X2) p) + end + | ?X1 => constr:(fct_cte X1) + end. (**********) -Recursive Tactic Definition ConsProof trm pt := -Match trm With -| [(plus_fct ?1 ?2)] -> Let p1 = (ConsProof ?1 pt) And p2 = (ConsProof ?2 pt) In '(derivable_pt_plus ?1 ?2 pt p1 p2) -| [(minus_fct ?1 ?2)] -> Let p1 = (ConsProof ?1 pt) And p2 = (ConsProof ?2 pt) In '(derivable_pt_minus ?1 ?2 pt p1 p2) -| [(mult_fct ?1 ?2)] -> Let p1 = (ConsProof ?1 pt) And p2 = (ConsProof ?2 pt) In '(derivable_pt_mult ?1 ?2 pt p1 p2) -| [(div_fct ?1 ?2)] -> - (Match Context With - |[id:~((?2 pt)==R0) |- ?] -> Let p1 = (ConsProof ?1 pt) And p2 = (ConsProof ?2 pt) In '(derivable_pt_div ?1 ?2 pt p1 p2 id) - | _ -> 'False) -| [(inv_fct ?1)] -> - (Match Context With - |[id:~((?1 pt)==R0) |- ?] -> Let p1 = (ConsProof ?1 pt) In '(derivable_pt_inv ?1 pt p1 id) - | _ -> 'False) -| [(comp ?1 ?2)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In Let p1 = (ConsProof ?1 pt_f1) And p2 = (ConsProof ?2 pt) In '(derivable_pt_comp ?2 ?1 pt p2 p1) -| [(opp_fct ?1)] -> Let p1 = (ConsProof ?1 pt) In '(derivable_pt_opp ?1 pt p1) -| [sin] -> '(derivable_pt_sin pt) -| [cos] -> '(derivable_pt_cos pt) -| [sinh] -> '(derivable_pt_sinh pt) -| [cosh] -> '(derivable_pt_cosh pt) -| [exp] -> '(derivable_pt_exp pt) -| [id] -> '(derivable_pt_id pt) -| [Rsqr] -> '(derivable_pt_Rsqr pt) -| [sqrt] -> - (Match Context With - |[id:(Rlt R0 pt) |- ?] -> '(derivable_pt_sqrt pt id) - | _ -> 'False) -| [(fct_cte ?1)] -> '(derivable_pt_const ?1 pt) -| [?1] -> Let aux = ?1 In - (Match Context With - [ id : (derivable_pt aux pt) |- ?] -> 'id - |[ id : (derivable aux) |- ?] -> '(id pt) - | _ -> 'False). +Ltac deriv_proof trm pt := + match constr:trm with + | (?X1 + ?X2)%F => + let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in + constr:(derivable_pt_plus X1 X2 pt p1 p2) + | (?X1 - ?X2)%F => + let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in + constr:(derivable_pt_minus X1 X2 pt p1 p2) + | (?X1 * ?X2)%F => + let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in + constr:(derivable_pt_mult X1 X2 pt p1 p2) + | (?X1 / ?X2)%F => + match goal with + | id:(?X2 pt <> 0) |- _ => + let p1 := deriv_proof X1 pt with p2 := deriv_proof X2 pt in + constr:(derivable_pt_div X1 X2 pt p1 p2 id) + | _ => constr:False + end + | (/ ?X1)%F => + match goal with + | id:(?X1 pt <> 0) |- _ => + let p1 := deriv_proof X1 pt in + constr:(derivable_pt_inv X1 pt p1 id) + | _ => constr:False + end + | (comp ?X1 ?X2) => + let pt_f1 := eval cbv beta in (X2 pt) in + let p1 := deriv_proof X1 pt_f1 with p2 := deriv_proof X2 pt in + constr:(derivable_pt_comp X2 X1 pt p2 p1) + | (- ?X1)%F => + let p1 := deriv_proof X1 pt in + constr:(derivable_pt_opp X1 pt p1) + | sin => constr:(derivable_pt_sin pt) + | cos => constr:(derivable_pt_cos pt) + | sinh => constr:(derivable_pt_sinh pt) + | cosh => constr:(derivable_pt_cosh pt) + | exp => constr:(derivable_pt_exp pt) + | id => constr:(derivable_pt_id pt) + | Rsqr => constr:(derivable_pt_Rsqr pt) + | sqrt => + match goal with + | id:(0 < pt) |- _ => constr:(derivable_pt_sqrt pt id) + | _ => constr:False + end + | (fct_cte ?X1) => constr:(derivable_pt_const X1 pt) + | ?X1 => + let aux := constr:X1 in + match goal with + | id:(derivable_pt aux pt) |- _ => constr:id + | id:(derivable aux) |- _ => constr:(id pt) + | _ => constr:False + end + end. (**********) -Recursive Tactic Definition SimplifyDerive trm pt := -Match trm With -| [(plus_fct ?1 ?2)] -> Try Rewrite derive_pt_plus; SimplifyDerive ?1 pt; SimplifyDerive ?2 pt -| [(minus_fct ?1 ?2)] -> Try Rewrite derive_pt_minus; SimplifyDerive ?1 pt; SimplifyDerive ?2 pt -| [(mult_fct ?1 ?2)] -> Try Rewrite derive_pt_mult; SimplifyDerive ?1 pt; SimplifyDerive ?2 pt -| [(div_fct ?1 ?2)] -> Try Rewrite derive_pt_div; SimplifyDerive ?1 pt; SimplifyDerive ?2 pt -| [(comp ?1 ?2)] -> Let pt_f1 = (Eval Cbv Beta in (?2 pt)) In Try Rewrite derive_pt_comp; SimplifyDerive ?1 pt_f1; SimplifyDerive ?2 pt -| [(opp_fct ?1)] -> Try Rewrite derive_pt_opp; SimplifyDerive ?1 pt -| [(inv_fct ?1)] -> Try Rewrite derive_pt_inv; SimplifyDerive ?1 pt -| [(fct_cte ?1)] -> Try Rewrite derive_pt_const -| [id] -> Try Rewrite derive_pt_id -| [sin] -> Try Rewrite derive_pt_sin -| [cos] -> Try Rewrite derive_pt_cos -| [sinh] -> Try Rewrite derive_pt_sinh -| [cosh] -> Try Rewrite derive_pt_cosh -| [exp] -> Try Rewrite derive_pt_exp -| [Rsqr] -> Try Rewrite derive_pt_Rsqr -| [sqrt] -> Try Rewrite derive_pt_sqrt -| [?1] -> Let aux = ?1 In - (Match Context With - [ id : (eqT ? (derive_pt aux pt ?2) ?); H : (derivable aux) |- ? ] -> Try Replace (derive_pt aux pt (H pt)) with (derive_pt aux pt ?2); [Rewrite id | Apply pr_nu] - |[ id : (eqT ? (derive_pt aux pt ?2) ?); H : (derivable_pt aux pt) |- ? ] -> Try Replace (derive_pt aux pt H) with (derive_pt aux pt ?2); [Rewrite id | Apply pr_nu] - | _ -> Idtac ) -| _ -> Idtac. +Ltac simplify_derive trm pt := + match constr:trm with + | (?X1 + ?X2)%F => + try rewrite derive_pt_plus; simplify_derive X1 pt; + simplify_derive X2 pt + | (?X1 - ?X2)%F => + try rewrite derive_pt_minus; simplify_derive X1 pt; + simplify_derive X2 pt + | (?X1 * ?X2)%F => + try rewrite derive_pt_mult; simplify_derive X1 pt; + simplify_derive X2 pt + | (?X1 / ?X2)%F => + try rewrite derive_pt_div; simplify_derive X1 pt; simplify_derive X2 pt + | (comp ?X1 ?X2) => + let pt_f1 := eval cbv beta in (X2 pt) in + (try rewrite derive_pt_comp; simplify_derive X1 pt_f1; + simplify_derive X2 pt) + | (- ?X1)%F => try rewrite derive_pt_opp; simplify_derive X1 pt + | (/ ?X1)%F => + try rewrite derive_pt_inv; simplify_derive X1 pt + | (fct_cte ?X1) => try rewrite derive_pt_const + | id => try rewrite derive_pt_id + | sin => try rewrite derive_pt_sin + | cos => try rewrite derive_pt_cos + | sinh => try rewrite derive_pt_sinh + | cosh => try rewrite derive_pt_cosh + | exp => try rewrite derive_pt_exp + | Rsqr => try rewrite derive_pt_Rsqr + | sqrt => try rewrite derive_pt_sqrt + | ?X1 => + let aux := constr:X1 in + match goal with + | id:(derive_pt aux pt ?X2 = _),H:(derivable aux) |- _ => + try replace (derive_pt aux pt (H pt)) with (derive_pt aux pt X2); + [ rewrite id | apply pr_nu ] + | id:(derive_pt aux pt ?X2 = _),H:(derivable_pt aux pt) |- _ => + try replace (derive_pt aux pt H) with (derive_pt aux pt X2); + [ rewrite id | apply pr_nu ] + | _ => idtac + end + | _ => idtac + end. (**********) -Tactic Definition Reg := -Match Context With -| [|-(derivable_pt ?1 ?2)] -> -Let trm = Eval Cbv Beta in (?1 AppVar) In -Let aux = (RewTerm trm) In IntroHypL aux ?2; Try (Change (derivable_pt aux ?2); IsDiff_pt) Orelse IsDiff_pt -| [|-(derivable ?1)] -> -Let trm = Eval Cbv Beta in (?1 AppVar) In -Let aux = (RewTerm trm) In IntroHypG aux; Try (Change (derivable aux); IsDiff_glob) Orelse IsDiff_glob -| [|-(continuity ?1)] -> -Let trm = Eval Cbv Beta in (?1 AppVar) In -Let aux = (RewTerm trm) In IntroHypG aux; Try (Change (continuity aux); IsCont_glob) Orelse IsCont_glob -| [|-(continuity_pt ?1 ?2)] -> -Let trm = Eval Cbv Beta in (?1 AppVar) In -Let aux = (RewTerm trm) In IntroHypL aux ?2; Try (Change (continuity_pt aux ?2); IsCont_pt) Orelse IsCont_pt -| [|-(eqT ? (derive_pt ?1 ?2 ?3) ?4)] -> -Let trm = Eval Cbv Beta in (?1 AppVar) In -Let aux = (RewTerm trm) In -IntroHypL aux ?2; Let aux2 = (ConsProof aux ?2) In Try (Replace (derive_pt ?1 ?2 ?3) with (derive_pt aux ?2 aux2); [SimplifyDerive aux ?2; Try Unfold plus_fct minus_fct mult_fct div_fct id fct_cte inv_fct opp_fct; Try Ring | Try Apply pr_nu]) Orelse IsDiff_pt. +Ltac reg := + match goal with + | |- (derivable_pt ?X1 ?X2) => + let trm := eval cbv beta in (X1 AppVar) in + let aux := rew_term trm in + (intro_hyp_pt aux X2; + try (change (derivable_pt aux X2) in |- *; is_diff_pt) || is_diff_pt) + | |- (derivable ?X1) => + let trm := eval cbv beta in (X1 AppVar) in + let aux := rew_term trm in + (intro_hyp_glob aux; + try (change (derivable aux) in |- *; is_diff_glob) || is_diff_glob) + | |- (continuity ?X1) => + let trm := eval cbv beta in (X1 AppVar) in + let aux := rew_term trm in + (intro_hyp_glob aux; + try (change (continuity aux) in |- *; is_cont_glob) || is_cont_glob) + | |- (continuity_pt ?X1 ?X2) => + let trm := eval cbv beta in (X1 AppVar) in + let aux := rew_term trm in + (intro_hyp_pt aux X2; + try (change (continuity_pt aux X2) in |- *; is_cont_pt) || is_cont_pt) + | |- (derive_pt ?X1 ?X2 ?X3 = ?X4) => + let trm := eval cbv beta in (X1 AppVar) in + let aux := rew_term trm in + (intro_hyp_pt aux X2; + let aux2 := deriv_proof aux X2 in + (try + (replace (derive_pt X1 X2 X3) with (derive_pt aux X2 aux2); + [ simplify_derive aux X2; + try + unfold plus_fct, minus_fct, mult_fct, div_fct, id, fct_cte, + inv_fct, opp_fct in |- *; try ring + | try apply pr_nu ]) || is_diff_pt)) + end.
\ No newline at end of file diff --git a/theories/Reals/Ranalysis1.v b/theories/Reals/Ranalysis1.v index b8c5c2f4c..f60c609a0 100644 --- a/theories/Reals/Ranalysis1.v +++ b/theories/Reals/Ranalysis1.v @@ -8,177 +8,222 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. +Require Import Rbase. +Require Import Rfunctions. Require Export Rlimit. -Require Export Rderiv. -V7only [Import R_scope.]. Open Local Scope R_scope. -Implicit Variable Type f:R->R. +Require Export Rderiv. Open Local Scope R_scope. +Implicit Type f : R -> R. (****************************************************) (** 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))``. -Definition inv_fct [f:R->R] : R->R := [x:R]``/(f x)``. - -V8Infix "+" plus_fct : Rfun_scope. -V8Notation "- x" := (opp_fct x) : Rfun_scope. -V8Infix "*" mult_fct : Rfun_scope. -V8Infix "-" minus_fct : Rfun_scope. -V8Infix "/" div_fct : Rfun_scope. -Notation Local "f1 'o' f2" := (comp f1 f2) (at level 2, right associativity) - : Rfun_scope - V8only (at level 20, right associativity). -V8Notation "/ x" := (inv_fct x) : Rfun_scope. - -Delimits Scope Rfun_scope with F. - -Definition fct_cte [a:R] : R->R := [x:R]a. -Definition id := [x:R]x. +Definition plus_fct f1 f2 (x:R) : R := f1 x + f2 x. +Definition opp_fct f (x:R) : R := - f x. +Definition mult_fct f1 f2 (x:R) : R := f1 x * f2 x. +Definition mult_real_fct (a:R) f (x:R) : R := a * f x. +Definition minus_fct f1 f2 (x:R) : R := f1 x - f2 x. +Definition div_fct f1 f2 (x:R) : R := f1 x / f2 x. +Definition div_real_fct (a:R) f (x:R) : R := a / f x. +Definition comp f1 f2 (x:R) : R := f1 (f2 x). +Definition inv_fct f (x:R) : R := / f x. + +Infix "+" := plus_fct : Rfun_scope. +Notation "- x" := (opp_fct x) : Rfun_scope. +Infix "*" := mult_fct : Rfun_scope. +Infix "-" := minus_fct : Rfun_scope. +Infix "/" := div_fct : Rfun_scope. +Notation Local "f1 'o' f2" := (comp f1 f2) + (at level 20, right associativity) : Rfun_scope. +Notation "/ x" := (inv_fct x) : Rfun_scope. + +Delimit Scope Rfun_scope with F. + +Definition fct_cte (a x:R) : R := a. +Definition id (x:R) := 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)``. +Definition increasing f : Prop := forall x y:R, x <= y -> f x <= f y. +Definition decreasing f : Prop := forall x y:R, x <= y -> f y <= f x. +Definition strict_increasing f : Prop := forall x y:R, x < y -> f x < f y. +Definition strict_decreasing f : Prop := forall x y:R, x < y -> f y < f x. +Definition constant f : Prop := forall x y:R, f x = f y. (**********) -Definition no_cond : R->Prop := [x:R] True. +Definition no_cond (x:R) : Prop := True. (**********) -Definition constant_D_eq [f:R->R;D:R->Prop;c:R] : Prop := (x:R) (D x) -> (f x)==c. +Definition constant_D_eq f (D:R -> Prop) (c:R) : Prop := + forall x:R, D x -> f x = c. (***************************************************) (** Definition of continuity as a limit *) (***************************************************) (**********) -Definition continuity_pt [f:R->R; x0:R] : Prop := (continue_in f no_cond x0). -Definition continuity [f:R->R] : Prop := (x:R) (continuity_pt f x). +Definition continuity_pt f (x0:R) : Prop := continue_in f no_cond x0. +Definition continuity f : Prop := forall x:R, continuity_pt f x. Arguments Scope continuity_pt [Rfun_scope R_scope]. Arguments Scope continuity [Rfun_scope]. (**********) -Lemma continuity_pt_plus : (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. +Lemma continuity_pt_plus : + forall f1 f2 (x0:R), + continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 + f2) x0. +unfold continuity_pt, plus_fct in |- *; unfold continue_in in |- *; intros; + apply limit_plus; assumption. Qed. -Lemma continuity_pt_opp : (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. +Lemma continuity_pt_opp : + forall f (x0:R), continuity_pt f x0 -> continuity_pt (- f) x0. +unfold continuity_pt, opp_fct in |- *; unfold continue_in in |- *; intros; + apply limit_Ropp; assumption. Qed. -Lemma continuity_pt_minus : (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 continuity_pt_mult : (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 continuity_pt_const : (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 continuity_pt_scal : (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 continuity_pt_inv : (f:R->R; x0:R) (continuity_pt f x0) -> ~``(f x0)==0`` -> (continuity_pt (inv_fct f) x0). -Intros. -Replace (inv_fct f) with [x:R]``/(f x)``. -Unfold continuity_pt; Unfold continue_in; Intros; Apply limit_inv; Assumption. -Unfold inv_fct; Reflexivity. +Lemma continuity_pt_minus : + forall f1 f2 (x0:R), + continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 - f2) x0. +unfold continuity_pt, minus_fct in |- *; unfold continue_in in |- *; intros; + apply limit_minus; assumption. +Qed. + +Lemma continuity_pt_mult : + forall f1 f2 (x0:R), + continuity_pt f1 x0 -> continuity_pt f2 x0 -> continuity_pt (f1 * f2) x0. +unfold continuity_pt, mult_fct in |- *; unfold continue_in in |- *; intros; + apply limit_mul; assumption. +Qed. + +Lemma continuity_pt_const : forall f (x0:R), constant f -> continuity_pt f x0. +unfold constant, continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + intros; exists 1; split; + [ apply Rlt_0_1 + | intros; generalize (H x x0); intro; rewrite H2; simpl in |- *; + rewrite R_dist_eq; assumption ]. +Qed. + +Lemma continuity_pt_scal : + forall f (a x0:R), + continuity_pt f x0 -> continuity_pt (mult_real_fct a f) x0. +unfold continuity_pt, mult_real_fct in |- *; unfold continue_in in |- *; + intros; apply (limit_mul (fun x:R => a) f (D_x no_cond x0) a (f x0) x0). +unfold limit1_in in |- *; unfold limit_in in |- *; intros; exists 1; split. +apply Rlt_0_1. +intros; rewrite R_dist_eq; assumption. +assumption. +Qed. + +Lemma continuity_pt_inv : + forall f (x0:R), continuity_pt f x0 -> f x0 <> 0 -> continuity_pt (/ f) x0. +intros. +replace (/ f)%F with (fun x:R => / f x). +unfold continuity_pt in |- *; unfold continue_in in |- *; intros; + apply limit_inv; assumption. +unfold inv_fct in |- *; reflexivity. Qed. -Lemma div_eq_inv : (f1,f2:R->R) (div_fct f1 f2)==(mult_fct f1 (inv_fct f2)). -Intros; Reflexivity. +Lemma div_eq_inv : forall f1 f2, (f1 / f2)%F = (f1 * / f2)%F. +intros; reflexivity. Qed. -Lemma continuity_pt_div : (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 continuity_pt_mult; [Assumption | Apply continuity_pt_inv; Assumption]. -Qed. - -Lemma continuity_pt_comp : (f1,f2:R->R;x:R) (continuity_pt f1 x) -> (continuity_pt f2 (f1 x)) -> (continuity_pt (comp f2 f1) x). -Unfold continuity_pt; Unfold continue_in; Intros; Unfold comp. -Cut (limit1_in [x0:R](f2 (f1 x0)) (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1) -(f2 (f1 x)) x) -> (limit1_in [x0:R](f2 (f1 x0)) (D_x no_cond x) (f2 (f1 x)) x). -Intro; Apply H1. -EApply limit_comp. -Apply H. -Apply H0. -Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. -Assert H3 := (H1 eps H2). -Elim H3; Intros. -Exists x0. -Split. -Elim H4; Intros; Assumption. -Intros; Case (Req_EM (f1 x) (f1 x1)); Intro. -Rewrite H6; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Elim H4; Intros; Apply H8. -Split. -Unfold Dgf D_x no_cond. -Split. -Split. -Trivial. -Elim H5; Unfold D_x no_cond; Intros. -Elim H9; Intros; Assumption. -Split. -Trivial. -Assumption. -Elim H5; Intros; Assumption. +Lemma continuity_pt_div : + forall f1 f2 (x0:R), + continuity_pt f1 x0 -> + continuity_pt f2 x0 -> f2 x0 <> 0 -> continuity_pt (f1 / f2) x0. +intros; rewrite (div_eq_inv f1 f2); apply continuity_pt_mult; + [ assumption | apply continuity_pt_inv; assumption ]. +Qed. + +Lemma continuity_pt_comp : + forall f1 f2 (x:R), + continuity_pt f1 x -> continuity_pt f2 (f1 x) -> continuity_pt (f2 o f1) x. +unfold continuity_pt in |- *; unfold continue_in in |- *; intros; + unfold comp in |- *. +cut + (limit1_in (fun x0:R => f2 (f1 x0)) + (Dgf (D_x no_cond x) (D_x no_cond (f1 x)) f1) ( + f2 (f1 x)) x -> + limit1_in (fun x0:R => f2 (f1 x0)) (D_x no_cond x) (f2 (f1 x)) x). +intro; apply H1. +eapply limit_comp. +apply H. +apply H0. +unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +assert (H3 := H1 eps H2). +elim H3; intros. +exists x0. +split. +elim H4; intros; assumption. +intros; case (Req_dec (f1 x) (f1 x1)); intro. +rewrite H6; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + assumption. +elim H4; intros; apply H8. +split. +unfold Dgf, D_x, no_cond in |- *. +split. +split. +trivial. +elim H5; unfold D_x, no_cond in |- *; intros. +elim H9; intros; assumption. +split. +trivial. +assumption. +elim H5; intros; assumption. Qed. (**********) -Lemma continuity_plus : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (plus_fct f1 f2)). -Unfold continuity; Intros; Apply (continuity_pt_plus f1 f2 x (H x) (H0 x)). +Lemma continuity_plus : + forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 + f2). +unfold continuity in |- *; intros; + apply (continuity_pt_plus f1 f2 x (H x) (H0 x)). Qed. -Lemma continuity_opp : (f:R->R) (continuity f)->(continuity (opp_fct f)). -Unfold continuity; Intros; Apply (continuity_pt_opp f x (H x)). +Lemma continuity_opp : forall f, continuity f -> continuity (- f). +unfold continuity in |- *; intros; apply (continuity_pt_opp f x (H x)). Qed. -Lemma continuity_minus : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (minus_fct f1 f2)). -Unfold continuity; Intros; Apply (continuity_pt_minus f1 f2 x (H x) (H0 x)). +Lemma continuity_minus : + forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 - f2). +unfold continuity in |- *; intros; + apply (continuity_pt_minus f1 f2 x (H x) (H0 x)). Qed. -Lemma continuity_mult : (f1,f2:R->R) (continuity f1)->(continuity f2)->(continuity (mult_fct f1 f2)). -Unfold continuity; Intros; Apply (continuity_pt_mult f1 f2 x (H x) (H0 x)). +Lemma continuity_mult : + forall f1 f2, continuity f1 -> continuity f2 -> continuity (f1 * f2). +unfold continuity in |- *; intros; + apply (continuity_pt_mult f1 f2 x (H x) (H0 x)). Qed. -Lemma continuity_const : (f:R->R) (constant f) -> (continuity f). -Unfold continuity; Intros; Apply (continuity_pt_const f x H). +Lemma continuity_const : forall f, constant f -> continuity f. +unfold continuity in |- *; intros; apply (continuity_pt_const f x H). Qed. -Lemma continuity_scal : (f:R->R;a:R) (continuity f) -> (continuity (mult_real_fct a f)). -Unfold continuity; Intros; Apply (continuity_pt_scal f a x (H x)). +Lemma continuity_scal : + forall f (a:R), continuity f -> continuity (mult_real_fct a f). +unfold continuity in |- *; intros; apply (continuity_pt_scal f a x (H x)). Qed. -Lemma continuity_inv : (f:R->R) (continuity f)->((x:R) ~``(f x)==0``)->(continuity (inv_fct f)). -Unfold continuity; Intros; Apply (continuity_pt_inv f x (H x) (H0 x)). +Lemma continuity_inv : + forall f, continuity f -> (forall x:R, f x <> 0) -> continuity (/ f). +unfold continuity in |- *; intros; apply (continuity_pt_inv f x (H x) (H0 x)). Qed. -Lemma continuity_div : (f1,f2:R->R) (continuity f1)->(continuity f2)->((x:R) ~``(f2 x)==0``)->(continuity (div_fct f1 f2)). -Unfold continuity; Intros; Apply (continuity_pt_div f1 f2 x (H x) (H0 x) (H1 x)). +Lemma continuity_div : + forall f1 f2, + continuity f1 -> + continuity f2 -> (forall x:R, f2 x <> 0) -> continuity (f1 / f2). +unfold continuity in |- *; intros; + apply (continuity_pt_div f1 f2 x (H x) (H0 x) (H1 x)). Qed. -Lemma continuity_comp : (f1,f2:R->R) (continuity f1) -> (continuity f2) -> (continuity (comp f2 f1)). -Unfold continuity; Intros. -Apply (continuity_pt_comp f1 f2 x (H x) (H0 (f1 x))). +Lemma continuity_comp : + forall f1 f2, continuity f1 -> continuity f2 -> continuity (f2 o f1). +unfold continuity in |- *; intros. +apply (continuity_pt_comp f1 f2 x (H x) (H0 (f1 x))). Qed. @@ -186,15 +231,20 @@ Qed. (** Derivative's definition using Landau's kernel *) (*****************************************************) -Definition derivable_pt_lim [f:R->R;x,l:R] : Prop := ((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_pt_lim f (x l:R) : Prop := + forall eps:R, + 0 < eps -> + exists delta : posreal + | (forall h:R, + h <> 0 -> Rabs h < delta -> Rabs ((f (x + h) - f x) / h - l) < eps). -Definition derivable_pt_abs [f:R->R;x:R] : R -> Prop := [l:R](derivable_pt_lim f x l). +Definition derivable_pt_abs f (x l:R) : Prop := derivable_pt_lim f x l. -Definition derivable_pt [f:R->R;x:R] := (SigT R (derivable_pt_abs f x)). -Definition derivable [f:R->R] := (x:R)(derivable_pt f x). +Definition derivable_pt f (x:R) := sigT (derivable_pt_abs f x). +Definition derivable f := forall x:R, derivable_pt f x. -Definition derive_pt [f:R->R;x:R;pr:(derivable_pt f x)] := (projT1 ? ? pr). -Definition derive [f:R->R;pr:(derivable f)] := [x:R](derive_pt f x (pr x)). +Definition derive_pt f (x:R) (pr:derivable_pt f x) := projT1 pr. +Definition derive f (pr:derivable f) (x:R) := derive_pt f x (pr x). Arguments Scope derivable_pt_lim [Rfun_scope R_scope]. Arguments Scope derivable_pt_abs [Rfun_scope R_scope R_scope]. @@ -203,125 +253,191 @@ Arguments Scope derivable [Rfun_scope]. Arguments Scope derive_pt [Rfun_scope R_scope _]. Arguments Scope derive [Rfun_scope _]. -Definition antiderivative [f,g:R->R;a,b:R] : Prop := ((x:R)``a<=x<=b``->(EXT pr : (derivable_pt g x) | (f x)==(derive_pt g x pr)))/\``a<=b``. +Definition antiderivative f (g:R -> R) (a b:R) : Prop := + (forall x:R, + a <= x <= b -> exists pr : derivable_pt g x | f x = derive_pt g x pr) /\ + a <= b. (************************************) (** Class of differential functions *) (************************************) -Record Differential : Type := mkDifferential { -d1 :> R->R; -cond_diff : (derivable d1) }. +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 cond_D1)) }. +Record Differential_D2 : Type := mkDifferential_D2 + {d2 :> R -> R; + cond_D1 : derivable d2; + cond_D2 : derivable (derive d2 cond_D1)}. (**********) -Lemma unicite_step1 : (f:R->R;x,l1,l2:R) (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l1 R0) -> (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l2 R0) -> l1 == l2. -Intros; Apply (single_limit [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l1 l2 R0); Try Assumption. -Unfold adhDa; Intros; Exists ``alp/2``. -Split. -Unfold Rdiv; Apply prod_neq_R0. -Red; Intro; Rewrite H2 in H1; Elim (Rlt_antirefl ? H1). -Apply Rinv_neq_R0; DiscrR. -Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Unfold Rdiv; Rewrite Rabsolu_mult. -Replace ``(Rabsolu (/2))`` with ``/2``. -Replace (Rabsolu alp) with alp. -Apply Rlt_monotony_contra with ``2``. -Sup0. -Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]; Rewrite Rmult_1r; Rewrite double; Pattern 1 alp; Replace alp with ``alp+0``; [Idtac | Ring]; Apply Rlt_compatibility; Assumption. -Symmetry; Apply Rabsolu_right; Left; Assumption. -Symmetry; Apply Rabsolu_right; Left; Change ``0</2``; Apply Rlt_Rinv; Sup0. -Qed. - -Lemma unicite_step2 : (f:R->R;x,l:R) (derivable_pt_lim f x l) -> (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l R0). -Unfold derivable_pt_lim; Intros; Unfold limit1_in; Unfold limit_in; Intros. -Assert H1 := (H eps H0). -Elim H1 ; Intros. -Exists (pos x0). -Split. -Apply (cond_pos x0). -Simpl; Unfold R_dist; Intros. -Elim H3; Intros. -Apply H2; [Assumption |Unfold Rminus in H5; Rewrite Ropp_O in H5; Rewrite Rplus_Or in H5; Assumption]. -Qed. - -Lemma unicite_step3 : (f:R->R;x,l:R) (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h<>0`` l R0) -> (derivable_pt_lim f x l). -Unfold limit1_in derivable_pt_lim; Unfold limit_in; Unfold dist; Simpl; Intros. -Elim (H eps H0). -Intros; Elim H1; Intros. -Exists (mkposreal x0 H2). -Simpl; Intros; Unfold R_dist in H3; Apply (H3 h). -Split; [Assumption | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Assumption]. -Qed. - -Lemma unicite_limite : (f:R->R;x,l1,l2:R) (derivable_pt_lim f x l1) -> (derivable_pt_lim f x l2) -> l1==l2. -Intros. -Assert H1 := (unicite_step2 ? ? ? H). -Assert H2 := (unicite_step2 ? ? ? H0). -Assert H3 := (unicite_step1 ? ? ? ? H1 H2). -Assumption. -Qed. - -Lemma derive_pt_eq : (f:R->R;x,l:R;pr:(derivable_pt f x)) (derive_pt f x pr)==l <-> (derivable_pt_lim f x l). -Intros; Split. -Intro; Assert H1 := (projT2 ? ? pr); Unfold derive_pt in H; Rewrite H in H1; Assumption. -Intro; Assert H1 := (projT2 ? ? pr); Unfold derivable_pt_abs in H1. -Assert H2 := (unicite_limite ? ? ? ? H H1). -Unfold derive_pt; Unfold derivable_pt_abs. -Symmetry; Assumption. +Lemma uniqueness_step1 : + forall f (x l1 l2:R), + limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l1 0 -> + limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l2 0 -> + l1 = l2. +intros; + apply + (single_limit (fun h:R => (f (x + h) - f x) / h) ( + fun h:R => h <> 0) l1 l2 0); try assumption. +unfold adhDa in |- *; intros; exists (alp / 2). +split. +unfold Rdiv in |- *; apply prod_neq_R0. +red in |- *; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1). +apply Rinv_neq_0_compat; discrR. +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; unfold Rdiv in |- *; rewrite Rabs_mult. +replace (Rabs (/ 2)) with (/ 2). +replace (Rabs alp) with alp. +apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite (Rmult_comm 2); rewrite Rmult_assoc; rewrite <- Rinv_l_sym; + [ idtac | discrR ]; rewrite Rmult_1_r; rewrite double; + pattern alp at 1 in |- *; replace alp with (alp + 0); + [ idtac | ring ]; apply Rplus_lt_compat_l; assumption. +symmetry in |- *; apply Rabs_right; left; assumption. +symmetry in |- *; apply Rabs_right; left; change (0 < / 2) in |- *; + apply Rinv_0_lt_compat; prove_sup0. +Qed. + +Lemma uniqueness_step2 : + forall f (x l:R), + derivable_pt_lim f x l -> + limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0. +unfold derivable_pt_lim in |- *; intros; unfold limit1_in in |- *; + unfold limit_in in |- *; intros. +assert (H1 := H eps H0). +elim H1; intros. +exists (pos x0). +split. +apply (cond_pos x0). +simpl in |- *; unfold R_dist in |- *; intros. +elim H3; intros. +apply H2; + [ assumption + | unfold Rminus in H5; rewrite Ropp_0 in H5; rewrite Rplus_0_r in H5; + assumption ]. +Qed. + +Lemma uniqueness_step3 : + forall f (x l:R), + limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 -> + derivable_pt_lim f x l. +unfold limit1_in, derivable_pt_lim in |- *; unfold limit_in in |- *; + unfold dist in |- *; simpl in |- *; intros. +elim (H eps H0). +intros; elim H1; intros. +exists (mkposreal x0 H2). +simpl in |- *; intros; unfold R_dist in H3; apply (H3 h). +split; + [ assumption + | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; assumption ]. +Qed. + +Lemma uniqueness_limite : + forall f (x l1 l2:R), + derivable_pt_lim f x l1 -> derivable_pt_lim f x l2 -> l1 = l2. +intros. +assert (H1 := uniqueness_step2 _ _ _ H). +assert (H2 := uniqueness_step2 _ _ _ H0). +assert (H3 := uniqueness_step1 _ _ _ _ H1 H2). +assumption. +Qed. + +Lemma derive_pt_eq : + forall f (x l:R) (pr:derivable_pt f x), + derive_pt f x pr = l <-> derivable_pt_lim f x l. +intros; split. +intro; assert (H1 := projT2 pr); unfold derive_pt in H; rewrite H in H1; + assumption. +intro; assert (H1 := projT2 pr); unfold derivable_pt_abs in H1. +assert (H2 := uniqueness_limite _ _ _ _ H H1). +unfold derive_pt in |- *; unfold derivable_pt_abs in |- *. +symmetry in |- *; assumption. Qed. (**********) -Lemma derive_pt_eq_0 : (f:R->R;x,l:R;pr:(derivable_pt f x)) (derivable_pt_lim f x l) -> (derive_pt f x pr)==l. -Intros; Elim (derive_pt_eq f x l pr); Intros. -Apply (H1 H). +Lemma derive_pt_eq_0 : + forall f (x l:R) (pr:derivable_pt f x), + derivable_pt_lim f x l -> derive_pt f x pr = l. +intros; elim (derive_pt_eq f x l pr); intros. +apply (H1 H). Qed. (**********) -Lemma derive_pt_eq_1 : (f:R->R;x,l:R;pr:(derivable_pt f x)) (derive_pt f x pr)==l -> (derivable_pt_lim f x l). -Intros; Elim (derive_pt_eq f x l pr); Intros. -Apply (H0 H). +Lemma derive_pt_eq_1 : + forall f (x l:R) (pr:derivable_pt f x), + derive_pt f x pr = l -> derivable_pt_lim f x l. +intros; elim (derive_pt_eq f x l pr); intros. +apply (H0 H). Qed. (********************************************************************) (** Equivalence of this definition with the one using limit concept *) (********************************************************************) -Lemma derive_pt_D_in : (f,df:R->R;x:R;pr:(derivable_pt f x)) (D_in f df no_cond x) <-> (derive_pt f x pr)==(df x). -Intros; Split. -Unfold D_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. -Apply derive_pt_eq_0. -Unfold derivable_pt_lim. -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. -Assert H0 := (derive_pt_eq_1 f x (df x) pr H). -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. - -Lemma derivable_pt_lim_D_in : (f,df:R->R;x:R) (D_in f df no_cond x) <-> (derivable_pt_lim f x (df x)). -Intros; Split. -Unfold D_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. -Unfold derivable_pt_lim. -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. -Unfold derivable_pt_lim in H. -Unfold D_in; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. -Elim (H eps H0); Intros alpha H2; Exists (pos alpha); Split. -Apply (cond_pos alpha). -Intros. -Elim H1; Intros; Unfold D_x in H3; Elim H3; Intros; Cut ``x0-x<>0``. -Intro; Generalize (H2 ``x0-x`` H7 H4); Replace ``x+(x0-x)`` with x0. -Intro; Assumption. -Ring. -Auto with real. +Lemma derive_pt_D_in : + forall f (df:R -> R) (x:R) (pr:derivable_pt f x), + D_in f df no_cond x <-> derive_pt f x pr = df x. +intros; split. +unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +apply derive_pt_eq_0. +unfold derivable_pt_lim in |- *. +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) /\ Rabs (x + h - x) < alpha); + [ intro; generalize (H6 H8); rewrite H7; intro; assumption + | split; + [ unfold D_x in |- *; split; + [ unfold no_cond in |- *; trivial + | apply Rminus_not_eq_right; rewrite H7; assumption ] + | rewrite H7; assumption ] ] + | ring ]. +intro. +assert (H0 := derive_pt_eq_1 f x (df x) pr H). +unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *; + unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + 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. + +Lemma derivable_pt_lim_D_in : + forall f (df:R -> R) (x:R), + D_in f df no_cond x <-> derivable_pt_lim f x (df x). +intros; split. +unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +unfold derivable_pt_lim in |- *. +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) /\ Rabs (x + h - x) < alpha); + [ intro; generalize (H6 H8); rewrite H7; intro; assumption + | split; + [ unfold D_x in |- *; split; + [ unfold no_cond in |- *; trivial + | apply Rminus_not_eq_right; rewrite H7; assumption ] + | rewrite H7; assumption ] ] + | ring ]. +intro. +unfold derivable_pt_lim in H. +unfold D_in in |- *; unfold limit1_in in |- *; unfold limit_in in |- *; + unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + intros. +elim (H eps H0); intros alpha H2; exists (pos alpha); split. +apply (cond_pos alpha). +intros. +elim H1; intros; unfold D_x in H3; elim H3; intros; cut (x0 - x <> 0). +intro; generalize (H2 (x0 - x) H7 H4); replace (x + (x0 - x)) with x0. +intro; assumption. +ring. +auto with real. Qed. @@ -329,457 +445,555 @@ Qed. (** derivability -> continuity *) (***********************************) (**********) -Lemma derivable_derive : (f:R->R;x:R;pr:(derivable_pt f x)) (EXT l : R | (derive_pt f x pr)==l). -Intros; Exists (projT1 ? ? pr). -Unfold derive_pt; Reflexivity. +Lemma derivable_derive : + forall f (x:R) (pr:derivable_pt f x), exists l : R | derive_pt f x pr = l. +intros; exists (projT1 pr). +unfold derive_pt in |- *; reflexivity. Qed. -Theorem derivable_continuous_pt : (f:R->R;x:R) (derivable_pt f x) -> (continuity_pt f x). -Intros. -Generalize (derivable_derive f x X); Intro. -Elim H; Intros l H1. -Cut l==((fct_cte l) x). -Intro. -Rewrite H0 in H1. -Generalize (derive_pt_D_in f (fct_cte l) x); Intro. -Elim (H2 X); Intros. -Generalize (H4 H1); Intro. -Unfold continuity_pt. -Apply (cont_deriv f (fct_cte l) no_cond x H5). -Unfold fct_cte; Reflexivity. +Theorem derivable_continuous_pt : + forall f (x:R), derivable_pt f x -> continuity_pt f x. +intros. +generalize (derivable_derive f x X); intro. +elim H; intros l H1. +cut (l = fct_cte l x). +intro. +rewrite H0 in H1. +generalize (derive_pt_D_in f (fct_cte l) x); intro. +elim (H2 X); intros. +generalize (H4 H1); intro. +unfold continuity_pt in |- *. +apply (cont_deriv f (fct_cte l) no_cond x H5). +unfold fct_cte in |- *; reflexivity. Qed. -Theorem derivable_continuous : (f:R->R) (derivable f) -> (continuity f). -Unfold derivable continuity; Intros. -Apply (derivable_continuous_pt f x (X x)). +Theorem derivable_continuous : forall f, derivable f -> continuity f. +unfold derivable, continuity in |- *; intros. +apply (derivable_continuous_pt f x (X x)). Qed. (****************************************************************) (** Main rules *) (****************************************************************) -Lemma derivable_pt_lim_plus : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 x l2) -> (derivable_pt_lim (plus_fct f1 f2) x ``l1+l2``). -Intros. -Apply unicite_step3. -Assert H1 := (unicite_step2 ? ? ? H). -Assert H2 := (unicite_step2 ? ? ? H0). -Unfold plus_fct. -Cut (h:R)``((f1 (x+h))+(f2 (x+h))-((f1 x)+(f2 x)))/h``==``((f1 (x+h))-(f1 x))/h+((f2 (x+h))-(f2 x))/h``. -Intro. -Generalize(limit_plus [h':R]``((f1 (x+h'))-(f1 x))/h'`` [h':R]``((f2 (x+h'))-(f2 x))/h'`` [h:R]``h <> 0`` l1 l2 ``0`` H1 H2). -Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. -Elim (H4 eps H5); Intros. -Exists x0. -Elim H6; Intros. -Split. -Assumption. -Intros; Rewrite H3; Apply H8; Assumption. -Intro; Unfold Rdiv; Ring. -Qed. - -Lemma derivable_pt_lim_opp : (f:R->R;x,l:R) (derivable_pt_lim f x l) -> (derivable_pt_lim (opp_fct f) x (Ropp l)). -Intros. -Apply unicite_step3. -Assert H1 := (unicite_step2 ? ? ? H). -Unfold opp_fct. -Cut (h:R) ``( -(f (x+h))- -(f x))/h``==(Ropp ``((f (x+h))-(f x))/h``). -Intro. -Generalize (limit_Ropp [h:R]``((f (x+h))-(f x))/h``[h:R]``h <> 0`` l ``0`` H1). -Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. -Elim (H2 eps H3); Intros. -Exists x0. -Elim H4; Intros. -Split. -Assumption. -Intros; Rewrite H0; Apply H6; Assumption. -Intro; Unfold Rdiv; Ring. -Qed. - -Lemma derivable_pt_lim_minus : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 x l2) -> (derivable_pt_lim (minus_fct f1 f2) x ``l1-l2``). -Intros. -Apply unicite_step3. -Assert H1 := (unicite_step2 ? ? ? H). -Assert H2 := (unicite_step2 ? ? ? H0). -Unfold minus_fct. -Cut (h:R)``((f1 (x+h))-(f1 x))/h-((f2 (x+h))-(f2 x))/h``==``((f1 (x+h))-(f2 (x+h))-((f1 x)-(f2 x)))/h``. -Intro. -Generalize (limit_minus [h':R]``((f1 (x+h'))-(f1 x))/h'`` [h':R]``((f2 (x+h'))-(f2 x))/h'`` [h:R]``h <> 0`` l1 l2 ``0`` H1 H2). -Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. -Elim (H4 eps H5); Intros. -Exists x0. -Elim H6; Intros. -Split. -Assumption. -Intros; Rewrite <- H3; Apply H8; Assumption. -Intro; Unfold Rdiv; Ring. -Qed. - -Lemma derivable_pt_lim_mult : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 x l2) -> (derivable_pt_lim (mult_fct f1 f2) x ``l1*(f2 x)+(f1 x)*l2``). -Intros. -Assert H1 := (derivable_pt_lim_D_in f1 [y:R]l1 x). -Elim H1; Intros. -Assert H4 := (H3 H). -Assert H5 := (derivable_pt_lim_D_in f2 [y:R]l2 x). -Elim H5; Intros. -Assert H8 := (H7 H0). -Clear H1 H2 H3 H5 H6 H7. -Assert H1 := (derivable_pt_lim_D_in (mult_fct f1 f2) [y:R]``l1*(f2 x)+(f1 x)*l2`` x). -Elim H1; Intros. -Clear H1 H3. -Apply H2. -Unfold mult_fct. -Apply (Dmult no_cond [y:R]l1 [y:R]l2 f1 f2 x); Assumption. -Qed. - -Lemma derivable_pt_lim_const : (a,x:R) (derivable_pt_lim (fct_cte a) x ``0``). -Intros; Unfold fct_cte derivable_pt_lim. -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 derivable_pt_lim_scal : (f:R->R;a,x,l:R) (derivable_pt_lim f x l) -> (derivable_pt_lim (mult_real_fct a f) x ``a*l``). -Intros. -Assert H0 := (derivable_pt_lim_const a x). -Replace (mult_real_fct a f) with (mult_fct (fct_cte a) f). -Replace ``a*l`` with ``0*(f x)+a*l``; [Idtac | Ring]. -Apply (derivable_pt_lim_mult (fct_cte a) f x ``0`` l); Assumption. -Unfold mult_real_fct mult_fct fct_cte; Reflexivity. -Qed. - -Lemma derivable_pt_lim_id : (x:R) (derivable_pt_lim id x ``1``). -Intro; Unfold derivable_pt_lim. -Intros eps Heps; Exists (mkposreal eps Heps); Intros h H1 H2; Unfold id; 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 derivable_pt_lim_Rsqr : (x:R) (derivable_pt_lim Rsqr x ``2*x``). -Intro; Unfold derivable_pt_lim. -Unfold Rsqr; 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)-x*x`` with ``2*x*h+h*h``; [Idtac | Ring]. -Unfold Rdiv; Rewrite Rmult_Rplus_distrl. -Repeat Rewrite Rmult_assoc. -Repeat Rewrite <- Rinv_r_sym; [Idtac | Assumption]. -Ring. -Qed. - -Lemma derivable_pt_lim_comp : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 (f1 x) l2) -> (derivable_pt_lim (comp f2 f1) x ``l2*l1``). -Intros; Assert H1 := (derivable_pt_lim_D_in f1 [y:R]l1 x). -Elim H1; Intros. -Assert H4 := (H3 H). -Assert H5 := (derivable_pt_lim_D_in f2 [y:R]l2 (f1 x)). -Elim H5; Intros. -Assert H8 := (H7 H0). -Clear H1 H2 H3 H5 H6 H7. -Assert H1 := (derivable_pt_lim_D_in (comp f2 f1) [y:R]``l2*l1`` x). -Elim H1; Intros. -Clear H1 H3; Apply H2. -Unfold comp; Cut (D_in [x0:R](f2 (f1 x0)) [y:R]``l2*l1`` (Dgf no_cond no_cond f1) x) -> (D_in [x0:R](f2 (f1 x0)) [y:R]``l2*l1`` no_cond x). -Intro; Apply H1. -Rewrite Rmult_sym; Apply (Dcomp no_cond no_cond [y:R]l1 [y:R]l2 f1 f2 x); Assumption. -Unfold Dgf D_in no_cond; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. -Elim (H1 eps H3); Intros. -Exists x0; Intros; Split. -Elim H5; Intros; Assumption. -Intros; Elim H5; Intros; Apply H9; Split. -Unfold D_x; Split. -Split; Trivial. -Elim H6; Intros; Unfold D_x in H10; Elim H10; Intros; Assumption. -Elim H6; Intros; Assumption. -Qed. - -Lemma derivable_pt_plus : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> (derivable_pt (plus_fct f1 f2) x). -Unfold derivable_pt; Intros. -Elim X; Intros. -Elim X0; Intros. -Apply Specif.existT with ``x0+x1``. -Apply derivable_pt_lim_plus; Assumption. -Qed. - -Lemma derivable_pt_opp : (f:R->R;x:R) (derivable_pt f x) -> (derivable_pt (opp_fct f) x). -Unfold derivable_pt; Intros. -Elim X; Intros. -Apply Specif.existT with ``-x0``. -Apply derivable_pt_lim_opp; Assumption. -Qed. - -Lemma derivable_pt_minus : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> (derivable_pt (minus_fct f1 f2) x). -Unfold derivable_pt; Intros. -Elim X; Intros. -Elim X0; Intros. -Apply Specif.existT with ``x0-x1``. -Apply derivable_pt_lim_minus; Assumption. -Qed. - -Lemma derivable_pt_mult : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> (derivable_pt (mult_fct f1 f2) x). -Unfold derivable_pt; Intros. -Elim X; Intros. -Elim X0; Intros. -Apply Specif.existT with ``x0*(f2 x)+(f1 x)*x1``. -Apply derivable_pt_lim_mult; Assumption. -Qed. - -Lemma derivable_pt_const : (a,x:R) (derivable_pt (fct_cte a) x). -Intros; Unfold derivable_pt. -Apply Specif.existT with ``0``. -Apply derivable_pt_lim_const. -Qed. - -Lemma derivable_pt_scal : (f:R->R;a,x:R) (derivable_pt f x) -> (derivable_pt (mult_real_fct a f) x). -Unfold derivable_pt; Intros. -Elim X; Intros. -Apply Specif.existT with ``a*x0``. -Apply derivable_pt_lim_scal; Assumption. -Qed. - -Lemma derivable_pt_id : (x:R) (derivable_pt id x). -Unfold derivable_pt; Intro. -Exists ``1``. -Apply derivable_pt_lim_id. -Qed. - -Lemma derivable_pt_Rsqr : (x:R) (derivable_pt Rsqr x). -Unfold derivable_pt; Intro; Apply Specif.existT with ``2*x``. -Apply derivable_pt_lim_Rsqr. -Qed. - -Lemma derivable_pt_comp : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 (f1 x)) -> (derivable_pt (comp f2 f1) x). -Unfold derivable_pt; Intros. -Elim X; Intros. -Elim X0 ;Intros. -Apply Specif.existT with ``x1*x0``. -Apply derivable_pt_lim_comp; Assumption. -Qed. - -Lemma derivable_plus : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (plus_fct f1 f2)). -Unfold derivable; Intros. -Apply (derivable_pt_plus ? ? x (X ?) (X0 ?)). -Qed. - -Lemma derivable_opp : (f:R->R) (derivable f) -> (derivable (opp_fct f)). -Unfold derivable; Intros. -Apply (derivable_pt_opp ? x (X ?)). -Qed. - -Lemma derivable_minus : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (minus_fct f1 f2)). -Unfold derivable; Intros. -Apply (derivable_pt_minus ? ? x (X ?) (X0 ?)). -Qed. - -Lemma derivable_mult : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (mult_fct f1 f2)). -Unfold derivable; Intros. -Apply (derivable_pt_mult ? ? x (X ?) (X0 ?)). -Qed. - -Lemma derivable_const : (a:R) (derivable (fct_cte a)). -Unfold derivable; Intros. -Apply derivable_pt_const. -Qed. - -Lemma derivable_scal : (f:R->R;a:R) (derivable f) -> (derivable (mult_real_fct a f)). -Unfold derivable; Intros. -Apply (derivable_pt_scal ? a x (X ?)). -Qed. - -Lemma derivable_id : (derivable id). -Unfold derivable; Intro; Apply derivable_pt_id. -Qed. - -Lemma derivable_Rsqr : (derivable Rsqr). -Unfold derivable; Intro; Apply derivable_pt_Rsqr. -Qed. - -Lemma derivable_comp : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> (derivable (comp f2 f1)). -Unfold derivable; Intros. -Apply (derivable_pt_comp ? ? x (X ?) (X0 ?)). -Qed. - -Lemma derive_pt_plus : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 x)) ``(derive_pt (plus_fct f1 f2) x (derivable_pt_plus ? ? ? pr1 pr2)) == (derive_pt f1 x pr1) + (derive_pt f2 x pr2)``. -Intros. -Assert H := (derivable_derive f1 x pr1). -Assert H0 := (derivable_derive f2 x pr2). -Assert H1 := (derivable_derive (plus_fct f1 f2) x (derivable_pt_plus ? ? ? pr1 pr2)). -Elim H; Clear H; Intros l1 H. -Elim H0; Clear H0; Intros l2 H0. -Elim H1; Clear H1; Intros l H1. -Rewrite H; Rewrite H0; Apply derive_pt_eq_0. -Assert H3 := (projT2 ? ? pr1). -Unfold derive_pt in H; Rewrite H in H3. -Assert H4 := (projT2 ? ? pr2). -Unfold derive_pt in H0; Rewrite H0 in H4. -Apply derivable_pt_lim_plus; Assumption. -Qed. - -Lemma derive_pt_opp : (f:R->R;x:R;pr1:(derivable_pt f x)) ``(derive_pt (opp_fct f) x (derivable_pt_opp ? ? pr1)) == -(derive_pt f x pr1)``. -Intros. -Assert H := (derivable_derive f x pr1). -Assert H0 := (derivable_derive (opp_fct f) x (derivable_pt_opp ? ? pr1)). -Elim H; Clear H; Intros l1 H. -Elim H0; Clear H0; Intros l2 H0. -Rewrite H; Apply derive_pt_eq_0. -Assert H3 := (projT2 ? ? pr1). -Unfold derive_pt in H; Rewrite H in H3. -Apply derivable_pt_lim_opp; Assumption. -Qed. - -Lemma derive_pt_minus : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 x)) ``(derive_pt (minus_fct f1 f2) x (derivable_pt_minus ? ? ? pr1 pr2)) == (derive_pt f1 x pr1) - (derive_pt f2 x pr2)``. -Intros. -Assert H := (derivable_derive f1 x pr1). -Assert H0 := (derivable_derive f2 x pr2). -Assert H1 := (derivable_derive (minus_fct f1 f2) x (derivable_pt_minus ? ? ? pr1 pr2)). -Elim H; Clear H; Intros l1 H. -Elim H0; Clear H0; Intros l2 H0. -Elim H1; Clear H1; Intros l H1. -Rewrite H; Rewrite H0; Apply derive_pt_eq_0. -Assert H3 := (projT2 ? ? pr1). -Unfold derive_pt in H; Rewrite H in H3. -Assert H4 := (projT2 ? ? pr2). -Unfold derive_pt in H0; Rewrite H0 in H4. -Apply derivable_pt_lim_minus; Assumption. -Qed. - -Lemma derive_pt_mult : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 x)) ``(derive_pt (mult_fct f1 f2) x (derivable_pt_mult ? ? ? pr1 pr2)) == (derive_pt f1 x pr1)*(f2 x) + (f1 x)*(derive_pt f2 x pr2)``. -Intros. -Assert H := (derivable_derive f1 x pr1). -Assert H0 := (derivable_derive f2 x pr2). -Assert H1 := (derivable_derive (mult_fct f1 f2) x (derivable_pt_mult ? ? ? pr1 pr2)). -Elim H; Clear H; Intros l1 H. -Elim H0; Clear H0; Intros l2 H0. -Elim H1; Clear H1; Intros l H1. -Rewrite H; Rewrite H0; Apply derive_pt_eq_0. -Assert H3 := (projT2 ? ? pr1). -Unfold derive_pt in H; Rewrite H in H3. -Assert H4 := (projT2 ? ? pr2). -Unfold derive_pt in H0; Rewrite H0 in H4. -Apply derivable_pt_lim_mult; Assumption. -Qed. - -Lemma derive_pt_const : (a,x:R) (derive_pt (fct_cte a) x (derivable_pt_const a x)) == R0. -Intros. -Apply derive_pt_eq_0. -Apply derivable_pt_lim_const. -Qed. - -Lemma derive_pt_scal : (f:R->R;a,x:R;pr:(derivable_pt f x)) ``(derive_pt (mult_real_fct a f) x (derivable_pt_scal ? ? ? pr)) == a * (derive_pt f x pr)``. -Intros. -Assert H := (derivable_derive f x pr). -Assert H0 := (derivable_derive (mult_real_fct a f) x (derivable_pt_scal ? ? ? pr)). -Elim H; Clear H; Intros l1 H. -Elim H0; Clear H0; Intros l2 H0. -Rewrite H; Apply derive_pt_eq_0. -Assert H3 := (projT2 ? ? pr). -Unfold derive_pt in H; Rewrite H in H3. -Apply derivable_pt_lim_scal; Assumption. -Qed. - -Lemma derive_pt_id : (x:R) (derive_pt id x (derivable_pt_id ?))==R1. -Intros. -Apply derive_pt_eq_0. -Apply derivable_pt_lim_id. -Qed. - -Lemma derive_pt_Rsqr : (x:R) (derive_pt Rsqr x (derivable_pt_Rsqr ?)) == ``2*x``. -Intros. -Apply derive_pt_eq_0. -Apply derivable_pt_lim_Rsqr. -Qed. - -Lemma derive_pt_comp : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 (f1 x))) ``(derive_pt (comp f2 f1) x (derivable_pt_comp ? ? ? pr1 pr2)) == (derive_pt f2 (f1 x) pr2) * (derive_pt f1 x pr1)``. -Intros. -Assert H := (derivable_derive f1 x pr1). -Assert H0 := (derivable_derive f2 (f1 x) pr2). -Assert H1 := (derivable_derive (comp f2 f1) x (derivable_pt_comp ? ? ? pr1 pr2)). -Elim H; Clear H; Intros l1 H. -Elim H0; Clear H0; Intros l2 H0. -Elim H1; Clear H1; Intros l H1. -Rewrite H; Rewrite H0; Apply derive_pt_eq_0. -Assert H3 := (projT2 ? ? pr1). -Unfold derive_pt in H; Rewrite H in H3. -Assert H4 := (projT2 ? ? pr2). -Unfold derive_pt in H0; Rewrite H0 in H4. -Apply derivable_pt_lim_comp; Assumption. +Lemma derivable_pt_lim_plus : + forall f1 f2 (x l1 l2:R), + derivable_pt_lim f1 x l1 -> + derivable_pt_lim f2 x l2 -> derivable_pt_lim (f1 + f2) x (l1 + l2). +intros. +apply uniqueness_step3. +assert (H1 := uniqueness_step2 _ _ _ H). +assert (H2 := uniqueness_step2 _ _ _ H0). +unfold plus_fct in |- *. +cut + (forall h:R, + (f1 (x + h) + f2 (x + h) - (f1 x + f2 x)) / h = + (f1 (x + h) - f1 x) / h + (f2 (x + h) - f2 x) / h). +intro. +generalize + (limit_plus (fun h':R => (f1 (x + h') - f1 x) / h') + (fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2). +unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +elim (H4 eps H5); intros. +exists x0. +elim H6; intros. +split. +assumption. +intros; rewrite H3; apply H8; assumption. +intro; unfold Rdiv in |- *; ring. +Qed. + +Lemma derivable_pt_lim_opp : + forall f (x l:R), derivable_pt_lim f x l -> derivable_pt_lim (- f) x (- l). +intros. +apply uniqueness_step3. +assert (H1 := uniqueness_step2 _ _ _ H). +unfold opp_fct in |- *. +cut (forall h:R, (- f (x + h) - - f x) / h = - ((f (x + h) - f x) / h)). +intro. +generalize + (limit_Ropp (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) l 0 H1). +unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +elim (H2 eps H3); intros. +exists x0. +elim H4; intros. +split. +assumption. +intros; rewrite H0; apply H6; assumption. +intro; unfold Rdiv in |- *; ring. +Qed. + +Lemma derivable_pt_lim_minus : + forall f1 f2 (x l1 l2:R), + derivable_pt_lim f1 x l1 -> + derivable_pt_lim f2 x l2 -> derivable_pt_lim (f1 - f2) x (l1 - l2). +intros. +apply uniqueness_step3. +assert (H1 := uniqueness_step2 _ _ _ H). +assert (H2 := uniqueness_step2 _ _ _ H0). +unfold minus_fct in |- *. +cut + (forall h:R, + (f1 (x + h) - f1 x) / h - (f2 (x + h) - f2 x) / h = + (f1 (x + h) - f2 (x + h) - (f1 x - f2 x)) / h). +intro. +generalize + (limit_minus (fun h':R => (f1 (x + h') - f1 x) / h') + (fun h':R => (f2 (x + h') - f2 x) / h') (fun h:R => h <> 0) l1 l2 0 H1 H2). +unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +elim (H4 eps H5); intros. +exists x0. +elim H6; intros. +split. +assumption. +intros; rewrite <- H3; apply H8; assumption. +intro; unfold Rdiv in |- *; ring. +Qed. + +Lemma derivable_pt_lim_mult : + forall f1 f2 (x l1 l2:R), + derivable_pt_lim f1 x l1 -> + derivable_pt_lim f2 x l2 -> + derivable_pt_lim (f1 * f2) x (l1 * f2 x + f1 x * l2). +intros. +assert (H1 := derivable_pt_lim_D_in f1 (fun y:R => l1) x). +elim H1; intros. +assert (H4 := H3 H). +assert (H5 := derivable_pt_lim_D_in f2 (fun y:R => l2) x). +elim H5; intros. +assert (H8 := H7 H0). +clear H1 H2 H3 H5 H6 H7. +assert + (H1 := + derivable_pt_lim_D_in (f1 * f2)%F (fun y:R => l1 * f2 x + f1 x * l2) x). +elim H1; intros. +clear H1 H3. +apply H2. +unfold mult_fct in |- *. +apply (Dmult no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x); assumption. +Qed. + +Lemma derivable_pt_lim_const : forall a x:R, derivable_pt_lim (fct_cte a) x 0. +intros; unfold fct_cte, derivable_pt_lim in |- *. +intros; exists (mkposreal 1 Rlt_0_1); intros; unfold Rminus in |- *; + rewrite Rplus_opp_r; unfold Rdiv in |- *; rewrite Rmult_0_l; + rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. +Qed. + +Lemma derivable_pt_lim_scal : + forall f (a x l:R), + derivable_pt_lim f x l -> derivable_pt_lim (mult_real_fct a f) x (a * l). +intros. +assert (H0 := derivable_pt_lim_const a x). +replace (mult_real_fct a f) with (fct_cte a * f)%F. +replace (a * l) with (0 * f x + a * l); [ idtac | ring ]. +apply (derivable_pt_lim_mult (fct_cte a) f x 0 l); assumption. +unfold mult_real_fct, mult_fct, fct_cte in |- *; reflexivity. +Qed. + +Lemma derivable_pt_lim_id : forall x:R, derivable_pt_lim id x 1. +intro; unfold derivable_pt_lim in |- *. +intros eps Heps; exists (mkposreal eps Heps); intros h H1 H2; + unfold id in |- *; replace ((x + h - x) / h - 1) with 0. +rewrite Rabs_R0; apply Rle_lt_trans with (Rabs h). +apply Rabs_pos. +assumption. +unfold Rminus in |- *; rewrite Rplus_assoc; rewrite (Rplus_comm x); + rewrite Rplus_assoc. +rewrite Rplus_opp_l; rewrite Rplus_0_r; unfold Rdiv in |- *; + rewrite <- Rinv_r_sym. +symmetry in |- *; apply Rplus_opp_r. +assumption. +Qed. + +Lemma derivable_pt_lim_Rsqr : forall x:R, derivable_pt_lim Rsqr x (2 * x). +intro; unfold derivable_pt_lim in |- *. +unfold Rsqr in |- *; 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) - x * x) with (2 * x * h + h * h); + [ idtac | ring ]. +unfold Rdiv in |- *; rewrite Rmult_plus_distr_r. +repeat rewrite Rmult_assoc. +repeat rewrite <- Rinv_r_sym; [ idtac | assumption ]. +ring. +Qed. + +Lemma derivable_pt_lim_comp : + forall f1 f2 (x l1 l2:R), + derivable_pt_lim f1 x l1 -> + derivable_pt_lim f2 (f1 x) l2 -> derivable_pt_lim (f2 o f1) x (l2 * l1). +intros; assert (H1 := derivable_pt_lim_D_in f1 (fun y:R => l1) x). +elim H1; intros. +assert (H4 := H3 H). +assert (H5 := derivable_pt_lim_D_in f2 (fun y:R => l2) (f1 x)). +elim H5; intros. +assert (H8 := H7 H0). +clear H1 H2 H3 H5 H6 H7. +assert (H1 := derivable_pt_lim_D_in (f2 o f1)%F (fun y:R => l2 * l1) x). +elim H1; intros. +clear H1 H3; apply H2. +unfold comp in |- *; + cut + (D_in (fun x0:R => f2 (f1 x0)) (fun y:R => l2 * l1) + (Dgf no_cond no_cond f1) x -> + D_in (fun x0:R => f2 (f1 x0)) (fun y:R => l2 * l1) no_cond x). +intro; apply H1. +rewrite Rmult_comm; + apply (Dcomp no_cond no_cond (fun y:R => l1) (fun y:R => l2) f1 f2 x); + assumption. +unfold Dgf, D_in, no_cond in |- *; unfold limit1_in in |- *; + unfold limit_in in |- *; unfold dist in |- *; simpl in |- *; + unfold R_dist in |- *; intros. +elim (H1 eps H3); intros. +exists x0; intros; split. +elim H5; intros; assumption. +intros; elim H5; intros; apply H9; split. +unfold D_x in |- *; split. +split; trivial. +elim H6; intros; unfold D_x in H10; elim H10; intros; assumption. +elim H6; intros; assumption. +Qed. + +Lemma derivable_pt_plus : + forall f1 f2 (x:R), + derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 + f2) x. +unfold derivable_pt in |- *; intros. +elim X; intros. +elim X0; intros. +apply existT with (x0 + x1). +apply derivable_pt_lim_plus; assumption. +Qed. + +Lemma derivable_pt_opp : + forall f (x:R), derivable_pt f x -> derivable_pt (- f) x. +unfold derivable_pt in |- *; intros. +elim X; intros. +apply existT with (- x0). +apply derivable_pt_lim_opp; assumption. +Qed. + +Lemma derivable_pt_minus : + forall f1 f2 (x:R), + derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 - f2) x. +unfold derivable_pt in |- *; intros. +elim X; intros. +elim X0; intros. +apply existT with (x0 - x1). +apply derivable_pt_lim_minus; assumption. +Qed. + +Lemma derivable_pt_mult : + forall f1 f2 (x:R), + derivable_pt f1 x -> derivable_pt f2 x -> derivable_pt (f1 * f2) x. +unfold derivable_pt in |- *; intros. +elim X; intros. +elim X0; intros. +apply existT with (x0 * f2 x + f1 x * x1). +apply derivable_pt_lim_mult; assumption. +Qed. + +Lemma derivable_pt_const : forall a x:R, derivable_pt (fct_cte a) x. +intros; unfold derivable_pt in |- *. +apply existT with 0. +apply derivable_pt_lim_const. +Qed. + +Lemma derivable_pt_scal : + forall f (a x:R), derivable_pt f x -> derivable_pt (mult_real_fct a f) x. +unfold derivable_pt in |- *; intros. +elim X; intros. +apply existT with (a * x0). +apply derivable_pt_lim_scal; assumption. +Qed. + +Lemma derivable_pt_id : forall x:R, derivable_pt id x. +unfold derivable_pt in |- *; intro. +exists 1. +apply derivable_pt_lim_id. +Qed. + +Lemma derivable_pt_Rsqr : forall x:R, derivable_pt Rsqr x. +unfold derivable_pt in |- *; intro; apply existT with (2 * x). +apply derivable_pt_lim_Rsqr. +Qed. + +Lemma derivable_pt_comp : + forall f1 f2 (x:R), + derivable_pt f1 x -> derivable_pt f2 (f1 x) -> derivable_pt (f2 o f1) x. +unfold derivable_pt in |- *; intros. +elim X; intros. +elim X0; intros. +apply existT with (x1 * x0). +apply derivable_pt_lim_comp; assumption. +Qed. + +Lemma derivable_plus : + forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 + f2). +unfold derivable in |- *; intros. +apply (derivable_pt_plus _ _ x (X _) (X0 _)). +Qed. + +Lemma derivable_opp : forall f, derivable f -> derivable (- f). +unfold derivable in |- *; intros. +apply (derivable_pt_opp _ x (X _)). +Qed. + +Lemma derivable_minus : + forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 - f2). +unfold derivable in |- *; intros. +apply (derivable_pt_minus _ _ x (X _) (X0 _)). +Qed. + +Lemma derivable_mult : + forall f1 f2, derivable f1 -> derivable f2 -> derivable (f1 * f2). +unfold derivable in |- *; intros. +apply (derivable_pt_mult _ _ x (X _) (X0 _)). +Qed. + +Lemma derivable_const : forall a:R, derivable (fct_cte a). +unfold derivable in |- *; intros. +apply derivable_pt_const. +Qed. + +Lemma derivable_scal : + forall f (a:R), derivable f -> derivable (mult_real_fct a f). +unfold derivable in |- *; intros. +apply (derivable_pt_scal _ a x (X _)). +Qed. + +Lemma derivable_id : derivable id. +unfold derivable in |- *; intro; apply derivable_pt_id. +Qed. + +Lemma derivable_Rsqr : derivable Rsqr. +unfold derivable in |- *; intro; apply derivable_pt_Rsqr. +Qed. + +Lemma derivable_comp : + forall f1 f2, derivable f1 -> derivable f2 -> derivable (f2 o f1). +unfold derivable in |- *; intros. +apply (derivable_pt_comp _ _ x (X _) (X0 _)). +Qed. + +Lemma derive_pt_plus : + forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x), + derive_pt (f1 + f2) x (derivable_pt_plus _ _ _ pr1 pr2) = + derive_pt f1 x pr1 + derive_pt f2 x pr2. +intros. +assert (H := derivable_derive f1 x pr1). +assert (H0 := derivable_derive f2 x pr2). +assert + (H1 := derivable_derive (f1 + f2)%F x (derivable_pt_plus _ _ _ pr1 pr2)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +elim H1; clear H1; intros l H1. +rewrite H; rewrite H0; apply derive_pt_eq_0. +assert (H3 := projT2 pr1). +unfold derive_pt in H; rewrite H in H3. +assert (H4 := projT2 pr2). +unfold derive_pt in H0; rewrite H0 in H4. +apply derivable_pt_lim_plus; assumption. +Qed. + +Lemma derive_pt_opp : + forall f (x:R) (pr1:derivable_pt f x), + derive_pt (- f) x (derivable_pt_opp _ _ pr1) = - derive_pt f x pr1. +intros. +assert (H := derivable_derive f x pr1). +assert (H0 := derivable_derive (- f)%F x (derivable_pt_opp _ _ pr1)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +rewrite H; apply derive_pt_eq_0. +assert (H3 := projT2 pr1). +unfold derive_pt in H; rewrite H in H3. +apply derivable_pt_lim_opp; assumption. +Qed. + +Lemma derive_pt_minus : + forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x), + derive_pt (f1 - f2) x (derivable_pt_minus _ _ _ pr1 pr2) = + derive_pt f1 x pr1 - derive_pt f2 x pr2. +intros. +assert (H := derivable_derive f1 x pr1). +assert (H0 := derivable_derive f2 x pr2). +assert + (H1 := derivable_derive (f1 - f2)%F x (derivable_pt_minus _ _ _ pr1 pr2)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +elim H1; clear H1; intros l H1. +rewrite H; rewrite H0; apply derive_pt_eq_0. +assert (H3 := projT2 pr1). +unfold derive_pt in H; rewrite H in H3. +assert (H4 := projT2 pr2). +unfold derive_pt in H0; rewrite H0 in H4. +apply derivable_pt_lim_minus; assumption. +Qed. + +Lemma derive_pt_mult : + forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 x), + derive_pt (f1 * f2) x (derivable_pt_mult _ _ _ pr1 pr2) = + derive_pt f1 x pr1 * f2 x + f1 x * derive_pt f2 x pr2. +intros. +assert (H := derivable_derive f1 x pr1). +assert (H0 := derivable_derive f2 x pr2). +assert + (H1 := derivable_derive (f1 * f2)%F x (derivable_pt_mult _ _ _ pr1 pr2)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +elim H1; clear H1; intros l H1. +rewrite H; rewrite H0; apply derive_pt_eq_0. +assert (H3 := projT2 pr1). +unfold derive_pt in H; rewrite H in H3. +assert (H4 := projT2 pr2). +unfold derive_pt in H0; rewrite H0 in H4. +apply derivable_pt_lim_mult; assumption. +Qed. + +Lemma derive_pt_const : + forall a x:R, derive_pt (fct_cte a) x (derivable_pt_const a x) = 0. +intros. +apply derive_pt_eq_0. +apply derivable_pt_lim_const. +Qed. + +Lemma derive_pt_scal : + forall f (a x:R) (pr:derivable_pt f x), + derive_pt (mult_real_fct a f) x (derivable_pt_scal _ _ _ pr) = + a * derive_pt f x pr. +intros. +assert (H := derivable_derive f x pr). +assert + (H0 := derivable_derive (mult_real_fct a f) x (derivable_pt_scal _ _ _ pr)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +rewrite H; apply derive_pt_eq_0. +assert (H3 := projT2 pr). +unfold derive_pt in H; rewrite H in H3. +apply derivable_pt_lim_scal; assumption. +Qed. + +Lemma derive_pt_id : forall x:R, derive_pt id x (derivable_pt_id _) = 1. +intros. +apply derive_pt_eq_0. +apply derivable_pt_lim_id. +Qed. + +Lemma derive_pt_Rsqr : + forall x:R, derive_pt Rsqr x (derivable_pt_Rsqr _) = 2 * x. +intros. +apply derive_pt_eq_0. +apply derivable_pt_lim_Rsqr. +Qed. + +Lemma derive_pt_comp : + forall f1 f2 (x:R) (pr1:derivable_pt f1 x) (pr2:derivable_pt f2 (f1 x)), + derive_pt (f2 o f1) x (derivable_pt_comp _ _ _ pr1 pr2) = + derive_pt f2 (f1 x) pr2 * derive_pt f1 x pr1. +intros. +assert (H := derivable_derive f1 x pr1). +assert (H0 := derivable_derive f2 (f1 x) pr2). +assert + (H1 := derivable_derive (f2 o f1)%F x (derivable_pt_comp _ _ _ pr1 pr2)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +elim H1; clear H1; intros l H1. +rewrite H; rewrite H0; apply derive_pt_eq_0. +assert (H3 := projT2 pr1). +unfold derive_pt in H; rewrite H in H3. +assert (H4 := projT2 pr2). +unfold derive_pt in H0; rewrite H0 in H4. +apply derivable_pt_lim_comp; assumption. Qed. (* Pow *) -Definition pow_fct [n:nat] : R->R := [y:R](pow y n). - -Lemma derivable_pt_lim_pow_pos : (x:R;n:nat) (lt O n) -> (derivable_pt_lim [y:R](pow y n) x ``(INR n)*(pow x (pred n))``). -Intros. -Induction n. -Elim (lt_n_n ? H). -Cut n=O\/(lt O n). -Intro; Elim H0; Intro. -Rewrite H1; Simpl. -Replace [y:R]``y*1`` with (mult_fct id (fct_cte R1)). -Replace ``1*1`` with ``1*(fct_cte R1 x)+(id x)*0``. -Apply derivable_pt_lim_mult. -Apply derivable_pt_lim_id. -Apply derivable_pt_lim_const. -Unfold fct_cte id; Ring. -Reflexivity. -Replace [y:R](pow y (S n)) with [y:R]``y*(pow y n)``. -Replace (pred (S n)) with n; [Idtac | Reflexivity]. -Replace [y:R]``y*(pow y n)`` with (mult_fct id [y:R](pow y n)). -Pose f := [y:R](pow y n). -Replace ``(INR (S n))*(pow x n)`` with (Rplus (Rmult R1 (f x)) (Rmult (id x) (Rmult (INR n) (pow x (pred n))))). -Apply derivable_pt_lim_mult. -Apply derivable_pt_lim_id. -Unfold f; Apply Hrecn; Assumption. -Unfold f. -Pattern 1 5 n; Replace n with (S (pred n)). -Unfold id; Rewrite S_INR; Simpl. -Ring. -Symmetry; Apply S_pred with O; Assumption. -Unfold mult_fct id; Reflexivity. -Reflexivity. -Inversion H. -Left; Reflexivity. -Right. -Apply lt_le_trans with (1). -Apply lt_O_Sn. -Assumption. -Qed. - -Lemma derivable_pt_lim_pow : (x:R; n:nat) (derivable_pt_lim [y:R](pow y n) x ``(INR n)*(pow x (pred n))``). -Intros. -Induction n. -Simpl. -Rewrite Rmult_Ol. -Replace [_:R]``1`` with (fct_cte R1); [Apply derivable_pt_lim_const | Reflexivity]. -Apply derivable_pt_lim_pow_pos. -Apply lt_O_Sn. -Qed. - -Lemma derivable_pt_pow : (n:nat;x:R) (derivable_pt [y:R](pow y n) x). -Intros; Unfold derivable_pt. -Apply Specif.existT with ``(INR n)*(pow x (pred n))``. -Apply derivable_pt_lim_pow. -Qed. - -Lemma derivable_pow : (n:nat) (derivable [y:R](pow y n)). -Intro; Unfold derivable; Intro; Apply derivable_pt_pow. -Qed. - -Lemma derive_pt_pow : (n:nat;x:R) (derive_pt [y:R](pow y n) x (derivable_pt_pow n x))==``(INR n)*(pow x (pred n))``. -Intros; Apply derive_pt_eq_0. -Apply derivable_pt_lim_pow. -Qed. - -Lemma pr_nu : (f:R->R;x:R;pr1,pr2:(derivable_pt f x)) (derive_pt f x pr1)==(derive_pt f x pr2). -Intros. -Unfold derivable_pt in pr1. -Unfold derivable_pt in pr2. -Elim pr1; Intros. -Elim pr2; Intros. -Unfold derivable_pt_abs in p. -Unfold derivable_pt_abs in p0. -Simpl. -Apply (unicite_limite f x x0 x1 p p0). +Definition pow_fct (n:nat) (y:R) : R := y ^ n. + +Lemma derivable_pt_lim_pow_pos : + forall (x:R) (n:nat), + (0 < n)%nat -> derivable_pt_lim (fun y:R => y ^ n) x (INR n * x ^ pred n). +intros. +induction n as [| n Hrecn]. +elim (lt_irrefl _ H). +cut (n = 0%nat \/ (0 < n)%nat). +intro; elim H0; intro. +rewrite H1; simpl in |- *. +replace (fun y:R => y * 1) with (id * fct_cte 1)%F. +replace (1 * 1) with (1 * fct_cte 1 x + id x * 0). +apply derivable_pt_lim_mult. +apply derivable_pt_lim_id. +apply derivable_pt_lim_const. +unfold fct_cte, id in |- *; ring. +reflexivity. +replace (fun y:R => y ^ S n) with (fun y:R => y * y ^ n). +replace (pred (S n)) with n; [ idtac | reflexivity ]. +replace (fun y:R => y * y ^ n) with (id * (fun y:R => y ^ n))%F. +pose (f := fun y:R => y ^ n). +replace (INR (S n) * x ^ n) with (1 * f x + id x * (INR n * x ^ pred n)). +apply derivable_pt_lim_mult. +apply derivable_pt_lim_id. +unfold f in |- *; apply Hrecn; assumption. +unfold f in |- *. +pattern n at 1 5 in |- *; replace n with (S (pred n)). +unfold id in |- *; rewrite S_INR; simpl in |- *. +ring. +symmetry in |- *; apply S_pred with 0%nat; assumption. +unfold mult_fct, id in |- *; reflexivity. +reflexivity. +inversion H. +left; reflexivity. +right. +apply lt_le_trans with 1%nat. +apply lt_O_Sn. +assumption. +Qed. + +Lemma derivable_pt_lim_pow : + forall (x:R) (n:nat), + derivable_pt_lim (fun y:R => y ^ n) x (INR n * x ^ pred n). +intros. +induction n as [| n Hrecn]. +simpl in |- *. +rewrite Rmult_0_l. +replace (fun _:R => 1) with (fct_cte 1); + [ apply derivable_pt_lim_const | reflexivity ]. +apply derivable_pt_lim_pow_pos. +apply lt_O_Sn. +Qed. + +Lemma derivable_pt_pow : + forall (n:nat) (x:R), derivable_pt (fun y:R => y ^ n) x. +intros; unfold derivable_pt in |- *. +apply existT with (INR n * x ^ pred n). +apply derivable_pt_lim_pow. +Qed. + +Lemma derivable_pow : forall n:nat, derivable (fun y:R => y ^ n). +intro; unfold derivable in |- *; intro; apply derivable_pt_pow. +Qed. + +Lemma derive_pt_pow : + forall (n:nat) (x:R), + derive_pt (fun y:R => y ^ n) x (derivable_pt_pow n x) = INR n * x ^ pred n. +intros; apply derive_pt_eq_0. +apply derivable_pt_lim_pow. +Qed. + +Lemma pr_nu : + forall f (x:R) (pr1 pr2:derivable_pt f x), + derive_pt f x pr1 = derive_pt f x pr2. +intros. +unfold derivable_pt in pr1. +unfold derivable_pt in pr2. +elim pr1; intros. +elim pr2; intros. +unfold derivable_pt_abs in p. +unfold derivable_pt_abs in p0. +simpl in |- *. +apply (uniqueness_limite f x x0 x1 p p0). Qed. @@ -787,260 +1001,479 @@ Qed. (** Local extremum's condition *) (************************************************************) -Theorem deriv_maximum : (f:R->R;a,b,c:R;pr:(derivable_pt f c)) ``a<c``->``c<b``->((x:R) ``a<x``->``x<b``->``(f x)<=(f c)``)->``(derive_pt f c pr)==0``. -Intros; Case (total_order R0 (derive_pt f c pr)); Intro. -Assert H3 := (derivable_derive f c pr). -Elim H3; Intros l H4; Rewrite H4 in H2. -Assert H5 := (derive_pt_eq_1 f c l pr H4). -Cut ``0<l/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. -Elim (H5 ``l/2`` H6); Intros delta H7. -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. -Assert H11 := (H7 ``(Rmin delta/2 ((b-c)/2))`` H9 H10). -Cut ``0<(Rmin (delta/2) ((b-c)/2))``. -Intro; Cut ``a<c+(Rmin (delta/2) ((b-c)/2))``. -Intro; Cut ``c+(Rmin (delta/2) ((b-c)/2))<b``. -Intro; Assert H15 := (H1 ``c+(Rmin (delta/2) ((b-c)/2))`` H13 H14). -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 H11. -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`` H19); 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)`` H20); 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))`` H6 H21); 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`` H22 H16)). -Pattern 2 l; Rewrite double_var. -Ring. -Ring. -Intro. -Assert H20 := (Rle_sym2 ``0`` ``((f (c+(Rmin (delta/2) ((b+ -c)/2))))+ -(f c))/(Rmin (delta/2) ((b+ -c)/2))+ -l`` r). -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H20 H18)). -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) H15); 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 H12; Rewrite H16 in H12; Elim (Rlt_antirefl ``0`` H12). -Red; Intro. -Unfold Rdiv in H12; Rewrite H16 in H12; Elim (Rlt_antirefl ``0`` H12). -Assert H14 := (Rmin_r ``(delta/2)`` ``((b-c)/2)``). -Assert H15 := (Rle_compatibility ``c`` ``(Rmin (delta/2) ((b-c)/2))`` ``(b-c)/2`` H14). -Apply Rle_lt_trans with ``c+(b-c)/2``. -Assumption. -Apply Rlt_monotony_contra with ``2``. -Sup0. -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. -DiscrR. -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`` H12) (mkposreal ``(b-c)/2`` H8)). -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Sup0]. -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`` H10) (mkposreal ``(b-c)/2`` H8)); Simpl; Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``(Rmin (delta/2) ((b-c)/2))`` ``0`` H11 r)). -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Sup0]. -Intro; Apply Rle_lt_trans with ``delta/2``. -Apply Rmin_l. -Unfold Rdiv; Apply Rlt_monotony_contra with ``2``. -Sup0. -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. -DiscrR. -Cut ``0<delta/2``. -Intro; Generalize (Rmin_stable_in_posreal (mkposreal ``delta/2`` H9) (mkposreal ``(b-c)/2`` H8)); Simpl; Intro; Red; Intro; Rewrite H11 in H10; Elim (Rlt_antirefl ``0`` H10). -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Sup0]. -Unfold Rdiv; Apply Rmult_lt_pos. -Generalize (Rlt_compatibility_r ``-c`` c b H0); Rewrite Rplus_Ropp_r; Intro; Assumption. -Apply Rlt_Rinv; Sup0. -Elim H2; Intro. -Symmetry; Assumption. -Generalize (derivable_derive f c pr); Intro; Elim H4; Intros l H5. -Rewrite H5 in H3; Generalize (derive_pt_eq_1 f c l pr H5); Intro; Cut ``0< -(l/2)``. -Intro; Elim (H6 ``-(l/2)`` H7); 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))`` H11 H12); Intro; Cut ``a<c+(Rmax (-(delta/2)) ((a-c)/2))``. -Cut ``c+(Rmax (-(delta/2)) ((a-c)/2))<b``. -Intros; Generalize (H1 ``c+(Rmax (-(delta/2)) ((a-c)/2))`` H15 H14); 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 H13; 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`` H19 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)`` H20); 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`` H22 H21); 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`` H17 H23)). -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) H16); 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 H11; Assumption. -Generalize (Rlt_compatibility c ``(Rmax ( -(delta/2)) ((a-c)/2))`` ``0`` H10); 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))`` H14); Intro; Apply Rlt_le_trans with ``c+(a-c)/2``. -Apply Rlt_monotony_contra with ``2``. -Sup0. -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))`` H12); Rewrite Ropp_Ropp; Intro; Generalize (Rle_sym2 ``-(Rmax ( -(delta/2)) ((a-c)/2))`` ``delta/2`` H13); Intro; Apply Rle_lt_trans with ``delta/2``. -Assumption. -Apply Rlt_monotony_contra with ``2``. -Sup0. -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). -DiscrR. -Cut ``-(delta/2) < 0``. -Cut ``(a-c)/2<0``. -Intros; Generalize (Rmax_stable_in_negreal (mknegreal ``-(delta/2)`` H13) (mknegreal ``(a-c)/2`` H12)); 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`` H15 H14)). -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) | Assert Hyp : ``0<2``; [Sup0 | Apply (Rlt_Rinv ``2`` Hyp)]]. -Red; Intro; Rewrite H11 in H10; Elim (Rlt_antirefl ``0`` H10). -Cut ``(a-c)/2<0``. -Intro; Cut ``-(delta/2)<0``. -Intro; Apply (Rmax_stable_in_negreal (mknegreal ``-(delta/2)`` H11) (mknegreal ``(a-c)/2`` H10)). -Rewrite <- Ropp_O; Apply Rlt_Ropp; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Assert Hyp : ``0<2``; [Sup0 | Apply (Rlt_Rinv ``2`` Hyp)]]. -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 | Assert Hyp : ``0<2``; [Sup0 | Apply (Rlt_Rinv ``2`` Hyp)]]. -Replace ``-(l/2)`` with ``(-l)/2``. -Unfold Rdiv; Apply Rmult_lt_pos. -Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. -Assert Hyp : ``0<2``; [Sup0 | Apply (Rlt_Rinv ``2`` Hyp)]. -Unfold Rdiv; Apply Ropp_mul1. -Qed. - -Theorem deriv_minimum : (f:R->R;a,b,c:R;pr:(derivable_pt f c)) ``a<c``->``c<b``->((x:R) ``a<x``->``x<b``->``(f c)<=(f x)``)->``(derive_pt f c pr)==0``. -Intros. -Rewrite <- (Ropp_Ropp (derive_pt f c pr)). -Apply eq_RoppO. -Rewrite <- (derive_pt_opp f c pr). -Cut (x:R)(``a<x``->``x<b``->``((opp_fct f) x)<=((opp_fct f) c)``). -Intro. -Apply (deriv_maximum (opp_fct f) a b c (derivable_pt_opp ? ? pr) H H0 H2). -Intros; Unfold opp_fct; Apply Rge_Ropp; Apply Rle_sym1. -Apply (H1 x H2 H3). +Theorem deriv_maximum : + forall f (a b c:R) (pr:derivable_pt f c), + a < c -> + c < b -> + (forall x:R, a < x -> x < b -> f x <= f c) -> derive_pt f c pr = 0. +intros; case (Rtotal_order 0 (derive_pt f c pr)); intro. +assert (H3 := derivable_derive f c pr). +elim H3; intros l H4; rewrite H4 in H2. +assert (H5 := derive_pt_eq_1 f c l pr H4). +cut (0 < l / 2); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H5 (l / 2) H6); intros delta H7. +cut (0 < (b - c) / 2). +intro; cut (Rmin (delta / 2) ((b - c) / 2) <> 0). +intro; cut (Rabs (Rmin (delta / 2) ((b - c) / 2)) < delta). +intro. +assert (H11 := H7 (Rmin (delta / 2) ((b - c) / 2)) H9 H10). +cut (0 < Rmin (delta / 2) ((b - c) / 2)). +intro; cut (a < c + Rmin (delta / 2) ((b - c) / 2)). +intro; cut (c + Rmin (delta / 2) ((b - c) / 2) < b). +intro; assert (H15 := H1 (c + Rmin (delta / 2) ((b - c) / 2)) H13 H14). +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 H11. +cut + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2) + - l < 0). +intro; + cut + (Rabs + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2) + - l) < l / 2). +unfold Rabs in |- *; + case + (Rcase_abs + ((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 + (Rplus_lt_compat_l (- l) + (l + + - + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2))) (l / 2) H19); + repeat rewrite <- Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_l; replace (- l + l / 2) with (- (l / 2)). +intro; + generalize + (Ropp_lt_gt_contravar + (- + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2))) (- (l / 2)) H20); + repeat rewrite Ropp_involutive; intro; + generalize + (Rlt_trans 0 (l / 2) + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2)) H6 H21); intro; + elim + (Rlt_irrefl 0 + (Rlt_le_trans 0 + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2)) 0 H22 H16)). +pattern l at 2 in |- *; rewrite double_var. +ring. +ring. +intro. +assert + (H20 := + Rge_le + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) + - f c) / + Rmin (delta / 2) ((b + - c) / 2) + - l) 0 r). +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H20 H18)). +assumption. +rewrite <- Ropp_0; + 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 Ropp_gt_lt_contravar; + change + (0 < + l + + - + ((f (c + Rmin (delta / 2) ((b + - c) / 2)) - f c) / + Rmin (delta / 2) ((b + - c) / 2))) in |- *; apply Rplus_lt_le_0_compat; + [ assumption + | rewrite <- Ropp_0; apply Ropp_ge_le_contravar; apply Rle_ge; assumption ]. +ring. +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; 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_0; apply Ropp_ge_le_contravar; apply Rle_ge; + unfold Rdiv in |- *; apply Rmult_le_pos; + [ generalize + (Rplus_le_compat_r (- f (c + Rmin (delta * / 2) ((b - c) * / 2))) + (f (c + Rmin (delta * / 2) ((b - c) * / 2))) ( + f c) H15); rewrite Rplus_opp_r; intro; assumption + | left; apply Rinv_0_lt_compat; assumption ]. +unfold Rdiv in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +repeat rewrite <- (Rmult_comm (/ Rmin (delta * / 2) ((b - c) * / 2))). +apply Rmult_eq_reg_l with (Rmin (delta * / 2) ((b - c) * / 2)). +repeat rewrite <- Rmult_assoc. +rewrite <- Rinv_r_sym. +repeat rewrite Rmult_1_l. +ring. +red in |- *; intro. +unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12). +red in |- *; intro. +unfold Rdiv in H12; rewrite H16 in H12; elim (Rlt_irrefl 0 H12). +assert (H14 := Rmin_r (delta / 2) ((b - c) / 2)). +assert + (H15 := + Rplus_le_compat_l c (Rmin (delta / 2) ((b - c) / 2)) ((b - c) / 2) H14). +apply Rle_lt_trans with (c + (b - c) / 2). +assumption. +apply Rmult_lt_reg_l with 2. +prove_sup0. +replace (2 * (c + (b - c) / 2)) with (c + b). +replace (2 * b) with (b + b). +apply Rplus_lt_compat_r; assumption. +ring. +unfold Rdiv in |- *; rewrite Rmult_plus_distr_l. +repeat rewrite (Rmult_comm 2). +rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +ring. +discrR. +apply Rlt_trans with c. +assumption. +pattern c at 1 in |- *; rewrite <- (Rplus_0_r c); apply Rplus_lt_compat_l; + assumption. +cut (0 < delta / 2). +intro; + apply + (Rmin_stable_in_posreal (mkposreal (delta / 2) H12) + (mkposreal ((b - c) / 2) H8)). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. +unfold Rabs in |- *; case (Rcase_abs (Rmin (delta / 2) ((b - c) / 2))). +intro. +cut (0 < delta / 2). +intro. +generalize + (Rmin_stable_in_posreal (mkposreal (delta / 2) H10) + (mkposreal ((b - c) / 2) H8)); simpl in |- *; intro; + elim (Rlt_irrefl 0 (Rlt_trans 0 (Rmin (delta / 2) ((b - c) / 2)) 0 H11 r)). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. +intro; apply Rle_lt_trans with (delta / 2). +apply Rmin_l. +unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_l. +replace (2 * delta) with (delta + delta). +pattern delta at 2 in |- *; rewrite <- (Rplus_0_r delta); + apply Rplus_lt_compat_l. +rewrite Rplus_0_r; apply (cond_pos delta). +symmetry in |- *; apply double. +discrR. +cut (0 < delta / 2). +intro; + generalize + (Rmin_stable_in_posreal (mkposreal (delta / 2) H9) + (mkposreal ((b - c) / 2) H8)); simpl in |- *; + intro; red in |- *; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +generalize (Rplus_lt_compat_r (- c) c b H0); rewrite Rplus_opp_r; intro; + assumption. +apply Rinv_0_lt_compat; prove_sup0. +elim H2; intro. +symmetry in |- *; assumption. +generalize (derivable_derive f c pr); intro; elim H4; intros l H5. +rewrite H5 in H3; generalize (derive_pt_eq_1 f c l pr H5); intro; + cut (0 < - (l / 2)). +intro; elim (H6 (- (l / 2)) H7); 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 (Rabs (Rmax (- (delta / 2)) ((a - c) / 2)) < delta). +intro; generalize (H9 (Rmax (- (delta / 2)) ((a - c) / 2)) H11 H12); intro; + cut (a < c + Rmax (- (delta / 2)) ((a - c) / 2)). +cut (c + Rmax (- (delta / 2)) ((a - c) / 2) < b). +intros; generalize (H1 (c + Rmax (- (delta / 2)) ((a - c) / 2)) H15 H14); + 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 H13; + cut + (0 < + (f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / + Rmax (- (delta / 2)) ((a + - c) / 2) + - l). +intro; + cut + (Rabs + ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / + Rmax (- (delta / 2)) ((a + - c) / 2) + - l) < + - (l / 2)). +unfold Rabs in |- *; + case + (Rcase_abs + ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / + Rmax (- (delta / 2)) ((a + - c) / 2) + - l)). +intro; + elim + (Rlt_irrefl 0 + (Rlt_trans 0 + ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / + Rmax (- (delta / 2)) ((a + - c) / 2) + - l) 0 H19 r)). +intros; + generalize + (Rplus_lt_compat_r l + ((f (c + Rmax (- (delta / 2)) ((a + - c) / 2)) + - f c) / + Rmax (- (delta / 2)) ((a + - c) / 2) + - l) ( + - (l / 2)) H20); repeat rewrite Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_r; 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 H22 H21); + intro; + elim + (Rlt_irrefl 0 + (Rle_lt_trans 0 + ((f (c + Rmax (- (delta / 2)) ((a - c) / 2)) - f c) / + Rmax (- (delta / 2)) ((a - c) / 2)) 0 H17 H23)). +rewrite <- (Ropp_involutive (l / 2)); rewrite <- Ropp_0; + apply Ropp_lt_gt_contravar; assumption. +pattern l at 3 in |- *; rewrite double_var. +ring. +assumption. +apply Rplus_le_lt_0_compat; assumption. +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption. +unfold Rdiv in |- *; + 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 + (Rplus_le_compat_l (- f (c + Rmax (- (delta * / 2)) ((a - c) * / 2))) + (f (c + Rmax (- (delta * / 2)) ((a - c) * / 2))) ( + f c) H16); rewrite Rplus_opp_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 Rinv_0_lt_compat; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; + assumption. +unfold Rdiv in |- *. +rewrite <- Ropp_inv_permute. +rewrite Rmult_opp_opp. +reflexivity. +unfold Rdiv in H11; assumption. +generalize (Rplus_lt_compat_l c (Rmax (- (delta / 2)) ((a - c) / 2)) 0 H10); + rewrite Rplus_0_r; intro; apply Rlt_trans with c; + assumption. +generalize (RmaxLess2 (- (delta / 2)) ((a - c) / 2)); intro; + generalize + (Rplus_le_compat_l c ((a - c) / 2) (Rmax (- (delta / 2)) ((a - c) / 2)) H14); + intro; apply Rlt_le_trans with (c + (a - c) / 2). +apply Rmult_lt_reg_l with 2. +prove_sup0. +replace (2 * (c + (a - c) / 2)) with (a + c). +rewrite double. +apply Rplus_lt_compat_l; assumption. +ring. +rewrite <- Rplus_assoc. +rewrite <- double_var. +ring. +assumption. +unfold Rabs in |- *; case (Rcase_abs (Rmax (- (delta / 2)) ((a - c) / 2))). +intro; generalize (RmaxLess1 (- (delta / 2)) ((a - c) / 2)); intro; + generalize + (Ropp_le_ge_contravar (- (delta / 2)) (Rmax (- (delta / 2)) ((a - c) / 2)) + H12); rewrite Ropp_involutive; intro; + generalize (Rge_le (delta / 2) (- Rmax (- (delta / 2)) ((a - c) / 2)) H13); + intro; apply Rle_lt_trans with (delta / 2). +assumption. +apply Rmult_lt_reg_l with 2. +prove_sup0. +unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite double. +pattern delta at 2 in |- *; rewrite <- (Rplus_0_r delta); + apply Rplus_lt_compat_l; rewrite Rplus_0_r; apply (cond_pos delta). +discrR. +cut (- (delta / 2) < 0). +cut ((a - c) / 2 < 0). +intros; + generalize + (Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H13) + (mknegreal ((a - c) / 2) H12)); simpl in |- *; + intro; generalize (Rge_le (Rmax (- (delta / 2)) ((a - c) / 2)) 0 r); + intro; + elim + (Rlt_irrefl 0 + (Rle_lt_trans 0 (Rmax (- (delta / 2)) ((a - c) / 2)) 0 H15 H14)). +rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2)); + apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2). +assumption. +unfold Rdiv in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +rewrite (Ropp_minus_distr a c). +reflexivity. +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv in |- *; + apply Rmult_lt_0_compat; + [ apply (cond_pos delta) + | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ]. +red in |- *; intro; rewrite H11 in H10; elim (Rlt_irrefl 0 H10). +cut ((a - c) / 2 < 0). +intro; cut (- (delta / 2) < 0). +intro; + apply + (Rmax_stable_in_negreal (mknegreal (- (delta / 2)) H11) + (mknegreal ((a - c) / 2) H10)). +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; unfold Rdiv in |- *; + apply Rmult_lt_0_compat; + [ apply (cond_pos delta) + | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ]. +rewrite <- Ropp_0; rewrite <- (Ropp_involutive ((a - c) / 2)); + apply Ropp_lt_gt_contravar; replace (- ((a - c) / 2)) with ((c - a) / 2). +assumption. +unfold Rdiv in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +rewrite (Ropp_minus_distr a c). +reflexivity. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ generalize (Rplus_lt_compat_r (- a) a c H); rewrite Rplus_opp_r; intro; + assumption + | assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ] ]. +replace (- (l / 2)) with (- l / 2). +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption. +assert (Hyp : 0 < 2); [ prove_sup0 | apply (Rinv_0_lt_compat 2 Hyp) ]. +unfold Rdiv in |- *; apply Ropp_mult_distr_l_reverse. +Qed. + +Theorem deriv_minimum : + forall f (a b c:R) (pr:derivable_pt f c), + a < c -> + c < b -> + (forall x:R, a < x -> x < b -> f c <= f x) -> derive_pt f c pr = 0. +intros. +rewrite <- (Ropp_involutive (derive_pt f c pr)). +apply Ropp_eq_0_compat. +rewrite <- (derive_pt_opp f c pr). +cut (forall x:R, a < x -> x < b -> (- f)%F x <= (- f)%F c). +intro. +apply (deriv_maximum (- f)%F a b c (derivable_pt_opp _ _ pr) H H0 H2). +intros; unfold opp_fct in |- *; apply Ropp_ge_le_contravar; apply Rle_ge. +apply (H1 x H2 H3). Qed. -Theorem deriv_constant2 : (f:R->R;a,b,c:R;pr:(derivable_pt f c)) ``a<c``->``c<b``->((x:R) ``a<x``->``x<b``->``(f x)==(f c)``)->``(derive_pt f c pr)==0``. -Intros. -EApply deriv_maximum with a b; Try Assumption. -Intros; Right; Apply (H1 x H2 H3). +Theorem deriv_constant2 : + forall f (a b c:R) (pr:derivable_pt f c), + a < c -> + c < b -> (forall x:R, a < x -> x < b -> f x = f c) -> derive_pt f c pr = 0. +intros. +eapply deriv_maximum with a b; try assumption. +intros; right; apply (H1 x H2 H3). Qed. (**********) -Lemma nonneg_derivative_0 : (f:R->R;pr:(derivable f)) (increasing f) -> ((x:R) ``0<=(derive_pt f x (pr x))``). -Intros; Unfold increasing in H. -Assert H0 := (derivable_derive f x (pr x)). -Elim H0; Intros l H1. -Rewrite H1; Case (total_order R0 l); Intro. -Left; Assumption. -Elim H2; Intro. -Right; Assumption. -Assert H4 := (derive_pt_eq_1 f x l (pr x) H1). -Cut ``0< -(l/2)``. -Intro; Elim (H4 ``-(l/2)`` H5); Intros delta H6. -Cut ``delta/2<>0``/\``0<delta/2``/\``(Rabsolu delta/2)<delta``. -Intro; Decompose [and] H7; Intros; Generalize (H6 ``delta/2`` H8 H11); 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`` H12 r)). -Intros; Generalize (Rlt_compatibility_r l ``((f (x+delta/2))-(f x))/(delta/2)-l`` ``-(l/2)`` H13); 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`` H9 H14); Intro; Cut ``l/2<0``. -Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``l/2`` ``0`` H15 H16)). -Rewrite <- Ropp_O in H5; Generalize (Rlt_Ropp ``-0`` ``-(l/2)`` H5); 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 (H x ``x+(delta*/2)`` H12); 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. -Left; Rewrite <- Ropp_O; Apply Rlt_Ropp; Assumption. -Unfold Rdiv; Apply Rmult_le_pos. -Cut ``x<=(x+(delta*/2))``. -Intro; Generalize (H x ``x+(delta*/2)`` H9); Intro; Generalize (Rle_compatibility ``-(f x)`` ``(f x)`` ``(f (x+delta*/2))`` H12); 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 H7; Elim (Rlt_antirefl ``0`` H7). -Apply Rinv_neq_R0; DiscrR. -Split. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Sup0]. -Replace ``(Rabsolu delta/2)`` with ``delta/2``. -Unfold Rdiv; Apply Rlt_monotony_contra with ``2``. -Sup0. -Rewrite (Rmult_sym ``2``). -Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]. -Rewrite Rmult_1r. -Rewrite double. -Pattern 1 (pos delta); Rewrite <- Rplus_Or. -Apply Rlt_compatibility; Apply (cond_pos delta). -Symmetry; Apply Rabsolu_right. -Left; Change ``0<delta/2``; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos delta) | Apply Rlt_Rinv; Sup0]. -Unfold Rdiv; Rewrite <- Ropp_mul1; Apply Rmult_lt_pos. -Apply Rlt_anti_compatibility with l. -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Assumption. -Apply Rlt_Rinv; Sup0. -Qed. +Lemma nonneg_derivative_0 : + forall f (pr:derivable f), + increasing f -> forall x:R, 0 <= derive_pt f x (pr x). +intros; unfold increasing in H. +assert (H0 := derivable_derive f x (pr x)). +elim H0; intros l H1. +rewrite H1; case (Rtotal_order 0 l); intro. +left; assumption. +elim H2; intro. +right; assumption. +assert (H4 := derive_pt_eq_1 f x l (pr x) H1). +cut (0 < - (l / 2)). +intro; elim (H4 (- (l / 2)) H5); intros delta H6. +cut (delta / 2 <> 0 /\ 0 < delta / 2 /\ Rabs (delta / 2) < delta). +intro; decompose [and] H7; intros; generalize (H6 (delta / 2) H8 H11); + cut (0 <= (f (x + delta / 2) - f x) / (delta / 2)). +intro; cut (0 <= (f (x + delta / 2) - f x) / (delta / 2) - l). +intro; unfold Rabs in |- *; + case (Rcase_abs ((f (x + delta / 2) - f x) / (delta / 2) - l)). +intro; + elim + (Rlt_irrefl 0 + (Rle_lt_trans 0 ((f (x + delta / 2) - f x) / (delta / 2) - l) 0 H12 r)). +intros; + generalize + (Rplus_lt_compat_r l ((f (x + delta / 2) - f x) / (delta / 2) - l) + (- (l / 2)) H13); unfold Rminus in |- *; + replace (- (l / 2) + l) with (l / 2). +rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; intro; + generalize + (Rle_lt_trans 0 ((f (x + delta / 2) - f x) / (delta / 2)) (l / 2) H9 H14); + intro; cut (l / 2 < 0). +intro; elim (Rlt_irrefl 0 (Rlt_trans 0 (l / 2) 0 H15 H16)). +rewrite <- Ropp_0 in H5; + generalize (Ropp_lt_gt_contravar (-0) (- (l / 2)) H5); + repeat rewrite Ropp_involutive; intro; assumption. +pattern l at 3 in |- *; rewrite double_var. +ring. +unfold Rminus in |- *; apply Rplus_le_le_0_compat. +unfold Rdiv in |- *; apply Rmult_le_pos. +cut (x <= x + delta * / 2). +intro; generalize (H x (x + delta * / 2) H12); intro; + generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H13); + rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption. +pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + left; assumption. +left; apply Rinv_0_lt_compat; assumption. +left; rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; assumption. +unfold Rdiv in |- *; apply Rmult_le_pos. +cut (x <= x + delta * / 2). +intro; generalize (H x (x + delta * / 2) H9); intro; + generalize (Rplus_le_compat_l (- f x) (f x) (f (x + delta * / 2)) H12); + rewrite Rplus_opp_l; rewrite Rplus_comm; intro; assumption. +pattern x at 1 in |- *; rewrite <- (Rplus_0_r x); apply Rplus_le_compat_l; + left; assumption. +left; apply Rinv_0_lt_compat; assumption. +split. +unfold Rdiv in |- *; apply prod_neq_R0. +generalize (cond_pos delta); intro; red in |- *; intro H9; rewrite H9 in H7; + elim (Rlt_irrefl 0 H7). +apply Rinv_neq_0_compat; discrR. +split. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. +replace (Rabs (delta / 2)) with (delta / 2). +unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite (Rmult_comm 2). +rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]. +rewrite Rmult_1_r. +rewrite double. +pattern (pos delta) at 1 in |- *; rewrite <- Rplus_0_r. +apply Rplus_lt_compat_l; apply (cond_pos delta). +symmetry in |- *; apply Rabs_right. +left; change (0 < delta / 2) in |- *; unfold Rdiv in |- *; + apply Rmult_lt_0_compat; + [ apply (cond_pos delta) | apply Rinv_0_lt_compat; prove_sup0 ]. +unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse; + apply Rmult_lt_0_compat. +apply Rplus_lt_reg_r with l. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r; assumption. +apply Rinv_0_lt_compat; prove_sup0. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Ranalysis2.v b/theories/Reals/Ranalysis2.v index 70f7adb1f..a02c5da6c 100644 --- a/theories/Reals/Ranalysis2.v +++ b/theories/Reals/Ranalysis2.v @@ -8,295 +8,443 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Ranalysis1. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rfunctions. +Require Import Ranalysis1. Open Local Scope R_scope. (**********) -Lemma formule : (x,h,l1,l2:R;f1,f2:R->R) ``h<>0`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ``((f1 (x+h))/(f2 (x+h))-(f1 x)/(f2 x))/h-(l1*(f2 x)-l2*(f1 x))/(Rsqr (f2 x))`` == ``/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1) + l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))) - (f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2) + (l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))``. -Intros; Unfold Rdiv Rminus Rsqr. -Repeat Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_Rplus_distr; Repeat Rewrite Rinv_Rmult; Try Assumption. -Replace ``l1*(f2 x)*(/(f2 x)*/(f2 x))`` with ``l1*/(f2 x)*((f2 x)*/(f2 x))``; [Idtac | Ring]. -Replace ``l1*(/(f2 x)*/(f2 (x+h)))*(f2 x)`` with ``l1*/(f2 (x+h))*((f2 x)*/(f2 x))``; [Idtac | Ring]. -Replace ``l1*(/(f2 x)*/(f2 (x+h)))* -(f2 (x+h))`` with ``-(l1*/(f2 x)*((f2 (x+h))*/(f2 (x+h))))``; [Idtac | Ring]. -Replace ``(f1 x)*(/(f2 x)*/(f2 (x+h)))*((f2 (x+h))*/h)`` with ``(f1 x)*/(f2 x)*/h*((f2 (x+h))*/(f2 (x+h)))``; [Idtac | Ring]. -Replace ``(f1 x)*(/(f2 x)*/(f2 (x+h)))*( -(f2 x)*/h)`` with ``-((f1 x)*/(f2 (x+h))*/h*((f2 x)*/(f2 x)))``; [Idtac | Ring]. -Replace ``(l2*(f1 x)*(/(f2 x)*/(f2 x)*/(f2 (x+h)))*(f2 (x+h)))`` with ``l2*(f1 x)*/(f2 x)*/(f2 x)*((f2 (x+h))*/(f2 (x+h)))``; [Idtac | Ring]. -Replace ``l2*(f1 x)*(/(f2 x)*/(f2 x)*/(f2 (x+h)))* -(f2 x)`` with ``-(l2*(f1 x)*/(f2 x)*/(f2 (x+h))*((f2 x)*/(f2 x)))``; [Idtac | Ring]. -Repeat Rewrite <- Rinv_r_sym; Try Assumption Orelse Ring. -Apply prod_neq_R0; Assumption. +Lemma formule : + forall (x h l1 l2:R) (f1 f2:R -> R), + h <> 0 -> + f2 x <> 0 -> + f2 (x + h) <> 0 -> + (f1 (x + h) / f2 (x + h) - f1 x / f2 x) / h - + (l1 * f2 x - l2 * f1 x) / Rsqr (f2 x) = + / f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1) + + l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h)) - + f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2) + + l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x). +intros; unfold Rdiv, Rminus, Rsqr in |- *. +repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l; + repeat rewrite Rinv_mult_distr; try assumption. +replace (l1 * f2 x * (/ f2 x * / f2 x)) with (l1 * / f2 x * (f2 x * / f2 x)); + [ idtac | ring ]. +replace (l1 * (/ f2 x * / f2 (x + h)) * f2 x) with + (l1 * / f2 (x + h) * (f2 x * / f2 x)); [ idtac | ring ]. +replace (l1 * (/ f2 x * / f2 (x + h)) * - f2 (x + h)) with + (- (l1 * / f2 x * (f2 (x + h) * / f2 (x + h)))); [ idtac | ring ]. +replace (f1 x * (/ f2 x * / f2 (x + h)) * (f2 (x + h) * / h)) with + (f1 x * / f2 x * / h * (f2 (x + h) * / f2 (x + h))); + [ idtac | ring ]. +replace (f1 x * (/ f2 x * / f2 (x + h)) * (- f2 x * / h)) with + (- (f1 x * / f2 (x + h) * / h * (f2 x * / f2 x))); + [ idtac | ring ]. +replace (l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * f2 (x + h)) with + (l2 * f1 x * / f2 x * / f2 x * (f2 (x + h) * / f2 (x + h))); + [ idtac | ring ]. +replace (l2 * f1 x * (/ f2 x * / f2 x * / f2 (x + h)) * - f2 x) with + (- (l2 * f1 x * / f2 x * / f2 (x + h) * (f2 x * / f2 x))); + [ idtac | ring ]. +repeat rewrite <- Rinv_r_sym; try assumption || ring. +apply prod_neq_R0; assumption. Qed. -Lemma Rmin_pos : (x,y:R) ``0<x`` -> ``0<y`` -> ``0 < (Rmin x y)``. -Intros; Unfold Rmin. -Case (total_order_Rle x y); Intro; Assumption. +Lemma Rmin_pos : forall x y:R, 0 < x -> 0 < y -> 0 < Rmin x y. +intros; unfold Rmin in |- *. +case (Rle_dec x y); intro; assumption. Qed. -Lemma maj_term1 : (x,h,eps,l1,alp_f2:R;eps_f2,alp_f1d:posreal;f1,f2:R->R) ``0 < eps`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ((h:R)``h <> 0``->``(Rabsolu h) < alp_f1d``->``(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < (Rabsolu ((eps*(f2 x))/8))``) -> ((a:R)``(Rabsolu a) < (Rmin eps_f2 alp_f2)``->``/(Rabsolu (f2 (x+a))) < 2/(Rabsolu (f2 x))``) -> ``h<>0`` -> ``(Rabsolu h)<alp_f1d`` -> ``(Rabsolu h) < (Rmin eps_f2 alp_f2)`` -> ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) < eps/4``. -Intros. -Assert H7 := (H3 h H6). -Assert H8 := (H2 h H4 H5). -Apply Rle_lt_trans with ``2/(Rabsolu (f2 x))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1))``. -Rewrite Rabsolu_mult. -Apply Rle_monotony_r. -Apply Rabsolu_pos. -Rewrite Rabsolu_Rinv; [Left; Exact H7 | Assumption]. -Apply Rlt_le_trans with ``2/(Rabsolu (f2 x))*(Rabsolu ((eps*(f2 x))/8))``. -Apply Rlt_monotony. -Unfold Rdiv; Apply Rmult_lt_pos; [Sup0 | Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption]. -Exact H8. -Right; Unfold Rdiv. -Repeat Rewrite Rabsolu_mult. -Rewrite Rabsolu_Rinv; DiscrR. -Replace ``(Rabsolu 8)`` with ``8``. -Replace ``8`` with ``2*4``; [Idtac | Ring]. -Rewrite Rinv_Rmult; [Idtac | DiscrR | DiscrR]. -Replace ``2*/(Rabsolu (f2 x))*((Rabsolu eps)*(Rabsolu (f2 x))*(/2*/4))`` with ``(Rabsolu eps)*/4*(2*/2)*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))``; [Idtac | Ring]. -Replace (Rabsolu eps) with eps. -Repeat Rewrite <- Rinv_r_sym; Try DiscrR Orelse (Apply Rabsolu_no_R0; Assumption). -Ring. -Symmetry; Apply Rabsolu_right; Left; Assumption. -Symmetry; Apply Rabsolu_right; Left; Sup. +Lemma maj_term1 : + forall (x h eps l1 alp_f2:R) (eps_f2 alp_f1d:posreal) + (f1 f2:R -> R), + 0 < eps -> + f2 x <> 0 -> + f2 (x + h) <> 0 -> + (forall h:R, + h <> 0 -> + Rabs h < alp_f1d -> + Rabs ((f1 (x + h) - f1 x) / h - l1) < Rabs (eps * f2 x / 8)) -> + (forall a:R, + Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) -> + h <> 0 -> + Rabs h < alp_f1d -> + Rabs h < Rmin eps_f2 alp_f2 -> + Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) < eps / 4. +intros. +assert (H7 := H3 h H6). +assert (H8 := H2 h H4 H5). +apply Rle_lt_trans with + (2 / Rabs (f2 x) * Rabs ((f1 (x + h) - f1 x) / h - l1)). +rewrite Rabs_mult. +apply Rmult_le_compat_r. +apply Rabs_pos. +rewrite Rabs_Rinv; [ left; exact H7 | assumption ]. +apply Rlt_le_trans with (2 / Rabs (f2 x) * Rabs (eps * f2 x / 8)). +apply Rmult_lt_compat_l. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ prove_sup0 | apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption ]. +exact H8. +right; unfold Rdiv in |- *. +repeat rewrite Rabs_mult. +rewrite Rabs_Rinv; discrR. +replace (Rabs 8) with 8. +replace 8 with 8; [ idtac | ring ]. +rewrite Rinv_mult_distr; [ idtac | discrR | discrR ]. +replace (2 * / Rabs (f2 x) * (Rabs eps * Rabs (f2 x) * (/ 2 * / 4))) with + (Rabs eps * / 4 * (2 * / 2) * (Rabs (f2 x) * / Rabs (f2 x))); + [ idtac | ring ]. +replace (Rabs eps) with eps. +repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption). +ring. +symmetry in |- *; apply Rabs_right; left; assumption. +symmetry in |- *; apply Rabs_right; left; prove_sup. Qed. -Lemma maj_term2 : (x,h,eps,l1,alp_f2,alp_f2t2:R;eps_f2:posreal;f2:R->R) ``0 < eps`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ((a:R)``(Rabsolu a) < alp_f2t2``->``(Rabsolu ((f2 (x+a))-(f2 x))) < (Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``)-> ((a:R)``(Rabsolu a) < (Rmin eps_f2 alp_f2)``->``/(Rabsolu (f2 (x+a))) < 2/(Rabsolu (f2 x))``) -> ``h<>0`` -> ``(Rabsolu h)<alp_f2t2`` -> ``(Rabsolu h) < (Rmin eps_f2 alp_f2)`` -> ``l1<>0`` -> ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) < eps/4``. -Intros. -Assert H8 := (H3 h H6). -Assert H9 := (H2 h H5). -Apply Rle_lt_trans with ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. -Rewrite Rabsolu_mult; Apply Rle_monotony. -Apply Rabsolu_pos. -Rewrite <- (Rabsolu_Ropp ``(f2 x)-(f2 (x+h))``); Rewrite Ropp_distr2. -Left; Apply H9. -Apply Rlt_le_trans with ``(Rabsolu (2*l1/((f2 x)*(f2 x))))*(Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. -Apply Rlt_monotony_r. -Apply Rabsolu_pos_lt. -Unfold Rdiv; Unfold Rsqr; Repeat Apply prod_neq_R0; Try Assumption Orelse DiscrR. -Red; Intro H10; Rewrite H10 in H; Elim (Rlt_antirefl ? H). -Apply Rinv_neq_R0; Apply prod_neq_R0; Try Assumption Orelse DiscrR. -Unfold Rdiv. -Repeat Rewrite Rinv_Rmult; Try Assumption. -Repeat Rewrite Rabsolu_mult. -Replace ``(Rabsolu 2)`` with ``2``. -Rewrite (Rmult_sym ``2``). -Replace ``(Rabsolu l1)*((Rabsolu (/(f2 x)))*(Rabsolu (/(f2 x))))*2`` with ``(Rabsolu l1)*((Rabsolu (/(f2 x)))*((Rabsolu (/(f2 x)))*2))``; [Idtac | Ring]. -Repeat Apply Rlt_monotony. -Apply Rabsolu_pos_lt; Assumption. -Apply Rabsolu_pos_lt; Apply Rinv_neq_R0; Assumption. -Repeat Rewrite Rabsolu_Rinv; Try Assumption. -Rewrite <- (Rmult_sym ``2``). -Unfold Rdiv in H8; Exact H8. -Symmetry; Apply Rabsolu_right; Left; Sup0. -Right. -Unfold Rsqr Rdiv. -Do 1 Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. -Do 1 Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. -Repeat Rewrite Rabsolu_mult. -Repeat Rewrite Rabsolu_Rinv; Try Assumption Orelse DiscrR. -Replace (Rabsolu eps) with eps. -Replace ``(Rabsolu (8))`` with ``8``. -Replace ``(Rabsolu 2)`` with ``2``. -Replace ``8`` with ``4*2``; [Idtac | Ring]. -Rewrite Rinv_Rmult; DiscrR. -Replace ``2*((Rabsolu l1)*(/(Rabsolu (f2 x))*/(Rabsolu (f2 x))))*(eps*((Rabsolu (f2 x))*(Rabsolu (f2 x)))*(/4*/2*/(Rabsolu l1)))`` with ``eps*/4*((Rabsolu l1)*/(Rabsolu l1))*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*(2*/2)``; [Idtac | Ring]. -Repeat Rewrite <- Rinv_r_sym; Try (Apply Rabsolu_no_R0; Assumption) Orelse DiscrR. -Ring. -Symmetry; Apply Rabsolu_right; Left; Sup0. -Symmetry; Apply Rabsolu_right; Left; Sup. -Symmetry; Apply Rabsolu_right; Left; Assumption. +Lemma maj_term2 : + forall (x h eps l1 alp_f2 alp_f2t2:R) (eps_f2:posreal) + (f2:R -> R), + 0 < eps -> + f2 x <> 0 -> + f2 (x + h) <> 0 -> + (forall a:R, + Rabs a < alp_f2t2 -> + Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))) -> + (forall a:R, + Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) -> + h <> 0 -> + Rabs h < alp_f2t2 -> + Rabs h < Rmin eps_f2 alp_f2 -> + l1 <> 0 -> Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) < eps / 4. +intros. +assert (H8 := H3 h H6). +assert (H9 := H2 h H5). +apply Rle_lt_trans with + (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))). +rewrite Rabs_mult; apply Rmult_le_compat_l. +apply Rabs_pos. +rewrite <- (Rabs_Ropp (f2 x - f2 (x + h))); rewrite Ropp_minus_distr. +left; apply H9. +apply Rlt_le_trans with + (Rabs (2 * (l1 / (f2 x * f2 x))) * Rabs (eps * Rsqr (f2 x) / (8 * l1))). +apply Rmult_lt_compat_r. +apply Rabs_pos_lt. +unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0; + try assumption || discrR. +red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H). +apply Rinv_neq_0_compat; apply prod_neq_R0; try assumption || discrR. +unfold Rdiv in |- *. +repeat rewrite Rinv_mult_distr; try assumption. +repeat rewrite Rabs_mult. +replace (Rabs 2) with 2. +rewrite (Rmult_comm 2). +replace (Rabs l1 * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with + (Rabs l1 * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2))); + [ idtac | ring ]. +repeat apply Rmult_lt_compat_l. +apply Rabs_pos_lt; assumption. +apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption. +repeat rewrite Rabs_Rinv; try assumption. +rewrite <- (Rmult_comm 2). +unfold Rdiv in H8; exact H8. +symmetry in |- *; apply Rabs_right; left; prove_sup0. +right. +unfold Rsqr, Rdiv in |- *. +do 1 rewrite Rinv_mult_distr; try assumption || discrR. +do 1 rewrite Rinv_mult_distr; try assumption || discrR. +repeat rewrite Rabs_mult. +repeat rewrite Rabs_Rinv; try assumption || discrR. +replace (Rabs eps) with eps. +replace (Rabs 8) with 8. +replace (Rabs 2) with 2. +replace 8 with (4 * 2); [ idtac | ring ]. +rewrite Rinv_mult_distr; discrR. +replace + (2 * (Rabs l1 * (/ Rabs (f2 x) * / Rabs (f2 x))) * + (eps * (Rabs (f2 x) * Rabs (f2 x)) * (/ 4 * / 2 * / Rabs l1))) with + (eps * / 4 * (Rabs l1 * / Rabs l1) * (Rabs (f2 x) * / Rabs (f2 x)) * + (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ]. +repeat rewrite <- Rinv_r_sym; try (apply Rabs_no_R0; assumption) || discrR. +ring. +symmetry in |- *; apply Rabs_right; left; prove_sup0. +symmetry in |- *; apply Rabs_right; left; prove_sup. +symmetry in |- *; apply Rabs_right; left; assumption. Qed. -Lemma maj_term3 : (x,h,eps,l2,alp_f2:R;eps_f2,alp_f2d:posreal;f1,f2:R->R) ``0 < eps`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ((h:R)``h <> 0``->``(Rabsolu h) < alp_f2d``->``(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < (Rabsolu (((Rsqr (f2 x))*eps)/(8*(f1 x))))``) -> ((a:R)``(Rabsolu a) < (Rmin eps_f2 alp_f2)``->``/(Rabsolu (f2 (x+a))) < 2/(Rabsolu (f2 x))``) -> ``h<>0`` -> ``(Rabsolu h)<alp_f2d`` -> ``(Rabsolu h) < (Rmin eps_f2 alp_f2)`` -> ``(f1 x)<>0`` -> ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) < eps/4``. -Intros. -Assert H8 := (H2 h H4 H5). -Assert H9 := (H3 h H6). -Apply Rle_lt_trans with ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((Rsqr (f2 x))*eps)/(8*(f1 x))))``. -Rewrite Rabsolu_mult. -Apply Rle_monotony. -Apply Rabsolu_pos. -Left; Apply H8. -Apply Rlt_le_trans with ``(Rabsolu (2*(f1 x)/((f2 x)*(f2 x))))*(Rabsolu (((Rsqr (f2 x))*eps)/(8*(f1 x))))``. -Apply Rlt_monotony_r. -Apply Rabsolu_pos_lt. -Unfold Rdiv; Unfold Rsqr; Repeat Apply prod_neq_R0; Try Assumption. -Red; Intro H10; Rewrite H10 in H; Elim (Rlt_antirefl ? H). -Apply Rinv_neq_R0; Apply prod_neq_R0; DiscrR Orelse Assumption. -Unfold Rdiv. -Repeat Rewrite Rinv_Rmult; Try Assumption. -Repeat Rewrite Rabsolu_mult. -Replace ``(Rabsolu 2)`` with ``2``. -Rewrite (Rmult_sym ``2``). -Replace ``(Rabsolu (f1 x))*((Rabsolu (/(f2 x)))*(Rabsolu (/(f2 x))))*2`` with ``(Rabsolu (f1 x))*((Rabsolu (/(f2 x)))*((Rabsolu (/(f2 x)))*2))``; [Idtac | Ring]. -Repeat Apply Rlt_monotony. -Apply Rabsolu_pos_lt; Assumption. -Apply Rabsolu_pos_lt; Apply Rinv_neq_R0; Assumption. -Repeat Rewrite Rabsolu_Rinv; Assumption Orelse Idtac. -Rewrite <- (Rmult_sym ``2``). -Unfold Rdiv in H9; Exact H9. -Symmetry; Apply Rabsolu_right; Left; Sup0. -Right. -Unfold Rsqr Rdiv. -Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. -Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. -Repeat Rewrite Rabsolu_mult. -Repeat Rewrite Rabsolu_Rinv; Try Assumption Orelse DiscrR. -Replace (Rabsolu eps) with eps. -Replace ``(Rabsolu (8))`` with ``8``. -Replace ``(Rabsolu 2)`` with ``2``. -Replace ``8`` with ``4*2``; [Idtac | Ring]. -Rewrite Rinv_Rmult; DiscrR. -Replace ``2*((Rabsolu (f1 x))*(/(Rabsolu (f2 x))*/(Rabsolu (f2 x))))*((Rabsolu (f2 x))*(Rabsolu (f2 x))*eps*(/4*/2*/(Rabsolu (f1 x))))`` with ``eps*/4*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*((Rabsolu (f1 x))*/(Rabsolu (f1 x)))*(2*/2)``; [Idtac | Ring]. -Repeat Rewrite <- Rinv_r_sym; Try DiscrR Orelse (Apply Rabsolu_no_R0; Assumption). -Ring. -Symmetry; Apply Rabsolu_right; Left; Sup0. -Symmetry; Apply Rabsolu_right; Left; Sup. -Symmetry; Apply Rabsolu_right; Left; Assumption. +Lemma maj_term3 : + forall (x h eps l2 alp_f2:R) (eps_f2 alp_f2d:posreal) + (f1 f2:R -> R), + 0 < eps -> + f2 x <> 0 -> + f2 (x + h) <> 0 -> + (forall h:R, + h <> 0 -> + Rabs h < alp_f2d -> + Rabs ((f2 (x + h) - f2 x) / h - l2) < + Rabs (Rsqr (f2 x) * eps / (8 * f1 x))) -> + (forall a:R, + Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) -> + h <> 0 -> + Rabs h < alp_f2d -> + Rabs h < Rmin eps_f2 alp_f2 -> + f1 x <> 0 -> + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) < + eps / 4. +intros. +assert (H8 := H2 h H4 H5). +assert (H9 := H3 h H6). +apply Rle_lt_trans with + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))). +rewrite Rabs_mult. +apply Rmult_le_compat_l. +apply Rabs_pos. +left; apply H8. +apply Rlt_le_trans with + (Rabs (2 * (f1 x / (f2 x * f2 x))) * Rabs (Rsqr (f2 x) * eps / (8 * f1 x))). +apply Rmult_lt_compat_r. +apply Rabs_pos_lt. +unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0; + try assumption. +red in |- *; intro H10; rewrite H10 in H; elim (Rlt_irrefl _ H). +apply Rinv_neq_0_compat; apply prod_neq_R0; discrR || assumption. +unfold Rdiv in |- *. +repeat rewrite Rinv_mult_distr; try assumption. +repeat rewrite Rabs_mult. +replace (Rabs 2) with 2. +rewrite (Rmult_comm 2). +replace (Rabs (f1 x) * (Rabs (/ f2 x) * Rabs (/ f2 x)) * 2) with + (Rabs (f1 x) * (Rabs (/ f2 x) * (Rabs (/ f2 x) * 2))); + [ idtac | ring ]. +repeat apply Rmult_lt_compat_l. +apply Rabs_pos_lt; assumption. +apply Rabs_pos_lt; apply Rinv_neq_0_compat; assumption. +repeat rewrite Rabs_Rinv; assumption || idtac. +rewrite <- (Rmult_comm 2). +unfold Rdiv in H9; exact H9. +symmetry in |- *; apply Rabs_right; left; prove_sup0. +right. +unfold Rsqr, Rdiv in |- *. +rewrite Rinv_mult_distr; try assumption || discrR. +rewrite Rinv_mult_distr; try assumption || discrR. +repeat rewrite Rabs_mult. +repeat rewrite Rabs_Rinv; try assumption || discrR. +replace (Rabs eps) with eps. +replace (Rabs 8) with 8. +replace (Rabs 2) with 2. +replace 8 with (4 * 2); [ idtac | ring ]. +rewrite Rinv_mult_distr; discrR. +replace + (2 * (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x))) * + (Rabs (f2 x) * Rabs (f2 x) * eps * (/ 4 * / 2 * / Rabs (f1 x)))) with + (eps * / 4 * (Rabs (f2 x) * / Rabs (f2 x)) * (Rabs (f2 x) * / Rabs (f2 x)) * + (Rabs (f1 x) * / Rabs (f1 x)) * (2 * / 2)); [ idtac | ring ]. +repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption). +ring. +symmetry in |- *; apply Rabs_right; left; prove_sup0. +symmetry in |- *; apply Rabs_right; left; prove_sup. +symmetry in |- *; apply Rabs_right; left; assumption. Qed. -Lemma maj_term4 : (x,h,eps,l2,alp_f2,alp_f2c:R;eps_f2:posreal;f1,f2:R->R) ``0 < eps`` -> ``(f2 x)<>0`` -> ``(f2 (x+h))<>0`` -> ((a:R)``(Rabsolu a) < alp_f2c`` -> ``(Rabsolu ((f2 (x+a))-(f2 x))) < (Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``) -> ((a:R)``(Rabsolu a) < (Rmin eps_f2 alp_f2)``->``/(Rabsolu (f2 (x+a))) < 2/(Rabsolu (f2 x))``) -> ``h<>0`` -> ``(Rabsolu h)<alp_f2c`` -> ``(Rabsolu h) < (Rmin eps_f2 alp_f2)`` -> ``(f1 x)<>0`` -> ``l2<>0`` -> ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x)))) < eps/4``. -Intros. -Assert H9 := (H2 h H5). -Assert H10 := (H3 h H6). -Apply Rle_lt_trans with ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``. -Rewrite Rabsolu_mult. -Apply Rle_monotony. -Apply Rabsolu_pos. -Left; Apply H9. -Apply Rlt_le_trans with ``(Rabsolu (2*l2*(f1 x)/((Rsqr (f2 x))*(f2 x))))*(Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``. -Apply Rlt_monotony_r. -Apply Rabsolu_pos_lt. -Unfold Rdiv; Unfold Rsqr; Repeat Apply prod_neq_R0; Assumption Orelse Idtac. -Red; Intro H11; Rewrite H11 in H; Elim (Rlt_antirefl ? H). -Apply Rinv_neq_R0; Apply prod_neq_R0. -Apply prod_neq_R0. -DiscrR. -Assumption. -Assumption. -Unfold Rdiv. -Repeat Rewrite Rinv_Rmult; Try Assumption Orelse (Unfold Rsqr; Apply prod_neq_R0; Assumption). -Repeat Rewrite Rabsolu_mult. -Replace ``(Rabsolu 2)`` with ``2``. -Replace ``2*(Rabsolu l2)*((Rabsolu (f1 x))*((Rabsolu (/(Rsqr (f2 x))))*(Rabsolu (/(f2 x)))))`` with ``(Rabsolu l2)*((Rabsolu (f1 x))*((Rabsolu (/(Rsqr (f2 x))))*((Rabsolu (/(f2 x)))*2)))``; [Idtac | Ring]. -Replace ``(Rabsolu l2)*(Rabsolu (f1 x))*((Rabsolu (/(Rsqr (f2 x))))*(Rabsolu (/(f2 (x+h)))))`` with ``(Rabsolu l2)*((Rabsolu (f1 x))*(((Rabsolu (/(Rsqr (f2 x))))*(Rabsolu (/(f2 (x+h)))))))``; [Idtac | Ring]. -Repeat Apply Rlt_monotony. -Apply Rabsolu_pos_lt; Assumption. -Apply Rabsolu_pos_lt; Assumption. -Apply Rabsolu_pos_lt; Apply Rinv_neq_R0; Unfold Rsqr; Apply prod_neq_R0; Assumption. -Repeat Rewrite Rabsolu_Rinv; [Idtac | Assumption | Assumption]. -Rewrite <- (Rmult_sym ``2``). -Unfold Rdiv in H10; Exact H10. -Symmetry; Apply Rabsolu_right; Left; Sup0. -Right; Unfold Rsqr Rdiv. -Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. -Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. -Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. -Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. -Repeat Rewrite Rabsolu_mult. -Repeat Rewrite Rabsolu_Rinv; Try Assumption Orelse DiscrR. -Replace (Rabsolu eps) with eps. -Replace ``(Rabsolu (8))`` with ``8``. -Replace ``(Rabsolu 2)`` with ``2``. -Replace ``8`` with ``4*2``; [Idtac | Ring]. -Rewrite Rinv_Rmult; DiscrR. -Replace ``2*(Rabsolu l2)*((Rabsolu (f1 x))*(/(Rabsolu (f2 x))*/(Rabsolu (f2 x))*/(Rabsolu (f2 x))))*((Rabsolu (f2 x))*(Rabsolu (f2 x))*(Rabsolu (f2 x))*eps*(/4*/2*/(Rabsolu (f1 x))*/(Rabsolu l2)))`` with ``eps*/4*((Rabsolu l2)*/(Rabsolu l2))*((Rabsolu (f1 x))*/(Rabsolu (f1 x)))*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*((Rabsolu (f2 x))*/(Rabsolu (f2 x)))*(2*/2)``; [Idtac | Ring]. -Repeat Rewrite <- Rinv_r_sym; Try DiscrR Orelse (Apply Rabsolu_no_R0; Assumption). -Ring. -Symmetry; Apply Rabsolu_right; Left; Sup0. -Symmetry; Apply Rabsolu_right; Left; Sup. -Symmetry; Apply Rabsolu_right; Left; Assumption. -Apply prod_neq_R0; Assumption Orelse DiscrR. -Apply prod_neq_R0; Assumption. +Lemma maj_term4 : + forall (x h eps l2 alp_f2 alp_f2c:R) (eps_f2:posreal) + (f1 f2:R -> R), + 0 < eps -> + f2 x <> 0 -> + f2 (x + h) <> 0 -> + (forall a:R, + Rabs a < alp_f2c -> + Rabs (f2 (x + a) - f2 x) < + Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) -> + (forall a:R, + Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)) -> + h <> 0 -> + Rabs h < alp_f2c -> + Rabs h < Rmin eps_f2 alp_f2 -> + f1 x <> 0 -> + l2 <> 0 -> + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x)) < + eps / 4. +intros. +assert (H9 := H2 h H5). +assert (H10 := H3 h H6). +apply Rle_lt_trans with + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * + Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))). +rewrite Rabs_mult. +apply Rmult_le_compat_l. +apply Rabs_pos. +left; apply H9. +apply Rlt_le_trans with + (Rabs (2 * l2 * (f1 x / (Rsqr (f2 x) * f2 x))) * + Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))). +apply Rmult_lt_compat_r. +apply Rabs_pos_lt. +unfold Rdiv in |- *; unfold Rsqr in |- *; repeat apply prod_neq_R0; + assumption || idtac. +red in |- *; intro H11; rewrite H11 in H; elim (Rlt_irrefl _ H). +apply Rinv_neq_0_compat; apply prod_neq_R0. +apply prod_neq_R0. +discrR. +assumption. +assumption. +unfold Rdiv in |- *. +repeat rewrite Rinv_mult_distr; + try assumption || (unfold Rsqr in |- *; apply prod_neq_R0; assumption). +repeat rewrite Rabs_mult. +replace (Rabs 2) with 2. +replace + (2 * Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 x)))) with + (Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * (Rabs (/ f2 x) * 2)))); + [ idtac | ring ]. +replace + (Rabs l2 * Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 (x + h)))) with + (Rabs l2 * (Rabs (f1 x) * (Rabs (/ Rsqr (f2 x)) * Rabs (/ f2 (x + h))))); + [ idtac | ring ]. +repeat apply Rmult_lt_compat_l. +apply Rabs_pos_lt; assumption. +apply Rabs_pos_lt; assumption. +apply Rabs_pos_lt; apply Rinv_neq_0_compat; unfold Rsqr in |- *; + apply prod_neq_R0; assumption. +repeat rewrite Rabs_Rinv; [ idtac | assumption | assumption ]. +rewrite <- (Rmult_comm 2). +unfold Rdiv in H10; exact H10. +symmetry in |- *; apply Rabs_right; left; prove_sup0. +right; unfold Rsqr, Rdiv in |- *. +rewrite Rinv_mult_distr; try assumption || discrR. +rewrite Rinv_mult_distr; try assumption || discrR. +rewrite Rinv_mult_distr; try assumption || discrR. +rewrite Rinv_mult_distr; try assumption || discrR. +repeat rewrite Rabs_mult. +repeat rewrite Rabs_Rinv; try assumption || discrR. +replace (Rabs eps) with eps. +replace (Rabs 8) with 8. +replace (Rabs 2) with 2. +replace 8 with (4 * 2); [ idtac | ring ]. +rewrite Rinv_mult_distr; discrR. +replace + (2 * Rabs l2 * + (Rabs (f1 x) * (/ Rabs (f2 x) * / Rabs (f2 x) * / Rabs (f2 x))) * + (Rabs (f2 x) * Rabs (f2 x) * Rabs (f2 x) * eps * + (/ 4 * / 2 * / Rabs (f1 x) * / Rabs l2))) with + (eps * / 4 * (Rabs l2 * / Rabs l2) * (Rabs (f1 x) * / Rabs (f1 x)) * + (Rabs (f2 x) * / Rabs (f2 x)) * (Rabs (f2 x) * / Rabs (f2 x)) * + (Rabs (f2 x) * / Rabs (f2 x)) * (2 * / 2)); [ idtac | ring ]. +repeat rewrite <- Rinv_r_sym; try discrR || (apply Rabs_no_R0; assumption). +ring. +symmetry in |- *; apply Rabs_right; left; prove_sup0. +symmetry in |- *; apply Rabs_right; left; prove_sup. +symmetry in |- *; apply Rabs_right; left; assumption. +apply prod_neq_R0; assumption || discrR. +apply prod_neq_R0; assumption. Qed. -Lemma D_x_no_cond : (x,a:R) ``a<>0`` -> (D_x no_cond x ``x+a``). -Intros. -Unfold D_x no_cond. -Split. -Trivial. -Apply Rminus_not_eq. -Unfold Rminus. -Rewrite Ropp_distr1. -Rewrite <- Rplus_assoc. -Rewrite Rplus_Ropp_r. -Rewrite Rplus_Ol. -Apply Ropp_neq; Assumption. +Lemma D_x_no_cond : forall x a:R, a <> 0 -> D_x no_cond x (x + a). +intros. +unfold D_x, no_cond in |- *. +split. +trivial. +apply Rminus_not_eq. +unfold Rminus in |- *. +rewrite Ropp_plus_distr. +rewrite <- Rplus_assoc. +rewrite Rplus_opp_r. +rewrite Rplus_0_l. +apply Ropp_neq_0_compat; assumption. Qed. -Lemma Rabsolu_4 : (a,b,c,d:R) ``(Rabsolu (a+b+c+d)) <= (Rabsolu a) + (Rabsolu b) + (Rabsolu c) + (Rabsolu d)``. -Intros. -Apply Rle_trans with ``(Rabsolu (a+b)) + (Rabsolu (c+d))``. -Replace ``a+b+c+d`` with ``(a+b)+(c+d)``; [Apply Rabsolu_triang | Ring]. -Apply Rle_trans with ``(Rabsolu a) + (Rabsolu b) + (Rabsolu (c+d))``. -Apply Rle_compatibility_r. -Apply Rabsolu_triang. -Repeat Rewrite Rplus_assoc; Repeat Apply Rle_compatibility. -Apply Rabsolu_triang. +Lemma Rabs_4 : + forall a b c d:R, Rabs (a + b + c + d) <= Rabs a + Rabs b + Rabs c + Rabs d. +intros. +apply Rle_trans with (Rabs (a + b) + Rabs (c + d)). +replace (a + b + c + d) with (a + b + (c + d)); [ apply Rabs_triang | ring ]. +apply Rle_trans with (Rabs a + Rabs b + Rabs (c + d)). +apply Rplus_le_compat_r. +apply Rabs_triang. +repeat rewrite Rplus_assoc; repeat apply Rplus_le_compat_l. +apply Rabs_triang. Qed. -Lemma Rlt_4 : (a,b,c,d,e,f,g,h:R) ``a < b`` -> ``c < d`` -> ``e < f `` -> ``g < h`` -> ``a+c+e+g < b+d+f+h``. -Intros; Apply Rlt_trans with ``b+c+e+g``. -Repeat Apply Rlt_compatibility_r; Assumption. -Repeat Rewrite Rplus_assoc; Apply Rlt_compatibility. -Apply Rlt_trans with ``d+e+g``. -Rewrite Rplus_assoc; Apply Rlt_compatibility_r; Assumption. -Rewrite Rplus_assoc; Apply Rlt_compatibility; Apply Rlt_trans with ``f+g``. -Apply Rlt_compatibility_r; Assumption. -Apply Rlt_compatibility; Assumption. +Lemma Rlt_4 : + forall a b c d e f g h:R, + a < b -> c < d -> e < f -> g < h -> a + c + e + g < b + d + f + h. +intros; apply Rlt_trans with (b + c + e + g). +repeat apply Rplus_lt_compat_r; assumption. +repeat rewrite Rplus_assoc; apply Rplus_lt_compat_l. +apply Rlt_trans with (d + e + g). +rewrite Rplus_assoc; apply Rplus_lt_compat_r; assumption. +rewrite Rplus_assoc; apply Rplus_lt_compat_l; apply Rlt_trans with (f + g). +apply Rplus_lt_compat_r; assumption. +apply Rplus_lt_compat_l; assumption. Qed. -Lemma Rmin_2 : (a,b,c:R) ``a < b`` -> ``a < c`` -> ``a < (Rmin b c)``. -Intros; Unfold Rmin; Case (total_order_Rle b c); Intro; Assumption. +Lemma Rmin_2 : forall a b c:R, a < b -> a < c -> a < Rmin b c. +intros; unfold Rmin in |- *; case (Rle_dec b c); intro; assumption. Qed. -Lemma quadruple : (x:R) ``4*x == x + x + x + x``. -Intro; Ring. +Lemma quadruple : forall x:R, 4 * x = x + x + x + x. +intro; ring. Qed. -Lemma quadruple_var : (x:R) `` x == x/4 + x/4 + x/4 + x/4``. -Intro; Rewrite <- quadruple. -Unfold Rdiv; Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m; DiscrR. -Reflexivity. +Lemma quadruple_var : forall x:R, x = x / 4 + x / 4 + x / 4 + x / 4. +intro; rewrite <- quadruple. +unfold Rdiv in |- *; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; discrR. +reflexivity. Qed. (**********) -Lemma continuous_neq_0 : (f:R->R; x0:R) (continuity_pt f x0) -> ~``(f x0)==0`` -> (EXT eps : posreal | (h:R) ``(Rabsolu h) < eps`` -> ~``(f (x0+h))==0``). -Intros; Unfold continuity_pt in H; Unfold continue_in in H; Unfold limit1_in in H; Unfold limit_in in H; Elim (H ``(Rabsolu ((f x0)/2))``). -Intros; Elim H1; Intros. -Exists (mkposreal x H2). -Intros; Assert H5 := (H3 ``x0+h``). -Cut ``(dist R_met (x0+h) x0) < x`` -> ``(dist R_met (f (x0+h)) (f x0)) < (Rabsolu ((f x0)/2))``. -Unfold dist; Simpl; Unfold R_dist; Replace ``x0+h-x0`` with h. -Intros; Assert H7 := (H6 H4). -Red; Intro. -Rewrite H8 in H7; Unfold Rminus in H7; Rewrite Rplus_Ol in H7; Rewrite Rabsolu_Ropp in H7; Unfold Rdiv in H7; Rewrite Rabsolu_mult in H7; Pattern 1 ``(Rabsolu (f x0)) `` in H7; Rewrite <- Rmult_1r in H7. -Cut ``0<(Rabsolu (f x0))``. -Intro; Assert H10 := (Rlt_monotony_contra ? ? ? H9 H7). -Cut ``(Rabsolu (/2))==/2``. -Assert Hyp:``0<2``. -Sup0. -Intro; Rewrite H11 in H10; Assert H12 := (Rlt_monotony ``2`` ? ? Hyp H10); Rewrite Rmult_1r in H12; Rewrite <- Rinv_r_sym in H12; [Idtac | DiscrR]. -Cut (Rlt (IZR `1`) (IZR `2`)). -Unfold IZR; Unfold INR convert; Simpl; Intro; Elim (Rlt_antirefl ``1`` (Rlt_trans ? ? ? H13 H12)). -Apply IZR_lt; Omega. -Unfold Rabsolu; Case (case_Rabsolu ``/2``); Intro. -Assert Hyp:``0<2``. -Sup0. -Assert H11 := (Rlt_monotony ``2`` ? ? Hyp r); Rewrite Rmult_Or in H11; Rewrite <- Rinv_r_sym in H11; [Idtac | DiscrR]. -Elim (Rlt_antirefl ``0`` (Rlt_trans ? ? ? Rlt_R0_R1 H11)). -Reflexivity. -Apply (Rabsolu_pos_lt ? H0). -Ring. -Assert H6 := (Req_EM ``x0`` ``x0+h``); Elim H6; Intro. -Intro; Rewrite <- H7; Unfold dist R_met; Unfold R_dist; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rabsolu_pos_lt. -Unfold Rdiv; Apply prod_neq_R0; [Assumption | Apply Rinv_neq_R0; DiscrR]. -Intro; Apply H5. -Split. -Unfold D_x no_cond. -Split; Trivial Orelse Assumption. -Assumption. -Change ``0 < (Rabsolu ((f x0)/2))``. -Apply Rabsolu_pos_lt; Unfold Rdiv; Apply prod_neq_R0. -Assumption. -Apply Rinv_neq_R0; DiscrR. -Qed. +Lemma continuous_neq_0 : + forall (f:R -> R) (x0:R), + continuity_pt f x0 -> + f x0 <> 0 -> + exists eps : posreal | (forall h:R, Rabs h < eps -> f (x0 + h) <> 0). +intros; unfold continuity_pt in H; unfold continue_in in H; + unfold limit1_in in H; unfold limit_in in H; elim (H (Rabs (f x0 / 2))). +intros; elim H1; intros. +exists (mkposreal x H2). +intros; assert (H5 := H3 (x0 + h)). +cut + (dist R_met (x0 + h) x0 < x -> + dist R_met (f (x0 + h)) (f x0) < Rabs (f x0 / 2)). +unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + replace (x0 + h - x0) with h. +intros; assert (H7 := H6 H4). +red in |- *; intro. +rewrite H8 in H7; unfold Rminus in H7; rewrite Rplus_0_l in H7; + rewrite Rabs_Ropp in H7; unfold Rdiv in H7; rewrite Rabs_mult in H7; + pattern (Rabs (f x0)) at 1 in H7; rewrite <- Rmult_1_r in H7. +cut (0 < Rabs (f x0)). +intro; assert (H10 := Rmult_lt_reg_l _ _ _ H9 H7). +cut (Rabs (/ 2) = / 2). +assert (Hyp : 0 < 2). +prove_sup0. +intro; rewrite H11 in H10; assert (H12 := Rmult_lt_compat_l 2 _ _ Hyp H10); + rewrite Rmult_1_r in H12; rewrite <- Rinv_r_sym in H12; + [ idtac | discrR ]. +cut (IZR 1 < IZR 2). +unfold IZR in |- *; unfold INR, nat_of_P in |- *; simpl in |- *; intro; + elim (Rlt_irrefl 1 (Rlt_trans _ _ _ H13 H12)). +apply IZR_lt; omega. +unfold Rabs in |- *; case (Rcase_abs (/ 2)); intro. +assert (Hyp : 0 < 2). +prove_sup0. +assert (H11 := Rmult_lt_compat_l 2 _ _ Hyp r); rewrite Rmult_0_r in H11; + rewrite <- Rinv_r_sym in H11; [ idtac | discrR ]. +elim (Rlt_irrefl 0 (Rlt_trans _ _ _ Rlt_0_1 H11)). +reflexivity. +apply (Rabs_pos_lt _ H0). +ring. +assert (H6 := Req_dec x0 (x0 + h)); elim H6; intro. +intro; rewrite <- H7; unfold dist, R_met in |- *; unfold R_dist in |- *; + unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + apply Rabs_pos_lt. +unfold Rdiv in |- *; apply prod_neq_R0; + [ assumption | apply Rinv_neq_0_compat; discrR ]. +intro; apply H5. +split. +unfold D_x, no_cond in |- *. +split; trivial || assumption. +assumption. +change (0 < Rabs (f x0 / 2)) in |- *. +apply Rabs_pos_lt; unfold Rdiv in |- *; apply prod_neq_R0. +assumption. +apply Rinv_neq_0_compat; discrR. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Ranalysis3.v b/theories/Reals/Ranalysis3.v index e8af542ac..1e0991e15 100644 --- a/theories/Reals/Ranalysis3.v +++ b/theories/Reals/Ranalysis3.v @@ -8,610 +8,786 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Ranalysis1. -Require Ranalysis2. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rfunctions. +Require Import Ranalysis1. +Require Import Ranalysis2. Open Local Scope R_scope. (* Division *) -Theorem derivable_pt_lim_div : (f1,f2:R->R;x,l1,l2:R) (derivable_pt_lim f1 x l1) -> (derivable_pt_lim f2 x l2) -> ~``(f2 x)==0``-> (derivable_pt_lim (div_fct f1 f2) x ``(l1*(f2 x)-l2*(f1 x))/(Rsqr (f2 x))``). -Intros. -Cut (derivable_pt f2 x); [Intro | Unfold derivable_pt; Apply Specif.existT with l2; Exact H0]. -Assert H2 := ((continuous_neq_0 ? ? (derivable_continuous_pt ? ? X)) H1). -Elim H2; Clear H2; Intros eps_f2 H2. -Unfold div_fct. -Assert H3 := (derivable_continuous_pt ? ? X). -Unfold continuity_pt in H3; Unfold continue_in in H3; Unfold limit1_in in H3; Unfold limit_in in H3; Unfold dist in H3. -Simpl in H3; Unfold R_dist in H3. -Elim (H3 ``(Rabsolu (f2 x))/2``); [Idtac | Unfold Rdiv; Change ``0 < (Rabsolu (f2 x))*/2``; Apply Rmult_lt_pos; [Apply Rabsolu_pos_lt; Assumption | Apply Rlt_Rinv; Sup0]]. -Clear H3; Intros alp_f2 H3. -Cut (x0:R) ``(Rabsolu (x0-x)) < alp_f2`` ->``(Rabsolu ((f2 x0)-(f2 x))) < (Rabsolu (f2 x))/2``. -Intro H4. -Cut (a:R) ``(Rabsolu (a-x)) < alp_f2``->``(Rabsolu (f2 x))/2 < (Rabsolu (f2 a))``. -Intro H5. -Cut (a:R) ``(Rabsolu (a)) < (Rmin eps_f2 alp_f2)`` -> ``/(Rabsolu (f2 (x+a))) < 2/(Rabsolu (f2 x))``. -Intro Maj. -Unfold derivable_pt_lim; Intros. -Elim (H ``(Rabsolu ((eps*(f2 x))/8))``); [Idtac | Unfold Rdiv; Change ``0 < (Rabsolu (eps*(f2 x)*/8))``; Apply Rabsolu_pos_lt; Repeat Apply prod_neq_R0; [Red; Intro H7; Rewrite H7 in H6; Elim (Rlt_antirefl ? H6) | Assumption | Apply Rinv_neq_R0; DiscrR]]. -Intros alp_f1d H7. -Case (Req_EM (f1 x) R0); Intro. -Case (Req_EM l1 R0); Intro. +Theorem derivable_pt_lim_div : + forall (f1 f2:R -> R) (x l1 l2:R), + derivable_pt_lim f1 x l1 -> + derivable_pt_lim f2 x l2 -> + f2 x <> 0 -> + derivable_pt_lim (f1 / f2) x ((l1 * f2 x - l2 * f1 x) / Rsqr (f2 x)). +intros. +cut (derivable_pt f2 x); + [ intro | unfold derivable_pt in |- *; apply existT with l2; exact H0 ]. +assert (H2 := continuous_neq_0 _ _ (derivable_continuous_pt _ _ X) H1). +elim H2; clear H2; intros eps_f2 H2. +unfold div_fct in |- *. +assert (H3 := derivable_continuous_pt _ _ X). +unfold continuity_pt in H3; unfold continue_in in H3; unfold limit1_in in H3; + unfold limit_in in H3; unfold dist in H3. +simpl in H3; unfold R_dist in H3. +elim (H3 (Rabs (f2 x) / 2)); + [ idtac + | unfold Rdiv in |- *; change (0 < Rabs (f2 x) * / 2) in |- *; + apply Rmult_lt_0_compat; + [ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +clear H3; intros alp_f2 H3. +cut + (forall x0:R, + Rabs (x0 - x) < alp_f2 -> Rabs (f2 x0 - f2 x) < Rabs (f2 x) / 2). +intro H4. +cut (forall a:R, Rabs (a - x) < alp_f2 -> Rabs (f2 x) / 2 < Rabs (f2 a)). +intro H5. +cut + (forall a:R, + Rabs a < Rmin eps_f2 alp_f2 -> / Rabs (f2 (x + a)) < 2 / Rabs (f2 x)). +intro Maj. +unfold derivable_pt_lim in |- *; intros. +elim (H (Rabs (eps * f2 x / 8))); + [ idtac + | unfold Rdiv in |- *; change (0 < Rabs (eps * f2 x * / 8)) in |- *; + apply Rabs_pos_lt; repeat apply prod_neq_R0; + [ red in |- *; intro H7; rewrite H7 in H6; elim (Rlt_irrefl _ H6) + | assumption + | apply Rinv_neq_0_compat; discrR ] ]. +intros alp_f1d H7. +case (Req_dec (f1 x) 0); intro. +case (Req_dec l1 0); intro. (***********************************) (* Cas n° 1 *) (* (f1 x)=0 l1 =0 *) (***********************************) -Cut ``0 < (Rmin eps_f2 (Rmin alp_f2 alp_f1d))``; [Intro | Repeat Apply Rmin_pos; [Apply (cond_pos eps_f2) | Elim H3; Intros; Assumption | Apply (cond_pos alp_f1d)]]. -Exists (mkposreal (Rmin eps_f2 (Rmin alp_f2 alp_f1d)) H10). -Simpl; Intros. -Assert H13 := (Rlt_le_trans ? ? ? H12 (Rmin_r ? ?)). -Assert H14 := (Rlt_le_trans ? ? ? H12 (Rmin_l ? ?)). -Assert H15 := (Rlt_le_trans ? ? ? H13 (Rmin_r ? ?)). -Assert H16 := (Rlt_le_trans ? ? ? H13 (Rmin_l ? ?)). -Assert H17 := (H7 ? H11 H15). -Rewrite formule; [Idtac | Assumption | Assumption | Apply H2; Apply H14]. -Apply Rle_lt_trans with ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) + (Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) + (Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) + (Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))))``. -Unfold Rminus. -Rewrite <- (Rabsolu_Ropp ``(f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))+ -(f2 x))/h+ -l2)``). -Apply Rabsolu_4. -Repeat Rewrite Rabsolu_mult. -Apply Rlt_le_trans with ``eps/4+eps/4+eps/4+eps/4``. -Cut ``(Rabsolu (/(f2 (x+h))))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < eps/4``. -Cut ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((f2 x)-(f2 (x+h)))) < eps/4``. -Cut ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < eps/4``. -Cut ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu ((f2 (x+h))-(f2 x))) < eps/4``. -Intros. -Apply Rlt_4; Assumption. -Rewrite H8. -Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. -Rewrite Rabsolu_R0; Rewrite Rmult_Ol. -Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. -Rewrite H8. -Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. -Rewrite Rabsolu_R0; Rewrite Rmult_Ol. -Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. -Rewrite H9. -Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. -Rewrite Rabsolu_R0; Rewrite Rmult_Ol. -Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. -Rewrite <- Rabsolu_mult. -Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Try Assumption Orelse Apply H2. -Apply H14. -Apply Rmin_2; Assumption. -Right; Symmetry; Apply quadruple_var. +cut (0 < Rmin eps_f2 (Rmin alp_f2 alp_f1d)); + [ intro + | repeat apply Rmin_pos; + [ apply (cond_pos eps_f2) + | elim H3; intros; assumption + | apply (cond_pos alp_f1d) ] ]. +exists (mkposreal (Rmin eps_f2 (Rmin alp_f2 alp_f1d)) H10). +simpl in |- *; intros. +assert (H13 := Rlt_le_trans _ _ _ H12 (Rmin_r _ _)). +assert (H14 := Rlt_le_trans _ _ _ H12 (Rmin_l _ _)). +assert (H15 := Rlt_le_trans _ _ _ H13 (Rmin_r _ _)). +assert (H16 := Rlt_le_trans _ _ _ H13 (Rmin_l _ _)). +assert (H17 := H7 _ H11 H15). +rewrite formule; [ idtac | assumption | assumption | apply H2; apply H14 ]. +apply Rle_lt_trans with + (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) + + Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). +unfold Rminus in |- *. +rewrite <- + (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) + . +apply Rabs_4. +repeat rewrite Rabs_mult. +apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4). +cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4). +cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4). +cut + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) < + eps / 4). +cut + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) < + eps / 4). +intros. +apply Rlt_4; assumption. +rewrite H8. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite H8. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite H9. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite <- Rabs_mult. +apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); + try assumption || apply H2. +apply H14. +apply Rmin_2; assumption. +right; symmetry in |- *; apply quadruple_var. (***********************************) (* Cas n° 2 *) (* (f1 x)=0 l1<>0 *) (***********************************) -Assert H10 := (derivable_continuous_pt ? ? X). -Unfold continuity_pt in H10. -Unfold continue_in in H10. -Unfold limit1_in in H10. -Unfold limit_in in H10. -Unfold dist in H10. -Simpl in H10. -Unfold R_dist in H10. -Elim (H10 ``(Rabsolu (eps*(Rsqr (f2 x)))/(8*l1))``). -Clear H10; Intros alp_f2t2 H10. -Cut (a:R) ``(Rabsolu a) < alp_f2t2`` -> ``(Rabsolu ((f2 (x+a)) - (f2 x))) < (Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. -Intro H11. -Cut ``0 < (Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2))``. -Intro. -Exists (mkposreal (Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)) H12). -Simpl. -Intros. -Assert H15 := (Rlt_le_trans ? ? ? H14 (Rmin_r ? ?)). -Assert H16 := (Rlt_le_trans ? ? ? H14 (Rmin_l ? ?)). -Assert H17 := (Rlt_le_trans ? ? ? H15 (Rmin_l ? ?)). -Assert H18 := (Rlt_le_trans ? ? ? H15 (Rmin_r ? ?)). -Assert H19 := (Rlt_le_trans ? ? ? H16 (Rmin_l ? ?)). -Assert H20 := (Rlt_le_trans ? ? ? H16 (Rmin_r ? ?)). -Clear H14 H15 H16. -Rewrite formule; Try Assumption. -Apply Rle_lt_trans with ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) + (Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) + (Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) + (Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))))``. -Unfold Rminus. -Rewrite <- (Rabsolu_Ropp ``(f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))+ -(f2 x))/h+ -l2)``). -Apply Rabsolu_4. -Repeat Rewrite Rabsolu_mult. -Apply Rlt_le_trans with ``eps/4+eps/4+eps/4+eps/4``. -Cut ``(Rabsolu (/(f2 (x+h))))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < eps/4``. -Cut ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((f2 x)-(f2 (x+h)))) < eps/4``. -Cut ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < eps/4``. -Cut ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu ((f2 (x+h))-(f2 x))) < eps/4``. -Intros. -Apply Rlt_4; Assumption. -Rewrite H8. -Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. -Rewrite Rabsolu_R0; Rewrite Rmult_Ol. -Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. -Rewrite H8. -Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. -Rewrite Rabsolu_R0; Rewrite Rmult_Ol. -Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. -Rewrite <- Rabsolu_mult. -Apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Rewrite <- Rabsolu_mult. -Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Right; Symmetry; Apply quadruple_var. -Apply H2; Assumption. -Repeat Apply Rmin_pos. -Apply (cond_pos eps_f2). -Apply (cond_pos alp_f1d). -Elim H3; Intros; Assumption. -Elim H10; Intros; Assumption. -Intros. -Elim H10; Intros. -Case (Req_EM a R0); Intro. -Rewrite H14; Rewrite Rplus_Or. -Unfold Rminus; Rewrite Rplus_Ropp_r. -Rewrite Rabsolu_R0. -Apply Rabsolu_pos_lt. -Unfold Rdiv Rsqr; Repeat Rewrite Rmult_assoc. -Repeat Apply prod_neq_R0; Try Assumption. -Red; Intro; Rewrite H15 in H6; Elim (Rlt_antirefl ? H6). -Apply Rinv_neq_R0; Repeat Apply prod_neq_R0; DiscrR Orelse Assumption. -Apply H13. -Split. -Apply D_x_no_cond; Assumption. -Replace ``x+a-x`` with a; [Assumption | Ring]. -Change ``0<(Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. -Apply Rabsolu_pos_lt; Unfold Rdiv Rsqr; Repeat Rewrite Rmult_assoc; Repeat Apply prod_neq_R0. -Red; Intro; Rewrite H11 in H6; Elim (Rlt_antirefl ? H6). -Assumption. -Assumption. -Apply Rinv_neq_R0; Repeat Apply prod_neq_R0; [DiscrR | DiscrR | DiscrR | Assumption]. +assert (H10 := derivable_continuous_pt _ _ X). +unfold continuity_pt in H10. +unfold continue_in in H10. +unfold limit1_in in H10. +unfold limit_in in H10. +unfold dist in H10. +simpl in H10. +unfold R_dist in H10. +elim (H10 (Rabs (eps * Rsqr (f2 x) / (8 * l1)))). +clear H10; intros alp_f2t2 H10. +cut + (forall a:R, + Rabs a < alp_f2t2 -> + Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))). +intro H11. +cut (0 < Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)). +intro. +exists (mkposreal (Rmin (Rmin eps_f2 alp_f1d) (Rmin alp_f2 alp_f2t2)) H12). +simpl in |- *. +intros. +assert (H15 := Rlt_le_trans _ _ _ H14 (Rmin_r _ _)). +assert (H16 := Rlt_le_trans _ _ _ H14 (Rmin_l _ _)). +assert (H17 := Rlt_le_trans _ _ _ H15 (Rmin_l _ _)). +assert (H18 := Rlt_le_trans _ _ _ H15 (Rmin_r _ _)). +assert (H19 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)). +assert (H20 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)). +clear H14 H15 H16. +rewrite formule; try assumption. +apply Rle_lt_trans with + (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) + + Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). +unfold Rminus in |- *. +rewrite <- + (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) + . +apply Rabs_4. +repeat rewrite Rabs_mult. +apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4). +cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4). +cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4). +cut + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) < + eps / 4). +cut + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) < + eps / 4). +intros. +apply Rlt_4; assumption. +rewrite H8. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite H8. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite <- Rabs_mult. +apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +right; symmetry in |- *; apply quadruple_var. +apply H2; assumption. +repeat apply Rmin_pos. +apply (cond_pos eps_f2). +apply (cond_pos alp_f1d). +elim H3; intros; assumption. +elim H10; intros; assumption. +intros. +elim H10; intros. +case (Req_dec a 0); intro. +rewrite H14; rewrite Rplus_0_r. +unfold Rminus in |- *; rewrite Rplus_opp_r. +rewrite Rabs_R0. +apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; repeat rewrite Rmult_assoc. +repeat apply prod_neq_R0; try assumption. +red in |- *; intro; rewrite H15 in H6; elim (Rlt_irrefl _ H6). +apply Rinv_neq_0_compat; repeat apply prod_neq_R0; discrR || assumption. +apply H13. +split. +apply D_x_no_cond; assumption. +replace (x + a - x) with a; [ assumption | ring ]. +change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))) in |- *. +apply Rabs_pos_lt; unfold Rdiv, Rsqr in |- *; repeat rewrite Rmult_assoc; + repeat apply prod_neq_R0. +red in |- *; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6). +assumption. +assumption. +apply Rinv_neq_0_compat; repeat apply prod_neq_R0; + [ discrR | discrR | discrR | assumption ]. (***********************************) (* Cas n° 3 *) (* (f1 x)<>0 l1=0 l2=0 *) (***********************************) -Case (Req_EM l1 R0); Intro. -Case (Req_EM l2 R0); Intro. -Elim (H0 ``(Rabsolu ((Rsqr (f2 x))*eps)/(8*(f1 x)))``); [Idtac | Apply Rabsolu_pos_lt; Unfold Rdiv Rsqr; Repeat Rewrite Rmult_assoc; Repeat Apply prod_neq_R0; [Assumption | Assumption | Red; Intro; Rewrite H11 in H6; Elim (Rlt_antirefl ? H6) | Apply Rinv_neq_R0; Repeat Apply prod_neq_R0; DiscrR Orelse Assumption]]. -Intros alp_f2d H12. -Cut ``0 < (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d))``. -Intro. -Exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) H11). -Simpl. -Intros. -Assert H15 := (Rlt_le_trans ? ? ? H14 (Rmin_l ? ?)). -Assert H16 := (Rlt_le_trans ? ? ? H14 (Rmin_r ? ?)). -Assert H17 := (Rlt_le_trans ? ? ? H15 (Rmin_l ? ?)). -Assert H18 := (Rlt_le_trans ? ? ? H15 (Rmin_r ? ?)). -Assert H19 := (Rlt_le_trans ? ? ? H16 (Rmin_l ? ?)). -Assert H20 := (Rlt_le_trans ? ? ? H16 (Rmin_r ? ?)). -Clear H15 H16. -Rewrite formule; Try Assumption. -Apply Rle_lt_trans with ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) + (Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) + (Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) + (Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))))``. -Unfold Rminus. -Rewrite <- (Rabsolu_Ropp ``(f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))+ -(f2 x))/h+ -l2)``). -Apply Rabsolu_4. -Repeat Rewrite Rabsolu_mult. -Apply Rlt_le_trans with ``eps/4+eps/4+eps/4+eps/4``. -Cut ``(Rabsolu (/(f2 (x+h))))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < eps/4``. -Cut ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((f2 x)-(f2 (x+h)))) < eps/4``. -Cut ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < eps/4``. -Cut ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu ((f2 (x+h))-(f2 x))) < eps/4``. -Intros. -Apply Rlt_4; Assumption. -Rewrite H10. -Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. -Rewrite Rabsolu_R0; Rewrite Rmult_Ol. -Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. -Rewrite <- Rabsolu_mult. -Apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Rewrite H9. -Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. -Rewrite Rabsolu_R0; Rewrite Rmult_Ol. -Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. -Rewrite <- Rabsolu_mult. -Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Assumption Orelse Idtac. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Right; Symmetry; Apply quadruple_var. -Apply H2; Assumption. -Repeat Apply Rmin_pos. -Apply (cond_pos eps_f2). -Elim H3; Intros; Assumption. -Apply (cond_pos alp_f1d). -Apply (cond_pos alp_f2d). +case (Req_dec l1 0); intro. +case (Req_dec l2 0); intro. +elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))); + [ idtac + | apply Rabs_pos_lt; unfold Rdiv, Rsqr in |- *; repeat rewrite Rmult_assoc; + repeat apply prod_neq_R0; + [ assumption + | assumption + | red in |- *; intro; rewrite H11 in H6; elim (Rlt_irrefl _ H6) + | apply Rinv_neq_0_compat; repeat apply prod_neq_R0; discrR || assumption ] ]. +intros alp_f2d H12. +cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)). +intro. +exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) H11). +simpl in |- *. +intros. +assert (H15 := Rlt_le_trans _ _ _ H14 (Rmin_l _ _)). +assert (H16 := Rlt_le_trans _ _ _ H14 (Rmin_r _ _)). +assert (H17 := Rlt_le_trans _ _ _ H15 (Rmin_l _ _)). +assert (H18 := Rlt_le_trans _ _ _ H15 (Rmin_r _ _)). +assert (H19 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)). +assert (H20 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)). +clear H15 H16. +rewrite formule; try assumption. +apply Rle_lt_trans with + (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) + + Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). +unfold Rminus in |- *. +rewrite <- + (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) + . +apply Rabs_4. +repeat rewrite Rabs_mult. +apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4). +cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4). +cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4). +cut + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) < + eps / 4). +cut + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) < + eps / 4). +intros. +apply Rlt_4; assumption. +rewrite H10. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite <- Rabs_mult. +apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite H9. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite <- Rabs_mult. +apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); assumption || idtac. +apply H2; assumption. +apply Rmin_2; assumption. +right; symmetry in |- *; apply quadruple_var. +apply H2; assumption. +repeat apply Rmin_pos. +apply (cond_pos eps_f2). +elim H3; intros; assumption. +apply (cond_pos alp_f1d). +apply (cond_pos alp_f2d). (***********************************) (* Cas n° 4 *) (* (f1 x)<>0 l1=0 l2<>0 *) (***********************************) -Elim (H0 ``(Rabsolu ((Rsqr (f2 x))*eps)/(8*(f1 x)))``); [Idtac | Apply Rabsolu_pos_lt; Unfold Rsqr Rdiv; Repeat Rewrite Rinv_Rmult; Repeat Apply prod_neq_R0; Try Assumption Orelse DiscrR]. -Intros alp_f2d H11. -Assert H12 := (derivable_continuous_pt ? ? X). -Unfold continuity_pt in H12. -Unfold continue_in in H12. -Unfold limit1_in in H12. -Unfold limit_in in H12. -Unfold dist in H12. -Simpl in H12. -Unfold R_dist in H12. -Elim (H12 ``(Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``). -Intros alp_f2c H13. -Cut ``0 < (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c)))``. -Intro. -Exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c))) H14). -Simpl; Intros. -Assert H17 := (Rlt_le_trans ? ? ? H16 (Rmin_l ? ?)). -Assert H18 := (Rlt_le_trans ? ? ? H16 (Rmin_r ? ?)). -Assert H19 := (Rlt_le_trans ? ? ? H18 (Rmin_r ? ?)). -Assert H20 := (Rlt_le_trans ? ? ? H19 (Rmin_l ? ?)). -Assert H21 := (Rlt_le_trans ? ? ? H19 (Rmin_r ? ?)). -Assert H22 := (Rlt_le_trans ? ? ? H18 (Rmin_l ? ?)). -Assert H23 := (Rlt_le_trans ? ? ? H17 (Rmin_l ? ?)). -Assert H24 := (Rlt_le_trans ? ? ? H17 (Rmin_r ? ?)). -Clear H16 H17 H18 H19. -Cut (a:R) ``(Rabsolu a) < alp_f2c`` -> ``(Rabsolu ((f2 (x+a))-(f2 x))) < (Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``. -Intro. -Rewrite formule; Try Assumption. -Apply Rle_lt_trans with ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) + (Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) + (Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) + (Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))))``. -Unfold Rminus. -Rewrite <- (Rabsolu_Ropp ``(f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))+ -(f2 x))/h+ -l2)``). -Apply Rabsolu_4. -Repeat Rewrite Rabsolu_mult. -Apply Rlt_le_trans with ``eps/4+eps/4+eps/4+eps/4``. -Cut ``(Rabsolu (/(f2 (x+h))))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < eps/4``. -Cut ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((f2 x)-(f2 (x+h)))) < eps/4``. -Cut ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < eps/4``. -Cut ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu ((f2 (x+h))-(f2 x))) < eps/4``. -Intros. -Apply Rlt_4; Assumption. -Rewrite <- Rabsolu_mult. -Apply (maj_term4 x h eps l2 alp_f2 alp_f2c eps_f2 f1 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Rewrite <- Rabsolu_mult. -Apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Rewrite H9. -Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. -Rewrite Rabsolu_R0; Rewrite Rmult_Ol. -Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. -Rewrite <- Rabsolu_mult. -Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Right; Symmetry; Apply quadruple_var. -Apply H2; Assumption. -Intros. -Case (Req_EM a R0); Intro. -Rewrite H17; Rewrite Rplus_Or. -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0. -Apply Rabsolu_pos_lt. -Unfold Rdiv Rsqr. -Repeat Rewrite Rinv_Rmult; Try Assumption. -Repeat Apply prod_neq_R0; Try Assumption. -Red; Intro H18; Rewrite H18 in H6; Elim (Rlt_antirefl ? H6). -Apply Rinv_neq_R0; DiscrR. -Apply Rinv_neq_R0; DiscrR. -Apply Rinv_neq_R0; DiscrR. -Apply Rinv_neq_R0; Assumption. -Apply Rinv_neq_R0; Assumption. -DiscrR. -DiscrR. -DiscrR. -DiscrR. -DiscrR. -Apply prod_neq_R0; [DiscrR | Assumption]. -Elim H13; Intros. -Apply H19. -Split. -Apply D_x_no_cond; Assumption. -Replace ``x+a-x`` with a; [Assumption | Ring]. -Repeat Apply Rmin_pos. -Apply (cond_pos eps_f2). -Elim H3; Intros; Assumption. -Apply (cond_pos alp_f1d). -Apply (cond_pos alp_f2d). -Elim H13; Intros; Assumption. -Change ``0 < (Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``. -Apply Rabsolu_pos_lt. -Unfold Rsqr Rdiv. -Repeat Rewrite Rinv_Rmult; Try Assumption Orelse DiscrR. -Repeat Apply prod_neq_R0; Try Assumption. -Red; Intro H13; Rewrite H13 in H6; Elim (Rlt_antirefl ? H6). -Apply Rinv_neq_R0; DiscrR. -Apply Rinv_neq_R0; DiscrR. -Apply Rinv_neq_R0; DiscrR. -Apply Rinv_neq_R0; Assumption. -Apply Rinv_neq_R0; Assumption. -Apply prod_neq_R0; [DiscrR | Assumption]. -Red; Intro H11; Rewrite H11 in H6; Elim (Rlt_antirefl ? H6). -Apply Rinv_neq_R0; DiscrR. -Apply Rinv_neq_R0; DiscrR. -Apply Rinv_neq_R0; DiscrR. -Apply Rinv_neq_R0; Assumption. +elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))); + [ idtac + | apply Rabs_pos_lt; unfold Rsqr, Rdiv in |- *; + repeat rewrite Rinv_mult_distr; repeat apply prod_neq_R0; + try assumption || discrR ]. +intros alp_f2d H11. +assert (H12 := derivable_continuous_pt _ _ X). +unfold continuity_pt in H12. +unfold continue_in in H12. +unfold limit1_in in H12. +unfold limit_in in H12. +unfold dist in H12. +simpl in H12. +unfold R_dist in H12. +elim (H12 (Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2)))). +intros alp_f2c H13. +cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c))). +intro. +exists + (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2c))) + H14). +simpl in |- *; intros. +assert (H17 := Rlt_le_trans _ _ _ H16 (Rmin_l _ _)). +assert (H18 := Rlt_le_trans _ _ _ H16 (Rmin_r _ _)). +assert (H19 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)). +assert (H20 := Rlt_le_trans _ _ _ H19 (Rmin_l _ _)). +assert (H21 := Rlt_le_trans _ _ _ H19 (Rmin_r _ _)). +assert (H22 := Rlt_le_trans _ _ _ H18 (Rmin_l _ _)). +assert (H23 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)). +assert (H24 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)). +clear H16 H17 H18 H19. +cut + (forall a:R, + Rabs a < alp_f2c -> + Rabs (f2 (x + a) - f2 x) < + Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))). +intro. +rewrite formule; try assumption. +apply Rle_lt_trans with + (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) + + Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). +unfold Rminus in |- *. +rewrite <- + (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) + . +apply Rabs_4. +repeat rewrite Rabs_mult. +apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4). +cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4). +cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4). +cut + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) < + eps / 4). +cut + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) < + eps / 4). +intros. +apply Rlt_4; assumption. +rewrite <- Rabs_mult. +apply (maj_term4 x h eps l2 alp_f2 alp_f2c eps_f2 f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite H9. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite <- Rabs_mult. +apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +right; symmetry in |- *; apply quadruple_var. +apply H2; assumption. +intros. +case (Req_dec a 0); intro. +rewrite H17; rewrite Rplus_0_r. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0. +apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *. +repeat rewrite Rinv_mult_distr; try assumption. +repeat apply prod_neq_R0; try assumption. +red in |- *; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6). +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; assumption. +apply Rinv_neq_0_compat; assumption. +discrR. +discrR. +discrR. +discrR. +discrR. +apply prod_neq_R0; [ discrR | assumption ]. +elim H13; intros. +apply H19. +split. +apply D_x_no_cond; assumption. +replace (x + a - x) with a; [ assumption | ring ]. +repeat apply Rmin_pos. +apply (cond_pos eps_f2). +elim H3; intros; assumption. +apply (cond_pos alp_f1d). +apply (cond_pos alp_f2d). +elim H13; intros; assumption. +change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) in |- *. +apply Rabs_pos_lt. +unfold Rsqr, Rdiv in |- *. +repeat rewrite Rinv_mult_distr; try assumption || discrR. +repeat apply prod_neq_R0; try assumption. +red in |- *; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6). +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; assumption. +apply Rinv_neq_0_compat; assumption. +apply prod_neq_R0; [ discrR | assumption ]. +red in |- *; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6). +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; discrR. +apply Rinv_neq_0_compat; assumption. (***********************************) (* Cas n° 5 *) (* (f1 x)<>0 l1<>0 l2=0 *) (***********************************) -Case (Req_EM l2 R0); Intro. -Assert H11 := (derivable_continuous_pt ? ? X). -Unfold continuity_pt in H11. -Unfold continue_in in H11. -Unfold limit1_in in H11. -Unfold limit_in in H11. -Unfold dist in H11. -Simpl in H11. -Unfold R_dist in H11. -Elim (H11 ``(Rabsolu (eps*(Rsqr (f2 x)))/(8*l1))``). -Clear H11; Intros alp_f2t2 H11. -Elim (H0 ``(Rabsolu ((Rsqr (f2 x))*eps)/(8*(f1 x)))``). -Intros alp_f2d H12. -Cut ``0 < (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2)))``. -Intro. -Exists (mkposreal (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))) H13). -Simpl. -Intros. -Cut (a:R) ``(Rabsolu a)<alp_f2t2`` -> ``(Rabsolu ((f2 (x+a))-(f2 x)))<(Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. -Intro. -Assert H17 := (Rlt_le_trans ? ? ? H15 (Rmin_l ? ?)). -Assert H18 := (Rlt_le_trans ? ? ? H15 (Rmin_r ? ?)). -Assert H19 := (Rlt_le_trans ? ? ? H17 (Rmin_r ? ?)). -Assert H20 := (Rlt_le_trans ? ? ? H17 (Rmin_l ? ?)). -Assert H21 := (Rlt_le_trans ? ? ? H18 (Rmin_r ? ?)). -Assert H22 := (Rlt_le_trans ? ? ? H18 (Rmin_l ? ?)). -Assert H23 := (Rlt_le_trans ? ? ? H21 (Rmin_l ? ?)). -Assert H24 := (Rlt_le_trans ? ? ? H21 (Rmin_r ? ?)). -Clear H15 H17 H18 H21. -Rewrite formule; Try Assumption. -Apply Rle_lt_trans with ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) + (Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) + (Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) + (Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))))``. -Unfold Rminus. -Rewrite <- (Rabsolu_Ropp ``(f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))+ -(f2 x))/h+ -l2)``). -Apply Rabsolu_4. -Repeat Rewrite Rabsolu_mult. -Apply Rlt_le_trans with ``eps/4+eps/4+eps/4+eps/4``. -Cut ``(Rabsolu (/(f2 (x+h))))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < eps/4``. -Cut ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((f2 x)-(f2 (x+h)))) < eps/4``. -Cut ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < eps/4``. -Cut ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu ((f2 (x+h))-(f2 x))) < eps/4``. -Intros. -Apply Rlt_4; Assumption. -Rewrite H10. -Unfold Rdiv; Repeat Rewrite Rmult_Or Orelse Rewrite Rmult_Ol. -Rewrite Rabsolu_R0; Rewrite Rmult_Ol. -Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup]. -Rewrite <- Rabsolu_mult. -Apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Rewrite <- Rabsolu_mult. -Apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Rewrite <- Rabsolu_mult. -Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Right; Symmetry; Apply quadruple_var. -Apply H2; Assumption. -Intros. -Case (Req_EM a R0); Intro. -Rewrite H17; Rewrite Rplus_Or; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0. -Apply Rabsolu_pos_lt. -Unfold Rdiv; Rewrite Rinv_Rmult; Try DiscrR Orelse Assumption. -Unfold Rsqr. -Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H18; Rewrite H18 in H6; Elim (Rlt_antirefl ? H6)). -Elim H11; Intros. -Apply H19. -Split. -Apply D_x_no_cond; Assumption. -Replace ``x+a-x`` with a; [Assumption | Ring]. -Repeat Apply Rmin_pos. -Apply (cond_pos eps_f2). -Elim H3; Intros; Assumption. -Apply (cond_pos alp_f1d). -Apply (cond_pos alp_f2d). -Elim H11; Intros; Assumption. -Apply Rabsolu_pos_lt. -Unfold Rdiv Rsqr; Rewrite Rinv_Rmult; Try DiscrR Orelse Assumption. -Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H12; Rewrite H12 in H6; Elim (Rlt_antirefl ? H6)). -Change ``0 < (Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. -Apply Rabsolu_pos_lt. -Unfold Rdiv Rsqr; Rewrite Rinv_Rmult; Try DiscrR Orelse Assumption. -Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H12; Rewrite H12 in H6; Elim (Rlt_antirefl ? H6)). +case (Req_dec l2 0); intro. +assert (H11 := derivable_continuous_pt _ _ X). +unfold continuity_pt in H11. +unfold continue_in in H11. +unfold limit1_in in H11. +unfold limit_in in H11. +unfold dist in H11. +simpl in H11. +unfold R_dist in H11. +elim (H11 (Rabs (eps * Rsqr (f2 x) / (8 * l1)))). +clear H11; intros alp_f2t2 H11. +elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))). +intros alp_f2d H12. +cut (0 < Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))). +intro. +exists + (mkposreal + (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d (Rmin alp_f2d alp_f2t2))) H13). +simpl in |- *. +intros. +cut + (forall a:R, + Rabs a < alp_f2t2 -> + Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))). +intro. +assert (H17 := Rlt_le_trans _ _ _ H15 (Rmin_l _ _)). +assert (H18 := Rlt_le_trans _ _ _ H15 (Rmin_r _ _)). +assert (H19 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)). +assert (H20 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)). +assert (H21 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)). +assert (H22 := Rlt_le_trans _ _ _ H18 (Rmin_l _ _)). +assert (H23 := Rlt_le_trans _ _ _ H21 (Rmin_l _ _)). +assert (H24 := Rlt_le_trans _ _ _ H21 (Rmin_r _ _)). +clear H15 H17 H18 H21. +rewrite formule; try assumption. +apply Rle_lt_trans with + (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) + + Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). +unfold Rminus in |- *. +rewrite <- + (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) + . +apply Rabs_4. +repeat rewrite Rabs_mult. +apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4). +cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4). +cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4). +cut + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) < + eps / 4). +cut + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) < + eps / 4). +intros. +apply Rlt_4; assumption. +rewrite H10. +unfold Rdiv in |- *; repeat rewrite Rmult_0_r || rewrite Rmult_0_l. +rewrite Rabs_R0; rewrite Rmult_0_l. +apply Rmult_lt_0_compat; [ assumption | apply Rinv_0_lt_compat; prove_sup ]. +rewrite <- Rabs_mult. +apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +right; symmetry in |- *; apply quadruple_var. +apply H2; assumption. +intros. +case (Req_dec a 0); intro. +rewrite H17; rewrite Rplus_0_r; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0. +apply Rabs_pos_lt. +unfold Rdiv in |- *; rewrite Rinv_mult_distr; try discrR || assumption. +unfold Rsqr in |- *. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H18; rewrite H18 in H6; elim (Rlt_irrefl _ H6)). +elim H11; intros. +apply H19. +split. +apply D_x_no_cond; assumption. +replace (x + a - x) with a; [ assumption | ring ]. +repeat apply Rmin_pos. +apply (cond_pos eps_f2). +elim H3; intros; assumption. +apply (cond_pos alp_f1d). +apply (cond_pos alp_f2d). +elim H11; intros; assumption. +apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; try discrR || assumption. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)). +change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))) in |- *. +apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; try discrR || assumption. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H12; rewrite H12 in H6; elim (Rlt_irrefl _ H6)). (***********************************) (* Cas n° 6 *) (* (f1 x)<>0 l1<>0 l2<>0 *) (***********************************) -Elim (H0 ``(Rabsolu ((Rsqr (f2 x))*eps)/(8*(f1 x)))``). -Intros alp_f2d H11. -Assert H12 := (derivable_continuous_pt ? ? X). -Unfold continuity_pt in H12. -Unfold continue_in in H12. -Unfold limit1_in in H12. -Unfold limit_in in H12. -Unfold dist in H12. -Simpl in H12. -Unfold R_dist in H12. -Elim (H12 ``(Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``). -Intros alp_f2c H13. -Elim (H12 ``(Rabsolu (eps*(Rsqr (f2 x)))/(8*l1))``). -Intros alp_f2t2 H14. -Cut ``0 < (Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) (Rmin alp_f2c alp_f2t2))``. -Intro. -Exists (mkposreal (Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) (Rmin alp_f2c alp_f2t2)) H15). -Simpl. -Intros. -Assert H18 := (Rlt_le_trans ? ? ? H17 (Rmin_l ? ?)). -Assert H19 := (Rlt_le_trans ? ? ? H17 (Rmin_r ? ?)). -Assert H20 := (Rlt_le_trans ? ? ? H18 (Rmin_l ? ?)). -Assert H21 := (Rlt_le_trans ? ? ? H18 (Rmin_r ? ?)). -Assert H22 := (Rlt_le_trans ? ? ? H19 (Rmin_l ? ?)). -Assert H23 := (Rlt_le_trans ? ? ? H19 (Rmin_r ? ?)). -Assert H24 := (Rlt_le_trans ? ? ? H20 (Rmin_l ? ?)). -Assert H25 := (Rlt_le_trans ? ? ? H20 (Rmin_r ? ?)). -Assert H26 := (Rlt_le_trans ? ? ? H21 (Rmin_l ? ?)). -Assert H27 := (Rlt_le_trans ? ? ? H21 (Rmin_r ? ?)). -Clear H17 H18 H19 H20 H21. -Cut (a:R) ``(Rabsolu a) < alp_f2t2`` -> ``(Rabsolu ((f2 (x+a))-(f2 x))) < (Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``. -Cut (a:R) ``(Rabsolu a) < alp_f2c`` -> ``(Rabsolu ((f2 (x+a))-(f2 x))) < (Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``. -Intros. -Rewrite formule; Try Assumption. -Apply Rle_lt_trans with ``(Rabsolu (/(f2 (x+h))*(((f1 (x+h))-(f1 x))/h-l1))) + (Rabsolu (l1/((f2 x)*(f2 (x+h)))*((f2 x)-(f2 (x+h))))) + (Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))-(f2 x))/h-l2))) + (Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))*((f2 (x+h))-(f2 x))))``. -Unfold Rminus. -Rewrite <- (Rabsolu_Ropp ``(f1 x)/((f2 x)*(f2 (x+h)))*(((f2 (x+h))+ -(f2 x))/h+ -l2)``). -Apply Rabsolu_4. -Repeat Rewrite Rabsolu_mult. -Apply Rlt_le_trans with ``eps/4+eps/4+eps/4+eps/4``. -Cut ``(Rabsolu (/(f2 (x+h))))*(Rabsolu (((f1 (x+h))-(f1 x))/h-l1)) < eps/4``. -Cut ``(Rabsolu (l1/((f2 x)*(f2 (x+h)))))*(Rabsolu ((f2 x)-(f2 (x+h)))) < eps/4``. -Cut ``(Rabsolu ((f1 x)/((f2 x)*(f2 (x+h)))))*(Rabsolu (((f2 (x+h))-(f2 x))/h-l2)) < eps/4``. -Cut ``(Rabsolu ((l2*(f1 x))/((Rsqr (f2 x))*(f2 (x+h)))))*(Rabsolu ((f2 (x+h))-(f2 x))) < eps/4``. -Intros. -Apply Rlt_4; Assumption. -Rewrite <- Rabsolu_mult. -Apply (maj_term4 x h eps l2 alp_f2 alp_f2c eps_f2 f1 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Rewrite <- Rabsolu_mult. -Apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Rewrite <- Rabsolu_mult. -Apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Rewrite <- Rabsolu_mult. -Apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); Try Assumption. -Apply H2; Assumption. -Apply Rmin_2; Assumption. -Right; Symmetry; Apply quadruple_var. -Apply H2; Assumption. -Intros. -Case (Req_EM a R0); Intro. -Rewrite H18; Rewrite Rplus_Or; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rabsolu_pos_lt. -Unfold Rdiv Rsqr; Rewrite Rinv_Rmult. -Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H28; Rewrite H28 in H6; Elim (Rlt_antirefl ? H6)). -Apply prod_neq_R0; [DiscrR | Assumption]. -Apply prod_neq_R0; [DiscrR | Assumption]. -Assumption. -Elim H13; Intros. -Apply H20. -Split. -Apply D_x_no_cond; Assumption. -Replace ``x+a-x`` with a; [Assumption | Ring]. -Intros. -Case (Req_EM a R0); Intro. -Rewrite H18; Rewrite Rplus_Or; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rabsolu_pos_lt. -Unfold Rdiv Rsqr; Rewrite Rinv_Rmult. -Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H28; Rewrite H28 in H6; Elim (Rlt_antirefl ? H6)). -DiscrR. -Assumption. -Elim H14; Intros. -Apply H20. -Split. -Unfold D_x no_cond; Split. -Trivial. -Apply Rminus_not_eq_right. -Replace ``x+a-x`` with a; [Assumption | Ring]. -Replace ``x+a-x`` with a; [Assumption | Ring]. -Repeat Apply Rmin_pos. -Apply (cond_pos eps_f2). -Elim H3; Intros; Assumption. -Apply (cond_pos alp_f1d). -Apply (cond_pos alp_f2d). -Elim H13; Intros; Assumption. -Elim H14; Intros; Assumption. -Change ``0 < (Rabsolu ((eps*(Rsqr (f2 x)))/(8*l1)))``; Apply Rabsolu_pos_lt. -Unfold Rdiv Rsqr; Rewrite Rinv_Rmult; Try DiscrR Orelse Assumption. -Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H14; Rewrite H14 in H6; Elim (Rlt_antirefl ? H6)). -Change ``0 < (Rabsolu (((Rsqr (f2 x))*(f2 x)*eps)/(8*(f1 x)*l2)))``; Apply Rabsolu_pos_lt. -Unfold Rdiv Rsqr; Rewrite Rinv_Rmult. -Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H13; Rewrite H13 in H6; Elim (Rlt_antirefl ? H6)). -Apply prod_neq_R0; [DiscrR | Assumption]. -Apply prod_neq_R0; [DiscrR | Assumption]. -Assumption. -Apply Rabsolu_pos_lt. -Unfold Rdiv Rsqr; Rewrite Rinv_Rmult; [Idtac | DiscrR | Assumption]. -Repeat Apply prod_neq_R0; Assumption Orelse (Apply Rinv_neq_R0; Assumption) Orelse (Apply Rinv_neq_R0; DiscrR) Orelse (Red; Intro H11; Rewrite H11 in H6; Elim (Rlt_antirefl ? H6)). -Intros. -Unfold Rdiv. -Apply Rlt_monotony_contra with ``(Rabsolu (f2 (x+a)))``. -Apply Rabsolu_pos_lt; Apply H2. -Apply Rlt_le_trans with (Rmin eps_f2 alp_f2). -Assumption. -Apply Rmin_l. -Rewrite <- Rinv_r_sym. -Apply Rlt_monotony_contra with (Rabsolu (f2 x)). -Apply Rabsolu_pos_lt; Assumption. -Rewrite Rmult_1r. -Rewrite (Rmult_sym (Rabsolu (f2 x))). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Apply Rlt_monotony_contra with ``/2``. -Apply Rlt_Rinv; Sup0. -Repeat Rewrite (Rmult_sym ``/2``). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r. -Unfold Rdiv in H5; Apply H5. -Replace ``x+a-x`` with a. -Assert H7 := (Rlt_le_trans ? ? ? H6 (Rmin_r ? ?)); Assumption. -Ring. -DiscrR. -Apply Rabsolu_no_R0; Assumption. -Apply Rabsolu_no_R0; Apply H2. -Assert H7 := (Rlt_le_trans ? ? ? H6 (Rmin_l ? ?)); Assumption. -Intros. -Assert H6 := (H4 a H5). -Rewrite <- (Rabsolu_Ropp ``(f2 a)-(f2 x)``) in H6. -Rewrite Ropp_distr2 in H6. -Assert H7 := (Rle_lt_trans ? ? ? (Rabsolu_triang_inv ? ?) H6). -Apply Rlt_anti_compatibility with ``-(Rabsolu (f2 a)) + (Rabsolu (f2 x))/2``. -Rewrite Rplus_assoc. -Rewrite <- double_var. -Do 2 Rewrite (Rplus_sym ``-(Rabsolu (f2 a))``). -Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or. -Unfold Rminus in H7; Assumption. -Intros. -Case (Req_EM x x0); Intro. -Rewrite <- H5; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Unfold Rdiv; Apply Rmult_lt_pos; [Apply Rabsolu_pos_lt; Assumption | Apply Rlt_Rinv; Sup0]. -Elim H3; Intros. -Apply H7. -Split. -Unfold D_x no_cond; Split. -Trivial. -Assumption. -Assumption. +elim (H0 (Rabs (Rsqr (f2 x) * eps / (8 * f1 x)))). +intros alp_f2d H11. +assert (H12 := derivable_continuous_pt _ _ X). +unfold continuity_pt in H12. +unfold continue_in in H12. +unfold limit1_in in H12. +unfold limit_in in H12. +unfold dist in H12. +simpl in H12. +unfold R_dist in H12. +elim (H12 (Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2)))). +intros alp_f2c H13. +elim (H12 (Rabs (eps * Rsqr (f2 x) / (8 * l1)))). +intros alp_f2t2 H14. +cut + (0 < + Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) + (Rmin alp_f2c alp_f2t2)). +intro. +exists + (mkposreal + (Rmin (Rmin (Rmin eps_f2 alp_f2) (Rmin alp_f1d alp_f2d)) + (Rmin alp_f2c alp_f2t2)) H15). +simpl in |- *. +intros. +assert (H18 := Rlt_le_trans _ _ _ H17 (Rmin_l _ _)). +assert (H19 := Rlt_le_trans _ _ _ H17 (Rmin_r _ _)). +assert (H20 := Rlt_le_trans _ _ _ H18 (Rmin_l _ _)). +assert (H21 := Rlt_le_trans _ _ _ H18 (Rmin_r _ _)). +assert (H22 := Rlt_le_trans _ _ _ H19 (Rmin_l _ _)). +assert (H23 := Rlt_le_trans _ _ _ H19 (Rmin_r _ _)). +assert (H24 := Rlt_le_trans _ _ _ H20 (Rmin_l _ _)). +assert (H25 := Rlt_le_trans _ _ _ H20 (Rmin_r _ _)). +assert (H26 := Rlt_le_trans _ _ _ H21 (Rmin_l _ _)). +assert (H27 := Rlt_le_trans _ _ _ H21 (Rmin_r _ _)). +clear H17 H18 H19 H20 H21. +cut + (forall a:R, + Rabs a < alp_f2t2 -> + Rabs (f2 (x + a) - f2 x) < Rabs (eps * Rsqr (f2 x) / (8 * l1))). +cut + (forall a:R, + Rabs a < alp_f2c -> + Rabs (f2 (x + a) - f2 x) < + Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))). +intros. +rewrite formule; try assumption. +apply Rle_lt_trans with + (Rabs (/ f2 (x + h) * ((f1 (x + h) - f1 x) / h - l1)) + + Rabs (l1 / (f2 x * f2 (x + h)) * (f2 x - f2 (x + h))) + + Rabs (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) - f2 x) / h - l2)) + + Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h)) * (f2 (x + h) - f2 x))). +unfold Rminus in |- *. +rewrite <- + (Rabs_Ropp (f1 x / (f2 x * f2 (x + h)) * ((f2 (x + h) + - f2 x) / h + - l2))) + . +apply Rabs_4. +repeat rewrite Rabs_mult. +apply Rlt_le_trans with (eps / 4 + eps / 4 + eps / 4 + eps / 4). +cut (Rabs (/ f2 (x + h)) * Rabs ((f1 (x + h) - f1 x) / h - l1) < eps / 4). +cut (Rabs (l1 / (f2 x * f2 (x + h))) * Rabs (f2 x - f2 (x + h)) < eps / 4). +cut + (Rabs (f1 x / (f2 x * f2 (x + h))) * Rabs ((f2 (x + h) - f2 x) / h - l2) < + eps / 4). +cut + (Rabs (l2 * f1 x / (Rsqr (f2 x) * f2 (x + h))) * Rabs (f2 (x + h) - f2 x) < + eps / 4). +intros. +apply Rlt_4; assumption. +rewrite <- Rabs_mult. +apply (maj_term4 x h eps l2 alp_f2 alp_f2c eps_f2 f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term3 x h eps l2 alp_f2 eps_f2 alp_f2d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term2 x h eps l1 alp_f2 alp_f2t2 eps_f2 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +rewrite <- Rabs_mult. +apply (maj_term1 x h eps l1 alp_f2 eps_f2 alp_f1d f1 f2); try assumption. +apply H2; assumption. +apply Rmin_2; assumption. +right; symmetry in |- *; apply quadruple_var. +apply H2; assumption. +intros. +case (Req_dec a 0); intro. +rewrite H18; rewrite Rplus_0_r; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)). +apply prod_neq_R0; [ discrR | assumption ]. +apply prod_neq_R0; [ discrR | assumption ]. +assumption. +elim H13; intros. +apply H20. +split. +apply D_x_no_cond; assumption. +replace (x + a - x) with a; [ assumption | ring ]. +intros. +case (Req_dec a 0); intro. +rewrite H18; rewrite Rplus_0_r; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H28; rewrite H28 in H6; elim (Rlt_irrefl _ H6)). +discrR. +assumption. +elim H14; intros. +apply H20. +split. +unfold D_x, no_cond in |- *; split. +trivial. +apply Rminus_not_eq_right. +replace (x + a - x) with a; [ assumption | ring ]. +replace (x + a - x) with a; [ assumption | ring ]. +repeat apply Rmin_pos. +apply (cond_pos eps_f2). +elim H3; intros; assumption. +apply (cond_pos alp_f1d). +apply (cond_pos alp_f2d). +elim H13; intros; assumption. +elim H14; intros; assumption. +change (0 < Rabs (eps * Rsqr (f2 x) / (8 * l1))) in |- *; apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; try discrR || assumption. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H14; rewrite H14 in H6; elim (Rlt_irrefl _ H6)). +change (0 < Rabs (Rsqr (f2 x) * f2 x * eps / (8 * f1 x * l2))) in |- *; + apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H13; rewrite H13 in H6; elim (Rlt_irrefl _ H6)). +apply prod_neq_R0; [ discrR | assumption ]. +apply prod_neq_R0; [ discrR | assumption ]. +assumption. +apply Rabs_pos_lt. +unfold Rdiv, Rsqr in |- *; rewrite Rinv_mult_distr; + [ idtac | discrR | assumption ]. +repeat apply prod_neq_R0; + assumption || + (apply Rinv_neq_0_compat; assumption) || + (apply Rinv_neq_0_compat; discrR) || + (red in |- *; intro H11; rewrite H11 in H6; elim (Rlt_irrefl _ H6)). +intros. +unfold Rdiv in |- *. +apply Rmult_lt_reg_l with (Rabs (f2 (x + a))). +apply Rabs_pos_lt; apply H2. +apply Rlt_le_trans with (Rmin eps_f2 alp_f2). +assumption. +apply Rmin_l. +rewrite <- Rinv_r_sym. +apply Rmult_lt_reg_l with (Rabs (f2 x)). +apply Rabs_pos_lt; assumption. +rewrite Rmult_1_r. +rewrite (Rmult_comm (Rabs (f2 x))). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +apply Rmult_lt_reg_l with (/ 2). +apply Rinv_0_lt_compat; prove_sup0. +repeat rewrite (Rmult_comm (/ 2)). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r. +unfold Rdiv in H5; apply H5. +replace (x + a - x) with a. +assert (H7 := Rlt_le_trans _ _ _ H6 (Rmin_r _ _)); assumption. +ring. +discrR. +apply Rabs_no_R0; assumption. +apply Rabs_no_R0; apply H2. +assert (H7 := Rlt_le_trans _ _ _ H6 (Rmin_l _ _)); assumption. +intros. +assert (H6 := H4 a H5). +rewrite <- (Rabs_Ropp (f2 a - f2 x)) in H6. +rewrite Ropp_minus_distr in H6. +assert (H7 := Rle_lt_trans _ _ _ (Rabs_triang_inv _ _) H6). +apply Rplus_lt_reg_r with (- Rabs (f2 a) + Rabs (f2 x) / 2). +rewrite Rplus_assoc. +rewrite <- double_var. +do 2 rewrite (Rplus_comm (- Rabs (f2 a))). +rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r. +unfold Rminus in H7; assumption. +intros. +case (Req_dec x x0); intro. +rewrite <- H5; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply Rabs_pos_lt; assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim H3; intros. +apply H7. +split. +unfold D_x, no_cond in |- *; split. +trivial. +assumption. +assumption. Qed. -Lemma derivable_pt_div : (f1,f2:R->R;x:R) (derivable_pt f1 x) -> (derivable_pt f2 x) -> ``(f2 x)<>0`` -> (derivable_pt (div_fct f1 f2) x). -Unfold derivable_pt. -Intros. -Elim X; Intros. -Elim X0; Intros. -Apply Specif.existT with ``(x0*(f2 x)-x1*(f1 x))/(Rsqr (f2 x))``. -Apply derivable_pt_lim_div; Assumption. +Lemma derivable_pt_div : + forall (f1 f2:R -> R) (x:R), + derivable_pt f1 x -> + derivable_pt f2 x -> f2 x <> 0 -> derivable_pt (f1 / f2) x. +unfold derivable_pt in |- *. +intros. +elim X; intros. +elim X0; intros. +apply existT with ((x0 * f2 x - x1 * f1 x) / Rsqr (f2 x)). +apply derivable_pt_lim_div; assumption. Qed. -Lemma derivable_div : (f1,f2:R->R) (derivable f1) -> (derivable f2) -> ((x:R)``(f2 x)<>0``) -> (derivable (div_fct f1 f2)). -Unfold derivable; Intros. -Apply (derivable_pt_div ? ? ? (X x) (X0 x) (H x)). +Lemma derivable_div : + forall f1 f2:R -> R, + derivable f1 -> + derivable f2 -> (forall x:R, f2 x <> 0) -> derivable (f1 / f2). +unfold derivable in |- *; intros. +apply (derivable_pt_div _ _ _ (X x) (X0 x) (H x)). Qed. -Lemma derive_pt_div : (f1,f2:R->R;x:R;pr1:(derivable_pt f1 x);pr2:(derivable_pt f2 x);na:``(f2 x)<>0``) ``(derive_pt (div_fct f1 f2) x (derivable_pt_div ? ? ? pr1 pr2 na)) == ((derive_pt f1 x pr1)*(f2 x)-(derive_pt f2 x pr2)*(f1 x))/(Rsqr (f2 x))``. -Intros. -Assert H := (derivable_derive f1 x pr1). -Assert H0 := (derivable_derive f2 x pr2). -Assert H1 := (derivable_derive (div_fct f1 f2) x (derivable_pt_div ? ? ? pr1 pr2 na)). -Elim H; Clear H; Intros l1 H. -Elim H0; Clear H0; Intros l2 H0. -Elim H1; Clear H1; Intros l H1. -Rewrite H; Rewrite H0; Apply derive_pt_eq_0. -Assert H3 := (projT2 ? ? pr1). -Unfold derive_pt in H; Rewrite H in H3. -Assert H4 := (projT2 ? ? pr2). -Unfold derive_pt in H0; Rewrite H0 in H4. -Apply derivable_pt_lim_div; Assumption. -Qed. +Lemma derive_pt_div : + forall (f1 f2:R -> R) (x:R) (pr1:derivable_pt f1 x) + (pr2:derivable_pt f2 x) (na:f2 x <> 0), + derive_pt (f1 / f2) x (derivable_pt_div _ _ _ pr1 pr2 na) = + (derive_pt f1 x pr1 * f2 x - derive_pt f2 x pr2 * f1 x) / Rsqr (f2 x). +intros. +assert (H := derivable_derive f1 x pr1). +assert (H0 := derivable_derive f2 x pr2). +assert + (H1 := derivable_derive (f1 / f2)%F x (derivable_pt_div _ _ _ pr1 pr2 na)). +elim H; clear H; intros l1 H. +elim H0; clear H0; intros l2 H0. +elim H1; clear H1; intros l H1. +rewrite H; rewrite H0; apply derive_pt_eq_0. +assert (H3 := projT2 pr1). +unfold derive_pt in H; rewrite H in H3. +assert (H4 := projT2 pr2). +unfold derive_pt in H0; rewrite H0 in H4. +apply derivable_pt_lim_div; assumption. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Ranalysis4.v b/theories/Reals/Ranalysis4.v index 6db2609a9..16d478fe4 100644 --- a/theories/Reals/Ranalysis4.v +++ b/theories/Reals/Ranalysis4.v @@ -8,306 +8,377 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo. -Require Ranalysis1. -Require Ranalysis3. -Require Exp_prop. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import Ranalysis1. +Require Import Ranalysis3. +Require Import Exp_prop. Open Local Scope R_scope. (**********) -Lemma derivable_pt_inv : (f:R->R;x:R) ``(f x)<>0`` -> (derivable_pt f x) -> (derivable_pt (inv_fct f) x). -Intros; Cut (derivable_pt (div_fct (fct_cte R1) f) x) -> (derivable_pt (inv_fct f) x). -Intro; Apply X0. -Apply derivable_pt_div. -Apply derivable_pt_const. -Assumption. -Assumption. -Unfold div_fct inv_fct fct_cte; Intro; Elim X0; Intros; Unfold derivable_pt; Apply Specif.existT with x0; Unfold derivable_pt_abs; Unfold derivable_pt_lim; Unfold derivable_pt_abs in p; Unfold derivable_pt_lim in p; Intros; Elim (p eps H0); Intros; Exists x1; Intros; Unfold Rdiv in H1; Unfold Rdiv; Rewrite <- (Rmult_1l ``/(f x)``); Rewrite <- (Rmult_1l ``/(f (x+h))``). -Apply H1; Assumption. +Lemma derivable_pt_inv : + forall (f:R -> R) (x:R), + f x <> 0 -> derivable_pt f x -> derivable_pt (/ f) x. +intros; cut (derivable_pt (fct_cte 1 / f) x -> derivable_pt (/ f) x). +intro; apply X0. +apply derivable_pt_div. +apply derivable_pt_const. +assumption. +assumption. +unfold div_fct, inv_fct, fct_cte in |- *; intro; elim X0; intros; + unfold derivable_pt in |- *; apply existT with x0; + unfold derivable_pt_abs in |- *; unfold derivable_pt_lim in |- *; + unfold derivable_pt_abs in p; unfold derivable_pt_lim in p; + intros; elim (p eps H0); intros; exists x1; intros; + unfold Rdiv in H1; unfold Rdiv in |- *; rewrite <- (Rmult_1_l (/ f x)); + rewrite <- (Rmult_1_l (/ f (x + h))). +apply H1; assumption. Qed. (**********) -Lemma pr_nu_var : (f,g:R->R;x:R;pr1:(derivable_pt f x);pr2:(derivable_pt g x)) f==g -> (derive_pt f x pr1) == (derive_pt g x pr2). -Unfold derivable_pt derive_pt; Intros. -Elim pr1; Intros. -Elim pr2; Intros. -Simpl. -Rewrite H in p. -Apply unicite_limite with g x; Assumption. +Lemma pr_nu_var : + forall (f g:R -> R) (x:R) (pr1:derivable_pt f x) (pr2:derivable_pt g x), + f = g -> derive_pt f x pr1 = derive_pt g x pr2. +unfold derivable_pt, derive_pt in |- *; intros. +elim pr1; intros. +elim pr2; intros. +simpl in |- *. +rewrite H in p. +apply uniqueness_limite with g x; assumption. Qed. (**********) -Lemma pr_nu_var2 : (f,g:R->R;x:R;pr1:(derivable_pt f x);pr2:(derivable_pt g x)) ((h:R)(f h)==(g h)) -> (derive_pt f x pr1) == (derive_pt g x pr2). -Unfold derivable_pt derive_pt; Intros. -Elim pr1; Intros. -Elim pr2; Intros. -Simpl. -Assert H0 := (unicite_step2 ? ? ? p). -Assert H1 := (unicite_step2 ? ? ? p0). -Cut (limit1_in [h:R]``((f (x+h))-(f x))/h`` [h:R]``h <> 0`` x1 ``0``). -Intro; Assert H3 := (unicite_step1 ? ? ? ? H0 H2). -Assumption. -Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Unfold limit1_in in H1; Unfold limit_in in H1; Unfold dist in H1; Simpl in H1; Unfold R_dist in H1. -Intros; Elim (H1 eps H2); Intros. -Elim H3; Intros. -Exists x2. -Split. -Assumption. -Intros; Do 2 Rewrite H; Apply H5; Assumption. +Lemma pr_nu_var2 : + forall (f g:R -> R) (x:R) (pr1:derivable_pt f x) (pr2:derivable_pt g x), + (forall h:R, f h = g h) -> derive_pt f x pr1 = derive_pt g x pr2. +unfold derivable_pt, derive_pt in |- *; intros. +elim pr1; intros. +elim pr2; intros. +simpl in |- *. +assert (H0 := uniqueness_step2 _ _ _ p). +assert (H1 := uniqueness_step2 _ _ _ p0). +cut (limit1_in (fun h:R => (f (x + h) - f x) / h) (fun h:R => h <> 0) x1 0). +intro; assert (H3 := uniqueness_step1 _ _ _ _ H0 H2). +assumption. +unfold limit1_in in |- *; unfold limit_in in |- *; unfold dist in |- *; + simpl in |- *; unfold R_dist in |- *; unfold limit1_in in H1; + unfold limit_in in H1; unfold dist in H1; simpl in H1; + unfold R_dist in H1. +intros; elim (H1 eps H2); intros. +elim H3; intros. +exists x2. +split. +assumption. +intros; do 2 rewrite H; apply H5; assumption. Qed. (**********) -Lemma derivable_inv : (f:R->R) ((x:R)``(f x)<>0``)->(derivable f)->(derivable (inv_fct f)). -Intros. -Unfold derivable; Intro. -Apply derivable_pt_inv. -Apply (H x). -Apply (X x). +Lemma derivable_inv : + forall f:R -> R, (forall x:R, f x <> 0) -> derivable f -> derivable (/ f). +intros. +unfold derivable in |- *; intro. +apply derivable_pt_inv. +apply (H x). +apply (X x). Qed. -Lemma derive_pt_inv : (f:R->R;x:R;pr:(derivable_pt f x);na:``(f x)<>0``) (derive_pt (inv_fct f) x (derivable_pt_inv f x na pr)) == ``-(derive_pt f x pr)/(Rsqr (f x))``. -Intros; Replace (derive_pt (inv_fct f) x (derivable_pt_inv f x na pr)) with (derive_pt (div_fct (fct_cte R1) f) x (derivable_pt_div (fct_cte R1) f x (derivable_pt_const R1 x) pr na)). -Rewrite derive_pt_div; Rewrite derive_pt_const; Unfold fct_cte; Rewrite Rmult_Ol; Rewrite Rmult_1r; Unfold Rminus; Rewrite Rplus_Ol; Reflexivity. -Apply pr_nu_var2. -Intro; Unfold div_fct fct_cte inv_fct. -Unfold Rdiv; Ring. +Lemma derive_pt_inv : + forall (f:R -> R) (x:R) (pr:derivable_pt f x) (na:f x <> 0), + derive_pt (/ f) x (derivable_pt_inv f x na pr) = + - derive_pt f x pr / Rsqr (f x). +intros; + replace (derive_pt (/ f) x (derivable_pt_inv f x na pr)) with + (derive_pt (fct_cte 1 / f) x + (derivable_pt_div (fct_cte 1) f x (derivable_pt_const 1 x) pr na)). +rewrite derive_pt_div; rewrite derive_pt_const; unfold fct_cte in |- *; + rewrite Rmult_0_l; rewrite Rmult_1_r; unfold Rminus in |- *; + rewrite Rplus_0_l; reflexivity. +apply pr_nu_var2. +intro; unfold div_fct, fct_cte, inv_fct in |- *. +unfold Rdiv in |- *; ring. Qed. (* Rabsolu *) -Lemma Rabsolu_derive_1 : (x:R) ``0<x`` -> (derivable_pt_lim Rabsolu x ``1``). -Intros. -Unfold derivable_pt_lim; Intros. -Exists (mkposreal x H); Intros. -Rewrite (Rabsolu_right x). -Rewrite (Rabsolu_right ``x+h``). -Rewrite Rplus_sym. -Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r. -Rewrite Rplus_Or; Unfold Rdiv; Rewrite <- Rinv_r_sym. -Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply H0. -Apply H1. -Apply Rle_sym1. -Case (case_Rabsolu h); Intro. -Rewrite (Rabsolu_left h r) in H2. -Left; Rewrite Rplus_sym; Apply Rlt_anti_compatibility with ``-h``; Rewrite Rplus_Or; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Apply H2. -Apply ge0_plus_ge0_is_ge0. -Left; Apply H. -Apply Rle_sym2; Apply r. -Left; Apply H. +Lemma Rabs_derive_1 : forall x:R, 0 < x -> derivable_pt_lim Rabs x 1. +intros. +unfold derivable_pt_lim in |- *; intros. +exists (mkposreal x H); intros. +rewrite (Rabs_right x). +rewrite (Rabs_right (x + h)). +rewrite Rplus_comm. +unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_r. +rewrite Rplus_0_r; unfold Rdiv in |- *; rewrite <- Rinv_r_sym. +rewrite Rplus_opp_r; rewrite Rabs_R0; apply H0. +apply H1. +apply Rle_ge. +case (Rcase_abs h); intro. +rewrite (Rabs_left h r) in H2. +left; rewrite Rplus_comm; apply Rplus_lt_reg_r with (- h); rewrite Rplus_0_r; + rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; + apply H2. +apply Rplus_le_le_0_compat. +left; apply H. +apply Rge_le; apply r. +left; apply H. Qed. -Lemma Rabsolu_derive_2 : (x:R) ``x<0`` -> (derivable_pt_lim Rabsolu x ``-1``). -Intros. -Unfold derivable_pt_lim; Intros. -Cut ``0< -x``. -Intro; Exists (mkposreal ``-x`` H1); Intros. -Rewrite (Rabsolu_left x). -Rewrite (Rabsolu_left ``x+h``). -Rewrite Rplus_sym. -Rewrite Ropp_distr1. -Unfold Rminus; Rewrite Ropp_Ropp; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l. -Rewrite Rplus_Or; Unfold Rdiv. -Rewrite Ropp_mul1. -Rewrite <- Rinv_r_sym. -Rewrite Ropp_Ropp; Rewrite Rplus_Ropp_l; Rewrite Rabsolu_R0; Apply H0. -Apply H2. -Case (case_Rabsolu h); Intro. -Apply Ropp_Rlt. -Rewrite Ropp_O; Rewrite Ropp_distr1; Apply gt0_plus_gt0_is_gt0. -Apply H1. -Apply Rgt_RO_Ropp; Apply r. -Rewrite (Rabsolu_right h r) in H3. -Apply Rlt_anti_compatibility with ``-x``; Rewrite Rplus_Or; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Apply H3. -Apply H. -Apply Rgt_RO_Ropp; Apply H. +Lemma Rabs_derive_2 : forall x:R, x < 0 -> derivable_pt_lim Rabs x (-1). +intros. +unfold derivable_pt_lim in |- *; intros. +cut (0 < - x). +intro; exists (mkposreal (- x) H1); intros. +rewrite (Rabs_left x). +rewrite (Rabs_left (x + h)). +rewrite Rplus_comm. +rewrite Ropp_plus_distr. +unfold Rminus in |- *; rewrite Ropp_involutive; rewrite Rplus_assoc; + rewrite Rplus_opp_l. +rewrite Rplus_0_r; unfold Rdiv in |- *. +rewrite Ropp_mult_distr_l_reverse. +rewrite <- Rinv_r_sym. +rewrite Ropp_involutive; rewrite Rplus_opp_l; rewrite Rabs_R0; apply H0. +apply H2. +case (Rcase_abs h); intro. +apply Ropp_lt_cancel. +rewrite Ropp_0; rewrite Ropp_plus_distr; apply Rplus_lt_0_compat. +apply H1. +apply Ropp_0_gt_lt_contravar; apply r. +rewrite (Rabs_right h r) in H3. +apply Rplus_lt_reg_r with (- x); rewrite Rplus_0_r; rewrite <- Rplus_assoc; + rewrite Rplus_opp_l; rewrite Rplus_0_l; apply H3. +apply H. +apply Ropp_0_gt_lt_contravar; apply H. Qed. (* Rabsolu is derivable for all x <> 0 *) -Lemma derivable_pt_Rabsolu : (x:R) ``x<>0`` -> (derivable_pt Rabsolu x). -Intros. -Case (total_order_T x R0); Intro. -Elim s; Intro. -Unfold derivable_pt; Apply Specif.existT with ``-1``. -Apply (Rabsolu_derive_2 x a). -Elim H; Exact b. -Unfold derivable_pt; Apply Specif.existT with ``1``. -Apply (Rabsolu_derive_1 x r). +Lemma Rderivable_pt_abs : forall x:R, x <> 0 -> derivable_pt Rabs x. +intros. +case (total_order_T x 0); intro. +elim s; intro. +unfold derivable_pt in |- *; apply existT with (-1). +apply (Rabs_derive_2 x a). +elim H; exact b. +unfold derivable_pt in |- *; apply existT with 1. +apply (Rabs_derive_1 x r). Qed. (* Rabsolu is continuous for all x *) -Lemma continuity_Rabsolu : (continuity Rabsolu). -Unfold continuity; Intro. -Case (Req_EM x R0); Intro. -Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Exists eps; Split. -Apply H0. -Intros; Rewrite H; Rewrite Rabsolu_R0; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Elim H1; Intros; Rewrite H in H3; Unfold Rminus in H3; Rewrite Ropp_O in H3; Rewrite Rplus_Or in H3; Apply H3. -Apply derivable_continuous_pt; Apply (derivable_pt_Rabsolu x H). +Lemma Rcontinuity_abs : continuity Rabs. +unfold continuity in |- *; intro. +case (Req_dec x 0); intro. +unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros; exists eps; + split. +apply H0. +intros; rewrite H; rewrite Rabs_R0; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; rewrite Rabs_Rabsolu; elim H1; + intros; rewrite H in H3; unfold Rminus in H3; rewrite Ropp_0 in H3; + rewrite Rplus_0_r in H3; apply H3. +apply derivable_continuous_pt; apply (Rderivable_pt_abs x H). Qed. (* Finite sums : Sum a_k x^k *) -Lemma continuity_finite_sum : (An:nat->R;N:nat) (continuity [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N)). -Intros; Unfold continuity; Intro. -Induction N. -Simpl. -Apply continuity_pt_const. -Unfold constant; Intros; Reflexivity. -Replace [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` (S N)) with (plus_fct [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) [y:R]``(An (S N))*(pow y (S N))``). -Apply continuity_pt_plus. -Apply HrecN. -Replace [y:R]``(An (S N))*(pow y (S N))`` with (mult_real_fct (An (S N)) [y:R](pow y (S N))). -Apply continuity_pt_scal. -Apply derivable_continuous_pt. -Apply derivable_pt_pow. -Reflexivity. -Reflexivity. +Lemma continuity_finite_sum : + forall (An:nat -> R) (N:nat), + continuity (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N). +intros; unfold continuity in |- *; intro. +induction N as [| N HrecN]. +simpl in |- *. +apply continuity_pt_const. +unfold constant in |- *; intros; reflexivity. +replace (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) (S N)) with + ((fun y:R => sum_f_R0 (fun k:nat => (An k * y ^ k)%R) N) + + (fun y:R => (An (S N) * y ^ S N)%R))%F. +apply continuity_pt_plus. +apply HrecN. +replace (fun y:R => An (S N) * y ^ S N) with + (mult_real_fct (An (S N)) (fun y:R => y ^ S N)). +apply continuity_pt_scal. +apply derivable_continuous_pt. +apply derivable_pt_pow. +reflexivity. +reflexivity. Qed. -Lemma derivable_pt_lim_fs : (An:nat->R;x:R;N:nat) (lt O N) -> (derivable_pt_lim [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) x (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred N))). -Intros; Induction N. -Elim (lt_n_n ? H). -Cut N=O\/(lt O N). -Intro; Elim H0; Intro. -Rewrite H1. -Simpl. -Replace [y:R]``(An O)*1+(An (S O))*(y*1)`` with (plus_fct (fct_cte ``(An O)*1``) (mult_real_fct ``(An (S O))`` (mult_fct id (fct_cte R1)))). -Replace ``1*(An (S O))*1`` with ``0+(An (S O))*(1*(fct_cte R1 x)+(id x)*0)``. -Apply derivable_pt_lim_plus. -Apply derivable_pt_lim_const. -Apply derivable_pt_lim_scal. -Apply derivable_pt_lim_mult. -Apply derivable_pt_lim_id. -Apply derivable_pt_lim_const. -Unfold fct_cte id; Ring. -Reflexivity. -Replace [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` (S N)) with (plus_fct [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) [y:R]``(An (S N))*(pow y (S N))``). -Replace (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred (S N))) with (Rplus (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred N)) ``(An (S N))*((INR (S (pred (S N))))*(pow x (pred (S N))))``). -Apply derivable_pt_lim_plus. -Apply HrecN. -Assumption. -Replace [y:R]``(An (S N))*(pow y (S N))`` with (mult_real_fct (An (S N)) [y:R](pow y (S N))). -Apply derivable_pt_lim_scal. -Replace (pred (S N)) with N; [Idtac | Reflexivity]. -Pattern 3 N; Replace N with (pred (S N)). -Apply derivable_pt_lim_pow. -Reflexivity. -Reflexivity. -Cut (pred (S N)) = (S (pred N)). -Intro; Rewrite H2. -Rewrite tech5. -Apply Rplus_plus_r. -Rewrite <- H2. -Replace (pred (S N)) with N; [Idtac | Reflexivity]. -Ring. -Simpl. -Apply S_pred with O; Assumption. -Unfold plus_fct. -Simpl; Reflexivity. -Inversion H. -Left; Reflexivity. -Right; Apply lt_le_trans with (1); [Apply lt_O_Sn | Assumption]. +Lemma derivable_pt_lim_fs : + forall (An:nat -> R) (x:R) (N:nat), + (0 < N)%nat -> + derivable_pt_lim (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N) x + (sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred N)). +intros; induction N as [| N HrecN]. +elim (lt_irrefl _ H). +cut (N = 0%nat \/ (0 < N)%nat). +intro; elim H0; intro. +rewrite H1. +simpl in |- *. +replace (fun y:R => An 0%nat * 1 + An 1%nat * (y * 1)) with + (fct_cte (An 0%nat * 1) + mult_real_fct (An 1%nat) (id * fct_cte 1))%F. +replace (1 * An 1%nat * 1) with (0 + An 1%nat * (1 * fct_cte 1 x + id x * 0)). +apply derivable_pt_lim_plus. +apply derivable_pt_lim_const. +apply derivable_pt_lim_scal. +apply derivable_pt_lim_mult. +apply derivable_pt_lim_id. +apply derivable_pt_lim_const. +unfold fct_cte, id in |- *; ring. +reflexivity. +replace (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) (S N)) with + ((fun y:R => sum_f_R0 (fun k:nat => (An k * y ^ k)%R) N) + + (fun y:R => (An (S N) * y ^ S N)%R))%F. +replace (sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred (S N))) + with + (sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred N) + + An (S N) * (INR (S (pred (S N))) * x ^ pred (S N))). +apply derivable_pt_lim_plus. +apply HrecN. +assumption. +replace (fun y:R => An (S N) * y ^ S N) with + (mult_real_fct (An (S N)) (fun y:R => y ^ S N)). +apply derivable_pt_lim_scal. +replace (pred (S N)) with N; [ idtac | reflexivity ]. +pattern N at 3 in |- *; replace N with (pred (S N)). +apply derivable_pt_lim_pow. +reflexivity. +reflexivity. +cut (pred (S N) = S (pred N)). +intro; rewrite H2. +rewrite tech5. +apply Rplus_eq_compat_l. +rewrite <- H2. +replace (pred (S N)) with N; [ idtac | reflexivity ]. +ring. +simpl in |- *. +apply S_pred with 0%nat; assumption. +unfold plus_fct in |- *. +simpl in |- *; reflexivity. +inversion H. +left; reflexivity. +right; apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ]. Qed. -Lemma derivable_pt_lim_finite_sum : (An:(nat->R); x:R; N:nat) (derivable_pt_lim [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) x (Cases N of O => R0 | _ => (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred N)) end)). -Intros. -Induction N. -Simpl. -Rewrite Rmult_1r. -Replace [_:R]``(An O)`` with (fct_cte (An O)); [Apply derivable_pt_lim_const | Reflexivity]. -Apply derivable_pt_lim_fs; Apply lt_O_Sn. +Lemma derivable_pt_lim_finite_sum : + forall (An:nat -> R) (x:R) (N:nat), + derivable_pt_lim (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N) x + match N with + | O => 0 + | _ => sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred N) + end. +intros. +induction N as [| N HrecN]. +simpl in |- *. +rewrite Rmult_1_r. +replace (fun _:R => An 0%nat) with (fct_cte (An 0%nat)); + [ apply derivable_pt_lim_const | reflexivity ]. +apply derivable_pt_lim_fs; apply lt_O_Sn. Qed. -Lemma derivable_pt_finite_sum : (An:nat->R;N:nat;x:R) (derivable_pt [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N) x). -Intros. -Unfold derivable_pt. -Assert H := (derivable_pt_lim_finite_sum An x N). -Induction N. -Apply Specif.existT with R0; Apply H. -Apply Specif.existT with (sum_f_R0 [k:nat]``(INR (S k))*(An (S k))*(pow x k)`` (pred (S N))); Apply H. +Lemma derivable_pt_finite_sum : + forall (An:nat -> R) (N:nat) (x:R), + derivable_pt (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N) x. +intros. +unfold derivable_pt in |- *. +assert (H := derivable_pt_lim_finite_sum An x N). +induction N as [| N HrecN]. +apply existT with 0; apply H. +apply existT with + (sum_f_R0 (fun k:nat => INR (S k) * An (S k) * x ^ k) (pred (S N))); + apply H. Qed. -Lemma derivable_finite_sum : (An:nat->R;N:nat) (derivable [y:R](sum_f_R0 [k:nat]``(An k)*(pow y k)`` N)). -Intros; Unfold derivable; Intro; Apply derivable_pt_finite_sum. +Lemma derivable_finite_sum : + forall (An:nat -> R) (N:nat), + derivable (fun y:R => sum_f_R0 (fun k:nat => An k * y ^ k) N). +intros; unfold derivable in |- *; intro; apply derivable_pt_finite_sum. Qed. (* Regularity of hyperbolic functions *) -Lemma derivable_pt_lim_cosh : (x:R) (derivable_pt_lim cosh x ``(sinh x)``). -Intro. -Unfold cosh sinh; Unfold Rdiv. -Replace [x0:R]``((exp x0)+(exp ( -x0)))*/2`` with (mult_fct (plus_fct exp (comp exp (opp_fct id))) (fct_cte ``/2``)); [Idtac | Reflexivity]. -Replace ``((exp x)-(exp ( -x)))*/2`` with ``((exp x)+((exp (-x))*-1))*((fct_cte (Rinv 2)) x)+((plus_fct exp (comp exp (opp_fct id))) x)*0``. -Apply derivable_pt_lim_mult. -Apply derivable_pt_lim_plus. -Apply derivable_pt_lim_exp. -Apply derivable_pt_lim_comp. -Apply derivable_pt_lim_opp. -Apply derivable_pt_lim_id. -Apply derivable_pt_lim_exp. -Apply derivable_pt_lim_const. -Unfold plus_fct mult_real_fct comp opp_fct id fct_cte; Ring. +Lemma derivable_pt_lim_cosh : forall x:R, derivable_pt_lim cosh x (sinh x). +intro. +unfold cosh, sinh in |- *; unfold Rdiv in |- *. +replace (fun x0:R => (exp x0 + exp (- x0)) * / 2) with + ((exp + comp exp (- id)) * fct_cte (/ 2))%F; [ idtac | reflexivity ]. +replace ((exp x - exp (- x)) * / 2) with + ((exp x + exp (- x) * -1) * fct_cte (/ 2) x + + (exp + comp exp (- id))%F x * 0). +apply derivable_pt_lim_mult. +apply derivable_pt_lim_plus. +apply derivable_pt_lim_exp. +apply derivable_pt_lim_comp. +apply derivable_pt_lim_opp. +apply derivable_pt_lim_id. +apply derivable_pt_lim_exp. +apply derivable_pt_lim_const. +unfold plus_fct, mult_real_fct, comp, opp_fct, id, fct_cte in |- *; ring. Qed. -Lemma derivable_pt_lim_sinh : (x:R) (derivable_pt_lim sinh x ``(cosh x)``). -Intro. -Unfold cosh sinh; Unfold Rdiv. -Replace [x0:R]``((exp x0)-(exp ( -x0)))*/2`` with (mult_fct (minus_fct exp (comp exp (opp_fct id))) (fct_cte ``/2``)); [Idtac | Reflexivity]. -Replace ``((exp x)+(exp ( -x)))*/2`` with ``((exp x)-((exp (-x))*-1))*((fct_cte (Rinv 2)) x)+((minus_fct exp (comp exp (opp_fct id))) x)*0``. -Apply derivable_pt_lim_mult. -Apply derivable_pt_lim_minus. -Apply derivable_pt_lim_exp. -Apply derivable_pt_lim_comp. -Apply derivable_pt_lim_opp. -Apply derivable_pt_lim_id. -Apply derivable_pt_lim_exp. -Apply derivable_pt_lim_const. -Unfold plus_fct mult_real_fct comp opp_fct id fct_cte; Ring. +Lemma derivable_pt_lim_sinh : forall x:R, derivable_pt_lim sinh x (cosh x). +intro. +unfold cosh, sinh in |- *; unfold Rdiv in |- *. +replace (fun x0:R => (exp x0 - exp (- x0)) * / 2) with + ((exp - comp exp (- id)) * fct_cte (/ 2))%F; [ idtac | reflexivity ]. +replace ((exp x + exp (- x)) * / 2) with + ((exp x - exp (- x) * -1) * fct_cte (/ 2) x + + (exp - comp exp (- id))%F x * 0). +apply derivable_pt_lim_mult. +apply derivable_pt_lim_minus. +apply derivable_pt_lim_exp. +apply derivable_pt_lim_comp. +apply derivable_pt_lim_opp. +apply derivable_pt_lim_id. +apply derivable_pt_lim_exp. +apply derivable_pt_lim_const. +unfold plus_fct, mult_real_fct, comp, opp_fct, id, fct_cte in |- *; ring. Qed. -Lemma derivable_pt_exp : (x:R) (derivable_pt exp x). -Intro. -Unfold derivable_pt. -Apply Specif.existT with (exp x). -Apply derivable_pt_lim_exp. +Lemma derivable_pt_exp : forall x:R, derivable_pt exp x. +intro. +unfold derivable_pt in |- *. +apply existT with (exp x). +apply derivable_pt_lim_exp. Qed. -Lemma derivable_pt_cosh : (x:R) (derivable_pt cosh x). -Intro. -Unfold derivable_pt. -Apply Specif.existT with (sinh x). -Apply derivable_pt_lim_cosh. +Lemma derivable_pt_cosh : forall x:R, derivable_pt cosh x. +intro. +unfold derivable_pt in |- *. +apply existT with (sinh x). +apply derivable_pt_lim_cosh. Qed. -Lemma derivable_pt_sinh : (x:R) (derivable_pt sinh x). -Intro. -Unfold derivable_pt. -Apply Specif.existT with (cosh x). -Apply derivable_pt_lim_sinh. +Lemma derivable_pt_sinh : forall x:R, derivable_pt sinh x. +intro. +unfold derivable_pt in |- *. +apply existT with (cosh x). +apply derivable_pt_lim_sinh. Qed. -Lemma derivable_exp : (derivable exp). -Unfold derivable; Apply derivable_pt_exp. +Lemma derivable_exp : derivable exp. +unfold derivable in |- *; apply derivable_pt_exp. Qed. -Lemma derivable_cosh : (derivable cosh). -Unfold derivable; Apply derivable_pt_cosh. +Lemma derivable_cosh : derivable cosh. +unfold derivable in |- *; apply derivable_pt_cosh. Qed. -Lemma derivable_sinh : (derivable sinh). -Unfold derivable; Apply derivable_pt_sinh. +Lemma derivable_sinh : derivable sinh. +unfold derivable in |- *; apply derivable_pt_sinh. Qed. -Lemma derive_pt_exp : (x:R) (derive_pt exp x (derivable_pt_exp x))==(exp x). -Intro; Apply derive_pt_eq_0. -Apply derivable_pt_lim_exp. +Lemma derive_pt_exp : + forall x:R, derive_pt exp x (derivable_pt_exp x) = exp x. +intro; apply derive_pt_eq_0. +apply derivable_pt_lim_exp. Qed. -Lemma derive_pt_cosh : (x:R) (derive_pt cosh x (derivable_pt_cosh x))==(sinh x). -Intro; Apply derive_pt_eq_0. -Apply derivable_pt_lim_cosh. +Lemma derive_pt_cosh : + forall x:R, derive_pt cosh x (derivable_pt_cosh x) = sinh x. +intro; apply derive_pt_eq_0. +apply derivable_pt_lim_cosh. Qed. -Lemma derive_pt_sinh : (x:R) (derive_pt sinh x (derivable_pt_sinh x))==(cosh x). -Intro; Apply derive_pt_eq_0. -Apply derivable_pt_lim_sinh. -Qed. +Lemma derive_pt_sinh : + forall x:R, derive_pt sinh x (derivable_pt_sinh x) = cosh x. +intro; apply derive_pt_eq_0. +apply derivable_pt_lim_sinh. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Raxioms.v b/theories/Reals/Raxioms.v index 4516a206f..a047c78c0 100644 --- a/theories/Reals/Raxioms.v +++ b/theories/Reals/Raxioms.v @@ -13,23 +13,8 @@ (*********************************************************) Require Export ZArith_base. -Require Export Rsyntax. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Export Rsyntax. Open Local Scope R_scope. -V7only [ -(*********************************************************) -(* Compatibility *) -(*********************************************************) -Notation sumboolT := Specif.sumbool. -Notation leftT := Specif.left. -Notation rightT := Specif.right. -Notation sumorT := Specif.sumor. -Notation inleftT := Specif.inleft. -Notation inrightT := Specif.inright. -Notation sigTT := Specif.sigT. -Notation existTT := Specif.existT. -Notation SigT := Specif.sigT. -]. (*********************************************************) (* Field axioms *) @@ -40,52 +25,53 @@ Notation SigT := Specif.sigT. (*********************************************************) (**********) -Axiom Rplus_sym:(r1,r2:R)``r1+r2==r2+r1``. -Hints Resolve Rplus_sym : real. +Axiom Rplus_comm : forall r1 r2:R, r1 + r2 = r2 + r1. +Hint Resolve Rplus_comm: real. (**********) -Axiom Rplus_assoc:(r1,r2,r3:R)``(r1+r2)+r3==r1+(r2+r3)``. -Hints Resolve Rplus_assoc : real. +Axiom Rplus_assoc : forall r1 r2 r3:R, r1 + r2 + r3 = r1 + (r2 + r3). +Hint Resolve Rplus_assoc: real. (**********) -Axiom Rplus_Ropp_r:(r:R)``r+(-r)==0``. -Hints Resolve Rplus_Ropp_r : real v62. +Axiom Rplus_opp_r : forall r:R, r + - r = 0. +Hint Resolve Rplus_opp_r: real v62. (**********) -Axiom Rplus_Ol:(r:R)``0+r==r``. -Hints Resolve Rplus_Ol : real. +Axiom Rplus_0_l : forall r:R, 0 + r = r. +Hint Resolve Rplus_0_l: real. (***********************************************************) (** Multiplication *) (***********************************************************) (**********) -Axiom Rmult_sym:(r1,r2:R)``r1*r2==r2*r1``. -Hints Resolve Rmult_sym : real v62. +Axiom Rmult_comm : forall r1 r2:R, r1 * r2 = r2 * r1. +Hint Resolve Rmult_comm: real v62. (**********) -Axiom Rmult_assoc:(r1,r2,r3:R)``(r1*r2)*r3==r1*(r2*r3)``. -Hints Resolve Rmult_assoc : real v62. +Axiom Rmult_assoc : forall r1 r2 r3:R, r1 * r2 * r3 = r1 * (r2 * r3). +Hint Resolve Rmult_assoc: real v62. (**********) -Axiom Rinv_l:(r:R)``r<>0``->``(/r)*r==1``. -Hints Resolve Rinv_l : real. +Axiom Rinv_l : forall r:R, r <> 0 -> / r * r = 1. +Hint Resolve Rinv_l: real. (**********) -Axiom Rmult_1l:(r:R)``1*r==r``. -Hints Resolve Rmult_1l : real. +Axiom Rmult_1_l : forall r:R, 1 * r = r. +Hint Resolve Rmult_1_l: real. (**********) -Axiom R1_neq_R0:``1<>0``. -Hints Resolve R1_neq_R0 : real. +Axiom R1_neq_R0 : 1 <> 0. +Hint Resolve R1_neq_R0: real. (*********************************************************) (** Distributivity *) (*********************************************************) (**********) -Axiom Rmult_Rplus_distr:(r1,r2,r3:R)``r1*(r2+r3)==(r1*r2)+(r1*r3)``. -Hints Resolve Rmult_Rplus_distr : real v62. +Axiom + Rmult_plus_distr_l : forall r1 r2 r3:R, r1 * (r2 + r3) = r1 * r2 + r1 * r3. +Hint Resolve Rmult_plus_distr_l: real v62. (*********************************************************) (** Order axioms *) @@ -95,37 +81,38 @@ Hints Resolve Rmult_Rplus_distr : real v62. (*********************************************************) (**********) -Axiom total_order_T:(r1,r2:R)(sumorT (sumboolT ``r1<r2`` r1==r2) ``r1>r2``). +Axiom total_order_T : forall r1 r2:R, {r1 < r2} + {r1 = r2} + {r1 > r2}. (*********************************************************) (** Lower *) (*********************************************************) (**********) -Axiom Rlt_antisym:(r1,r2:R)``r1<r2`` -> ~ ``r2<r1``. +Axiom Rlt_asym : forall r1 r2:R, r1 < r2 -> ~ r2 < r1. (**********) -Axiom Rlt_trans:(r1,r2,r3:R) - ``r1<r2``->``r2<r3``->``r1<r3``. +Axiom Rlt_trans : forall r1 r2 r3:R, r1 < r2 -> r2 < r3 -> r1 < r3. (**********) -Axiom Rlt_compatibility:(r,r1,r2:R)``r1<r2``->``r+r1<r+r2``. +Axiom Rplus_lt_compat_l : forall r r1 r2:R, r1 < r2 -> r + r1 < r + r2. (**********) -Axiom Rlt_monotony:(r,r1,r2:R)``0<r``->``r1<r2``->``r*r1<r*r2``. +Axiom + Rmult_lt_compat_l : forall r r1 r2:R, 0 < r -> r1 < r2 -> r * r1 < r * r2. -Hints Resolve Rlt_antisym Rlt_compatibility Rlt_monotony : real. +Hint Resolve Rlt_asym Rplus_lt_compat_l Rmult_lt_compat_l: real. (**********************************************************) (** Injection from N to R *) (**********************************************************) (**********) -Fixpoint INR [n:nat]:R:=(Cases n of - O => ``0`` - |(S O) => ``1`` - |(S n) => ``(INR n)+1`` - end). +Fixpoint INR (n:nat) : R := + match n with + | O => 0 + | S O => 1 + | S n => INR n + 1 + end. Arguments Scope INR [nat_scope]. @@ -134,11 +121,12 @@ Arguments Scope INR [nat_scope]. (**********************************************************) (**********) -Definition IZR:Z->R:=[z:Z](Cases z of - ZERO => ``0`` - |(POS n) => (INR (convert n)) - |(NEG n) => ``-(INR (convert n))`` - end). +Definition IZR (z:Z) : R := + match z with + | Z0 => 0 + | Zpos n => INR (nat_of_P n) + | Zneg n => - INR (nat_of_P n) + end. Arguments Scope IZR [Z_scope]. (**********************************************************) @@ -146,24 +134,24 @@ Arguments Scope IZR [Z_scope]. (**********************************************************) (**********) -Axiom archimed:(r:R)``(IZR (up r)) > r``/\``(IZR (up r))-r <= 1``. +Axiom archimed : forall r:R, IZR (up r) > r /\ IZR (up r) - r <= 1. (**********************************************************) (** [R] Complete *) (**********************************************************) (**********) -Definition is_upper_bound:=[E:R->Prop][m:R](x:R)(E x)->``x <= m``. +Definition is_upper_bound (E:R -> Prop) (m:R) := forall x:R, E x -> x <= m. (**********) -Definition bound:=[E:R->Prop](ExT [m:R](is_upper_bound E m)). +Definition bound (E:R -> Prop) := exists m : R | is_upper_bound E m. (**********) -Definition is_lub:=[E:R->Prop][m:R] - (is_upper_bound E m)/\(b:R)(is_upper_bound E b)->``m <= b``. +Definition is_lub (E:R -> Prop) (m:R) := + is_upper_bound E m /\ (forall b:R, is_upper_bound E b -> m <= b). (**********) -Axiom complet:(E:R->Prop)(bound E)-> - (ExT [x:R] (E x))-> - (sigTT R [m:R](is_lub E m)). - +Axiom + completeness : + forall E:R -> Prop, + bound E -> ( exists x : R | E x) -> sigT (fun m:R => is_lub E m). diff --git a/theories/Reals/Rbase.v b/theories/Reals/Rbase.v index 1df44bbf5..f1e17e305 100644 --- a/theories/Reals/Rbase.v +++ b/theories/Reals/Rbase.v @@ -11,4 +11,4 @@ Require Export Rdefinitions. Require Export Raxioms. Require Export RIneq. -Require Export DiscrR. +Require Export DiscrR.
\ No newline at end of file diff --git a/theories/Reals/Rbasic_fun.v b/theories/Reals/Rbasic_fun.v index c586acdca..d5b090677 100644 --- a/theories/Reals/Rbasic_fun.v +++ b/theories/Reals/Rbasic_fun.v @@ -13,69 +13,68 @@ (* *) (*********************************************************) -Require Rbase. -Require R_Ifp. -Require Fourier. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import R_Ifp. +Require Import Fourier. Open Local Scope R_scope. -Implicit Variable Type r:R. +Implicit Type r : R. (*******************************) (** Rmin *) (*******************************) (*********) -Definition Rmin :R->R->R:=[x,y:R] - Cases (total_order_Rle x y) of - (leftT _) => x - | (rightT _) => y +Definition Rmin (x y:R) : R := + match Rle_dec x y with + | left _ => x + | right _ => 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. +Lemma Rmin_Rgt_l : forall r1 r2 r, Rmin r1 r2 > r -> r1 > r /\ r2 > r. +intros r1 r2 r; unfold Rmin in |- *; case (Rle_dec r1 r2); intros. +split. +assumption. +unfold Rgt in |- *; unfold Rgt in H; exact (Rlt_le_trans r r1 r2 H r0). +split. +generalize (Rnot_le_lt 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. +Lemma Rmin_Rgt_r : forall r1 r2 r, r1 > r /\ r2 > r -> Rmin r1 r2 > r. +intros; unfold Rmin in |- *; case (Rle_dec 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). +Lemma Rmin_Rgt : forall r1 r2 r, Rmin r1 r2 > r <-> r1 > r /\ 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]. +Lemma Rmin_l : forall x y:R, Rmin x y <= x. +intros; unfold Rmin in |- *; case (Rle_dec 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]. +Lemma Rmin_r : forall x y:R, Rmin x y <= y. +intros; unfold Rmin in |- *; case (Rle_dec 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). +Lemma Rmin_comm : forall a b:R, Rmin a b = Rmin b a. +intros; unfold Rmin in |- *; case (Rle_dec a b); case (Rle_dec b a); intros; + try reflexivity || (apply Rle_antisym; assumption || 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)]. +Lemma Rmin_stable_in_posreal : forall x y:posreal, 0 < Rmin x y. +intros; apply Rmin_Rgt_r; split; [ apply (cond_pos x) | apply (cond_pos y) ]. Qed. (*******************************) @@ -83,54 +82,52 @@ Qed. (*******************************) (*********) -Definition Rmax :R->R->R:=[x,y:R] - Cases (total_order_Rle x y) of - (leftT _) => y - | (rightT _) => x +Definition Rmax (x y:R) : R := + match Rle_dec x y with + | left _ => y + | right _ => 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)). +Lemma Rmax_Rle : forall r1 r2 r, r <= Rmax r1 r2 <-> r <= r1 \/ r <= r2. +intros; split. +unfold Rmax in |- *; case (Rle_dec r1 r2); intros; auto. +intro; unfold Rmax in |- *; case (Rle_dec r1 r2); elim H; clear H; intros; + auto. +apply (Rle_trans r r1 r2); auto. +generalize (Rnot_le_lt 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. +Lemma RmaxLess1 : forall r1 r2, r1 <= Rmax r1 r2. +intros r1 r2; unfold Rmax in |- *; case (Rle_dec 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. +Lemma RmaxLess2 : forall r1 r2, r2 <= Rmax r1 r2. +intros r1 r2; unfold Rmax in |- *; case (Rle_dec 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. +Lemma RmaxSym : forall p q:R, Rmax p q = Rmax q p. +intros p q; unfold Rmax in |- *; case (Rle_dec p q); case (Rle_dec 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. +Lemma RmaxRmult : + forall (p q:R) r, 0 <= r -> Rmax (r * p) (r * q) = r * Rmax p q. +intros p q r H; unfold Rmax in |- *. +case (Rle_dec p q); case (Rle_dec (r * p) (r * q)); auto; intros H1 H2; auto. +case H; intros E1. +case H1; auto with real. +rewrite <- E1; repeat rewrite Rmult_0_l; auto. +case H; intros E1. +case H2; auto with real. +apply Rmult_le_reg_l with (r := r); auto. +rewrite <- E1; repeat rewrite Rmult_0_l; 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)]. +Lemma Rmax_stable_in_negreal : forall x y:negreal, Rmax x y < 0. +intros; unfold Rmax in |- *; case (Rle_dec x y); intro; + [ apply (cond_neg y) | apply (cond_neg x) ]. Qed. (*******************************) @@ -138,339 +135,336 @@ Qed. (*******************************) (*********) -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). +Lemma Rcase_abs : forall r, {r < 0} + {r >= 0}. +intro; generalize (Rle_dec 0 r); intro X; elim X; intro; clear X. +right; apply (Rle_ge 0 r a). +left; fold (0 > r) in |- *; apply (Rnot_le_lt 0 r b). Qed. (*********) -Definition Rabsolu:R->R:= - [r:R](Cases (case_Rabsolu r) of - (leftT _) => (Ropp r) - |(rightT _) => r - end). +Definition Rabs r : R := + match Rcase_abs r with + | left _ => - r + | right _ => r + end. (*********) -Lemma Rabsolu_R0:(Rabsolu R0)==R0. -Unfold Rabsolu;Case (case_Rabsolu R0);Auto;Intro. -Generalize (Rlt_antirefl R0);Intro;ElimType False;Auto. +Lemma Rabs_R0 : Rabs 0 = 0. +unfold Rabs in |- *; case (Rcase_abs 0); auto; intro. +generalize (Rlt_irrefl 0); 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. +Lemma Rabs_R1 : Rabs 1 = 1. +unfold Rabs in |- *; case (Rcase_abs 1); 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. +Lemma Rabs_no_R0 : forall r, r <> 0 -> Rabs r <> 0. +intros; unfold Rabs in |- *; case (Rcase_abs r); intro; auto. +apply Ropp_neq_0_compat; 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. +Lemma Rabs_left : forall r, r < 0 -> Rabs r = - r. +intros; unfold Rabs in |- *; case (Rcase_abs r); trivial; intro; + absurd (r >= 0). +exact (Rlt_not_ge r 0 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. +Lemma Rabs_right : forall r, r >= 0 -> Rabs r = r. +intros; unfold Rabs in |- *; case (Rcase_abs r); intro. +absurd (r >= 0). +exact (Rlt_not_ge r 0 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. +Lemma Rabs_left1 : forall a:R, a <= 0 -> Rabs a = - a. +intros a H; case H; intros H1. +apply Rabs_left; auto. +rewrite H1; simpl in |- *; rewrite Rabs_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. +Lemma Rabs_pos : forall x:R, 0 <= Rabs x. +intros; unfold Rabs in |- *; case (Rcase_abs x); intro. +generalize (Ropp_lt_gt_contravar x 0 r); intro; unfold Rgt in H; + rewrite Ropp_0 in H; unfold Rle in |- *; left; assumption. +apply Rge_le; assumption. Qed. -Lemma Rle_Rabsolu: - (x:R) (Rle x (Rabsolu x)). -Intro; Unfold Rabsolu;Case (case_Rabsolu x);Intros;Fourier. +Lemma RRle_abs : forall x:R, x <= Rabs x. +intro; unfold Rabs in |- *; case (Rcase_abs 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]. +Lemma Rabs_pos_eq : forall x:R, 0 <= x -> Rabs x = x. +intros; unfold Rabs in |- *; case (Rcase_abs x); intro; + [ generalize (Rgt_not_le 0 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)). +Lemma Rabs_Rabsolu : forall x:R, Rabs (Rabs x) = Rabs x. +intro; apply (Rabs_pos_eq (Rabs x) (Rabs_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. +Lemma Rabs_pos_lt : forall x:R, x <> 0 -> 0 < Rabs x. +intros; generalize (Rabs_pos x); intro; unfold Rle in H0; elim H0; intro; + auto. +elimtype False; clear H0; elim H; clear H; generalize H1; unfold Rabs in |- *; + case (Rcase_abs x); intros; auto. +clear r H1; generalize (Rplus_eq_compat_l x 0 (- x) H0); + rewrite (let (H1, H2) := Rplus_ne x in H1); rewrite (Rplus_opp_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. +Lemma Rabs_minus_sym : forall x y:R, Rabs (x - y) = Rabs (y - x). +intros; unfold Rabs in |- *; case (Rcase_abs (x - y)); + case (Rcase_abs (y - x)); intros. + generalize (Rminus_lt y x r); generalize (Rminus_lt x y r0); intros; + generalize (Rlt_asym x y H); intro; elimtype False; + auto. +rewrite (Ropp_minus_distr x y); trivial. +rewrite (Ropp_minus_distr y x); trivial. +unfold Rge in r, r0; elim r; elim r0; intros; clear r r0. +generalize (Ropp_lt_gt_0_contravar (x - y) H); rewrite (Ropp_minus_distr x y); + intro; unfold Rgt in H0; generalize (Rlt_asym 0 (y - x) H0); + intro; elimtype False; auto. +rewrite (Rminus_diag_uniq x y H); trivial. +rewrite (Rminus_diag_uniq y x H0); trivial. +rewrite (Rminus_diag_uniq 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. +Lemma Rabs_mult : forall x y:R, Rabs (x * y) = Rabs x * Rabs y. +intros; unfold Rabs in |- *; case (Rcase_abs (x * y)); case (Rcase_abs x); + case (Rcase_abs y); intros; auto. +generalize (Rmult_lt_gt_compat_neg_l y x 0 r r0); intro; + rewrite (Rmult_0_r y) in H; generalize (Rlt_asym (x * y) 0 r1); + intro; unfold Rgt in H; elimtype False; rewrite (Rmult_comm y x) in H; + auto. +rewrite (Ropp_mult_distr_l_reverse x y); trivial. +rewrite (Rmult_comm x (- y)); rewrite (Ropp_mult_distr_l_reverse y x); + rewrite (Rmult_comm x y); trivial. +unfold Rge in r, r0; elim r; elim r0; clear r r0; intros; unfold Rgt in H, H0. +generalize (Rmult_lt_compat_l x 0 y H H0); intro; rewrite (Rmult_0_r x) in H1; + generalize (Rlt_asym (x * y) 0 r1); intro; elimtype False; + auto. +rewrite H in r1; rewrite (Rmult_0_l y) in r1; generalize (Rlt_irrefl 0); + intro; elimtype False; auto. +rewrite H0 in r1; rewrite (Rmult_0_r x) in r1; generalize (Rlt_irrefl 0); + intro; elimtype False; auto. +rewrite H0 in r1; rewrite (Rmult_0_r x) in r1; generalize (Rlt_irrefl 0); + intro; elimtype False; auto. +rewrite (Rmult_opp_opp x y); trivial. +unfold Rge in r, r1; elim r; elim r1; clear r r1; intros; unfold Rgt in H0, H. +generalize (Rmult_lt_compat_l y x 0 H0 r0); intro; + rewrite (Rmult_0_r y) in H1; rewrite (Rmult_comm y x) in H1; + generalize (Rlt_asym (x * y) 0 H1); intro; elimtype False; + auto. +generalize (Rlt_dichotomy_converse x 0 (or_introl (x > 0) r0)); + generalize (Rlt_dichotomy_converse y 0 (or_intror (y < 0) H0)); + intros; generalize (Rmult_integral x y H); intro; + elim H3; intro; elimtype False; auto. +rewrite H0 in H; rewrite (Rmult_0_r x) in H; unfold Rgt in H; + generalize (Rlt_irrefl 0); intro; elimtype False; + auto. +rewrite H0; rewrite (Rmult_0_r x); rewrite (Rmult_0_r (- x)); trivial. +unfold Rge in r0, r1; elim r0; elim r1; clear r0 r1; intros; + unfold Rgt in H0, H. +generalize (Rmult_lt_compat_l x y 0 H0 r); intro; rewrite (Rmult_0_r x) in H1; + generalize (Rlt_asym (x * y) 0 H1); intro; elimtype False; + auto. +generalize (Rlt_dichotomy_converse y 0 (or_introl (y > 0) r)); + generalize (Rlt_dichotomy_converse 0 x (or_introl (0 > x) H0)); + intros; generalize (Rmult_integral x y H); intro; + elim H3; intro; elimtype False; auto. +rewrite H0 in H; rewrite (Rmult_0_l y) in H; unfold Rgt in H; + generalize (Rlt_irrefl 0); intro; elimtype False; + auto. +rewrite H0; rewrite (Rmult_0_l y); rewrite (Rmult_0_l (- 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. +Lemma Rabs_Rinv : forall r, r <> 0 -> Rabs (/ r) = / Rabs r. +intro; unfold Rabs in |- *; case (Rcase_abs r); case (Rcase_abs (/ r)); auto; + intros. +apply Ropp_inv_permute; auto. +generalize (Rinv_lt_0_compat r r1); intro; unfold Rge in r0; elim r0; intros. +unfold Rgt in H1; generalize (Rlt_asym 0 (/ r) H1); intro; elimtype False; + auto. +generalize (Rlt_dichotomy_converse (/ r) 0 (or_introl (/ r > 0) H0)); intro; + elimtype False; auto. +unfold Rge in r1; elim r1; clear r1; intro. +unfold Rgt in H0; generalize (Rlt_asym 0 (/ r) (Rinv_0_lt_compat 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. +Lemma Rabs_Ropp : forall x:R, Rabs (- x) = Rabs x. +intro; cut (- x = -1 * x). +intros; rewrite H. +rewrite Rabs_mult. +cut (Rabs (-1) = 1). +intros; rewrite H0. +ring. +unfold Rabs in |- *; case (Rcase_abs (-1)). +intro; ring. +intro H0; generalize (Rge_le (-1) 0 H0); intros. +generalize (Ropp_le_ge_contravar 0 (-1) H1). +rewrite Ropp_involutive; rewrite Ropp_0. +intro; generalize (Rgt_not_le 1 0 Rlt_0_1); intro; generalize (Rge_le 0 1 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. +Lemma Rabs_triang : forall a b:R, Rabs (a + b) <= Rabs a + Rabs b. +intros a b; unfold Rabs in |- *; case (Rcase_abs (a + b)); case (Rcase_abs a); + case (Rcase_abs b); intros. +apply (Req_le (- (a + b)) (- a + - b)); rewrite (Ropp_plus_distr 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_plus_distr a b); apply (Rplus_le_compat_l (- a) (- b) b); + unfold Rle in |- *; unfold Rge in r; elim r; intro. +left; unfold Rgt in H; generalize (Rplus_lt_compat_l (- b) 0 b H); intro; + elim (Rplus_ne (- b)); intros v w; rewrite v in H0; + clear v w; rewrite (Rplus_opp_l b) in H0; apply (Rlt_trans (- b) 0 b H0 H). +right; rewrite H; apply Ropp_0. (**) -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. +rewrite (Ropp_plus_distr a b); rewrite (Rplus_comm (- a) (- b)); + rewrite (Rplus_comm a (- b)); apply (Rplus_le_compat_l (- b) (- a) a); + unfold Rle in |- *; unfold Rge in r0; elim r0; intro. +left; unfold Rgt in H; generalize (Rplus_lt_compat_l (- a) 0 a H); intro; + elim (Rplus_ne (- a)); intros v w; rewrite v in H0; + clear v w; rewrite (Rplus_opp_l a) in H0; apply (Rlt_trans (- a) 0 a H0 H). +right; rewrite H; apply Ropp_0. (**) -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 (Rplus_ge_compat_l a b 0 r); intro; + elim (Rplus_ne a); intros v w; rewrite v in H; clear v w; + generalize (Rge_trans (a + b) a 0 H r0); intro; clear H; + unfold Rge in H0; elim H0; intro; clear H0. +unfold Rgt in H; generalize (Rlt_asym (a + b) 0 r1); intro; auto. +absurd (a + b = 0); auto. +apply (Rlt_dichotomy_converse (a + b) 0); 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. +elimtype False; generalize (Rplus_lt_compat_l a b 0 r); intro; + elim (Rplus_ne a); intros v w; rewrite v in H; clear v w; + generalize (Rlt_trans (a + b) a 0 H r0); intro; clear H; + unfold Rge in r1; elim r1; clear r1; intro. +unfold Rgt in H; generalize (Rlt_trans (a + b) 0 (a + b) H0 H); intro; + apply (Rlt_irrefl (a + b)); assumption. +rewrite H in H0; apply (Rlt_irrefl 0); 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). +rewrite (Rplus_comm a b); rewrite (Rplus_comm (- a) b); + apply (Rplus_le_compat_l b a (- a)); apply (Rminus_le a (- a)); + unfold Rminus in |- *; rewrite (Ropp_involutive a); + generalize (Rplus_lt_compat_l a a 0 r0); clear r r1; + intro; elim (Rplus_ne a); intros v w; rewrite v in H; + clear v w; generalize (Rlt_trans (a + a) a 0 H r0); + intro; apply (Rlt_le (a + a) 0 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). +apply (Rplus_le_compat_l a b (- b)); apply (Rminus_le b (- b)); + unfold Rminus in |- *; rewrite (Ropp_involutive b); + generalize (Rplus_lt_compat_l b b 0 r); clear r0 r1; + intro; elim (Rplus_ne b); intros v w; rewrite v in H; + clear v w; generalize (Rlt_trans (b + b) b 0 H r); + intro; apply (Rlt_le (b + b) 0 H0). (**) -Unfold Rle;Right;Reflexivity. +unfold Rle in |- *; 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. +Lemma Rabs_triang_inv : forall a b:R, Rabs a - Rabs b <= Rabs (a - b). +intros; apply (Rplus_le_reg_l (Rabs b) (Rabs a - Rabs b) (Rabs (a - b))); + unfold Rminus in |- *; rewrite <- (Rplus_assoc (Rabs b) (Rabs a) (- Rabs b)); + rewrite (Rplus_comm (Rabs b) (Rabs a)); + rewrite (Rplus_assoc (Rabs a) (Rabs b) (- Rabs b)); + rewrite (Rplus_opp_r (Rabs b)); rewrite (proj1 (Rplus_ne (Rabs a))); + replace (Rabs a) with (Rabs (a + 0)). + rewrite <- (Rplus_opp_r b); rewrite <- (Rplus_assoc a b (- b)); + rewrite (Rplus_comm a b); rewrite (Rplus_assoc b a (- b)). + exact (Rabs_triang b (a + - 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]]. +Lemma Rabs_triang_inv2 : forall a b:R, Rabs (Rabs a - Rabs b) <= Rabs (a - b). +cut + (forall a b:R, Rabs b <= Rabs a -> Rabs (Rabs a - Rabs b) <= Rabs (a - b)). +intros; destruct (Rtotal_order (Rabs a) (Rabs b)) as [Hlt| [Heq| Hgt]]. +rewrite <- (Rabs_Ropp (Rabs a - Rabs b)); rewrite <- (Rabs_Ropp (a - b)); + do 2 rewrite Ropp_minus_distr. +apply H; left; assumption. +rewrite Heq; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + apply Rabs_pos. +apply H; left; assumption. +intros; replace (Rabs (Rabs a - Rabs b)) with (Rabs a - Rabs b). +apply Rabs_triang_inv. +rewrite (Rabs_right (Rabs a - Rabs b)); + [ reflexivity + | apply Rle_ge; apply Rplus_le_reg_l with (Rabs b); rewrite Rplus_0_r; + replace (Rabs b + (Rabs a - Rabs b)) with (Rabs 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. +Lemma Rabs_def1 : forall x a:R, x < a -> - a < x -> Rabs x < a. +unfold Rabs in |- *; intros; case (Rcase_abs x); intro. +generalize (Ropp_lt_gt_contravar (- a) x H0); unfold Rgt in |- *; + rewrite Ropp_involutive; 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. - +Lemma Rabs_def2 : forall x a:R, Rabs x < a -> x < a /\ - a < x. +unfold Rabs in |- *; intro x; case (Rcase_abs x); intros. +generalize (Ropp_gt_lt_0_contravar x r); unfold Rgt in |- *; intro; + generalize (Rlt_trans 0 (- x) a H0 H); intro; split. +apply (Rlt_trans x 0 a r H1). +generalize (Ropp_lt_gt_contravar (- x) a H); rewrite (Ropp_involutive x); + unfold Rgt in |- *; trivial. +fold (a > x) in H; generalize (Rgt_ge_trans a x 0 H r); intro; + generalize (Ropp_lt_gt_0_contravar a H0); intro; fold (0 > - a) in |- *; + generalize (Rge_gt_trans x 0 (- a) r H1); unfold Rgt in |- *; + intro; split; assumption. +Qed. + +Lemma RmaxAbs : + forall (p q:R) r, p <= q -> q <= r -> Rabs q <= Rmax (Rabs p) (Rabs r). +intros p q r H' H'0; case (Rle_or_lt 0 p); intros H'1. +repeat rewrite Rabs_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 (Rabs_left p); auto. +case (Rle_or_lt 0 q); intros H'2. +repeat rewrite Rabs_right; auto with real. +apply Rle_trans with r; auto. +apply RmaxLess2; auto. +apply Rge_trans with q; auto with real. +rewrite (Rabs_left q); auto. +case (Rle_or_lt 0 r); intros H'3. +repeat rewrite Rabs_right; auto with real. +apply Rle_trans with (- p); auto with real. +apply RmaxLess1; auto. +rewrite (Rabs_left r); auto. +apply Rle_trans with (- p); auto with real. +apply RmaxLess1; auto. +Qed. + +Lemma Rabs_Zabs : forall z:Z, Rabs (IZR z) = IZR (Zabs z). +intros z; case z; simpl in |- *; auto with real. +apply Rabs_right; auto with real. +intros p0; apply Rabs_right; auto with real zarith. +intros p0; rewrite Rabs_Ropp. +apply Rabs_right; auto with real zarith. +Qed. +
\ No newline at end of file diff --git a/theories/Reals/Rcomplete.v b/theories/Reals/Rcomplete.v index 5dca3068c..53624cbb2 100644 --- a/theories/Reals/Rcomplete.v +++ b/theories/Reals/Rcomplete.v @@ -8,12 +8,11 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Rseries. -Require SeqProp. -Require Max. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import Rseries. +Require Import SeqProp. +Require Import Max. Open Local Scope R_scope. (****************************************************) @@ -24,152 +23,176 @@ Open Local Scope R_scope. (* Proof with adjacent sequences (Vn and Wn) *) (****************************************************) -Theorem R_complete : (Un:nat->R) (Cauchy_crit Un) -> (sigTT R [l:R](Un_cv Un l)). -Intros. -Pose Vn := (sequence_minorant Un (cauchy_min Un H)). -Pose Wn := (sequence_majorant Un (cauchy_maj Un H)). -Assert H0 := (maj_cv Un H). -Fold Wn in H0. -Assert H1 := (min_cv Un H). -Fold Vn in H1. -Elim H0; Intros. -Elim H1; Intros. -Cut x==x0. -Intros. -Apply existTT with x. -Rewrite <- H2 in p0. -Unfold Un_cv. -Intros. -Unfold Un_cv in p; Unfold Un_cv in p0. -Cut ``0<eps/3``. -Intro. -Elim (p ``eps/3`` H4); Intros. -Elim (p0 ``eps/3`` H4); Intros. -Exists (max x1 x2). -Intros. -Unfold R_dist. -Apply Rle_lt_trans with ``(Rabsolu ((Un n)-(Vn n)))+(Rabsolu ((Vn n)-x))``. -Replace ``(Un n)-x`` with ``((Un n)-(Vn n))+((Vn n)-x)``; [Apply Rabsolu_triang | Ring]. -Apply Rle_lt_trans with ``(Rabsolu ((Wn n)-(Vn n)))+(Rabsolu ((Vn n)-x))``. -Do 2 Rewrite <- (Rplus_sym ``(Rabsolu ((Vn n)-x))``). -Apply Rle_compatibility. -Repeat Rewrite Rabsolu_right. -Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-(Vn n)``); Apply Rle_compatibility. -Assert H8 := (Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)). -Fold Vn Wn in H8. -Elim (H8 n); Intros. -Assumption. -Apply Rle_sym1. -Unfold Rminus; Apply Rle_anti_compatibility with (Vn n). -Rewrite Rplus_Or. -Replace ``(Vn n)+((Wn n)+ -(Vn n))`` with (Wn n); [Idtac | Ring]. -Assert H8 := (Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)). -Fold Vn Wn in H8. -Elim (H8 n); Intros. -Apply Rle_trans with (Un n); Assumption. -Apply Rle_sym1. -Unfold Rminus; Apply Rle_anti_compatibility with (Vn n). -Rewrite Rplus_Or. -Replace ``(Vn n)+((Un n)+ -(Vn n))`` with (Un n); [Idtac | Ring]. -Assert H8 := (Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)). -Fold Vn Wn in H8. -Elim (H8 n); Intros. -Assumption. -Apply Rle_lt_trans with ``(Rabsolu ((Wn n)-x))+(Rabsolu (x-(Vn n)))+(Rabsolu ((Vn n)-x))``. -Do 2 Rewrite <- (Rplus_sym ``(Rabsolu ((Vn n)-x))``). -Apply Rle_compatibility. -Replace ``(Wn n)-(Vn n)`` with ``((Wn n)-x)+(x-(Vn n))``; [Apply Rabsolu_triang | Ring]. -Apply Rlt_le_trans with ``eps/3+eps/3+eps/3``. -Repeat Apply Rplus_lt. -Unfold R_dist in H5. -Apply H5. -Unfold ge; Apply le_trans with (max x1 x2). -Apply le_max_l. -Assumption. -Rewrite <- Rabsolu_Ropp. -Replace ``-(x-(Vn n))`` with ``(Vn n)-x``; [Idtac | Ring]. -Unfold R_dist in H6. -Apply H6. -Unfold ge; Apply le_trans with (max x1 x2). -Apply le_max_r. -Assumption. -Unfold R_dist in H6. -Apply H6. -Unfold ge; Apply le_trans with (max x1 x2). -Apply le_max_r. -Assumption. -Right. -Pattern 4 eps; Replace ``eps`` with ``3*eps/3``. -Ring. -Unfold Rdiv; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m; DiscrR. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Apply cond_eq. -Intros. -Cut ``0<eps/5``. -Intro. -Unfold Un_cv in p; Unfold Un_cv in p0. -Unfold R_dist in p; Unfold R_dist in p0. -Elim (p ``eps/5`` H3); Intros N1 H4. -Elim (p0 ``eps/5`` H3); Intros N2 H5. -Unfold Cauchy_crit in H. -Unfold R_dist in H. -Elim (H ``eps/5`` H3); Intros N3 H6. -Pose N := (max (max N1 N2) N3). -Apply Rle_lt_trans with ``(Rabsolu (x-(Wn N)))+(Rabsolu ((Wn N)-x0))``. -Replace ``x-x0`` with ``(x-(Wn N))+((Wn N)-x0)``; [Apply Rabsolu_triang | Ring]. -Apply Rle_lt_trans with ``(Rabsolu (x-(Wn N)))+(Rabsolu ((Wn N)-(Vn N)))+(Rabsolu (((Vn N)-x0)))``. -Rewrite Rplus_assoc. -Apply Rle_compatibility. -Replace ``(Wn N)-x0`` with ``((Wn N)-(Vn N))+((Vn N)-x0)``; [Apply Rabsolu_triang | Ring]. -Replace ``eps`` with ``eps/5+3*eps/5+eps/5``. -Repeat Apply Rplus_lt. -Rewrite <- Rabsolu_Ropp. -Replace ``-(x-(Wn N))`` with ``(Wn N)-x``; [Apply H4 | Ring]. -Unfold ge N. -Apply le_trans with (max N1 N2); Apply le_max_l. -Unfold Wn Vn. -Unfold sequence_majorant sequence_minorant. -Assert H7 := (approx_maj [k:nat](Un (plus N k)) (maj_ss Un N (cauchy_maj Un H))). -Assert H8 := (approx_min [k:nat](Un (plus N k)) (min_ss Un N (cauchy_min Un H))). -Cut (Wn N)==(majorant ([k:nat](Un (plus N k))) (maj_ss Un N (cauchy_maj Un H))). -Cut (Vn N)==(minorant ([k:nat](Un (plus N k))) (min_ss Un N (cauchy_min Un H))). -Intros. -Rewrite <- H9; Rewrite <- H10. -Rewrite <- H9 in H8. -Rewrite <- H10 in H7. -Elim (H7 ``eps/5`` H3); Intros k2 H11. -Elim (H8 ``eps/5`` H3); Intros k1 H12. -Apply Rle_lt_trans with ``(Rabsolu ((Wn N)-(Un (plus N k2))))+(Rabsolu ((Un (plus N k2))-(Vn N)))``. -Replace ``(Wn N)-(Vn N)`` with ``((Wn N)-(Un (plus N k2)))+((Un (plus N k2))-(Vn N))``; [Apply Rabsolu_triang | Ring]. -Apply Rle_lt_trans with ``(Rabsolu ((Wn N)-(Un (plus N k2))))+(Rabsolu ((Un (plus N k2))-(Un (plus N k1))))+(Rabsolu ((Un (plus N k1))-(Vn N)))``. -Rewrite Rplus_assoc. -Apply Rle_compatibility. -Replace ``(Un (plus N k2))-(Vn N)`` with ``((Un (plus N k2))-(Un (plus N k1)))+((Un (plus N k1))-(Vn N))``; [Apply Rabsolu_triang | Ring]. -Replace ``3*eps/5`` with ``eps/5+eps/5+eps/5``; [Repeat Apply Rplus_lt | Ring]. -Assumption. -Apply H6. -Unfold ge. -Apply le_trans with N. -Unfold N; Apply le_max_r. -Apply le_plus_l. -Unfold ge. -Apply le_trans with N. -Unfold N; Apply le_max_r. -Apply le_plus_l. -Rewrite <- Rabsolu_Ropp. -Replace ``-((Un (plus N k1))-(Vn N))`` with ``(Vn N)-(Un (plus N k1))``; [Assumption | Ring]. -Reflexivity. -Reflexivity. -Apply H5. -Unfold ge; Apply le_trans with (max N1 N2). -Apply le_max_r. -Unfold N; Apply le_max_l. -Pattern 4 eps; Replace ``eps`` with ``5*eps/5``. -Ring. -Unfold Rdiv; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m. -DiscrR. -Unfold Rdiv; Apply Rmult_lt_pos. -Assumption. -Apply Rlt_Rinv. -Sup0; Try Apply lt_O_Sn. -Qed. +Theorem R_complete : + forall Un:nat -> R, Cauchy_crit Un -> sigT (fun l:R => Un_cv Un l). +intros. +pose (Vn := sequence_minorant Un (cauchy_min Un H)). +pose (Wn := sequence_majorant Un (cauchy_maj Un H)). +assert (H0 := maj_cv Un H). +fold Wn in H0. +assert (H1 := min_cv Un H). +fold Vn in H1. +elim H0; intros. +elim H1; intros. +cut (x = x0). +intros. +apply existT with x. +rewrite <- H2 in p0. +unfold Un_cv in |- *. +intros. +unfold Un_cv in p; unfold Un_cv in p0. +cut (0 < eps / 3). +intro. +elim (p (eps / 3) H4); intros. +elim (p0 (eps / 3) H4); intros. +exists (max x1 x2). +intros. +unfold R_dist in |- *. +apply Rle_lt_trans with (Rabs (Un n - Vn n) + Rabs (Vn n - x)). +replace (Un n - x) with (Un n - Vn n + (Vn n - x)); + [ apply Rabs_triang | ring ]. +apply Rle_lt_trans with (Rabs (Wn n - Vn n) + Rabs (Vn n - x)). +do 2 rewrite <- (Rplus_comm (Rabs (Vn n - x))). +apply Rplus_le_compat_l. +repeat rewrite Rabs_right. +unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- Vn n)); + apply Rplus_le_compat_l. +assert (H8 := Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)). +fold Vn Wn in H8. +elim (H8 n); intros. +assumption. +apply Rle_ge. +unfold Rminus in |- *; apply Rplus_le_reg_l with (Vn n). +rewrite Rplus_0_r. +replace (Vn n + (Wn n + - Vn n)) with (Wn n); [ idtac | ring ]. +assert (H8 := Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)). +fold Vn Wn in H8. +elim (H8 n); intros. +apply Rle_trans with (Un n); assumption. +apply Rle_ge. +unfold Rminus in |- *; apply Rplus_le_reg_l with (Vn n). +rewrite Rplus_0_r. +replace (Vn n + (Un n + - Vn n)) with (Un n); [ idtac | ring ]. +assert (H8 := Vn_Un_Wn_order Un (cauchy_maj Un H) (cauchy_min Un H)). +fold Vn Wn in H8. +elim (H8 n); intros. +assumption. +apply Rle_lt_trans with (Rabs (Wn n - x) + Rabs (x - Vn n) + Rabs (Vn n - x)). +do 2 rewrite <- (Rplus_comm (Rabs (Vn n - x))). +apply Rplus_le_compat_l. +replace (Wn n - Vn n) with (Wn n - x + (x - Vn n)); + [ apply Rabs_triang | ring ]. +apply Rlt_le_trans with (eps / 3 + eps / 3 + eps / 3). +repeat apply Rplus_lt_compat. +unfold R_dist in H5. +apply H5. +unfold ge in |- *; apply le_trans with (max x1 x2). +apply le_max_l. +assumption. +rewrite <- Rabs_Ropp. +replace (- (x - Vn n)) with (Vn n - x); [ idtac | ring ]. +unfold R_dist in H6. +apply H6. +unfold ge in |- *; apply le_trans with (max x1 x2). +apply le_max_r. +assumption. +unfold R_dist in H6. +apply H6. +unfold ge in |- *; apply le_trans with (max x1 x2). +apply le_max_r. +assumption. +right. +pattern eps at 4 in |- *; replace eps with (3 * (eps / 3)). +ring. +unfold Rdiv in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m; discrR. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +apply cond_eq. +intros. +cut (0 < eps / 5). +intro. +unfold Un_cv in p; unfold Un_cv in p0. +unfold R_dist in p; unfold R_dist in p0. +elim (p (eps / 5) H3); intros N1 H4. +elim (p0 (eps / 5) H3); intros N2 H5. +unfold Cauchy_crit in H. +unfold R_dist in H. +elim (H (eps / 5) H3); intros N3 H6. +pose (N := max (max N1 N2) N3). +apply Rle_lt_trans with (Rabs (x - Wn N) + Rabs (Wn N - x0)). +replace (x - x0) with (x - Wn N + (Wn N - x0)); [ apply Rabs_triang | ring ]. +apply Rle_lt_trans with + (Rabs (x - Wn N) + Rabs (Wn N - Vn N) + Rabs (Vn N - x0)). +rewrite Rplus_assoc. +apply Rplus_le_compat_l. +replace (Wn N - x0) with (Wn N - Vn N + (Vn N - x0)); + [ apply Rabs_triang | ring ]. +replace eps with (eps / 5 + 3 * (eps / 5) + eps / 5). +repeat apply Rplus_lt_compat. +rewrite <- Rabs_Ropp. +replace (- (x - Wn N)) with (Wn N - x); [ apply H4 | ring ]. +unfold ge, N in |- *. +apply le_trans with (max N1 N2); apply le_max_l. +unfold Wn, Vn in |- *. +unfold sequence_majorant, sequence_minorant in |- *. +assert + (H7 := + approx_maj (fun k:nat => Un (N + k)%nat) (maj_ss Un N (cauchy_maj Un H))). +assert + (H8 := + approx_min (fun k:nat => Un (N + k)%nat) (min_ss Un N (cauchy_min Un H))). +cut + (Wn N = + majorant (fun k:nat => Un (N + k)%nat) (maj_ss Un N (cauchy_maj Un H))). +cut + (Vn N = + minorant (fun k:nat => Un (N + k)%nat) (min_ss Un N (cauchy_min Un H))). +intros. +rewrite <- H9; rewrite <- H10. +rewrite <- H9 in H8. +rewrite <- H10 in H7. +elim (H7 (eps / 5) H3); intros k2 H11. +elim (H8 (eps / 5) H3); intros k1 H12. +apply Rle_lt_trans with + (Rabs (Wn N - Un (N + k2)%nat) + Rabs (Un (N + k2)%nat - Vn N)). +replace (Wn N - Vn N) with + (Wn N - Un (N + k2)%nat + (Un (N + k2)%nat - Vn N)); + [ apply Rabs_triang | ring ]. +apply Rle_lt_trans with + (Rabs (Wn N - Un (N + k2)%nat) + Rabs (Un (N + k2)%nat - Un (N + k1)%nat) + + Rabs (Un (N + k1)%nat - Vn N)). +rewrite Rplus_assoc. +apply Rplus_le_compat_l. +replace (Un (N + k2)%nat - Vn N) with + (Un (N + k2)%nat - Un (N + k1)%nat + (Un (N + k1)%nat - Vn N)); + [ apply Rabs_triang | ring ]. +replace (3 * (eps / 5)) with (eps / 5 + eps / 5 + eps / 5); + [ repeat apply Rplus_lt_compat | ring ]. +assumption. +apply H6. +unfold ge in |- *. +apply le_trans with N. +unfold N in |- *; apply le_max_r. +apply le_plus_l. +unfold ge in |- *. +apply le_trans with N. +unfold N in |- *; apply le_max_r. +apply le_plus_l. +rewrite <- Rabs_Ropp. +replace (- (Un (N + k1)%nat - Vn N)) with (Vn N - Un (N + k1)%nat); + [ assumption | ring ]. +reflexivity. +reflexivity. +apply H5. +unfold ge in |- *; apply le_trans with (max N1 N2). +apply le_max_r. +unfold N in |- *; apply le_max_l. +pattern eps at 4 in |- *; replace eps with (5 * (eps / 5)). +ring. +unfold Rdiv in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. +discrR. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +assumption. +apply Rinv_0_lt_compat. +prove_sup0; try apply lt_O_Sn. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Rdefinitions.v b/theories/Reals/Rdefinitions.v index 75a082cfc..a862a0ac3 100644 --- a/theories/Reals/Rdefinitions.v +++ b/theories/Reals/Rdefinitions.v @@ -15,55 +15,55 @@ Require Export ZArith_base. -Parameter R:Set. +Parameter R : Set. (* Declare Scope positive_scope with Key R *) -Delimits Scope R_scope with R. +Delimit Scope R_scope with R. (* Automatically open scope R_scope for arguments of type R *) Bind Scope R_scope with R. -Parameter R0:R. -Parameter R1:R. -Parameter Rplus:R->R->R. -Parameter Rmult:R->R->R. -Parameter Ropp:R->R. -Parameter Rinv:R->R. -Parameter Rlt:R->R->Prop. -Parameter up:R->Z. +Parameter R0 : R. +Parameter R1 : R. +Parameter Rplus : R -> R -> R. +Parameter Rmult : R -> R -> R. +Parameter Ropp : R -> R. +Parameter Rinv : R -> R. +Parameter Rlt : R -> R -> Prop. +Parameter up : R -> Z. -V8Infix "+" Rplus : R_scope. -V8Infix "*" Rmult : R_scope. -V8Notation "- x" := (Ropp x) : R_scope. -V8Notation "/ x" := (Rinv x) : R_scope. +Infix "+" := Rplus : R_scope. +Infix "*" := Rmult : R_scope. +Notation "- x" := (Ropp x) : R_scope. +Notation "/ x" := (Rinv x) : R_scope. -V8Infix "<" Rlt : R_scope. +Infix "<" := Rlt : R_scope. (*i*******************************************************i*) (**********) -Definition Rgt:R->R->Prop:=[r1,r2:R](Rlt r2 r1). +Definition Rgt (r1 r2:R) : Prop := (r2 < r1)%R. (**********) -Definition Rle:R->R->Prop:=[r1,r2:R]((Rlt r1 r2)\/(r1==r2)). +Definition Rle (r1 r2:R) : Prop := (r1 < r2)%R \/ r1 = r2. (**********) -Definition Rge:R->R->Prop:=[r1,r2:R]((Rgt r1 r2)\/(r1==r2)). +Definition Rge (r1 r2:R) : Prop := Rgt r1 r2 \/ r1 = r2. (**********) -Definition Rminus:R->R->R:=[r1,r2:R](Rplus r1 (Ropp r2)). +Definition Rminus (r1 r2:R) : R := (r1 + - r2)%R. (**********) -Definition Rdiv:R->R->R:=[r1,r2:R](Rmult r1 (Rinv r2)). +Definition Rdiv (r1 r2:R) : R := (r1 * / r2)%R. -V8Infix "-" Rminus : R_scope. -V8Infix "/" Rdiv : R_scope. +Infix "-" := Rminus : R_scope. +Infix "/" := Rdiv : R_scope. -V8Infix "<=" Rle : R_scope. -V8Infix ">=" Rge : R_scope. -V8Infix ">" Rgt : R_scope. +Infix "<=" := Rle : R_scope. +Infix ">=" := Rge : R_scope. +Infix ">" := Rgt : R_scope. -V8Notation "x <= y <= z" := (Rle x y)/\(Rle y z) : R_scope. -V8Notation "x <= y < z" := (Rle x y)/\(Rlt y z) : R_scope. -V8Notation "x < y < z" := (Rlt x y)/\(Rlt y z) : R_scope. -V8Notation "x < y <= z" := (Rlt x y)/\(Rle y z) : R_scope. +Notation "x <= y <= z" := ((x <= y)%R /\ (y <= z)%R) : R_scope. +Notation "x <= y < z" := ((x <= y)%R /\ (y < z)%R) : R_scope. +Notation "x < y < z" := ((x < y)%R /\ (y < z)%R) : R_scope. +Notation "x < y <= z" := ((x < y)%R /\ (y <= z)%R) : R_scope.
\ No newline at end of file diff --git a/theories/Reals/Rderiv.v b/theories/Reals/Rderiv.v index 4f7420306..3f56ccdf1 100644 --- a/theories/Reals/Rderiv.v +++ b/theories/Reals/Rderiv.v @@ -13,441 +13,419 @@ (* *) (*********************************************************) -Require Rbase. -Require Rfunctions. -Require Rlimit. -Require Fourier. -Require Classical_Prop. -Require Classical_Pred_Type. -Require Omega. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rfunctions. +Require Import Rlimit. +Require Import Fourier. +Require Import Classical_Prop. +Require Import Classical_Pred_Type. +Require Import Omega. Open Local Scope R_scope. (*********) -Definition D_x:(R->Prop)->R->R->Prop:=[D:R->Prop][y:R][x:R] - (D x)/\(~y==x). +Definition D_x (D:R -> Prop) (y x:R) : Prop := D x /\ y <> x. (*********) -Definition continue_in:(R->R)->(R->Prop)->R->Prop:= - [f:R->R; D:R->Prop; x0:R](limit1_in f (D_x D x0) (f x0) x0). +Definition continue_in (f:R -> R) (D:R -> Prop) (x0:R) : Prop := + limit1_in f (D_x D x0) (f x0) x0. (*********) -Definition D_in:(R->R)->(R->R)->(R->Prop)->R->Prop:= - [f:R->R; d:R->R; D:R->Prop; x0:R](limit1_in - [x:R] (Rdiv (Rminus (f x) (f x0)) (Rminus x x0)) - (D_x D x0) (d x0) x0). +Definition D_in (f d:R -> R) (D:R -> Prop) (x0:R) : Prop := + limit1_in (fun x:R => (f x - f x0) / (x - x0)) (D_x D x0) (d x0) x0. (*********) -Lemma cont_deriv:(f,d:R->R;D:R->Prop;x0:R) - (D_in f d D x0)->(continue_in f D x0). -Unfold continue_in;Unfold D_in;Unfold limit1_in;Unfold limit_in; - Unfold Rdiv;Simpl;Intros;Elim (H eps H0); Clear H;Intros; - Elim H;Clear H;Intros; Elim (Req_EM (d x0) R0);Intro. -Split with (Rmin R1 x);Split. -Elim (Rmin_Rgt R1 x R0);Intros a b; - Apply (b (conj (Rgt R1 R0) (Rgt x R0) Rlt_R0_R1 H)). -Intros;Elim H3;Clear H3;Intros; -Generalize (let (H1,H2)=(Rmin_Rgt R1 x (R_dist x1 x0)) in H1); - Unfold Rgt;Intro;Elim (H5 H4);Clear H5;Intros; - Generalize (H1 x1 (conj (D_x D x0 x1) (Rlt (R_dist x1 x0) x) H3 H6)); - Clear H1;Intro;Unfold D_x in H3;Elim H3;Intros. -Rewrite H2 in H1;Unfold R_dist; Unfold R_dist in H1; - Cut (Rlt (Rabsolu (Rminus (f x1) (f x0))) - (Rmult eps (Rabsolu (Rminus x1 x0)))). -Intro;Unfold R_dist in H5; - Generalize (Rlt_monotony eps ``(Rabsolu (x1-x0))`` ``1`` H0 H5); -Rewrite Rmult_1r;Intro;Apply Rlt_trans with r2:=``eps*(Rabsolu (x1-x0))``; - Assumption. -Rewrite (minus_R0 ``((f x1)-(f x0))*/(x1-x0)``) in H1; - Rewrite Rabsolu_mult in H1; Cut ``x1-x0 <> 0``. -Intro;Rewrite (Rabsolu_Rinv (Rminus x1 x0) H9) in H1; - Generalize (Rlt_monotony ``(Rabsolu (x1-x0))`` - ``(Rabsolu ((f x1)-(f x0)))*/(Rabsolu (x1-x0))`` eps - (Rabsolu_pos_lt ``x1-x0`` H9) H1);Intro; Rewrite Rmult_sym in H10; - Rewrite Rmult_assoc in H10;Rewrite Rinv_l in H10. -Rewrite Rmult_1r in H10;Rewrite Rmult_sym;Assumption. -Apply Rabsolu_no_R0;Auto. -Apply Rminus_eq_contra;Auto. +Lemma cont_deriv : + forall (f d:R -> R) (D:R -> Prop) (x0:R), + D_in f d D x0 -> continue_in f D x0. +unfold continue_in in |- *; unfold D_in in |- *; unfold limit1_in in |- *; + unfold limit_in in |- *; unfold Rdiv in |- *; simpl in |- *; + intros; elim (H eps H0); clear H; intros; elim H; + clear H; intros; elim (Req_dec (d x0) 0); intro. +split with (Rmin 1 x); split. +elim (Rmin_Rgt 1 x 0); intros a b; apply (b (conj Rlt_0_1 H)). +intros; elim H3; clear H3; intros; + generalize (let (H1, H2) := Rmin_Rgt 1 x (R_dist x1 x0) in H1); + unfold Rgt in |- *; intro; elim (H5 H4); clear H5; + intros; generalize (H1 x1 (conj H3 H6)); clear H1; + intro; unfold D_x in H3; elim H3; intros. +rewrite H2 in H1; unfold R_dist in |- *; unfold R_dist in H1; + cut (Rabs (f x1 - f x0) < eps * Rabs (x1 - x0)). +intro; unfold R_dist in H5; + generalize (Rmult_lt_compat_l eps (Rabs (x1 - x0)) 1 H0 H5); + rewrite Rmult_1_r; intro; apply Rlt_trans with (r2 := eps * Rabs (x1 - x0)); + assumption. +rewrite (Rminus_0_r ((f x1 - f x0) * / (x1 - x0))) in H1; + rewrite Rabs_mult in H1; cut (x1 - x0 <> 0). +intro; rewrite (Rabs_Rinv (x1 - x0) H9) in H1; + generalize + (Rmult_lt_compat_l (Rabs (x1 - x0)) (Rabs (f x1 - f x0) * / Rabs (x1 - x0)) + eps (Rabs_pos_lt (x1 - x0) H9) H1); intro; rewrite Rmult_comm in H10; + rewrite Rmult_assoc in H10; rewrite Rinv_l in H10. +rewrite Rmult_1_r in H10; rewrite Rmult_comm; assumption. +apply Rabs_no_R0; auto. +apply Rminus_eq_contra; auto. (**) - Split with (Rmin (Rmin (Rinv (Rplus R1 R1)) x) - (Rmult eps (Rinv (Rabsolu (Rmult (Rplus R1 R1) (d x0)))))); - Split. -Cut (Rgt (Rmin (Rinv (Rplus R1 R1)) x) R0). -Cut (Rgt (Rmult eps (Rinv (Rabsolu (Rmult (Rplus R1 R1) (d x0))))) R0). -Intros;Elim (Rmin_Rgt (Rmin (Rinv (Rplus R1 R1)) x) - (Rmult eps (Rinv (Rabsolu (Rmult (Rplus R1 R1) (d x0))))) R0); - Intros a b; - Apply (b (conj (Rgt (Rmin (Rinv (Rplus R1 R1)) x) R0) - (Rgt (Rmult eps (Rinv (Rabsolu (Rmult (Rplus R1 R1) (d x0))))) R0) - H4 H3)). -Apply Rmult_gt;Auto. -Unfold Rgt;Apply Rlt_Rinv;Apply Rabsolu_pos_lt;Apply mult_non_zero; - Split. -DiscrR. -Assumption. -Elim (Rmin_Rgt (Rinv (Rplus R1 R1)) x R0);Intros a b; - Cut (Rlt R0 (Rplus R1 R1)). -Intro;Generalize (Rlt_Rinv (Rplus R1 R1) H3);Intro; - Fold (Rgt (Rinv (Rplus R1 R1)) R0) in H4; - Apply (b (conj (Rgt (Rinv (Rplus R1 R1)) R0) (Rgt x R0) H4 H)). -Fourier. -Intros;Elim H3;Clear H3;Intros; - Generalize (let (H1,H2)=(Rmin_Rgt (Rmin (Rinv (Rplus R1 R1)) x) - (Rmult eps (Rinv (Rabsolu (Rmult (Rplus R1 R1) (d x0))))) - (R_dist x1 x0)) in H1);Unfold Rgt;Intro;Elim (H5 H4);Clear H5; - Intros; - Generalize (let (H1,H2)=(Rmin_Rgt (Rinv (Rplus R1 R1)) x - (R_dist x1 x0)) in H1);Unfold Rgt;Intro;Elim (H7 H5);Clear H7; - Intros;Clear H4 H5; - Generalize (H1 x1 (conj (D_x D x0 x1) (Rlt (R_dist x1 x0) x) H3 H8)); - Clear H1;Intro;Unfold D_x in H3;Elim H3;Intros; - Generalize (sym_not_eqT R x0 x1 H5);Clear H5;Intro H5; - Generalize (Rminus_eq_contra x1 x0 H5); - Intro;Generalize H1;Pattern 1 (d x0); - Rewrite <-(let (H1,H2)=(Rmult_ne (d x0)) in H2); - Rewrite <-(Rinv_l (Rminus x1 x0) H9); Unfold R_dist;Unfold 1 Rminus; - Rewrite (Rmult_sym (Rminus (f x1) (f x0)) (Rinv (Rminus x1 x0))); - Rewrite (Rmult_sym (Rmult (Rinv (Rminus x1 x0)) (Rminus x1 x0)) (d x0)); - Rewrite <-(Ropp_mul1 (d x0) (Rmult (Rinv (Rminus x1 x0)) (Rminus x1 x0))); - Rewrite (Rmult_sym (Ropp (d x0)) - (Rmult (Rinv (Rminus x1 x0)) (Rminus x1 x0))); - Rewrite (Rmult_assoc (Rinv (Rminus x1 x0)) (Rminus x1 x0) (Ropp (d x0))); - Rewrite <-(Rmult_Rplus_distr (Rinv (Rminus x1 x0)) (Rminus (f x1) (f x0)) - (Rmult (Rminus x1 x0) (Ropp (d x0)))); - Rewrite (Rabsolu_mult (Rinv (Rminus x1 x0)) - (Rplus (Rminus (f x1) (f x0)) - (Rmult (Rminus x1 x0) (Ropp (d x0))))); - Clear H1;Intro;Generalize (Rlt_monotony (Rabsolu (Rminus x1 x0)) - (Rmult (Rabsolu (Rinv (Rminus x1 x0))) - (Rabsolu - (Rplus (Rminus (f x1) (f x0)) - (Rmult (Rminus x1 x0) (Ropp (d x0)))))) eps - (Rabsolu_pos_lt (Rminus x1 x0) H9) H1); - Rewrite <-(Rmult_assoc (Rabsolu (Rminus x1 x0)) - (Rabsolu (Rinv (Rminus x1 x0))) - (Rabsolu - (Rplus (Rminus (f x1) (f x0)) - (Rmult (Rminus x1 x0) (Ropp (d x0)))))); - Rewrite (Rabsolu_Rinv (Rminus x1 x0) H9); - Rewrite (Rinv_r (Rabsolu (Rminus x1 x0)) - (Rabsolu_no_R0 (Rminus x1 x0) H9)); - Rewrite (let (H1,H2)=(Rmult_ne (Rabsolu - (Rplus (Rminus (f x1) (f x0)) - (Rmult (Rminus x1 x0) (Ropp (d x0)))))) in H2); - Generalize (Rabsolu_triang_inv (Rminus (f x1) (f x0)) - (Rmult (Rminus x1 x0) (d x0)));Intro; - Rewrite (Rmult_sym (Rminus x1 x0) (Ropp (d x0))); - Rewrite (Ropp_mul1 (d x0) (Rminus x1 x0)); - Fold (Rminus (Rminus (f x1) (f x0)) (Rmult (d x0) (Rminus x1 x0))); - Rewrite (Rmult_sym (Rminus x1 x0) (d x0)) in H10; - Clear H1;Intro;Generalize (Rle_lt_trans - (Rminus (Rabsolu (Rminus (f x1) (f x0))) - (Rabsolu (Rmult (d x0) (Rminus x1 x0)))) - (Rabsolu - (Rminus (Rminus (f x1) (f x0)) (Rmult (d x0) (Rminus x1 x0)))) - (Rmult (Rabsolu (Rminus x1 x0)) eps) H10 H1); - Clear H1;Intro; - Generalize (Rlt_compatibility (Rabsolu (Rmult (d x0) (Rminus x1 x0))) - (Rminus (Rabsolu (Rminus (f x1) (f x0))) - (Rabsolu (Rmult (d x0) (Rminus x1 x0)))) - (Rmult (Rabsolu (Rminus x1 x0)) eps) H1); - Unfold 2 Rminus;Rewrite (Rplus_sym (Rabsolu (Rminus (f x1) (f x0))) - (Ropp (Rabsolu (Rmult (d x0) (Rminus x1 x0))))); - Rewrite <-(Rplus_assoc (Rabsolu (Rmult (d x0) (Rminus x1 x0))) - (Ropp (Rabsolu (Rmult (d x0) (Rminus x1 x0)))) - (Rabsolu (Rminus (f x1) (f x0)))); - Rewrite (Rplus_Ropp_r (Rabsolu (Rmult (d x0) (Rminus x1 x0)))); - Rewrite (let (H1,H2)=(Rplus_ne (Rabsolu (Rminus (f x1) (f x0)))) in H2); - Clear H1;Intro;Cut (Rlt (Rplus (Rabsolu (Rmult (d x0) (Rminus x1 x0))) - (Rmult (Rabsolu (Rminus x1 x0)) eps)) eps). -Intro;Apply (Rlt_trans (Rabsolu (Rminus (f x1) (f x0))) - (Rplus (Rabsolu (Rmult (d x0) (Rminus x1 x0))) - (Rmult (Rabsolu (Rminus x1 x0)) eps)) eps H1 H11). -Clear H1 H5 H3 H10;Generalize (Rabsolu_pos_lt (d x0) H2); - Intro;Unfold Rgt in H0;Generalize (Rlt_monotony eps (R_dist x1 x0) - (Rinv (Rplus R1 R1)) H0 H7);Clear H7;Intro; - Generalize (Rlt_monotony (Rabsolu (d x0)) (R_dist x1 x0) - (Rmult eps (Rinv (Rabsolu (Rmult (Rplus R1 R1) (d x0))))) H1 H6); - Clear H6;Intro;Rewrite (Rmult_sym eps (R_dist x1 x0)) in H3; - Unfold R_dist in H3 H5; - Rewrite <-(Rabsolu_mult (d x0) (Rminus x1 x0)) in H5; - Rewrite (Rabsolu_mult (Rplus R1 R1) (d x0)) in H5; - Cut ~(Rabsolu (Rplus R1 R1))==R0. -Intro;Fold (Rgt (Rabsolu (d x0)) R0) in H1; - Rewrite (Rinv_Rmult (Rabsolu (Rplus R1 R1)) (Rabsolu (d x0)) - H6 (imp_not_Req (Rabsolu (d x0)) R0 - (or_intror (Rlt (Rabsolu (d x0)) R0) (Rgt (Rabsolu (d x0)) R0) H1))) - in H5; - Rewrite (Rmult_sym (Rabsolu (d x0)) (Rmult eps - (Rmult (Rinv (Rabsolu (Rplus R1 R1))) - (Rinv (Rabsolu (d x0)))))) in H5; - Rewrite <-(Rmult_assoc eps (Rinv (Rabsolu (Rplus R1 R1))) - (Rinv (Rabsolu (d x0)))) in H5; - Rewrite (Rmult_assoc (Rmult eps (Rinv (Rabsolu (Rplus R1 R1)))) - (Rinv (Rabsolu (d x0))) (Rabsolu (d x0))) in H5; - Rewrite (Rinv_l (Rabsolu (d x0)) (imp_not_Req (Rabsolu (d x0)) R0 - (or_intror (Rlt (Rabsolu (d x0)) R0) (Rgt (Rabsolu (d x0)) R0) H1))) - in H5; - Rewrite (let (H1,H2)=(Rmult_ne (Rmult eps (Rinv (Rabsolu (Rplus R1 R1))))) - in H1) in H5;Cut (Rabsolu (Rplus R1 R1))==(Rplus R1 R1). -Intro;Rewrite H7 in H5; - Generalize (Rplus_lt (Rabsolu (Rmult (d x0) (Rminus x1 x0))) - (Rmult eps (Rinv (Rplus R1 R1))) - (Rmult (Rabsolu (Rminus x1 x0)) eps) - (Rmult eps (Rinv (Rplus R1 R1))) H5 H3);Intro; - Rewrite eps2 in H10;Assumption. -Unfold Rabsolu;Case (case_Rabsolu (Rplus R1 R1));Auto. - Intro;Cut (Rlt R0 (Rplus R1 R1)). -Intro;Generalize (Rlt_antisym R0 (Rplus R1 R1) H7);Intro;ElimType False; - Auto. -Fourier. -Apply Rabsolu_no_R0. -DiscrR. + split with (Rmin (Rmin (/ 2) x) (eps * / Rabs (2 * d x0))); split. +cut (Rmin (/ 2) x > 0). +cut (eps * / Rabs (2 * d x0) > 0). +intros; elim (Rmin_Rgt (Rmin (/ 2) x) (eps * / Rabs (2 * d x0)) 0); + intros a b; apply (b (conj H4 H3)). +apply Rmult_gt_0_compat; auto. +unfold Rgt in |- *; apply Rinv_0_lt_compat; apply Rabs_pos_lt; + apply Rmult_integral_contrapositive; split. +discrR. +assumption. +elim (Rmin_Rgt (/ 2) x 0); intros a b; cut (0 < 2). +intro; generalize (Rinv_0_lt_compat 2 H3); intro; fold (/ 2 > 0) in H4; + apply (b (conj H4 H)). +fourier. +intros; elim H3; clear H3; intros; + generalize + (let (H1, H2) := + Rmin_Rgt (Rmin (/ 2) x) (eps * / Rabs (2 * d x0)) (R_dist x1 x0) in + H1); unfold Rgt in |- *; intro; elim (H5 H4); clear H5; + intros; generalize (let (H1, H2) := Rmin_Rgt (/ 2) x (R_dist x1 x0) in H1); + unfold Rgt in |- *; intro; elim (H7 H5); clear H7; + intros; clear H4 H5; generalize (H1 x1 (conj H3 H8)); + clear H1; intro; unfold D_x in H3; elim H3; intros; + generalize (sym_not_eq H5); clear H5; intro H5; + generalize (Rminus_eq_contra x1 x0 H5); intro; generalize H1; + pattern (d x0) at 1 in |- *; + rewrite <- (let (H1, H2) := Rmult_ne (d x0) in H2); + rewrite <- (Rinv_l (x1 - x0) H9); unfold R_dist in |- *; + unfold Rminus at 1 in |- *; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))); + rewrite (Rmult_comm (/ (x1 - x0) * (x1 - x0)) (d x0)); + rewrite <- (Ropp_mult_distr_l_reverse (d x0) (/ (x1 - x0) * (x1 - x0))); + rewrite (Rmult_comm (- d x0) (/ (x1 - x0) * (x1 - x0))); + rewrite (Rmult_assoc (/ (x1 - x0)) (x1 - x0) (- d x0)); + rewrite <- + (Rmult_plus_distr_l (/ (x1 - x0)) (f x1 - f x0) ((x1 - x0) * - d x0)) + ; rewrite (Rabs_mult (/ (x1 - x0)) (f x1 - f x0 + (x1 - x0) * - d x0)); + clear H1; intro; + generalize + (Rmult_lt_compat_l (Rabs (x1 - x0)) + (Rabs (/ (x1 - x0)) * Rabs (f x1 - f x0 + (x1 - x0) * - d x0)) eps + (Rabs_pos_lt (x1 - x0) H9) H1); + rewrite <- + (Rmult_assoc (Rabs (x1 - x0)) (Rabs (/ (x1 - x0))) + (Rabs (f x1 - f x0 + (x1 - x0) * - d x0))); + rewrite (Rabs_Rinv (x1 - x0) H9); + rewrite (Rinv_r (Rabs (x1 - x0)) (Rabs_no_R0 (x1 - x0) H9)); + rewrite + (let (H1, H2) := Rmult_ne (Rabs (f x1 - f x0 + (x1 - x0) * - d x0)) in H2) + ; generalize (Rabs_triang_inv (f x1 - f x0) ((x1 - x0) * d x0)); + intro; rewrite (Rmult_comm (x1 - x0) (- d x0)); + rewrite (Ropp_mult_distr_l_reverse (d x0) (x1 - x0)); + fold (f x1 - f x0 - d x0 * (x1 - x0)) in |- *; + rewrite (Rmult_comm (x1 - x0) (d x0)) in H10; clear H1; + intro; + generalize + (Rle_lt_trans (Rabs (f x1 - f x0) - Rabs (d x0 * (x1 - x0))) + (Rabs (f x1 - f x0 - d x0 * (x1 - x0))) (Rabs (x1 - x0) * eps) H10 H1); + clear H1; intro; + generalize + (Rplus_lt_compat_l (Rabs (d x0 * (x1 - x0))) + (Rabs (f x1 - f x0) - Rabs (d x0 * (x1 - x0))) ( + Rabs (x1 - x0) * eps) H1); unfold Rminus at 2 in |- *; + rewrite (Rplus_comm (Rabs (f x1 - f x0)) (- Rabs (d x0 * (x1 - x0)))); + rewrite <- + (Rplus_assoc (Rabs (d x0 * (x1 - x0))) (- Rabs (d x0 * (x1 - x0))) + (Rabs (f x1 - f x0))); rewrite (Rplus_opp_r (Rabs (d x0 * (x1 - x0)))); + rewrite (let (H1, H2) := Rplus_ne (Rabs (f x1 - f x0)) in H2); + clear H1; intro; cut (Rabs (d x0 * (x1 - x0)) + Rabs (x1 - x0) * eps < eps). +intro; + apply + (Rlt_trans (Rabs (f x1 - f x0)) + (Rabs (d x0 * (x1 - x0)) + Rabs (x1 - x0) * eps) eps H1 H11). +clear H1 H5 H3 H10; generalize (Rabs_pos_lt (d x0) H2); intro; + unfold Rgt in H0; + generalize (Rmult_lt_compat_l eps (R_dist x1 x0) (/ 2) H0 H7); + clear H7; intro; + generalize + (Rmult_lt_compat_l (Rabs (d x0)) (R_dist x1 x0) ( + eps * / Rabs (2 * d x0)) H1 H6); clear H6; intro; + rewrite (Rmult_comm eps (R_dist x1 x0)) in H3; unfold R_dist in H3, H5; + rewrite <- (Rabs_mult (d x0) (x1 - x0)) in H5; + rewrite (Rabs_mult 2 (d x0)) in H5; cut (Rabs 2 <> 0). +intro; fold (Rabs (d x0) > 0) in H1; + rewrite + (Rinv_mult_distr (Rabs 2) (Rabs (d x0)) H6 + (Rlt_dichotomy_converse (Rabs (d x0)) 0 (or_intror (Rabs (d x0) < 0) H1))) + in H5; + rewrite (Rmult_comm (Rabs (d x0)) (eps * (/ Rabs 2 * / Rabs (d x0)))) in H5; + rewrite <- (Rmult_assoc eps (/ Rabs 2) (/ Rabs (d x0))) in H5; + rewrite (Rmult_assoc (eps * / Rabs 2) (/ Rabs (d x0)) (Rabs (d x0))) in H5; + rewrite + (Rinv_l (Rabs (d x0)) + (Rlt_dichotomy_converse (Rabs (d x0)) 0 (or_intror (Rabs (d x0) < 0) H1))) + in H5; rewrite (let (H1, H2) := Rmult_ne (eps * / Rabs 2) in H1) in H5; + cut (Rabs 2 = 2). +intro; rewrite H7 in H5; + generalize + (Rplus_lt_compat (Rabs (d x0 * (x1 - x0))) (eps * / 2) + (Rabs (x1 - x0) * eps) (eps * / 2) H5 H3); intro; + rewrite eps2 in H10; assumption. +unfold Rabs in |- *; case (Rcase_abs 2); auto. + intro; cut (0 < 2). +intro; generalize (Rlt_asym 0 2 H7); intro; elimtype False; auto. +fourier. +apply Rabs_no_R0. +discrR. Qed. (*********) -Lemma Dconst:(D:R->Prop)(y:R)(x0:R)(D_in [x:R]y [x:R]R0 D x0). -Unfold D_in;Intros;Unfold limit1_in;Unfold limit_in;Unfold Rdiv;Intros;Simpl; - Split with eps;Split;Auto. -Intros;Rewrite (eq_Rminus y y (refl_eqT R y)); - Rewrite Rmult_Ol;Unfold R_dist; - Rewrite (eq_Rminus R0 R0 (refl_eqT R R0));Unfold Rabsolu; - Case (case_Rabsolu R0);Intro. -Absurd (Rlt R0 R0);Auto. -Red;Intro;Apply (Rlt_antirefl R0 H1). -Unfold Rgt in H0;Assumption. +Lemma Dconst : + forall (D:R -> Prop) (y x0:R), D_in (fun x:R => y) (fun x:R => 0) D x0. +unfold D_in in |- *; intros; unfold limit1_in in |- *; + unfold limit_in in |- *; unfold Rdiv in |- *; intros; + simpl in |- *; split with eps; split; auto. +intros; rewrite (Rminus_diag_eq y y (refl_equal y)); rewrite Rmult_0_l; + unfold R_dist in |- *; rewrite (Rminus_diag_eq 0 0 (refl_equal 0)); + unfold Rabs in |- *; case (Rcase_abs 0); intro. +absurd (0 < 0); auto. +red in |- *; intro; apply (Rlt_irrefl 0 H1). +unfold Rgt in H0; assumption. Qed. (*********) -Lemma Dx:(D:R->Prop)(x0:R)(D_in [x:R]x [x:R]R1 D x0). -Unfold D_in;Unfold Rdiv;Intros;Unfold limit1_in;Unfold limit_in;Intros;Simpl; - Split with eps;Split;Auto. -Intros;Elim H0;Clear H0;Intros;Unfold D_x in H0; - Elim H0;Intros; - Rewrite (Rinv_r (Rminus x x0) (Rminus_eq_contra x x0 - (sym_not_eqT R x0 x H3))); - Unfold R_dist; - Rewrite (eq_Rminus R1 R1 (refl_eqT R R1));Unfold Rabsolu; - Case (case_Rabsolu R0);Intro. -Absurd (Rlt R0 R0);Auto. -Red;Intro;Apply (Rlt_antirefl R0 r). -Unfold Rgt in H;Assumption. +Lemma Dx : + forall (D:R -> Prop) (x0:R), D_in (fun x:R => x) (fun x:R => 1) D x0. +unfold D_in in |- *; unfold Rdiv in |- *; intros; unfold limit1_in in |- *; + unfold limit_in in |- *; intros; simpl in |- *; split with eps; + split; auto. +intros; elim H0; clear H0; intros; unfold D_x in H0; elim H0; intros; + rewrite (Rinv_r (x - x0) (Rminus_eq_contra x x0 (sym_not_eq H3))); + unfold R_dist in |- *; rewrite (Rminus_diag_eq 1 1 (refl_equal 1)); + unfold Rabs in |- *; case (Rcase_abs 0); intro. +absurd (0 < 0); auto. +red in |- *; intro; apply (Rlt_irrefl 0 r). +unfold Rgt in H; assumption. Qed. (*********) -Lemma Dadd:(D:R->Prop)(df,dg:R->R)(f,g:R->R)(x0:R) - (D_in f df D x0)->(D_in g dg D x0)-> - (D_in [x:R](Rplus (f x) (g x)) [x:R](Rplus (df x) (dg x)) D x0). -Unfold D_in;Intros;Generalize (limit_plus - [x:R](Rmult (Rminus (f x) (f x0)) (Rinv (Rminus x x0))) - [x:R](Rmult (Rminus (g x) (g x0)) (Rinv (Rminus x x0))) - (D_x D x0) (df x0) (dg x0) x0 H H0);Clear H H0; - 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; - Rewrite (Rmult_sym (Rminus (f x1) (f x0)) (Rinv (Rminus x1 x0))) in H1; - Rewrite (Rmult_sym (Rminus (g x1) (g x0)) (Rinv (Rminus x1 x0))) in H1; - Rewrite <-(Rmult_Rplus_distr (Rinv (Rminus x1 x0)) - (Rminus (f x1) (f x0)) - (Rminus (g x1) (g x0))) in H1; - Rewrite (Rmult_sym (Rinv (Rminus x1 x0)) - (Rplus (Rminus (f x1) (f x0)) (Rminus (g x1) (g x0)))) in H1; - Cut (Rplus (Rminus (f x1) (f x0)) (Rminus (g x1) (g x0)))== - (Rminus (Rplus (f x1) (g x1)) (Rplus (f x0) (g x0))). -Intro;Rewrite H3 in H1;Assumption. -Ring. +Lemma Dadd : + forall (D:R -> Prop) (df dg f g:R -> R) (x0:R), + D_in f df D x0 -> + D_in g dg D x0 -> + D_in (fun x:R => f x + g x) (fun x:R => df x + dg x) D x0. +unfold D_in in |- *; intros; + generalize + (limit_plus (fun x:R => (f x - f x0) * / (x - x0)) + (fun x:R => (g x - g x0) * / (x - x0)) (D_x D x0) ( + df x0) (dg x0) x0 H H0); clear H H0; unfold limit1_in in |- *; + unfold limit_in in |- *; simpl in |- *; 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; rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))) in H1; + rewrite (Rmult_comm (g x1 - g x0) (/ (x1 - x0))) in H1; + rewrite <- (Rmult_plus_distr_l (/ (x1 - x0)) (f x1 - f x0) (g x1 - g x0)) + in H1; + rewrite (Rmult_comm (/ (x1 - x0)) (f x1 - f x0 + (g x1 - g x0))) in H1; + cut (f x1 - f x0 + (g x1 - g x0) = f x1 + g x1 - (f x0 + g x0)). +intro; rewrite H3 in H1; assumption. +ring. Qed. (*********) -Lemma Dmult:(D:R->Prop)(df,dg:R->R)(f,g:R->R)(x0:R) - (D_in f df D x0)->(D_in g dg D x0)-> - (D_in [x:R](Rmult (f x) (g x)) - [x:R](Rplus (Rmult (df x) (g x)) (Rmult (f x) (dg x))) D x0). -Intros;Unfold D_in;Generalize H H0;Intros;Unfold D_in in H H0; - Generalize (cont_deriv f df D x0 H1);Unfold continue_in;Intro; - Generalize (limit_mul - [x:R](Rmult (Rminus (g x) (g x0)) (Rinv (Rminus x x0))) - [x:R](f x) (D_x D x0) (dg x0) (f x0) x0 H0 H3);Intro; - Cut (limit1_in [x:R](g x0) (D_x D x0) (g x0) x0). -Intro;Generalize (limit_mul - [x:R](Rmult (Rminus (f x) (f x0)) (Rinv (Rminus x x0))) - [_:R](g x0) (D_x D x0) (df x0) (g x0) x0 H H5);Clear H H0 H1 H2 H3 H5; - Intro;Generalize (limit_plus - [x:R](Rmult (Rmult (Rminus (f x) (f x0)) (Rinv (Rminus x x0))) (g x0)) - [x:R](Rmult (Rmult (Rminus (g x) (g x0)) (Rinv (Rminus x x0))) - (f x)) (D_x D x0) (Rmult (df x0) (g x0)) - (Rmult (dg x0) (f x0)) x0 H H4); - Clear H4 H;Intro;Unfold limit1_in in H;Unfold limit_in in H; - Simpl in H;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; - Rewrite (Rmult_sym (Rminus (f x1) (f x0)) (Rinv (Rminus x1 x0))) in H1; - Rewrite (Rmult_sym (Rminus (g x1) (g x0)) (Rinv (Rminus x1 x0))) in H1; - Rewrite (Rmult_assoc (Rinv (Rminus x1 x0)) (Rminus (f x1) (f x0)) - (g x0)) in H1; - Rewrite (Rmult_assoc (Rinv (Rminus x1 x0)) (Rminus (g x1) (g x0)) - (f x1)) in H1; - Rewrite <-(Rmult_Rplus_distr (Rinv (Rminus x1 x0)) - (Rmult (Rminus (f x1) (f x0)) (g x0)) - (Rmult (Rminus (g x1) (g x0)) (f x1))) in H1; - Rewrite (Rmult_sym (Rinv (Rminus x1 x0)) - (Rplus (Rmult (Rminus (f x1) (f x0)) (g x0)) - (Rmult (Rminus (g x1) (g x0)) (f x1)))) in H1; - Rewrite (Rmult_sym (dg x0) (f x0)) in H1; - Cut (Rplus (Rmult (Rminus (f x1) (f x0)) (g x0)) - (Rmult (Rminus (g x1) (g x0)) (f x1)))== - (Rminus (Rmult (f x1) (g x1)) (Rmult (f x0) (g x0))). -Intro;Rewrite H3 in H1;Assumption. -Ring. -Unfold limit1_in;Unfold limit_in;Simpl;Intros; - Split with eps;Split;Auto;Intros;Elim (R_dist_refl (g x0) (g x0)); - Intros a b;Rewrite (b (refl_eqT R (g x0)));Unfold Rgt in H;Assumption. +Lemma Dmult : + forall (D:R -> Prop) (df dg f g:R -> R) (x0:R), + D_in f df D x0 -> + D_in g dg D x0 -> + D_in (fun x:R => f x * g x) (fun x:R => df x * g x + f x * dg x) D x0. +intros; unfold D_in in |- *; generalize H H0; intros; unfold D_in in H, H0; + generalize (cont_deriv f df D x0 H1); unfold continue_in in |- *; + intro; + generalize + (limit_mul (fun x:R => (g x - g x0) * / (x - x0)) ( + fun x:R => f x) (D_x D x0) (dg x0) (f x0) x0 H0 H3); + intro; cut (limit1_in (fun x:R => g x0) (D_x D x0) (g x0) x0). +intro; + generalize + (limit_mul (fun x:R => (f x - f x0) * / (x - x0)) ( + fun _:R => g x0) (D_x D x0) (df x0) (g x0) x0 H H5); + clear H H0 H1 H2 H3 H5; intro; + generalize + (limit_plus (fun x:R => (f x - f x0) * / (x - x0) * g x0) + (fun x:R => (g x - g x0) * / (x - x0) * f x) ( + D_x D x0) (df x0 * g x0) (dg x0 * f x0) x0 H H4); + clear H4 H; intro; unfold limit1_in in H; unfold limit_in in H; + simpl in H; unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; 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; + rewrite (Rmult_comm (f x1 - f x0) (/ (x1 - x0))) in H1; + rewrite (Rmult_comm (g x1 - g x0) (/ (x1 - x0))) in H1; + rewrite (Rmult_assoc (/ (x1 - x0)) (f x1 - f x0) (g x0)) in H1; + rewrite (Rmult_assoc (/ (x1 - x0)) (g x1 - g x0) (f x1)) in H1; + rewrite <- + (Rmult_plus_distr_l (/ (x1 - x0)) ((f x1 - f x0) * g x0) + ((g x1 - g x0) * f x1)) in H1; + rewrite + (Rmult_comm (/ (x1 - x0)) ((f x1 - f x0) * g x0 + (g x1 - g x0) * f x1)) + in H1; rewrite (Rmult_comm (dg x0) (f x0)) in H1; + cut + ((f x1 - f x0) * g x0 + (g x1 - g x0) * f x1 = f x1 * g x1 - f x0 * g x0). +intro; rewrite H3 in H1; assumption. +ring. +unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros; + split with eps; split; auto; intros; elim (R_dist_refl (g x0) (g x0)); + intros a b; rewrite (b (refl_equal (g x0))); unfold Rgt in H; + assumption. Qed. (*********) -Lemma Dmult_const:(D:R->Prop)(f,df:R->R)(x0:R)(a:R)(D_in f df D x0)-> - (D_in [x:R](Rmult a (f x)) ([x:R](Rmult a (df x))) D x0). -Intros;Generalize (Dmult D [_:R]R0 df [_:R]a f x0 (Dconst D a x0) H); - Unfold D_in;Intros; - Rewrite (Rmult_Ol (f x0)) in H0; - Rewrite (let (H1,H2)=(Rplus_ne (Rmult a (df x0))) in H2) in H0; - Assumption. +Lemma Dmult_const : + forall (D:R -> Prop) (f df:R -> R) (x0 a:R), + D_in f df D x0 -> D_in (fun x:R => a * f x) (fun x:R => a * df x) D x0. +intros; + generalize (Dmult D (fun _:R => 0) df (fun _:R => a) f x0 (Dconst D a x0) H); + unfold D_in in |- *; intros; rewrite (Rmult_0_l (f x0)) in H0; + rewrite (let (H1, H2) := Rplus_ne (a * df x0) in H2) in H0; + assumption. Qed. (*********) -Lemma Dopp:(D:R->Prop)(f,df:R->R)(x0:R)(D_in f df D x0)-> - (D_in [x:R](Ropp (f x)) ([x:R](Ropp (df x))) D x0). -Intros;Generalize (Dmult_const D f df x0 (Ropp R1) H); Unfold D_in; - Unfold limit1_in;Unfold limit_in;Intros; - Generalize (H0 eps H1);Clear H0;Intro;Elim H0;Clear H0;Intros; - Elim H0;Clear H0;Simpl;Intros;Split with x;Split;Auto. -Intros;Generalize (H2 x1 H3);Clear H2;Intro;Rewrite Ropp_mul1 in H2; - Rewrite Ropp_mul1 in H2;Rewrite Ropp_mul1 in H2; - Rewrite (let (H1,H2)=(Rmult_ne (f x1)) in H2) in H2; - Rewrite (let (H1,H2)=(Rmult_ne (f x0)) in H2) in H2; - Rewrite (let (H1,H2)=(Rmult_ne (df x0)) in H2) in H2;Assumption. +Lemma Dopp : + forall (D:R -> Prop) (f df:R -> R) (x0:R), + D_in f df D x0 -> D_in (fun x:R => - f x) (fun x:R => - df x) D x0. +intros; generalize (Dmult_const D f df x0 (-1) H); unfold D_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + intros; generalize (H0 eps H1); clear H0; intro; elim H0; + clear H0; intros; elim H0; clear H0; simpl in |- *; + intros; split with x; split; auto. +intros; generalize (H2 x1 H3); clear H2; intro; + rewrite Ropp_mult_distr_l_reverse in H2; + rewrite Ropp_mult_distr_l_reverse in H2; + rewrite Ropp_mult_distr_l_reverse in H2; + rewrite (let (H1, H2) := Rmult_ne (f x1) in H2) in H2; + rewrite (let (H1, H2) := Rmult_ne (f x0) in H2) in H2; + rewrite (let (H1, H2) := Rmult_ne (df x0) in H2) in H2; + assumption. Qed. (*********) -Lemma Dminus:(D:R->Prop)(df,dg:R->R)(f,g:R->R)(x0:R) - (D_in f df D x0)->(D_in g dg D x0)-> - (D_in [x:R](Rminus (f x) (g x)) [x:R](Rminus (df x) (dg x)) D x0). -Unfold Rminus;Intros;Generalize (Dopp D g dg x0 H0);Intro; - Apply (Dadd D df [x:R](Ropp (dg x)) f [x:R](Ropp (g x)) x0);Assumption. +Lemma Dminus : + forall (D:R -> Prop) (df dg f g:R -> R) (x0:R), + D_in f df D x0 -> + D_in g dg D x0 -> + D_in (fun x:R => f x - g x) (fun x:R => df x - dg x) D x0. +unfold Rminus in |- *; intros; generalize (Dopp D g dg x0 H0); intro; + apply (Dadd D df (fun x:R => - dg x) f (fun x:R => - g x) x0); + assumption. Qed. (*********) -Lemma Dx_pow_n:(n:nat)(D:R->Prop)(x0:R) - (D_in [x:R](pow x n) - [x:R](Rmult (INR n) (pow x (minus n (1)))) D x0). -Induction n;Intros. -Simpl; Rewrite Rmult_Ol; Apply Dconst. -Intros;Cut n0=(minus (S n0) (1)); - [ Intro a; Rewrite <- a;Clear a | Simpl; Apply minus_n_O ]. -Generalize (Dmult D [_:R]R1 - [x:R](Rmult (INR n0) (pow x (minus n0 (1)))) [x:R]x [x:R](pow x n0) - x0 (Dx D x0) (H D x0));Unfold D_in;Unfold limit1_in;Unfold limit_in; - Simpl;Intros; - Elim (H0 eps H1);Clear H0;Intros;Elim H0;Clear H0;Intros; - Split with x;Split;Auto. -Intros;Generalize (H2 x1 H3);Clear H2 H3;Intro; - Rewrite (let (H1,H2)=(Rmult_ne (pow x0 n0)) in H2) in H2; - Rewrite (tech_pow_Rmult x1 n0) in H2; - Rewrite (tech_pow_Rmult x0 n0) in H2; - Rewrite (Rmult_sym (INR n0) (pow x0 (minus n0 (1)))) in H2; - Rewrite <-(Rmult_assoc x0 (pow x0 (minus n0 (1))) (INR n0)) in H2; - Rewrite (tech_pow_Rmult x0 (minus n0 (1))) in H2; - Elim (classic (n0=O));Intro cond. -Rewrite cond in H2;Rewrite cond;Simpl in H2;Simpl; - Cut (Rplus R1 (Rmult (Rmult x0 R1) R0))==(Rmult R1 R1); - [Intro A; Rewrite A in H2; Assumption|Ring]. -Cut ~(n0=O)->(S (minus n0 (1)))=n0;[Intro|Omega]; - Rewrite (H3 cond) in H2; Rewrite (Rmult_sym (pow x0 n0) (INR n0)) in H2; - Rewrite (tech_pow_Rplus x0 n0 n0) in H2; Assumption. +Lemma Dx_pow_n : + forall (n:nat) (D:R -> Prop) (x0:R), + D_in (fun x:R => x ^ n) (fun x:R => INR n * x ^ (n - 1)) D x0. +simple induction n; intros. +simpl in |- *; rewrite Rmult_0_l; apply Dconst. +intros; cut (n0 = (S n0 - 1)%nat); + [ intro a; rewrite <- a; clear a | simpl in |- *; apply minus_n_O ]. +generalize + (Dmult D (fun _:R => 1) (fun x:R => INR n0 * x ^ (n0 - 1)) ( + fun x:R => x) (fun x:R => x ^ n0) x0 (Dx D x0) ( + H D x0)); unfold D_in in |- *; unfold limit1_in in |- *; + unfold limit_in in |- *; simpl in |- *; intros; elim (H0 eps H1); + clear H0; intros; elim H0; clear H0; intros; split with x; + split; auto. +intros; generalize (H2 x1 H3); clear H2 H3; intro; + rewrite (let (H1, H2) := Rmult_ne (x0 ^ n0) in H2) in H2; + rewrite (tech_pow_Rmult x1 n0) in H2; rewrite (tech_pow_Rmult x0 n0) in H2; + rewrite (Rmult_comm (INR n0) (x0 ^ (n0 - 1))) in H2; + rewrite <- (Rmult_assoc x0 (x0 ^ (n0 - 1)) (INR n0)) in H2; + rewrite (tech_pow_Rmult x0 (n0 - 1)) in H2; elim (classic (n0 = 0%nat)); + intro cond. +rewrite cond in H2; rewrite cond; simpl in H2; simpl in |- *; + cut (1 + x0 * 1 * 0 = 1 * 1); + [ intro A; rewrite A in H2; assumption | ring ]. +cut (n0 <> 0%nat -> S (n0 - 1) = n0); [ intro | omega ]; + rewrite (H3 cond) in H2; rewrite (Rmult_comm (x0 ^ n0) (INR n0)) in H2; + rewrite (tech_pow_Rplus x0 n0 n0) in H2; assumption. Qed. (*********) -Lemma Dcomp:(Df,Dg:R->Prop)(df,dg:R->R)(f,g:R->R)(x0:R) - (D_in f df Df x0)->(D_in g dg Dg (f x0))-> - (D_in [x:R](g (f x)) [x:R](Rmult (df x) (dg (f x))) - (Dgf Df Dg f) x0). -Intros Df Dg df dg f g x0 H H0;Generalize H H0;Unfold D_in;Unfold Rdiv;Intros; -Generalize (limit_comp f [x:R](Rmult (Rminus (g x) (g (f x0))) - (Rinv (Rminus x (f x0)))) (D_x Df x0) - (D_x Dg (f x0)) - (f x0) (dg (f x0)) x0);Intro; - Generalize (cont_deriv f df Df x0 H);Intro;Unfold continue_in in H4; - Generalize (H3 H4 H2);Clear H3;Intro; - Generalize (limit_mul [x:R](Rmult (Rminus (g (f x)) (g (f x0))) - (Rinv (Rminus (f x) (f x0)))) - [x:R](Rmult (Rminus (f x) (f x0)) - (Rinv (Rminus x x0))) - (Dgf (D_x Df x0) (D_x Dg (f x0)) f) - (dg (f x0)) (df x0) x0 H3);Intro; - Cut (limit1_in - [x:R](Rmult (Rminus (f x) (f x0)) (Rinv (Rminus x x0))) - (Dgf (D_x Df x0) (D_x Dg (f x0)) f) (df x0) x0). -Intro;Generalize (H5 H6);Clear H5;Intro; - Generalize (limit_mul - [x:R](Rmult (Rminus (f x) (f x0)) (Rinv (Rminus x x0))) - [x:R](dg (f x0)) - (D_x Df x0) (df x0) (dg (f x0)) x0 H1 - (limit_free [x:R](dg (f x0)) (D_x Df x0) x0 x0)); - Intro; - Unfold limit1_in;Unfold limit_in;Simpl;Unfold limit1_in in H5 H7; - Unfold limit_in in H5 H7;Simpl in H5 H7;Intros;Elim (H5 eps H8); - Elim (H7 eps H8);Clear H5 H7;Intros;Elim H5;Elim H7;Clear H5 H7; - Intros;Split with (Rmin x x1);Split. -Elim (Rmin_Rgt x x1 R0);Intros a b; - Apply (b (conj (Rgt x R0) (Rgt x1 R0) H9 H5));Clear a b. -Intros;Elim H11;Clear H11;Intros;Elim (Rmin_Rgt x x1 (R_dist x2 x0)); - Intros a b;Clear b;Unfold Rgt in a;Elim (a H12);Clear H5 a;Intros; - Unfold D_x Dgf in H11 H7 H10;Clear H12; - Elim (classic (f x2)==(f x0));Intro. -Elim H11;Clear H11;Intros;Elim H11;Clear H11;Intros; - Generalize (H10 x2 (conj (Df x2)/\~x0==x2 (Rlt (R_dist x2 x0) x) - (conj (Df x2) ~x0==x2 H11 H14) H5));Intro; - Rewrite (eq_Rminus (f x2) (f x0) H12) in H16; - Rewrite (Rmult_Ol (Rinv (Rminus x2 x0))) in H16; - Rewrite (Rmult_Ol (dg (f x0))) in H16; - Rewrite H12; - Rewrite (eq_Rminus (g (f x0)) (g (f x0)) (refl_eqT R (g (f x0)))); - Rewrite (Rmult_Ol (Rinv (Rminus x2 x0)));Assumption. -Clear H10 H5;Elim H11;Clear H11;Intros;Elim H5;Clear H5;Intros; -Cut (((Df x2)/\~x0==x2)/\(Dg (f x2))/\~(f x0)==(f x2)) - /\(Rlt (R_dist x2 x0) x1);Auto;Intro; - Generalize (H7 x2 H14);Intro; - Generalize (Rminus_eq_contra (f x2) (f x0) H12);Intro; - Rewrite (Rmult_assoc (Rminus (g (f x2)) (g (f x0))) - (Rinv (Rminus (f x2) (f x0))) - (Rmult (Rminus (f x2) (f x0)) (Rinv (Rminus x2 x0)))) in H15; - Rewrite <-(Rmult_assoc (Rinv (Rminus (f x2) (f x0))) - (Rminus (f x2) (f x0)) (Rinv (Rminus x2 x0))) in H15; - Rewrite (Rinv_l (Rminus (f x2) (f x0)) H16) in H15; - Rewrite (let (H1,H2)=(Rmult_ne (Rinv (Rminus x2 x0))) in H2) in H15; - Rewrite (Rmult_sym (df x0) (dg (f x0)));Assumption. -Clear H5 H3 H4 H2;Unfold limit1_in;Unfold limit_in;Simpl; - Unfold limit1_in in H1;Unfold limit_in in H1;Simpl in H1;Intros; - Elim (H1 eps H2);Clear H1;Intros;Elim H1;Clear H1;Intros; - Split with x;Split;Auto;Intros;Unfold D_x Dgf in H4 H3; - Elim H4;Clear H4;Intros;Elim H4;Clear H4;Intros; - Exact (H3 x1 (conj (Df x1)/\~x0==x1 (Rlt (R_dist x1 x0) x) H4 H5)). +Lemma Dcomp : + forall (Df Dg:R -> Prop) (df dg f g:R -> R) (x0:R), + D_in f df Df x0 -> + D_in g dg Dg (f x0) -> + D_in (fun x:R => g (f x)) (fun x:R => df x * dg (f x)) (Dgf Df Dg f) x0. +intros Df Dg df dg f g x0 H H0; generalize H H0; unfold D_in in |- *; + unfold Rdiv in |- *; intros; + generalize + (limit_comp f (fun x:R => (g x - g (f x0)) * / (x - f x0)) ( + D_x Df x0) (D_x Dg (f x0)) (f x0) (dg (f x0)) x0); + intro; generalize (cont_deriv f df Df x0 H); intro; + unfold continue_in in H4; generalize (H3 H4 H2); clear H3; + intro; + generalize + (limit_mul (fun x:R => (g (f x) - g (f x0)) * / (f x - f x0)) + (fun x:R => (f x - f x0) * / (x - x0)) + (Dgf (D_x Df x0) (D_x Dg (f x0)) f) (dg (f x0)) ( + df x0) x0 H3); intro; + cut + (limit1_in (fun x:R => (f x - f x0) * / (x - x0)) + (Dgf (D_x Df x0) (D_x Dg (f x0)) f) (df x0) x0). +intro; generalize (H5 H6); clear H5; intro; + generalize + (limit_mul (fun x:R => (f x - f x0) * / (x - x0)) ( + fun x:R => dg (f x0)) (D_x Df x0) (df x0) (dg (f x0)) x0 H1 + (limit_free (fun x:R => dg (f x0)) (D_x Df x0) x0 x0)); + intro; unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold limit1_in in H5, H7; unfold limit_in in H5, H7; + simpl in H5, H7; intros; elim (H5 eps H8); elim (H7 eps H8); + clear H5 H7; intros; elim H5; elim H7; clear H5 H7; + intros; split with (Rmin x x1); split. +elim (Rmin_Rgt x x1 0); intros a b; apply (b (conj H9 H5)); clear a b. +intros; elim H11; clear H11; intros; elim (Rmin_Rgt x x1 (R_dist x2 x0)); + intros a b; clear b; unfold Rgt in a; elim (a H12); + clear H5 a; intros; unfold D_x, Dgf in H11, H7, H10; + clear H12; elim (classic (f x2 = f x0)); intro. +elim H11; clear H11; intros; elim H11; clear H11; intros; + generalize (H10 x2 (conj (conj H11 H14) H5)); intro; + rewrite (Rminus_diag_eq (f x2) (f x0) H12) in H16; + rewrite (Rmult_0_l (/ (x2 - x0))) in H16; + rewrite (Rmult_0_l (dg (f x0))) in H16; rewrite H12; + rewrite (Rminus_diag_eq (g (f x0)) (g (f x0)) (refl_equal (g (f x0)))); + rewrite (Rmult_0_l (/ (x2 - x0))); assumption. +clear H10 H5; elim H11; clear H11; intros; elim H5; clear H5; intros; + cut + (((Df x2 /\ x0 <> x2) /\ Dg (f x2) /\ f x0 <> f x2) /\ R_dist x2 x0 < x1); + auto; intro; generalize (H7 x2 H14); intro; + generalize (Rminus_eq_contra (f x2) (f x0) H12); intro; + rewrite + (Rmult_assoc (g (f x2) - g (f x0)) (/ (f x2 - f x0)) + ((f x2 - f x0) * / (x2 - x0))) in H15; + rewrite <- (Rmult_assoc (/ (f x2 - f x0)) (f x2 - f x0) (/ (x2 - x0))) + in H15; rewrite (Rinv_l (f x2 - f x0) H16) in H15; + rewrite (let (H1, H2) := Rmult_ne (/ (x2 - x0)) in H2) in H15; + rewrite (Rmult_comm (df x0) (dg (f x0))); assumption. +clear H5 H3 H4 H2; unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold limit1_in in H1; unfold limit_in in H1; + simpl in H1; intros; elim (H1 eps H2); clear H1; intros; + elim H1; clear H1; intros; split with x; split; auto; + intros; unfold D_x, Dgf in H4, H3; elim H4; clear H4; + intros; elim H4; clear H4; intros; exact (H3 x1 (conj H4 H5)). Qed. (*********) -Lemma D_pow_n:(n:nat)(D:R->Prop)(x0:R)(expr,dexpr:R->R) - (D_in expr dexpr D x0)-> (D_in [x:R](pow (expr x) n) - [x:R](Rmult (Rmult (INR n) (pow (expr x) (minus n (1)))) (dexpr x)) - (Dgf D D expr) x0). -Intros n D x0 expr dexpr H; - Generalize (Dcomp D D dexpr [x:R](Rmult (INR n) (pow x (minus n (1)))) - expr [x:R](pow x n) x0 H (Dx_pow_n n D (expr x0))); - Intro; Unfold D_in; Unfold limit1_in; Unfold limit_in;Simpl;Intros; - Unfold D_in in H0; Unfold limit1_in in H0; Unfold limit_in in H0;Simpl in H0; - Elim (H0 eps H1);Clear H0;Intros;Elim H0;Clear H0;Intros;Split with x;Split; - Intros; Auto. -Cut ``((dexpr x0)*((INR n)*(pow (expr x0) (minus n (S O)))))== - ((INR n)*(pow (expr x0) (minus n (S O)))*(dexpr x0))``; - [Intro Rew;Rewrite <- Rew;Exact (H2 x1 H3)|Ring]. +Lemma D_pow_n : + forall (n:nat) (D:R -> Prop) (x0:R) (expr dexpr:R -> R), + D_in expr dexpr D x0 -> + D_in (fun x:R => expr x ^ n) + (fun x:R => INR n * expr x ^ (n - 1) * dexpr x) ( + Dgf D D expr) x0. +intros n D x0 expr dexpr H; + generalize + (Dcomp D D dexpr (fun x:R => INR n * x ^ (n - 1)) expr ( + fun x:R => x ^ n) x0 H (Dx_pow_n n D (expr x0))); + intro; unfold D_in in |- *; unfold limit1_in in |- *; + unfold limit_in in |- *; simpl in |- *; intros; unfold D_in in H0; + unfold limit1_in in H0; unfold limit_in in H0; simpl in H0; + elim (H0 eps H1); clear H0; intros; elim H0; clear H0; + intros; split with x; split; intros; auto. +cut + (dexpr x0 * (INR n * expr x0 ^ (n - 1)) = + INR n * expr x0 ^ (n - 1) * dexpr x0); + [ intro Rew; rewrite <- Rew; exact (H2 x1 H3) | ring ]. Qed. - diff --git a/theories/Reals/Reals.v b/theories/Reals/Reals.v index db6df635c..6e10fa8f1 100644 --- a/theories/Reals/Reals.v +++ b/theories/Reals/Reals.v @@ -29,4 +29,4 @@ Require Export Rfunctions. Require Export SeqSeries. Require Export Rtrigo. Require Export Ranalysis. -Require Export Integration. +Require Export Integration.
\ No newline at end of file diff --git a/theories/Reals/Rfunctions.v b/theories/Reals/Rfunctions.v index b283b9fd8..30b4a5396 100644 --- a/theories/Reals/Rfunctions.v +++ b/theories/Reals/Rfunctions.v @@ -16,16 +16,15 @@ (* *) (********************************************************) -Require Rbase. +Require Import Rbase. Require Export R_Ifp. Require Export Rbasic_fun. Require Export R_sqr. Require Export SplitAbsolu. Require Export SplitRmult. Require Export ArithProp. -Require Omega. -Require Zpower. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Omega. +Require Import Zpower. Open Local Scope nat_scope. Open Local Scope R_scope. @@ -33,522 +32,491 @@ Open Local Scope R_scope. (** Lemmas about factorial *) (*******************************) (*********) -Lemma INR_fact_neq_0:(n:nat)~(INR (fact n))==R0. +Lemma INR_fact_neq_0 : forall n:nat, INR (fact n) <> 0. Proof. -Intro;Red;Intro;Apply (not_O_INR (fact n) (fact_neq_0 n));Assumption. +intro; red in |- *; intro; apply (not_O_INR (fact n) (fact_neq_0 n)); + assumption. Qed. (*********) -Lemma fact_simpl : (n:nat) (fact (S n))=(mult (S n) (fact n)). +Lemma fact_simpl : forall n:nat, fact (S n) = (S n * fact n)%nat. Proof. -Intro; Reflexivity. +intro; reflexivity. Qed. (*********) -Lemma simpl_fact:(n:nat)(Rmult (Rinv (INR (fact (S n)))) - (Rinv (Rinv (INR (fact n)))))== - (Rinv (INR (S n))). +Lemma simpl_fact : + forall n:nat, / INR (fact (S n)) * / / INR (fact n) = / INR (S n). Proof. -Intro;Rewrite (Rinv_Rinv (INR (fact n)) (INR_fact_neq_0 n)); - Unfold 1 fact;Cbv Beta Iota;Fold fact; - Rewrite (mult_INR (S n) (fact n)); - Rewrite (Rinv_Rmult (INR (S n)) (INR (fact n))). -Rewrite (Rmult_assoc (Rinv (INR (S n))) (Rinv (INR (fact n))) - (INR (fact n)));Rewrite (Rinv_l (INR (fact n)) (INR_fact_neq_0 n)); - Apply (let (H1,H2)=(Rmult_ne (Rinv (INR (S n)))) in H1). -Apply not_O_INR;Auto. -Apply INR_fact_neq_0. +intro; rewrite (Rinv_involutive (INR (fact n)) (INR_fact_neq_0 n)); + unfold fact at 1 in |- *; cbv beta iota in |- *; fold fact in |- *; + rewrite (mult_INR (S n) (fact n)); + rewrite (Rinv_mult_distr (INR (S n)) (INR (fact n))). +rewrite (Rmult_assoc (/ INR (S n)) (/ INR (fact n)) (INR (fact n))); + rewrite (Rinv_l (INR (fact n)) (INR_fact_neq_0 n)); + apply (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1). +apply not_O_INR; auto. +apply INR_fact_neq_0. Qed. (*******************************) (* Power *) (*******************************) (*********) -Fixpoint pow [r:R;n:nat]:R:= - Cases n of - O => R1 - |(S n) => (Rmult r (pow r n)) +Fixpoint pow (r:R) (n:nat) {struct n} : R := + match n with + | O => 1 + | S n => r * pow r n end. -Infix "^" pow (at level 2, left associativity) : R_scope V8only. +Infix "^" := pow : R_scope. -Lemma pow_O: (x : R) (pow x O) == R1. +Lemma pow_O : forall x:R, x ^ 0 = 1. Proof. -Reflexivity. +reflexivity. Qed. -Lemma pow_1: (x : R) (pow x (1)) == x. +Lemma pow_1 : forall x:R, x ^ 1 = x. Proof. -Simpl; Auto with real. +simpl in |- *; auto with real. Qed. -Lemma pow_add: - (x : R) (n, m : nat) (pow x (plus n m)) == (Rmult (pow x n) (pow x m)). +Lemma pow_add : forall (x:R) (n m:nat), x ^ (n + m) = x ^ n * x ^ m. Proof. -Intros x n; Elim n; Simpl; Auto with real. -Intros n0 H' m; Rewrite H'; Auto with real. +intros x n; elim n; simpl in |- *; auto with real. +intros n0 H' m; rewrite H'; auto with real. Qed. -Lemma pow_nonzero: - (x:R) (n:nat) ~(x==R0) -> ~((pow x n)==R0). +Lemma pow_nonzero : forall (x:R) (n:nat), x <> 0 -> x ^ n <> 0. Proof. -Intro; Induction n; Simpl. -Intro; Red;Intro;Apply R1_neq_R0;Assumption. -Intros;Red; Intro;Elim (without_div_Od x (pow x n0) H1). -Intro; Auto. -Apply H;Assumption. +intro; simple induction n; simpl in |- *. +intro; red in |- *; intro; apply R1_neq_R0; assumption. +intros; red in |- *; intro; elim (Rmult_integral x (x ^ n0) H1). +intro; auto. +apply H; assumption. Qed. -Hints Resolve pow_O pow_1 pow_add pow_nonzero:real. +Hint Resolve pow_O pow_1 pow_add pow_nonzero: real. -Lemma pow_RN_plus: - (x : R) - (n, m : nat) - ~ x == R0 -> (pow x n) == (Rmult (pow x (plus n m)) (Rinv (pow x m))). -Proof. -Intros x n; Elim n; Simpl; Auto with real. -Intros n0 H' m H'0. -Rewrite Rmult_assoc; Rewrite <- H'; Auto. -Qed. - -Lemma pow_lt: (x : R) (n : nat) (Rlt R0 x) -> (Rlt R0 (pow x n)). -Proof. -Intros x n; Elim n; Simpl; Auto with real. -Intros n0 H' H'0; Replace R0 with (Rmult x R0); Auto with real. -Qed. -Hints Resolve pow_lt :real. - -Lemma Rlt_pow_R1: - (x : R) (n : nat) (Rlt R1 x) -> (lt O n) -> (Rlt R1 (pow x n)). -Proof. -Intros x n; Elim n; Simpl; Auto with real. -Intros H' H'0; ElimType False; Omega. -Intros n0; Case n0. -Simpl; Rewrite Rmult_1r; Auto. -Intros n1 H' H'0 H'1. -Replace R1 with (Rmult R1 R1); Auto with real. -Apply Rlt_trans with r2 := (Rmult x R1); Auto with real. -Apply Rlt_monotony; Auto with real. -Apply Rlt_trans with r2 := R1; Auto with real. -Apply H'; Auto with arith. -Qed. -Hints Resolve Rlt_pow_R1 :real. - -Lemma Rlt_pow: - (x : R) (n, m : nat) (Rlt R1 x) -> (lt n m) -> (Rlt (pow x n) (pow x m)). -Proof. -Intros x n m H' H'0; Replace m with (plus (minus m n) n). -Rewrite pow_add. -Pattern 1 (pow x n); Replace (pow x n) with (Rmult R1 (pow x n)); - Auto with real. -Apply Rminus_lt. -Repeat Rewrite [y : R] (Rmult_sym y (pow x n)); Rewrite <- Rminus_distr. -Replace R0 with (Rmult (pow x n) R0); Auto with real. -Apply Rlt_monotony; Auto with real. -Apply pow_lt; Auto with real. -Apply Rlt_trans with r2 := R1; Auto with real. -Apply Rlt_minus; Auto with real. -Apply Rlt_pow_R1; Auto with arith. -Apply simpl_lt_plus_l with p := n; Auto with arith. -Rewrite le_plus_minus_r; Auto with arith; Rewrite <- plus_n_O; Auto. -Rewrite plus_sym; Auto with arith. -Qed. -Hints Resolve Rlt_pow :real. +Lemma pow_RN_plus : + forall (x:R) (n m:nat), x <> 0 -> x ^ n = x ^ (n + m) * / x ^ m. +Proof. +intros x n; elim n; simpl in |- *; auto with real. +intros n0 H' m H'0. +rewrite Rmult_assoc; rewrite <- H'; auto. +Qed. + +Lemma pow_lt : forall (x:R) (n:nat), 0 < x -> 0 < x ^ n. +Proof. +intros x n; elim n; simpl in |- *; auto with real. +intros n0 H' H'0; replace 0 with (x * 0); auto with real. +Qed. +Hint Resolve pow_lt: real. + +Lemma Rlt_pow_R1 : forall (x:R) (n:nat), 1 < x -> (0 < n)%nat -> 1 < x ^ n. +Proof. +intros x n; elim n; simpl in |- *; auto with real. +intros H' H'0; elimtype False; omega. +intros n0; case n0. +simpl in |- *; rewrite Rmult_1_r; auto. +intros n1 H' H'0 H'1. +replace 1 with (1 * 1); auto with real. +apply Rlt_trans with (r2 := x * 1); auto with real. +apply Rmult_lt_compat_l; auto with real. +apply Rlt_trans with (r2 := 1); auto with real. +apply H'; auto with arith. +Qed. +Hint Resolve Rlt_pow_R1: real. + +Lemma Rlt_pow : forall (x:R) (n m:nat), 1 < x -> (n < m)%nat -> x ^ n < x ^ m. +Proof. +intros x n m H' H'0; replace m with (m - n + n)%nat. +rewrite pow_add. +pattern (x ^ n) at 1 in |- *; replace (x ^ n) with (1 * x ^ n); + auto with real. +apply Rminus_lt. +repeat rewrite (fun y:R => Rmult_comm y (x ^ n)); + rewrite <- Rmult_minus_distr_l. +replace 0 with (x ^ n * 0); auto with real. +apply Rmult_lt_compat_l; auto with real. +apply pow_lt; auto with real. +apply Rlt_trans with (r2 := 1); auto with real. +apply Rlt_minus; auto with real. +apply Rlt_pow_R1; auto with arith. +apply plus_lt_reg_l with (p := n); auto with arith. +rewrite le_plus_minus_r; auto with arith; rewrite <- plus_n_O; auto. +rewrite plus_comm; auto with arith. +Qed. +Hint Resolve Rlt_pow: real. (*********) -Lemma tech_pow_Rmult:(x:R)(n:nat)(Rmult x (pow x n))==(pow x (S n)). +Lemma tech_pow_Rmult : forall (x:R) (n:nat), x * x ^ n = x ^ S n. Proof. -Induction n; Simpl; Trivial. +simple induction n; simpl in |- *; trivial. Qed. (*********) -Lemma tech_pow_Rplus:(x:R)(a,n:nat) - (Rplus (pow x a) (Rmult (INR n) (pow x a)))== - (Rmult (INR (S n)) (pow x a)). -Proof. -Intros; Pattern 1 (pow x a); - Rewrite <-(let (H1,H2)=(Rmult_ne (pow x a)) in H1); - Rewrite (Rmult_sym (INR n) (pow x a)); - Rewrite <- (Rmult_Rplus_distr (pow x a) R1 (INR n)); - Rewrite (Rplus_sym R1 (INR n)); Rewrite <-(S_INR n); - Apply Rmult_sym. -Qed. - -Lemma poly: (n:nat)(x:R)(Rlt R0 x)-> - (Rle (Rplus R1 (Rmult (INR n) x)) (pow (Rplus R1 x) n)). -Proof. -Intros;Elim n. -Simpl;Cut (Rplus R1 (Rmult R0 x))==R1. -Intro;Rewrite H0;Unfold Rle;Right; Reflexivity. -Ring. -Intros;Unfold pow; Fold pow; - Apply (Rle_trans (Rplus R1 (Rmult (INR (S n0)) x)) - (Rmult (Rplus R1 x) (Rplus R1 (Rmult (INR n0) x))) - (Rmult (Rplus R1 x) (pow (Rplus R1 x) n0))). -Cut (Rmult (Rplus R1 x) (Rplus R1 (Rmult (INR n0) x)))== - (Rplus (Rplus R1 (Rmult (INR (S n0)) x)) - (Rmult (INR n0) (Rmult x x))). -Intro;Rewrite H1;Pattern 1 (Rplus R1 (Rmult (INR (S n0)) x)); - Rewrite <-(let (H1,H2)= - (Rplus_ne (Rplus R1 (Rmult (INR (S n0)) x))) in H1); - Apply Rle_compatibility;Elim n0;Intros. -Simpl;Rewrite Rmult_Ol;Unfold Rle;Right;Auto. -Unfold Rle;Left;Generalize Rmult_gt;Unfold Rgt;Intro; - Fold (Rsqr x);Apply (H3 (INR (S n1)) (Rsqr x) - (lt_INR_0 (S n1) (lt_O_Sn n1)));Fold (Rgt x R0) in H; - Apply (pos_Rsqr1 x (imp_not_Req x R0 - (or_intror (Rlt x R0) (Rgt x R0) H))). -Rewrite (S_INR n0);Ring. -Unfold Rle in H0;Elim H0;Intro. -Unfold Rle;Left;Apply Rlt_monotony. -Rewrite Rplus_sym; - Apply (Rlt_r_plus_R1 x (Rlt_le R0 x H)). -Assumption. -Rewrite H1;Unfold Rle;Right;Trivial. -Qed. - -Lemma Power_monotonic: - (x:R) (m,n:nat) (Rgt (Rabsolu x) R1) - -> (le m n) - -> (Rle (Rabsolu (pow x m)) (Rabsolu (pow x n))). -Proof. -Intros x m n H;Induction n;Intros;Inversion H0. -Unfold Rle; Right; Reflexivity. -Unfold Rle; Right; Reflexivity. -Apply (Rle_trans (Rabsolu (pow x m)) - (Rabsolu (pow x n)) - (Rabsolu (pow x (S n)))). -Apply Hrecn; Assumption. -Simpl;Rewrite Rabsolu_mult. -Pattern 1 (Rabsolu (pow x n)). -Rewrite <-Rmult_1r. -Rewrite (Rmult_sym (Rabsolu x) (Rabsolu (pow x n))). -Apply Rle_monotony. -Apply Rabsolu_pos. -Unfold Rgt in H. -Apply Rlt_le; Assumption. -Qed. - -Lemma Pow_Rabsolu: (x:R) (n:nat) - (pow (Rabsolu x) n)==(Rabsolu (pow x n)). -Proof. -Intro;Induction n;Simpl. -Apply sym_eqT;Apply Rabsolu_pos_eq;Apply Rlt_le;Apply Rlt_R0_R1. -Intros; Rewrite H;Apply sym_eqT;Apply Rabsolu_mult. -Qed. - - -Lemma Pow_x_infinity: - (x:R) (Rgt (Rabsolu x) R1) - -> (b:R) (Ex [N:nat] ((n:nat) (ge n N) - -> (Rge (Rabsolu (pow x n)) b ))). -Proof. -Intros;Elim (archimed (Rmult b (Rinv (Rminus (Rabsolu x) R1))));Intros; - Clear H1; - Cut (Ex[N:nat] (Rge (INR N) (Rmult b (Rinv (Rminus (Rabsolu x) R1))))). -Intro; Elim H1;Clear H1;Intros;Exists x0;Intros; - Apply (Rge_trans (Rabsolu (pow x n)) (Rabsolu (pow x x0)) b). -Apply Rle_sym1;Apply Power_monotonic;Assumption. -Rewrite <- Pow_Rabsolu;Cut (Rabsolu x)==(Rplus R1 (Rminus (Rabsolu x) R1)). -Intro; Rewrite H3; - Apply (Rge_trans (pow (Rplus R1 (Rminus (Rabsolu x) R1)) x0) - (Rplus R1 (Rmult (INR x0) - (Rminus (Rabsolu x) R1))) - b). -Apply Rle_sym1;Apply poly;Fold (Rgt (Rminus (Rabsolu x) R1) R0); - Apply Rgt_minus;Assumption. -Apply (Rge_trans - (Rplus R1 (Rmult (INR x0) (Rminus (Rabsolu x) R1))) - (Rmult (INR x0) (Rminus (Rabsolu x) R1)) - b). -Apply Rle_sym1; Apply Rlt_le;Rewrite (Rplus_sym R1 - (Rmult (INR x0) (Rminus (Rabsolu x) R1))); - Pattern 1 (Rmult (INR x0) (Rminus (Rabsolu x) R1)); - Rewrite <- (let (H1,H2) = (Rplus_ne - (Rmult (INR x0) (Rminus (Rabsolu x) R1))) in - H1); - Apply Rlt_compatibility; - Apply Rlt_R0_R1. -Cut b==(Rmult (Rmult b (Rinv (Rminus (Rabsolu x) R1))) - (Rminus (Rabsolu x) R1)). -Intros; Rewrite H4;Apply Rge_monotony. -Apply Rge_minus;Unfold Rge; Left; Assumption. -Assumption. -Rewrite Rmult_assoc;Rewrite Rinv_l. -Ring. -Apply imp_not_Req; Right;Apply Rgt_minus;Assumption. -Ring. -Cut `0<= (up (Rmult b (Rinv (Rminus (Rabsolu x) R1))))`\/ - `(up (Rmult b (Rinv (Rminus (Rabsolu x) R1)))) <= 0`. -Intros;Elim H1;Intro. -Elim (IZN (up (Rmult b (Rinv (Rminus (Rabsolu x) R1)))) H2);Intros;Exists x0; - Apply (Rge_trans - (INR x0) - (IZR (up (Rmult b (Rinv (Rminus (Rabsolu x) R1))))) - (Rmult b (Rinv (Rminus (Rabsolu x) R1)))). -Rewrite INR_IZR_INZ;Apply IZR_ge;Omega. -Unfold Rge; Left; Assumption. -Exists O;Apply (Rge_trans (INR (0)) - (IZR (up (Rmult b (Rinv (Rminus (Rabsolu x) R1))))) - (Rmult b (Rinv (Rminus (Rabsolu x) R1)))). -Rewrite INR_IZR_INZ;Apply IZR_ge;Simpl;Omega. -Unfold Rge; Left; Assumption. -Omega. -Qed. - -Lemma pow_ne_zero: - (n:nat) ~(n=(0))-> (pow R0 n) == R0. -Proof. -Induction n. -Simpl;Auto. -Intros;Elim H;Reflexivity. -Intros; Simpl;Apply Rmult_Ol. -Qed. - -Lemma Rinv_pow: - (x:R) (n:nat) ~(x==R0) -> (Rinv (pow x n))==(pow (Rinv x) n). -Proof. -Intros; Elim n; Simpl. -Apply Rinv_R1. -Intro m;Intro;Rewrite Rinv_Rmult. -Rewrite H0; Reflexivity;Assumption. -Assumption. -Apply pow_nonzero;Assumption. -Qed. - -Lemma pow_lt_1_zero: - (x:R) (Rlt (Rabsolu x) R1) - -> (y:R) (Rlt R0 y) - -> (Ex[N:nat] (n:nat) (ge n N) - -> (Rlt (Rabsolu (pow x n)) y)). -Proof. -Intros;Elim (Req_EM x R0);Intro. -Exists (1);Rewrite H1;Intros n GE;Rewrite pow_ne_zero. -Rewrite Rabsolu_R0;Assumption. -Inversion GE;Auto. -Cut (Rgt (Rabsolu (Rinv x)) R1). -Intros;Elim (Pow_x_infinity (Rinv x) H2 (Rplus (Rinv y) R1));Intros N. -Exists N;Intros;Rewrite <- (Rinv_Rinv y). -Rewrite <- (Rinv_Rinv (Rabsolu (pow x n))). -Apply Rinv_lt. -Apply Rmult_lt_pos. -Apply Rlt_Rinv. -Assumption. -Apply Rlt_Rinv. -Apply Rabsolu_pos_lt. -Apply pow_nonzero. -Assumption. -Rewrite <- Rabsolu_Rinv. -Rewrite Rinv_pow. -Apply (Rlt_le_trans (Rinv y) - (Rplus (Rinv y) R1) - (Rabsolu (pow (Rinv x) n))). -Pattern 1 (Rinv y). -Rewrite <- (let (H1,H2) = - (Rplus_ne (Rinv y)) in H1). -Apply Rlt_compatibility. -Apply Rlt_R0_R1. -Apply Rle_sym2. -Apply H3. -Assumption. -Assumption. -Apply pow_nonzero. -Assumption. -Apply Rabsolu_no_R0. -Apply pow_nonzero. -Assumption. -Apply imp_not_Req. -Right; Unfold Rgt; Assumption. -Rewrite <- (Rinv_Rinv R1). -Rewrite Rabsolu_Rinv. -Unfold Rgt; Apply Rinv_lt. -Apply Rmult_lt_pos. -Apply Rabsolu_pos_lt. -Assumption. -Rewrite Rinv_R1; Apply Rlt_R0_R1. -Rewrite Rinv_R1; Assumption. -Assumption. -Red;Intro; Apply R1_neq_R0;Assumption. -Qed. - -Lemma pow_R1: - (r : R) (n : nat) (pow r n) == R1 -> (Rabsolu r) == R1 \/ n = O. -Proof. -Intros r n H'. -Case (Req_EM (Rabsolu r) R1); Auto; Intros H'1. -Case (not_Req ? ? H'1); Intros H'2. -Generalize H'; Case n; Auto. -Intros n0 H'0. -Cut ~ r == R0; [Intros Eq1 | Idtac]. -Cut ~ (Rabsolu r) == R0; [Intros Eq2 | Apply Rabsolu_no_R0]; Auto. -Absurd (Rlt (pow (Rabsolu (Rinv r)) O) (pow (Rabsolu (Rinv r)) (S n0))); Auto. -Replace (pow (Rabsolu (Rinv r)) (S n0)) with R1. -Simpl; Apply Rlt_antirefl; Auto. -Rewrite Rabsolu_Rinv; Auto. -Rewrite <- Rinv_pow; Auto. -Rewrite Pow_Rabsolu; Auto. -Rewrite H'0; Rewrite Rabsolu_right; Auto with real. -Apply Rle_ge; Auto with real. -Apply Rlt_pow; Auto with arith. -Rewrite Rabsolu_Rinv; Auto. -Apply Rlt_monotony_contra with z := (Rabsolu r). -Case (Rabsolu_pos r); Auto. -Intros H'3; Case Eq2; Auto. -Rewrite Rmult_1r; Rewrite Rinv_r; Auto with real. -Red;Intro;Absurd ``(pow r (S n0)) == 1``;Auto. -Simpl; Rewrite H; Rewrite Rmult_Ol; Auto with real. -Generalize H'; Case n; Auto. -Intros n0 H'0. -Cut ~ r == R0; [Intros Eq1 | Auto with real]. -Cut ~ (Rabsolu r) == R0; [Intros Eq2 | Apply Rabsolu_no_R0]; Auto. -Absurd (Rlt (pow (Rabsolu r) O) (pow (Rabsolu r) (S n0))); - Auto with real arith. -Repeat Rewrite Pow_Rabsolu; Rewrite H'0; Simpl; Auto with real. -Red;Intro;Absurd ``(pow r (S n0)) == 1``;Auto. -Simpl; Rewrite H; Rewrite Rmult_Ol; Auto with real. -Qed. - -Lemma pow_Rsqr : (x:R;n:nat) (pow x (mult (2) n))==(pow (Rsqr x) n). -Proof. -Intros; Induction n. -Reflexivity. -Replace (mult (2) (S n)) with (S (S (mult (2) n))). -Replace (pow x (S (S (mult (2) n)))) with ``x*x*(pow x (mult (S (S O)) n))``. -Rewrite Hrecn; Reflexivity. -Simpl; Ring. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Qed. - -Lemma pow_le : (a:R;n:nat) ``0<=a`` -> ``0<=(pow a n)``. -Proof. -Intros; Induction n. -Simpl; Left; Apply Rlt_R0_R1. -Simpl; Apply Rmult_le_pos; Assumption. +Lemma tech_pow_Rplus : + forall (x:R) (a n:nat), x ^ a + INR n * x ^ a = INR (S n) * x ^ a. +Proof. +intros; pattern (x ^ a) at 1 in |- *; + rewrite <- (let (H1, H2) := Rmult_ne (x ^ a) in H1); + rewrite (Rmult_comm (INR n) (x ^ a)); + rewrite <- (Rmult_plus_distr_l (x ^ a) 1 (INR n)); + rewrite (Rplus_comm 1 (INR n)); rewrite <- (S_INR n); + apply Rmult_comm. +Qed. + +Lemma poly : forall (n:nat) (x:R), 0 < x -> 1 + INR n * x <= (1 + x) ^ n. +Proof. +intros; elim n. +simpl in |- *; cut (1 + 0 * x = 1). +intro; rewrite H0; unfold Rle in |- *; right; reflexivity. +ring. +intros; unfold pow in |- *; fold pow in |- *; + apply + (Rle_trans (1 + INR (S n0) * x) ((1 + x) * (1 + INR n0 * x)) + ((1 + x) * (1 + x) ^ n0)). +cut ((1 + x) * (1 + INR n0 * x) = 1 + INR (S n0) * x + INR n0 * (x * x)). +intro; rewrite H1; pattern (1 + INR (S n0) * x) at 1 in |- *; + rewrite <- (let (H1, H2) := Rplus_ne (1 + INR (S n0) * x) in H1); + apply Rplus_le_compat_l; elim n0; intros. +simpl in |- *; rewrite Rmult_0_l; unfold Rle in |- *; right; auto. +unfold Rle in |- *; left; generalize Rmult_gt_0_compat; unfold Rgt in |- *; + intro; fold (Rsqr x) in |- *; + apply (H3 (INR (S n1)) (Rsqr x) (lt_INR_0 (S n1) (lt_O_Sn n1))); + fold (x > 0) in H; + apply (Rlt_0_sqr x (Rlt_dichotomy_converse x 0 (or_intror (x < 0) H))). +rewrite (S_INR n0); ring. +unfold Rle in H0; elim H0; intro. +unfold Rle in |- *; left; apply Rmult_lt_compat_l. +rewrite Rplus_comm; apply (Rle_lt_0_plus_1 x (Rlt_le 0 x H)). +assumption. +rewrite H1; unfold Rle in |- *; right; trivial. +Qed. + +Lemma Power_monotonic : + forall (x:R) (m n:nat), + Rabs x > 1 -> (m <= n)%nat -> Rabs (x ^ m) <= Rabs (x ^ n). +Proof. +intros x m n H; induction n as [| n Hrecn]; intros; inversion H0. +unfold Rle in |- *; right; reflexivity. +unfold Rle in |- *; right; reflexivity. +apply (Rle_trans (Rabs (x ^ m)) (Rabs (x ^ n)) (Rabs (x ^ S n))). +apply Hrecn; assumption. +simpl in |- *; rewrite Rabs_mult. +pattern (Rabs (x ^ n)) at 1 in |- *. +rewrite <- Rmult_1_r. +rewrite (Rmult_comm (Rabs x) (Rabs (x ^ n))). +apply Rmult_le_compat_l. +apply Rabs_pos. +unfold Rgt in H. +apply Rlt_le; assumption. +Qed. + +Lemma RPow_abs : forall (x:R) (n:nat), Rabs x ^ n = Rabs (x ^ n). +Proof. +intro; simple induction n; simpl in |- *. +apply sym_eq; apply Rabs_pos_eq; apply Rlt_le; apply Rlt_0_1. +intros; rewrite H; apply sym_eq; apply Rabs_mult. +Qed. + + +Lemma Pow_x_infinity : + forall x:R, + Rabs x > 1 -> + forall b:R, + exists N : nat | (forall n:nat, (n >= N)%nat -> Rabs (x ^ n) >= b). +Proof. +intros; elim (archimed (b * / (Rabs x - 1))); intros; clear H1; + cut ( exists N : nat | INR N >= b * / (Rabs x - 1)). +intro; elim H1; clear H1; intros; exists x0; intros; + apply (Rge_trans (Rabs (x ^ n)) (Rabs (x ^ x0)) b). +apply Rle_ge; apply Power_monotonic; assumption. +rewrite <- RPow_abs; cut (Rabs x = 1 + (Rabs x - 1)). +intro; rewrite H3; + apply (Rge_trans ((1 + (Rabs x - 1)) ^ x0) (1 + INR x0 * (Rabs x - 1)) b). +apply Rle_ge; apply poly; fold (Rabs x - 1 > 0) in |- *; apply Rgt_minus; + assumption. +apply (Rge_trans (1 + INR x0 * (Rabs x - 1)) (INR x0 * (Rabs x - 1)) b). +apply Rle_ge; apply Rlt_le; rewrite (Rplus_comm 1 (INR x0 * (Rabs x - 1))); + pattern (INR x0 * (Rabs x - 1)) at 1 in |- *; + rewrite <- (let (H1, H2) := Rplus_ne (INR x0 * (Rabs x - 1)) in H1); + apply Rplus_lt_compat_l; apply Rlt_0_1. +cut (b = b * / (Rabs x - 1) * (Rabs x - 1)). +intros; rewrite H4; apply Rmult_ge_compat_r. +apply Rge_minus; unfold Rge in |- *; left; assumption. +assumption. +rewrite Rmult_assoc; rewrite Rinv_l. +ring. +apply Rlt_dichotomy_converse; right; apply Rgt_minus; assumption. +ring. +cut ((0 <= up (b * / (Rabs x - 1)))%Z \/ (up (b * / (Rabs x - 1)) <= 0)%Z). +intros; elim H1; intro. +elim (IZN (up (b * / (Rabs x - 1))) H2); intros; exists x0; + apply + (Rge_trans (INR x0) (IZR (up (b * / (Rabs x - 1)))) (b * / (Rabs x - 1))). +rewrite INR_IZR_INZ; apply IZR_ge; omega. +unfold Rge in |- *; left; assumption. +exists 0%nat; + apply + (Rge_trans (INR 0) (IZR (up (b * / (Rabs x - 1)))) (b * / (Rabs x - 1))). +rewrite INR_IZR_INZ; apply IZR_ge; simpl in |- *; omega. +unfold Rge in |- *; left; assumption. +omega. +Qed. + +Lemma pow_ne_zero : forall n:nat, n <> 0%nat -> 0 ^ n = 0. +Proof. +simple induction n. +simpl in |- *; auto. +intros; elim H; reflexivity. +intros; simpl in |- *; apply Rmult_0_l. +Qed. + +Lemma Rinv_pow : forall (x:R) (n:nat), x <> 0 -> / x ^ n = (/ x) ^ n. +Proof. +intros; elim n; simpl in |- *. +apply Rinv_1. +intro m; intro; rewrite Rinv_mult_distr. +rewrite H0; reflexivity; assumption. +assumption. +apply pow_nonzero; assumption. +Qed. + +Lemma pow_lt_1_zero : + forall x:R, + Rabs x < 1 -> + forall y:R, + 0 < y -> + exists N : nat | (forall n:nat, (n >= N)%nat -> Rabs (x ^ n) < y). +Proof. +intros; elim (Req_dec x 0); intro. +exists 1%nat; rewrite H1; intros n GE; rewrite pow_ne_zero. +rewrite Rabs_R0; assumption. +inversion GE; auto. +cut (Rabs (/ x) > 1). +intros; elim (Pow_x_infinity (/ x) H2 (/ y + 1)); intros N. +exists N; intros; rewrite <- (Rinv_involutive y). +rewrite <- (Rinv_involutive (Rabs (x ^ n))). +apply Rinv_lt_contravar. +apply Rmult_lt_0_compat. +apply Rinv_0_lt_compat. +assumption. +apply Rinv_0_lt_compat. +apply Rabs_pos_lt. +apply pow_nonzero. +assumption. +rewrite <- Rabs_Rinv. +rewrite Rinv_pow. +apply (Rlt_le_trans (/ y) (/ y + 1) (Rabs ((/ x) ^ n))). +pattern (/ y) at 1 in |- *. +rewrite <- (let (H1, H2) := Rplus_ne (/ y) in H1). +apply Rplus_lt_compat_l. +apply Rlt_0_1. +apply Rge_le. +apply H3. +assumption. +assumption. +apply pow_nonzero. +assumption. +apply Rabs_no_R0. +apply pow_nonzero. +assumption. +apply Rlt_dichotomy_converse. +right; unfold Rgt in |- *; assumption. +rewrite <- (Rinv_involutive 1). +rewrite Rabs_Rinv. +unfold Rgt in |- *; apply Rinv_lt_contravar. +apply Rmult_lt_0_compat. +apply Rabs_pos_lt. +assumption. +rewrite Rinv_1; apply Rlt_0_1. +rewrite Rinv_1; assumption. +assumption. +red in |- *; intro; apply R1_neq_R0; assumption. +Qed. + +Lemma pow_R1 : forall (r:R) (n:nat), r ^ n = 1 -> Rabs r = 1 \/ n = 0%nat. +Proof. +intros r n H'. +case (Req_dec (Rabs r) 1); auto; intros H'1. +case (Rdichotomy _ _ H'1); intros H'2. +generalize H'; case n; auto. +intros n0 H'0. +cut (r <> 0); [ intros Eq1 | idtac ]. +cut (Rabs r <> 0); [ intros Eq2 | apply Rabs_no_R0 ]; auto. +absurd (Rabs (/ r) ^ 0 < Rabs (/ r) ^ S n0); auto. +replace (Rabs (/ r) ^ S n0) with 1. +simpl in |- *; apply Rlt_irrefl; auto. +rewrite Rabs_Rinv; auto. +rewrite <- Rinv_pow; auto. +rewrite RPow_abs; auto. +rewrite H'0; rewrite Rabs_right; auto with real. +apply Rle_ge; auto with real. +apply Rlt_pow; auto with arith. +rewrite Rabs_Rinv; auto. +apply Rmult_lt_reg_l with (r := Rabs r). +case (Rabs_pos r); auto. +intros H'3; case Eq2; auto. +rewrite Rmult_1_r; rewrite Rinv_r; auto with real. +red in |- *; intro; absurd (r ^ S n0 = 1); auto. +simpl in |- *; rewrite H; rewrite Rmult_0_l; auto with real. +generalize H'; case n; auto. +intros n0 H'0. +cut (r <> 0); [ intros Eq1 | auto with real ]. +cut (Rabs r <> 0); [ intros Eq2 | apply Rabs_no_R0 ]; auto. +absurd (Rabs r ^ 0 < Rabs r ^ S n0); auto with real arith. +repeat rewrite RPow_abs; rewrite H'0; simpl in |- *; auto with real. +red in |- *; intro; absurd (r ^ S n0 = 1); auto. +simpl in |- *; rewrite H; rewrite Rmult_0_l; auto with real. +Qed. + +Lemma pow_Rsqr : forall (x:R) (n:nat), x ^ (2 * n) = Rsqr x ^ n. +Proof. +intros; induction n as [| n Hrecn]. +reflexivity. +replace (2 * S n)%nat with (S (S (2 * n))). +replace (x ^ S (S (2 * n))) with (x * x * x ^ (2 * n)). +rewrite Hrecn; reflexivity. +simpl in |- *; ring. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +Qed. + +Lemma pow_le : forall (a:R) (n:nat), 0 <= a -> 0 <= a ^ n. +Proof. +intros; induction n as [| n Hrecn]. +simpl in |- *; left; apply Rlt_0_1. +simpl in |- *; apply Rmult_le_pos; assumption. Qed. (**********) -Lemma pow_1_even : (n:nat) ``(pow (-1) (mult (S (S O)) n))==1``. +Lemma pow_1_even : forall n:nat, (-1) ^ (2 * n) = 1. Proof. -Intro; Induction n. -Reflexivity. -Replace (mult (2) (S n)) with (plus (2) (mult (2) n)). -Rewrite pow_add; Rewrite Hrecn; Simpl; Ring. -Replace (S n) with (plus n (1)); [Ring | Ring]. +intro; induction n as [| n Hrecn]. +reflexivity. +replace (2 * S n)%nat with (2 + 2 * n)%nat. +rewrite pow_add; rewrite Hrecn; simpl in |- *; ring. +replace (S n) with (n + 1)%nat; [ ring | ring ]. Qed. (**********) -Lemma pow_1_odd : (n:nat) ``(pow (-1) (S (mult (S (S O)) n)))==-1``. +Lemma pow_1_odd : forall n:nat, (-1) ^ S (2 * n) = -1. Proof. -Intro; Replace (S (mult (2) n)) with (plus (mult (2) n) (1)); [Idtac | Ring]. -Rewrite pow_add; Rewrite pow_1_even; Simpl; Ring. +intro; replace (S (2 * n)) with (2 * n + 1)%nat; [ idtac | ring ]. +rewrite pow_add; rewrite pow_1_even; simpl in |- *; ring. Qed. (**********) -Lemma pow_1_abs : (n:nat) ``(Rabsolu (pow (-1) n))==1``. -Proof. -Intro; Induction n. -Simpl; Apply Rabsolu_R1. -Replace (S n) with (plus n (1)); [Rewrite pow_add | Ring]. -Rewrite Rabsolu_mult. -Rewrite Hrecn; Rewrite Rmult_1l; Simpl; Rewrite Rmult_1r; Rewrite Rabsolu_Ropp; Apply Rabsolu_R1. -Qed. - -Lemma pow_mult : (x:R;n1,n2:nat) (pow x (mult n1 n2))==(pow (pow x n1) n2). -Proof. -Intros; Induction n2. -Simpl; Replace (mult n1 O) with O; [Reflexivity | Ring]. -Replace (mult n1 (S n2)) with (plus (mult n1 n2) n1). -Replace (S n2) with (plus n2 (1)); [Idtac | Ring]. -Do 2 Rewrite pow_add. -Rewrite Hrecn2. -Simpl. -Ring. -Apply INR_eq; Rewrite plus_INR; Do 2 Rewrite mult_INR; Rewrite S_INR; Ring. -Qed. - -Lemma pow_incr : (x,y:R;n:nat) ``0<=x<=y`` -> ``(pow x n)<=(pow y n)``. -Proof. -Intros. -Induction n. -Right; Reflexivity. -Simpl. -Elim H; Intros. -Apply Rle_trans with ``y*(pow x n)``. -Do 2 Rewrite <- (Rmult_sym (pow x n)). -Apply Rle_monotony. -Apply pow_le; Assumption. -Assumption. -Apply Rle_monotony. -Apply Rle_trans with x; Assumption. -Apply Hrecn. -Qed. - -Lemma pow_R1_Rle : (x:R;k:nat) ``1<=x`` -> ``1<=(pow x k)``. -Proof. -Intros. -Induction k. -Right; Reflexivity. -Simpl. -Apply Rle_trans with ``x*1``. -Rewrite Rmult_1r; Assumption. -Apply Rle_monotony. -Left; Apply Rlt_le_trans with R1; [Apply Rlt_R0_R1 | Assumption]. -Exact Hreck. -Qed. - -Lemma Rle_pow : (x:R;m,n:nat) ``1<=x`` -> (le m n) -> ``(pow x m)<=(pow x n)``. -Proof. -Intros. -Replace n with (plus (minus n m) m). -Rewrite pow_add. -Rewrite Rmult_sym. -Pattern 1 (pow x m); Rewrite <- Rmult_1r. -Apply Rle_monotony. -Apply pow_le; Left; Apply Rlt_le_trans with R1; [Apply Rlt_R0_R1 | Assumption]. -Apply pow_R1_Rle; Assumption. -Rewrite plus_sym. -Symmetry; Apply le_plus_minus; Assumption. -Qed. - -Lemma pow1 : (n:nat) (pow R1 n)==R1. -Proof. -Intro; Induction n. -Reflexivity. -Simpl; Rewrite Hrecn; Rewrite Rmult_1r; Reflexivity. -Qed. - -Lemma pow_Rabs : (x:R;n:nat) ``(pow x n)<=(pow (Rabsolu x) n)``. -Proof. -Intros; Induction n. -Right; Reflexivity. -Simpl; Case (case_Rabsolu x); Intro. -Apply Rle_trans with (Rabsolu ``x*(pow x n)``). -Apply Rle_Rabsolu. -Rewrite Rabsolu_mult. -Apply Rle_monotony. -Apply Rabsolu_pos. -Right; Symmetry; Apply Pow_Rabsolu. -Pattern 1 (Rabsolu x); Rewrite (Rabsolu_right x r); Apply Rle_monotony. -Apply Rle_sym2; Exact r. -Apply Hrecn. -Qed. - -Lemma pow_maj_Rabs : (x,y:R;n:nat) ``(Rabsolu y)<=x`` -> ``(pow y n)<=(pow x n)``. -Proof. -Intros; Cut ``0<=x``. -Intro; Apply Rle_trans with (pow (Rabsolu y) n). -Apply pow_Rabs. -Induction n. -Right; Reflexivity. -Simpl; Apply Rle_trans with ``x*(pow (Rabsolu y) n)``. -Do 2 Rewrite <- (Rmult_sym (pow (Rabsolu y) n)). -Apply Rle_monotony. -Apply pow_le; Apply Rabsolu_pos. -Assumption. -Apply Rle_monotony. -Apply H0. -Apply Hrecn. -Apply Rle_trans with (Rabsolu y); [Apply Rabsolu_pos | Exact H]. +Lemma pow_1_abs : forall n:nat, Rabs ((-1) ^ n) = 1. +Proof. +intro; induction n as [| n Hrecn]. +simpl in |- *; apply Rabs_R1. +replace (S n) with (n + 1)%nat; [ rewrite pow_add | ring ]. +rewrite Rabs_mult. +rewrite Hrecn; rewrite Rmult_1_l; simpl in |- *; rewrite Rmult_1_r; + rewrite Rabs_Ropp; apply Rabs_R1. +Qed. + +Lemma pow_mult : forall (x:R) (n1 n2:nat), x ^ (n1 * n2) = x ^ n1 ^ n2. +Proof. +intros; induction n2 as [| n2 Hrecn2]. +simpl in |- *; replace (n1 * 0)%nat with 0%nat; [ reflexivity | ring ]. +replace (n1 * S n2)%nat with (n1 * n2 + n1)%nat. +replace (S n2) with (n2 + 1)%nat; [ idtac | ring ]. +do 2 rewrite pow_add. +rewrite Hrecn2. +simpl in |- *. +ring. +apply INR_eq; rewrite plus_INR; do 2 rewrite mult_INR; rewrite S_INR; ring. +Qed. + +Lemma pow_incr : forall (x y:R) (n:nat), 0 <= x <= y -> x ^ n <= y ^ n. +Proof. +intros. +induction n as [| n Hrecn]. +right; reflexivity. +simpl in |- *. +elim H; intros. +apply Rle_trans with (y * x ^ n). +do 2 rewrite <- (Rmult_comm (x ^ n)). +apply Rmult_le_compat_l. +apply pow_le; assumption. +assumption. +apply Rmult_le_compat_l. +apply Rle_trans with x; assumption. +apply Hrecn. +Qed. + +Lemma pow_R1_Rle : forall (x:R) (k:nat), 1 <= x -> 1 <= x ^ k. +Proof. +intros. +induction k as [| k Hreck]. +right; reflexivity. +simpl in |- *. +apply Rle_trans with (x * 1). +rewrite Rmult_1_r; assumption. +apply Rmult_le_compat_l. +left; apply Rlt_le_trans with 1; [ apply Rlt_0_1 | assumption ]. +exact Hreck. +Qed. + +Lemma Rle_pow : + forall (x:R) (m n:nat), 1 <= x -> (m <= n)%nat -> x ^ m <= x ^ n. +Proof. +intros. +replace n with (n - m + m)%nat. +rewrite pow_add. +rewrite Rmult_comm. +pattern (x ^ m) at 1 in |- *; rewrite <- Rmult_1_r. +apply Rmult_le_compat_l. +apply pow_le; left; apply Rlt_le_trans with 1; [ apply Rlt_0_1 | assumption ]. +apply pow_R1_Rle; assumption. +rewrite plus_comm. +symmetry in |- *; apply le_plus_minus; assumption. +Qed. + +Lemma pow1 : forall n:nat, 1 ^ n = 1. +Proof. +intro; induction n as [| n Hrecn]. +reflexivity. +simpl in |- *; rewrite Hrecn; rewrite Rmult_1_r; reflexivity. +Qed. + +Lemma pow_Rabs : forall (x:R) (n:nat), x ^ n <= Rabs x ^ n. +Proof. +intros; induction n as [| n Hrecn]. +right; reflexivity. +simpl in |- *; case (Rcase_abs x); intro. +apply Rle_trans with (Rabs (x * x ^ n)). +apply RRle_abs. +rewrite Rabs_mult. +apply Rmult_le_compat_l. +apply Rabs_pos. +right; symmetry in |- *; apply RPow_abs. +pattern (Rabs x) at 1 in |- *; rewrite (Rabs_right x r); + apply Rmult_le_compat_l. +apply Rge_le; exact r. +apply Hrecn. +Qed. + +Lemma pow_maj_Rabs : forall (x y:R) (n:nat), Rabs y <= x -> y ^ n <= x ^ n. +Proof. +intros; cut (0 <= x). +intro; apply Rle_trans with (Rabs y ^ n). +apply pow_Rabs. +induction n as [| n Hrecn]. +right; reflexivity. +simpl in |- *; apply Rle_trans with (x * Rabs y ^ n). +do 2 rewrite <- (Rmult_comm (Rabs y ^ n)). +apply Rmult_le_compat_l. +apply pow_le; apply Rabs_pos. +assumption. +apply Rmult_le_compat_l. +apply H0. +apply Hrecn. +apply Rle_trans with (Rabs y); [ apply Rabs_pos | exact H ]. Qed. (*******************************) @@ -556,207 +524,200 @@ Qed. (*******************************) (*i Due to L.Thery i*) -Tactic Definition CaseEqk name := -Generalize (refl_equal ? name); Pattern -1 name; Case name. +Ltac case_eq name := + generalize (refl_equal name); pattern name at -1 in |- *; case name. -Definition powerRZ := - [x : R] [n : Z] Cases n of - ZERO => R1 - | (POS p) => (pow x (convert p)) - | (NEG p) => (Rinv (pow x (convert p))) - end. +Definition powerRZ (x:R) (n:Z) := + match n with + | Z0 => 1 + | Zpos p => x ^ nat_of_P p + | Zneg p => / x ^ nat_of_P p + end. -Infix Local "^Z" powerRZ (at level 2, left associativity) : R_scope. +Infix Local "^Z" := powerRZ (at level 30, left associativity) : R_scope. -Lemma Zpower_NR0: - (x : Z) (n : nat) (Zle ZERO x) -> (Zle ZERO (Zpower_nat x n)). +Lemma Zpower_NR0 : + forall (x:Z) (n:nat), (0 <= x)%Z -> (0 <= Zpower_nat x n)%Z. Proof. -NewInduction n; Unfold Zpower_nat; Simpl; Auto with zarith. +induction n; unfold Zpower_nat in |- *; simpl in |- *; auto with zarith. Qed. -Lemma powerRZ_O: (x : R) (powerRZ x ZERO) == R1. +Lemma powerRZ_O : forall x:R, x ^Z 0 = 1. Proof. -Reflexivity. +reflexivity. Qed. -Lemma powerRZ_1: (x : R) (powerRZ x (Zs ZERO)) == x. +Lemma powerRZ_1 : forall x:R, x ^Z Zsucc 0 = x. Proof. -Simpl; Auto with real. +simpl in |- *; auto with real. Qed. -Lemma powerRZ_NOR: (x : R) (z : Z) ~ x == R0 -> ~ (powerRZ x z) == R0. +Lemma powerRZ_NOR : forall (x:R) (z:Z), x <> 0 -> x ^Z z <> 0. Proof. -NewDestruct z; Simpl; Auto with real. +destruct z; simpl in |- *; auto with real. Qed. -Lemma powerRZ_add: - (x : R) - (n, m : Z) - ~ x == R0 -> (powerRZ x (Zplus n m)) == (Rmult (powerRZ x n) (powerRZ x m)). +Lemma powerRZ_add : + forall (x:R) (n m:Z), x <> 0 -> x ^Z (n + m) = x ^Z n * x ^Z m. Proof. -Intro x; NewDestruct n as [|n1|n1]; NewDestruct m as [|m1|m1]; Simpl; - Auto with real. +intro x; destruct n as [| n1| n1]; destruct m as [| m1| m1]; simpl in |- *; + auto with real. (* POS/POS *) -Rewrite convert_add; Auto with real. +rewrite nat_of_P_plus_morphism; auto with real. (* POS/NEG *) -(CaseEqk '(compare n1 m1 EGAL)); Simpl; Auto with real. -Intros H' H'0; Rewrite compare_convert_EGAL with 1 := H'; Auto with real. -Intros H' H'0; Rewrite (true_sub_convert m1 n1); Auto with real. -Rewrite (pow_RN_plus x (minus (convert m1) (convert n1)) (convert n1)); - Auto with real. -Rewrite plus_sym; Rewrite le_plus_minus_r; Auto with real. -Rewrite Rinv_Rmult; Auto with real. -Rewrite Rinv_Rinv; Auto with real. -Apply lt_le_weak. -Apply compare_convert_INFERIEUR; Auto. -Apply ZC2; Auto. -Intros H' H'0; Rewrite (true_sub_convert n1 m1); Auto with real. -Rewrite (pow_RN_plus x (minus (convert n1) (convert m1)) (convert m1)); - Auto with real. -Rewrite plus_sym; Rewrite le_plus_minus_r; Auto with real. -Apply lt_le_weak. -Change (gt (convert n1) (convert m1)). -Apply compare_convert_SUPERIEUR; Auto. +case_eq ((n1 ?= m1)%positive Datatypes.Eq); simpl in |- *; auto with real. +intros H' H'0; rewrite Pcompare_Eq_eq with (1 := H'); auto with real. +intros H' H'0; rewrite (nat_of_P_minus_morphism m1 n1); auto with real. +rewrite (pow_RN_plus x (nat_of_P m1 - nat_of_P n1) (nat_of_P n1)); + auto with real. +rewrite plus_comm; rewrite le_plus_minus_r; auto with real. +rewrite Rinv_mult_distr; auto with real. +rewrite Rinv_involutive; auto with real. +apply lt_le_weak. +apply nat_of_P_lt_Lt_compare_morphism; auto. +apply ZC2; auto. +intros H' H'0; rewrite (nat_of_P_minus_morphism n1 m1); auto with real. +rewrite (pow_RN_plus x (nat_of_P n1 - nat_of_P m1) (nat_of_P m1)); + auto with real. +rewrite plus_comm; rewrite le_plus_minus_r; auto with real. +apply lt_le_weak. +change (nat_of_P n1 > nat_of_P m1)%nat in |- *. +apply nat_of_P_gt_Gt_compare_morphism; auto. (* NEG/POS *) -(CaseEqk '(compare n1 m1 EGAL)); Simpl; Auto with real. -Intros H' H'0; Rewrite compare_convert_EGAL with 1 := H'; Auto with real. -Intros H' H'0; Rewrite (true_sub_convert m1 n1); Auto with real. -Rewrite (pow_RN_plus x (minus (convert m1) (convert n1)) (convert n1)); - Auto with real. -Rewrite plus_sym; Rewrite le_plus_minus_r; Auto with real. -Apply lt_le_weak. -Apply compare_convert_INFERIEUR; Auto. -Apply ZC2; Auto. -Intros H' H'0; Rewrite (true_sub_convert n1 m1); Auto with real. -Rewrite (pow_RN_plus x (minus (convert n1) (convert m1)) (convert m1)); - Auto with real. -Rewrite plus_sym; Rewrite le_plus_minus_r; Auto with real. -Rewrite Rinv_Rmult; Auto with real. -Apply lt_le_weak. -Change (gt (convert n1) (convert m1)). -Apply compare_convert_SUPERIEUR; Auto. +case_eq ((n1 ?= m1)%positive Datatypes.Eq); simpl in |- *; auto with real. +intros H' H'0; rewrite Pcompare_Eq_eq with (1 := H'); auto with real. +intros H' H'0; rewrite (nat_of_P_minus_morphism m1 n1); auto with real. +rewrite (pow_RN_plus x (nat_of_P m1 - nat_of_P n1) (nat_of_P n1)); + auto with real. +rewrite plus_comm; rewrite le_plus_minus_r; auto with real. +apply lt_le_weak. +apply nat_of_P_lt_Lt_compare_morphism; auto. +apply ZC2; auto. +intros H' H'0; rewrite (nat_of_P_minus_morphism n1 m1); auto with real. +rewrite (pow_RN_plus x (nat_of_P n1 - nat_of_P m1) (nat_of_P m1)); + auto with real. +rewrite plus_comm; rewrite le_plus_minus_r; auto with real. +rewrite Rinv_mult_distr; auto with real. +apply lt_le_weak. +change (nat_of_P n1 > nat_of_P m1)%nat in |- *. +apply nat_of_P_gt_Gt_compare_morphism; auto. (* NEG/NEG *) -Rewrite convert_add; Auto with real. -Intros H'; Rewrite pow_add; Auto with real. -Apply Rinv_Rmult; Auto. -Apply pow_nonzero; Auto. -Apply pow_nonzero; Auto. +rewrite nat_of_P_plus_morphism; auto with real. +intros H'; rewrite pow_add; auto with real. +apply Rinv_mult_distr; auto. +apply pow_nonzero; auto. +apply pow_nonzero; auto. Qed. -Hints Resolve powerRZ_O powerRZ_1 powerRZ_NOR powerRZ_add :real. +Hint Resolve powerRZ_O powerRZ_1 powerRZ_NOR powerRZ_add: real. -Lemma Zpower_nat_powerRZ: - (n, m : nat) - (IZR (Zpower_nat (inject_nat n) m)) == (powerRZ (INR n) (inject_nat m)). -Proof. -Intros n m; Elim m; Simpl; Auto with real. -Intros m1 H'; Rewrite bij1; Simpl. -Replace (Zpower_nat (inject_nat n) (S m1)) - with (Zmult (inject_nat n) (Zpower_nat (inject_nat n) m1)). -Rewrite mult_IZR; Auto with real. -Repeat Rewrite <- INR_IZR_INZ; Simpl. -Rewrite H'; Simpl. -Case m1; Simpl; Auto with real. -Intros m2; Rewrite bij1; Auto. -Unfold Zpower_nat; Auto. +Lemma Zpower_nat_powerRZ : + forall n m:nat, IZR (Zpower_nat (Z_of_nat n) m) = INR n ^Z Z_of_nat m. +Proof. +intros n m; elim m; simpl in |- *; auto with real. +intros m1 H'; rewrite nat_of_P_o_P_of_succ_nat_eq_succ; simpl in |- *. +replace (Zpower_nat (Z_of_nat n) (S m1)) with + (Z_of_nat n * Zpower_nat (Z_of_nat n) m1)%Z. +rewrite mult_IZR; auto with real. +repeat rewrite <- INR_IZR_INZ; simpl in |- *. +rewrite H'; simpl in |- *. +case m1; simpl in |- *; auto with real. +intros m2; rewrite nat_of_P_o_P_of_succ_nat_eq_succ; auto. +unfold Zpower_nat in |- *; auto. Qed. -Lemma powerRZ_lt: (x : R) (z : Z) (Rlt R0 x) -> (Rlt R0 (powerRZ x z)). +Lemma powerRZ_lt : forall (x:R) (z:Z), 0 < x -> 0 < x ^Z z. Proof. -Intros x z; Case z; Simpl; Auto with real. +intros x z; case z; simpl in |- *; auto with real. Qed. -Hints Resolve powerRZ_lt :real. +Hint Resolve powerRZ_lt: real. -Lemma powerRZ_le: (x : R) (z : Z) (Rlt R0 x) -> (Rle R0 (powerRZ x z)). +Lemma powerRZ_le : forall (x:R) (z:Z), 0 < x -> 0 <= x ^Z z. Proof. -Intros x z H'; Apply Rlt_le; Auto with real. +intros x z H'; apply Rlt_le; auto with real. Qed. -Hints Resolve powerRZ_le :real. +Hint Resolve powerRZ_le: real. -Lemma Zpower_nat_powerRZ_absolu: - (n, m : Z) - (Zle ZERO m) -> (IZR (Zpower_nat n (absolu m))) == (powerRZ (IZR n) m). +Lemma Zpower_nat_powerRZ_absolu : + forall n m:Z, (0 <= m)%Z -> IZR (Zpower_nat n (Zabs_nat m)) = IZR n ^Z m. Proof. -Intros n m; Case m; Simpl; Auto with zarith. -Intros p H'; Elim (convert p); Simpl; Auto with zarith. -Intros n0 H'0; Rewrite <- H'0; Simpl; Auto with zarith. -Rewrite <- mult_IZR; Auto. -Intros p H'; Absurd `0 <= (NEG p)`;Auto with zarith. +intros n m; case m; simpl in |- *; auto with zarith. +intros p H'; elim (nat_of_P p); simpl in |- *; auto with zarith. +intros n0 H'0; rewrite <- H'0; simpl in |- *; auto with zarith. +rewrite <- mult_IZR; auto. +intros p H'; absurd (0 <= Zneg p)%Z; auto with zarith. Qed. -Lemma powerRZ_R1: (n : Z) (powerRZ R1 n) == R1. +Lemma powerRZ_R1 : forall n:Z, 1 ^Z n = 1. Proof. -Intros n; Case n; Simpl; Auto. -Intros p; Elim (convert p); Simpl; Auto; Intros n0 H'; Rewrite H'; Ring. -Intros p; Elim (convert p); Simpl. -Exact Rinv_R1. -Intros n1 H'; Rewrite Rinv_Rmult; Try Rewrite Rinv_R1; Try Rewrite H'; - Auto with real. +intros n; case n; simpl in |- *; auto. +intros p; elim (nat_of_P p); simpl in |- *; auto; intros n0 H'; rewrite H'; + ring. +intros p; elim (nat_of_P p); simpl in |- *. +exact Rinv_1. +intros n1 H'; rewrite Rinv_mult_distr; try rewrite Rinv_1; try rewrite H'; + auto with real. Qed. (*******************************) (** Sum of n first naturals *) (*******************************) (*********) -Fixpoint sum_nat_f_O [f:nat->nat;n:nat]:nat:= - Cases n of - O => (f O) - |(S n') => (plus (sum_nat_f_O f n') (f (S n'))) +Fixpoint sum_nat_f_O (f:nat -> nat) (n:nat) {struct n} : nat := + match n with + | O => f 0%nat + | S n' => (sum_nat_f_O f n' + f (S n'))%nat end. (*********) -Definition sum_nat_f [s,n:nat;f:nat->nat]:nat:= - (sum_nat_f_O [x:nat](f (plus x s)) (minus n s)). +Definition sum_nat_f (s n:nat) (f:nat -> nat) : nat := + sum_nat_f_O (fun x:nat => f (x + s)%nat) (n - s). (*********) -Definition sum_nat_O [n:nat]:nat:= - (sum_nat_f_O [x:nat]x n). +Definition sum_nat_O (n:nat) : nat := sum_nat_f_O (fun x:nat => x) n. (*********) -Definition sum_nat [s,n:nat]:nat:= - (sum_nat_f s n [x:nat]x). +Definition sum_nat (s n:nat) : nat := sum_nat_f s n (fun x:nat => x). (*******************************) (** Sum *) (*******************************) (*********) -Fixpoint sum_f_R0 [f:nat->R;N:nat]:R:= - Cases N of - O => (f O) - |(S i) => (Rplus (sum_f_R0 f i) (f (S i))) +Fixpoint sum_f_R0 (f:nat -> R) (N:nat) {struct N} : R := + match N with + | O => f 0%nat + | S i => sum_f_R0 f i + f (S i) end. (*********) -Definition sum_f [s,n:nat;f:nat->R]:R:= - (sum_f_R0 [x:nat](f (plus x s)) (minus n s)). - -Lemma GP_finite: - (x:R) (n:nat) (Rmult (sum_f_R0 [n:nat] (pow x n) n) - (Rminus x R1)) == - (Rminus (pow x (plus n (1))) R1). -Proof. -Intros; Induction n; Simpl. -Ring. -Rewrite Rmult_Rplus_distrl;Rewrite Hrecn;Cut (plus n (1))=(S n). -Intro H;Rewrite H;Simpl;Ring. -Omega. -Qed. - -Lemma sum_f_R0_triangle: - (x:nat->R)(n:nat) (Rle (Rabsolu (sum_f_R0 x n)) - (sum_f_R0 [i:nat] (Rabsolu (x i)) n)). -Proof. -Intro; Induction n; Simpl. -Unfold Rle; Right; Reflexivity. -Intro m; Intro;Apply (Rle_trans - (Rabsolu (Rplus (sum_f_R0 x m) (x (S m)))) - (Rplus (Rabsolu (sum_f_R0 x m)) - (Rabsolu (x (S m)))) - (Rplus (sum_f_R0 [i:nat](Rabsolu (x i)) m) - (Rabsolu (x (S m))))). -Apply Rabsolu_triang. -Rewrite Rplus_sym;Rewrite (Rplus_sym - (sum_f_R0 [i:nat](Rabsolu (x i)) m) (Rabsolu (x (S m)))); - Apply Rle_compatibility;Assumption. +Definition sum_f (s n:nat) (f:nat -> R) : R := + sum_f_R0 (fun x:nat => f (x + s)%nat) (n - s). + +Lemma GP_finite : + forall (x:R) (n:nat), + sum_f_R0 (fun n:nat => x ^ n) n * (x - 1) = x ^ (n + 1) - 1. +Proof. +intros; induction n as [| n Hrecn]; simpl in |- *. +ring. +rewrite Rmult_plus_distr_r; rewrite Hrecn; cut ((n + 1)%nat = S n). +intro H; rewrite H; simpl in |- *; ring. +omega. +Qed. + +Lemma sum_f_R0_triangle : + forall (x:nat -> R) (n:nat), + Rabs (sum_f_R0 x n) <= sum_f_R0 (fun i:nat => Rabs (x i)) n. +Proof. +intro; simple induction n; simpl in |- *. +unfold Rle in |- *; right; reflexivity. +intro m; intro; + apply + (Rle_trans (Rabs (sum_f_R0 x m + x (S m))) + (Rabs (sum_f_R0 x m) + Rabs (x (S m))) + (sum_f_R0 (fun i:nat => Rabs (x i)) m + Rabs (x (S m)))). +apply Rabs_triang. +rewrite Rplus_comm; + rewrite (Rplus_comm (sum_f_R0 (fun i:nat => Rabs (x i)) m) (Rabs (x (S m)))); + apply Rplus_le_compat_l; assumption. Qed. (*******************************) @@ -764,69 +725,69 @@ Qed. (*******************************) (*********) -Definition R_dist:R->R->R:=[x,y:R](Rabsolu (Rminus x y)). +Definition R_dist (x y:R) : R := Rabs (x - y). (*********) -Lemma R_dist_pos:(x,y:R)(Rge (R_dist x y) R0). +Lemma R_dist_pos : forall x y:R, R_dist x y >= 0. Proof. -Intros;Unfold R_dist;Unfold Rabsolu;Case (case_Rabsolu (Rminus x y));Intro l. -Unfold Rge;Left;Apply (Rlt_RoppO (Rminus x y) l). -Trivial. +intros; unfold R_dist in |- *; unfold Rabs in |- *; case (Rcase_abs (x - y)); + intro l. +unfold Rge in |- *; left; apply (Ropp_gt_lt_0_contravar (x - y) l). +trivial. Qed. (*********) -Lemma R_dist_sym:(x,y:R)(R_dist x y)==(R_dist y x). +Lemma R_dist_sym : forall x y:R, R_dist x y = R_dist y x. Proof. -Unfold R_dist;Intros;SplitAbsolu;Ring. -Generalize (Rlt_RoppO (Rminus y x) r); Intro; - Rewrite (Ropp_distr2 y x) in H; - Generalize (Rlt_antisym (Rminus x y) R0 r0); Intro;Unfold Rgt in H; - ElimType False; Auto. -Generalize (minus_Rge y x r); Intro; - Generalize (minus_Rge x y r0); Intro; - Generalize (Rge_ge_eq x y H0 H); Intro;Rewrite H1;Ring. +unfold R_dist in |- *; intros; split_Rabs; ring. +generalize (Ropp_gt_lt_0_contravar (y - x) r); intro; + rewrite (Ropp_minus_distr y x) in H; generalize (Rlt_asym (x - y) 0 r0); + intro; unfold Rgt in H; elimtype False; auto. +generalize (minus_Rge y x r); intro; generalize (minus_Rge x y r0); intro; + generalize (Rge_antisym x y H0 H); intro; rewrite H1; + ring. Qed. (*********) -Lemma R_dist_refl:(x,y:R)((R_dist x y)==R0<->x==y). +Lemma R_dist_refl : forall x y:R, R_dist x y = 0 <-> x = y. Proof. -Unfold R_dist;Intros;SplitAbsolu;Split;Intros. -Rewrite (Ropp_distr2 x y) in H;Apply sym_eqT; - Apply (Rminus_eq y x H). -Rewrite (Ropp_distr2 x y);Generalize (sym_eqT R x y H);Intro; - Apply (eq_Rminus y x H0). -Apply (Rminus_eq x y H). -Apply (eq_Rminus x y H). +unfold R_dist in |- *; intros; split_Rabs; split; intros. +rewrite (Ropp_minus_distr x y) in H; apply sym_eq; + apply (Rminus_diag_uniq y x H). +rewrite (Ropp_minus_distr x y); generalize (sym_eq H); intro; + apply (Rminus_diag_eq y x H0). +apply (Rminus_diag_uniq x y H). +apply (Rminus_diag_eq x y H). Qed. -Lemma R_dist_eq:(x:R)(R_dist x x)==R0. +Lemma R_dist_eq : forall x:R, R_dist x x = 0. Proof. -Unfold R_dist;Intros;SplitAbsolu;Intros;Ring. +unfold R_dist in |- *; intros; split_Rabs; intros; ring. Qed. (***********) -Lemma R_dist_tri:(x,y,z:R)(Rle (R_dist x y) - (Rplus (R_dist x z) (R_dist z y))). +Lemma R_dist_tri : forall x y z:R, R_dist x y <= R_dist x z + R_dist z y. Proof. -Intros;Unfold R_dist; Replace ``x-y`` with ``(x-z)+(z-y)``; - [Apply (Rabsolu_triang ``x-z`` ``z-y``)|Ring]. +intros; unfold R_dist in |- *; replace (x - y) with (x - z + (z - y)); + [ apply (Rabs_triang (x - z) (z - y)) | ring ]. Qed. (*********) -Lemma R_dist_plus: (a,b,c,d:R)(Rle (R_dist (Rplus a c) (Rplus b d)) - (Rplus (R_dist a b) (R_dist c d))). +Lemma R_dist_plus : + forall a b c d:R, R_dist (a + c) (b + d) <= R_dist a b + R_dist c d. Proof. -Intros;Unfold R_dist; - Replace (Rminus (Rplus a c) (Rplus b d)) - with (Rplus (Rminus a b) (Rminus c d)). -Exact (Rabsolu_triang (Rminus a b) (Rminus c d)). -Ring. +intros; unfold R_dist in |- *; + replace (a + c - (b + d)) with (a - b + (c - d)). +exact (Rabs_triang (a - b) (c - d)). +ring. Qed. (*******************************) (** Infinit Sum *) (*******************************) (*********) -Definition infinit_sum:(nat->R)->R->Prop:=[s:nat->R;l:R] - (eps:R)(Rgt eps R0)-> - (Ex[N:nat](n:nat)(ge n N)->(Rlt (R_dist (sum_f_R0 s n) l) eps)). +Definition infinit_sum (s:nat -> R) (l:R) : Prop := + forall eps:R, + eps > 0 -> + exists N : nat + | (forall n:nat, (n >= N)%nat -> R_dist (sum_f_R0 s n) l < eps).
\ No newline at end of file diff --git a/theories/Reals/Rgeom.v b/theories/Reals/Rgeom.v index 6e7a3bc67..522ae235c 100644 --- a/theories/Reals/Rgeom.v +++ b/theories/Reals/Rgeom.v @@ -8,77 +8,180 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo. -Require R_sqrt. -V7only [Import R_scope.]. Open Local Scope R_scope. - -Definition dist_euc [x0,y0,x1,y1:R] : R := ``(sqrt ((Rsqr (x0-x1))+(Rsqr (y0-y1))))``. - -Lemma distance_refl : (x0,y0:R) ``(dist_euc x0 y0 x0 y0)==0``. -Intros x0 y0; Unfold dist_euc; Apply Rsqr_inj; [Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0; [Apply pos_Rsqr | Apply pos_Rsqr] | Right; Reflexivity | Rewrite Rsqr_O; Rewrite Rsqr_sqrt; [Unfold Rsqr; Ring | Apply ge0_plus_ge0_is_ge0; [Apply pos_Rsqr | Apply pos_Rsqr]]]. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import R_sqrt. Open Local Scope R_scope. + +Definition dist_euc (x0 y0 x1 y1:R) : R := + sqrt (Rsqr (x0 - x1) + Rsqr (y0 - y1)). + +Lemma distance_refl : forall x0 y0:R, dist_euc x0 y0 x0 y0 = 0. +intros x0 y0; unfold dist_euc in |- *; apply Rsqr_inj; + [ apply sqrt_positivity; apply Rplus_le_le_0_compat; + [ apply Rle_0_sqr | apply Rle_0_sqr ] + | right; reflexivity + | rewrite Rsqr_0; rewrite Rsqr_sqrt; + [ unfold Rsqr in |- *; ring + | apply Rplus_le_le_0_compat; [ apply Rle_0_sqr | apply Rle_0_sqr ] ] ]. Qed. -Lemma distance_symm : (x0,y0,x1,y1:R) ``(dist_euc x0 y0 x1 y1) == (dist_euc x1 y1 x0 y0)``. -Intros x0 y0 x1 y1; Unfold dist_euc; Apply Rsqr_inj; [ Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0 | Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0 | Repeat Rewrite Rsqr_sqrt; [Unfold Rsqr; Ring | Apply ge0_plus_ge0_is_ge0 |Apply ge0_plus_ge0_is_ge0]]; Apply pos_Rsqr. +Lemma distance_symm : + forall x0 y0 x1 y1:R, dist_euc x0 y0 x1 y1 = dist_euc x1 y1 x0 y0. +intros x0 y0 x1 y1; unfold dist_euc in |- *; apply Rsqr_inj; + [ apply sqrt_positivity; apply Rplus_le_le_0_compat + | apply sqrt_positivity; apply Rplus_le_le_0_compat + | repeat rewrite Rsqr_sqrt; + [ unfold Rsqr in |- *; ring + | apply Rplus_le_le_0_compat + | apply Rplus_le_le_0_compat ] ]; apply Rle_0_sqr. Qed. -Lemma law_cosines : (x0,y0,x1,y1,x2,y2,ac:R) let a = (dist_euc x1 y1 x0 y0) in let b=(dist_euc x2 y2 x0 y0) in let c=(dist_euc x2 y2 x1 y1) in ( ``a*c*(cos ac) == ((x0-x1)*(x2-x1) + (y0-y1)*(y2-y1))`` -> ``(Rsqr b)==(Rsqr c)+(Rsqr a)-2*(a*c*(cos ac))`` ). -Unfold dist_euc; Intros; Repeat Rewrite -> Rsqr_sqrt; [ Rewrite H; Unfold Rsqr; Ring | Apply ge0_plus_ge0_is_ge0 | Apply ge0_plus_ge0_is_ge0 | Apply ge0_plus_ge0_is_ge0]; Apply pos_Rsqr. +Lemma law_cosines : + forall x0 y0 x1 y1 x2 y2 ac:R, + let a := dist_euc x1 y1 x0 y0 in + let b := dist_euc x2 y2 x0 y0 in + let c := dist_euc x2 y2 x1 y1 in + a * c * cos ac = (x0 - x1) * (x2 - x1) + (y0 - y1) * (y2 - y1) -> + Rsqr b = Rsqr c + Rsqr a - 2 * (a * c * cos ac). +unfold dist_euc in |- *; intros; repeat rewrite Rsqr_sqrt; + [ rewrite H; unfold Rsqr in |- *; ring + | apply Rplus_le_le_0_compat + | apply Rplus_le_le_0_compat + | apply Rplus_le_le_0_compat ]; apply Rle_0_sqr. Qed. -Lemma triangle : (x0,y0,x1,y1,x2,y2:R) ``(dist_euc x0 y0 x1 y1)<=(dist_euc x0 y0 x2 y2)+(dist_euc x2 y2 x1 y1)``. -Intros; Unfold dist_euc; Apply Rsqr_incr_0; [Rewrite Rsqr_plus; Repeat Rewrite Rsqr_sqrt; [Replace ``(Rsqr (x0-x1))`` with ``(Rsqr (x0-x2))+(Rsqr (x2-x1))+2*(x0-x2)*(x2-x1)``; [Replace ``(Rsqr (y0-y1))`` with ``(Rsqr (y0-y2))+(Rsqr (y2-y1))+2*(y0-y2)*(y2-y1)``; [Apply Rle_anti_compatibility with ``-(Rsqr (x0-x2))-(Rsqr (x2-x1))-(Rsqr (y0-y2))-(Rsqr (y2-y1))``; Replace `` -(Rsqr (x0-x2))-(Rsqr (x2-x1))-(Rsqr (y0-y2))-(Rsqr (y2-y1))+((Rsqr (x0-x2))+(Rsqr (x2-x1))+2*(x0-x2)*(x2-x1)+((Rsqr (y0-y2))+(Rsqr (y2-y1))+2*(y0-y2)*(y2-y1)))`` with ``2*((x0-x2)*(x2-x1)+(y0-y2)*(y2-y1))``; [Replace ``-(Rsqr (x0-x2))-(Rsqr (x2-x1))-(Rsqr (y0-y2))-(Rsqr (y2-y1))+((Rsqr (x0-x2))+(Rsqr (y0-y2))+((Rsqr (x2-x1))+(Rsqr (y2-y1)))+2*(sqrt ((Rsqr (x0-x2))+(Rsqr (y0-y2))))*(sqrt ((Rsqr (x2-x1))+(Rsqr (y2-y1)))))`` with ``2*((sqrt ((Rsqr (x0-x2))+(Rsqr (y0-y2))))*(sqrt ((Rsqr (x2-x1))+(Rsqr (y2-y1)))))``; [Apply Rle_monotony; [Left; Cut ~(O=(2)); [Intros; Generalize (lt_INR_0 (2) (neq_O_lt (2) H)); Intro H0; Assumption | Discriminate] | Apply sqrt_cauchy] | Ring] | Ring] | SqRing] | SqRing] | Apply ge0_plus_ge0_is_ge0; Apply pos_Rsqr | Apply ge0_plus_ge0_is_ge0; Apply pos_Rsqr | Apply ge0_plus_ge0_is_ge0; Apply pos_Rsqr] | Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0; Apply pos_Rsqr | Apply ge0_plus_ge0_is_ge0; Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0; Apply pos_Rsqr]. +Lemma triangle : + forall x0 y0 x1 y1 x2 y2:R, + dist_euc x0 y0 x1 y1 <= dist_euc x0 y0 x2 y2 + dist_euc x2 y2 x1 y1. +intros; unfold dist_euc in |- *; apply Rsqr_incr_0; + [ rewrite Rsqr_plus; repeat rewrite Rsqr_sqrt; + [ replace (Rsqr (x0 - x1)) with + (Rsqr (x0 - x2) + Rsqr (x2 - x1) + 2 * (x0 - x2) * (x2 - x1)); + [ replace (Rsqr (y0 - y1)) with + (Rsqr (y0 - y2) + Rsqr (y2 - y1) + 2 * (y0 - y2) * (y2 - y1)); + [ apply Rplus_le_reg_l with + (- Rsqr (x0 - x2) - Rsqr (x2 - x1) - Rsqr (y0 - y2) - + Rsqr (y2 - y1)); + replace + (- Rsqr (x0 - x2) - Rsqr (x2 - x1) - Rsqr (y0 - y2) - + Rsqr (y2 - y1) + + (Rsqr (x0 - x2) + Rsqr (x2 - x1) + 2 * (x0 - x2) * (x2 - x1) + + (Rsqr (y0 - y2) + Rsqr (y2 - y1) + 2 * (y0 - y2) * (y2 - y1)))) + with (2 * ((x0 - x2) * (x2 - x1) + (y0 - y2) * (y2 - y1))); + [ replace + (- Rsqr (x0 - x2) - Rsqr (x2 - x1) - Rsqr (y0 - y2) - + Rsqr (y2 - y1) + + (Rsqr (x0 - x2) + Rsqr (y0 - y2) + + (Rsqr (x2 - x1) + Rsqr (y2 - y1)) + + 2 * sqrt (Rsqr (x0 - x2) + Rsqr (y0 - y2)) * + sqrt (Rsqr (x2 - x1) + Rsqr (y2 - y1)))) with + (2 * + (sqrt (Rsqr (x0 - x2) + Rsqr (y0 - y2)) * + sqrt (Rsqr (x2 - x1) + Rsqr (y2 - y1)))); + [ apply Rmult_le_compat_l; + [ left; cut (0%nat <> 2%nat); + [ intros; generalize (lt_INR_0 2 (neq_O_lt 2 H)); + intro H0; assumption + | discriminate ] + | apply sqrt_cauchy ] + | ring ] + | ring ] + | ring_Rsqr ] + | ring_Rsqr ] + | apply Rplus_le_le_0_compat; apply Rle_0_sqr + | apply Rplus_le_le_0_compat; apply Rle_0_sqr + | apply Rplus_le_le_0_compat; apply Rle_0_sqr ] + | apply sqrt_positivity; apply Rplus_le_le_0_compat; apply Rle_0_sqr + | apply Rplus_le_le_0_compat; apply sqrt_positivity; + apply Rplus_le_le_0_compat; apply Rle_0_sqr ]. Qed. (******************************************************************) (** Translation *) (******************************************************************) -Definition xt[x,tx:R] : R := ``x+tx``. -Definition yt[y,ty:R] : R := ``y+ty``. +Definition xt (x tx:R) : R := x + tx. +Definition yt (y ty:R) : R := y + ty. -Lemma translation_0 : (x,y:R) ``(xt x 0)==x``/\``(yt y 0)==y``. -Intros x y; Split; [Unfold xt | Unfold yt]; Ring. +Lemma translation_0 : forall x y:R, xt x 0 = x /\ yt y 0 = y. +intros x y; split; [ unfold xt in |- * | unfold yt in |- * ]; ring. Qed. -Lemma isometric_translation : (x1,x2,y1,y2,tx,ty:R) ``(Rsqr (x1-x2))+(Rsqr (y1-y2))==(Rsqr ((xt x1 tx)-(xt x2 tx)))+(Rsqr ((yt y1 ty)-(yt y2 ty)))``. -Intros; Unfold Rsqr xt yt; Ring. +Lemma isometric_translation : + forall x1 x2 y1 y2 tx ty:R, + Rsqr (x1 - x2) + Rsqr (y1 - y2) = + Rsqr (xt x1 tx - xt x2 tx) + Rsqr (yt y1 ty - yt y2 ty). +intros; unfold Rsqr, xt, yt in |- *; ring. Qed. (******************************************************************) (** Rotation *) (******************************************************************) -Definition xr [x,y,theta:R] : R := ``x*(cos theta)+y*(sin theta)``. -Definition yr [x,y,theta:R] : R := ``-x*(sin theta)+y*(cos theta)``. +Definition xr (x y theta:R) : R := x * cos theta + y * sin theta. +Definition yr (x y theta:R) : R := - x * sin theta + y * cos theta. -Lemma rotation_0 : (x,y:R) ``(xr x y 0)==x`` /\ ``(yr x y 0)==y``. -Intros x y; Unfold xr yr; Split; Rewrite cos_0; Rewrite sin_0; Ring. +Lemma rotation_0 : forall x y:R, xr x y 0 = x /\ yr x y 0 = y. +intros x y; unfold xr, yr in |- *; split; rewrite cos_0; rewrite sin_0; ring. Qed. -Lemma rotation_PI2 : (x,y:R) ``(xr x y PI/2)==y`` /\ ``(yr x y PI/2)==-x``. -Intros x y; Unfold xr yr; Split; Rewrite cos_PI2; Rewrite sin_PI2; Ring. +Lemma rotation_PI2 : + forall x y:R, xr x y (PI / 2) = y /\ yr x y (PI / 2) = - x. +intros x y; unfold xr, yr in |- *; split; rewrite cos_PI2; rewrite sin_PI2; + ring. Qed. -Lemma isometric_rotation_0 : (x1,y1,x2,y2,theta:R) ``(Rsqr (x1-x2))+(Rsqr (y1-y2)) == (Rsqr ((xr x1 y1 theta))-(xr x2 y2 theta)) + (Rsqr ((yr x1 y1 theta))-(yr x2 y2 theta))``. -Intros; Unfold xr yr; Replace ``x1*(cos theta)+y1*(sin theta)-(x2*(cos theta)+y2*(sin theta))`` with ``(cos theta)*(x1-x2)+(sin theta)*(y1-y2)``; [Replace ``-x1*(sin theta)+y1*(cos theta)-( -x2*(sin theta)+y2*(cos theta))`` with ``(cos theta)*(y1-y2)+(sin theta)*(x2-x1)``; [Repeat Rewrite Rsqr_plus; Repeat Rewrite Rsqr_times; Repeat Rewrite cos2; Ring; Replace ``x2-x1`` with ``-(x1-x2)``; [Rewrite <- Rsqr_neg; Ring | Ring] |Ring] | Ring]. +Lemma isometric_rotation_0 : + forall x1 y1 x2 y2 theta:R, + Rsqr (x1 - x2) + Rsqr (y1 - y2) = + Rsqr (xr x1 y1 theta - xr x2 y2 theta) + + Rsqr (yr x1 y1 theta - yr x2 y2 theta). +intros; unfold xr, yr in |- *; + replace + (x1 * cos theta + y1 * sin theta - (x2 * cos theta + y2 * sin theta)) with + (cos theta * (x1 - x2) + sin theta * (y1 - y2)); + [ replace + (- x1 * sin theta + y1 * cos theta - (- x2 * sin theta + y2 * cos theta)) + with (cos theta * (y1 - y2) + sin theta * (x2 - x1)); + [ repeat rewrite Rsqr_plus; repeat rewrite Rsqr_mult; repeat rewrite cos2; + ring; replace (x2 - x1) with (- (x1 - x2)); + [ rewrite <- Rsqr_neg; ring | ring ] + | ring ] + | ring ]. Qed. -Lemma isometric_rotation : (x1,y1,x2,y2,theta:R) ``(dist_euc x1 y1 x2 y2) == (dist_euc (xr x1 y1 theta) (yr x1 y1 theta) (xr x2 y2 theta) (yr x2 y2 theta))``. -Unfold dist_euc; Intros; Apply Rsqr_inj; [Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0 | Apply sqrt_positivity; Apply ge0_plus_ge0_is_ge0 | Repeat Rewrite Rsqr_sqrt; [ Apply isometric_rotation_0 | Apply ge0_plus_ge0_is_ge0 | Apply ge0_plus_ge0_is_ge0]]; Apply pos_Rsqr. +Lemma isometric_rotation : + forall x1 y1 x2 y2 theta:R, + dist_euc x1 y1 x2 y2 = + dist_euc (xr x1 y1 theta) (yr x1 y1 theta) (xr x2 y2 theta) + (yr x2 y2 theta). +unfold dist_euc in |- *; intros; apply Rsqr_inj; + [ apply sqrt_positivity; apply Rplus_le_le_0_compat + | apply sqrt_positivity; apply Rplus_le_le_0_compat + | repeat rewrite Rsqr_sqrt; + [ apply isometric_rotation_0 + | apply Rplus_le_le_0_compat + | apply Rplus_le_le_0_compat ] ]; apply Rle_0_sqr. Qed. (******************************************************************) (** Similarity *) (******************************************************************) -Lemma isometric_rot_trans : (x1,y1,x2,y2,tx,ty,theta:R) ``(Rsqr (x1-x2))+(Rsqr (y1-y2)) == (Rsqr ((xr (xt x1 tx) (yt y1 ty) theta)-(xr (xt x2 tx) (yt y2 ty) theta))) + (Rsqr ((yr (xt x1 tx) (yt y1 ty) theta)-(yr (xt x2 tx) (yt y2 ty) theta)))``. -Intros; Rewrite <- isometric_rotation_0; Apply isometric_translation. +Lemma isometric_rot_trans : + forall x1 y1 x2 y2 tx ty theta:R, + Rsqr (x1 - x2) + Rsqr (y1 - y2) = + Rsqr (xr (xt x1 tx) (yt y1 ty) theta - xr (xt x2 tx) (yt y2 ty) theta) + + Rsqr (yr (xt x1 tx) (yt y1 ty) theta - yr (xt x2 tx) (yt y2 ty) theta). +intros; rewrite <- isometric_rotation_0; apply isometric_translation. Qed. -Lemma isometric_trans_rot : (x1,y1,x2,y2,tx,ty,theta:R) ``(Rsqr (x1-x2))+(Rsqr (y1-y2)) == (Rsqr ((xt (xr x1 y1 theta) tx)-(xt (xr x2 y2 theta) tx))) + (Rsqr ((yt (yr x1 y1 theta) ty)-(yt (yr x2 y2 theta) ty)))``. -Intros; Rewrite <- isometric_translation; Apply isometric_rotation_0. -Qed. +Lemma isometric_trans_rot : + forall x1 y1 x2 y2 tx ty theta:R, + Rsqr (x1 - x2) + Rsqr (y1 - y2) = + Rsqr (xt (xr x1 y1 theta) tx - xt (xr x2 y2 theta) tx) + + Rsqr (yt (yr x1 y1 theta) ty - yt (yr x2 y2 theta) ty). +intros; rewrite <- isometric_translation; apply isometric_rotation_0. +Qed.
\ No newline at end of file diff --git a/theories/Reals/RiemannInt.v b/theories/Reals/RiemannInt.v index a44f3c1b5..2766aa2fe 100644 --- a/theories/Reals/RiemannInt.v +++ b/theories/Reals/RiemannInt.v @@ -8,1692 +8,3256 @@ (*i $Id$ i*) -Require Rfunctions. -Require SeqSeries. -Require Ranalysis. -Require Rbase. -Require RiemannInt_SF. -Require Classical_Prop. -Require Classical_Pred_Type. -Require Max. -V7only [Import R_scope.]. Open Local Scope R_scope. - -Implicit Arguments On. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Ranalysis. +Require Import Rbase. +Require Import RiemannInt_SF. +Require Import Classical_Prop. +Require Import Classical_Pred_Type. +Require Import Max. Open Local Scope R_scope. + +Set Implicit Arguments. (********************************************) (* Riemann's Integral *) (********************************************) -Definition Riemann_integrable [f:R->R;a,b:R] : Type := (eps:posreal) (SigT ? [phi:(StepFun a b)](SigT ? [psi:(StepFun a b)]((t:R)``(Rmin a b)<=t<=(Rmax a b)``->``(Rabsolu ((f t)-(phi t)))<=(psi t)``)/\``(Rabsolu (RiemannInt_SF psi))<eps``)). - -Definition phi_sequence [un:nat->posreal;f:R->R;a,b:R;pr:(Riemann_integrable f a b)] := [n:nat](projT1 ? ? (pr (un n))). - -Lemma phi_sequence_prop : (un:nat->posreal;f:R->R;a,b:R;pr:(Riemann_integrable f a b);N:nat) (SigT ? [psi:(StepFun a b)]((t:R)``(Rmin a b)<=t<=(Rmax a b)``->``(Rabsolu ((f t)-[(phi_sequence un pr N t)]))<=(psi t)``)/\``(Rabsolu (RiemannInt_SF psi))<(un N)``). -Intros; Apply (projT2 ? ? (pr (un N))). +Definition Riemann_integrable (f:R -> R) (a b:R) : Type := + forall eps:posreal, + sigT + (fun phi:StepFun a b => + sigT + (fun psi:StepFun a b => + (forall t:R, + Rmin a b <= t <= Rmax a b -> Rabs (f t - phi t) <= psi t) /\ + Rabs (RiemannInt_SF psi) < eps)). + +Definition phi_sequence (un:nat -> posreal) (f:R -> R) + (a b:R) (pr:Riemann_integrable f a b) (n:nat) := + projT1 (pr (un n)). + +Lemma phi_sequence_prop : + forall (un:nat -> posreal) (f:R -> R) (a b:R) (pr:Riemann_integrable f a b) + (N:nat), + sigT + (fun psi:StepFun a b => + (forall t:R, + Rmin a b <= t <= Rmax a b -> + Rabs (f t - phi_sequence un pr N t) <= psi t) /\ + Rabs (RiemannInt_SF psi) < un N). +intros; apply (projT2 (pr (un N))). Qed. -Lemma RiemannInt_P1 : (f:R->R;a,b:R) (Riemann_integrable f a b) -> (Riemann_integrable f b a). -Unfold Riemann_integrable; Intros; Elim (X eps); Clear X; Intros; Elim p; Clear p; Intros; Apply Specif.existT with (mkStepFun (StepFun_P6 (pre x))); Apply Specif.existT with (mkStepFun (StepFun_P6 (pre x0))); Elim p; Clear p; Intros; Split. -Intros; Apply (H t); Elim H1; Clear H1; Intros; Split; [Apply Rle_trans with (Rmin b a); Try Assumption; Right; Unfold Rmin | Apply Rle_trans with (Rmax b a); Try Assumption; Right; Unfold Rmax]; (Case (total_order_Rle a b); Case (total_order_Rle b a); Intros; Try Reflexivity Orelse Apply Rle_antisym; [Assumption | Assumption | Auto with real | Auto with real]). -Generalize H0; Unfold RiemannInt_SF; Case (total_order_Rle a b); Case (total_order_Rle b a); Intros; (Replace (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre x0)))) (subdivision (mkStepFun (StepFun_P6 (pre x0))))) with (Int_SF (subdivision_val x0) (subdivision x0)); [Idtac | Apply StepFun_P17 with (fe x0) a b; [Apply StepFun_P1 | Apply StepFun_P2; Apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre x0))))]]). -Apply H1. -Rewrite Rabsolu_Ropp; Apply H1. -Rewrite Rabsolu_Ropp in H1; Apply H1. -Apply H1. +Lemma RiemannInt_P1 : + forall (f:R -> R) (a b:R), + Riemann_integrable f a b -> Riemann_integrable f b a. +unfold Riemann_integrable in |- *; intros; elim (X eps); clear X; intros; + elim p; clear p; intros; apply existT with (mkStepFun (StepFun_P6 (pre x))); + apply existT with (mkStepFun (StepFun_P6 (pre x0))); + elim p; clear p; intros; split. +intros; apply (H t); elim H1; clear H1; intros; split; + [ apply Rle_trans with (Rmin b a); try assumption; right; + unfold Rmin in |- * + | apply Rle_trans with (Rmax b a); try assumption; right; + unfold Rmax in |- * ]; + (case (Rle_dec a b); case (Rle_dec b a); intros; + try reflexivity || apply Rle_antisym; + [ assumption | assumption | auto with real | auto with real ]). +generalize H0; unfold RiemannInt_SF in |- *; case (Rle_dec a b); + case (Rle_dec b a); intros; + (replace + (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre x0)))) + (subdivision (mkStepFun (StepFun_P6 (pre x0))))) with + (Int_SF (subdivision_val x0) (subdivision x0)); + [ idtac + | apply StepFun_P17 with (fe x0) a b; + [ apply StepFun_P1 + | apply StepFun_P2; + apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre x0)))) ] ]). +apply H1. +rewrite Rabs_Ropp; apply H1. +rewrite Rabs_Ropp in H1; apply H1. +apply H1. Qed. -Lemma RiemannInt_P2 : (f:R->R;a,b:R;un:nat->posreal;vn,wn:nat->(StepFun a b)) (Un_cv un R0) -> ``a<=b`` -> ((n:nat)((t:R)``(Rmin a b)<=t<=(Rmax a b)``->``(Rabsolu ((f t)-(vn n t)))<=(wn n t)``)/\``(Rabsolu (RiemannInt_SF (wn n)))<(un n)``) -> (sigTT ? [l:R](Un_cv [N:nat](RiemannInt_SF (vn N)) l)). -Intros; Apply R_complete; Unfold Un_cv in H; Unfold Cauchy_crit; Intros; Assert H3 : ``0<eps/2``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Elim (H ? H3); Intros N0 H4; Exists N0; Intros; Unfold R_dist; Unfold R_dist in H4; Elim (H1 n); Elim (H1 m); Intros; Replace ``(RiemannInt_SF (vn n))-(RiemannInt_SF (vn m))`` with ``(RiemannInt_SF (vn n))+(-1)*(RiemannInt_SF (vn m))``; [Idtac | Ring]; Rewrite <- StepFun_P30; Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 ``-1`` (vn n) (vn m)))))). -Apply StepFun_P34; Assumption. -Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 R1 (wn n) (wn m)))). -Apply StepFun_P37; Try Assumption. -Intros; Simpl; Apply Rle_trans with ``(Rabsolu ((vn n x)-(f x)))+(Rabsolu ((f x)-(vn m x)))``. -Replace ``(vn n x)+-1*(vn m x)`` with ``((vn n x)-(f x))+((f x)-(vn m x))``; [Apply Rabsolu_triang | Ring]. -Assert H12 : (Rmin a b)==a. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Assert H13 : (Rmax a b)==b. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Rewrite <- H12 in H11; Pattern 2 b in H11; Rewrite <- H13 in H11; Rewrite Rmult_1l; Apply Rplus_le. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H9. -Elim H11; Intros; Split; Left; Assumption. -Apply H7. -Elim H11; Intros; Split; Left; Assumption. -Rewrite StepFun_P30; Rewrite Rmult_1l; Apply Rlt_trans with ``(un n)+(un m)``. -Apply Rle_lt_trans with ``(Rabsolu (RiemannInt_SF (wn n)))+(Rabsolu (RiemannInt_SF (wn m)))``. -Apply Rplus_le; Apply Rle_Rabsolu. -Apply Rplus_lt; Assumption. -Apply Rle_lt_trans with ``(Rabsolu (un n))+(Rabsolu (un m))``. -Apply Rplus_le; Apply Rle_Rabsolu. -Replace (pos (un n)) with ``(un n)-0``; [Idtac | Ring]; Replace (pos (un m)) with ``(un m)-0``; [Idtac | Ring]; Rewrite (double_var eps); Apply Rplus_lt; Apply H4; Assumption. +Lemma RiemannInt_P2 : + forall (f:R -> R) (a b:R) (un:nat -> posreal) (vn wn:nat -> StepFun a b), + Un_cv un 0 -> + a <= b -> + (forall n:nat, + (forall t:R, Rmin a b <= t <= Rmax a b -> Rabs (f t - vn n t) <= wn n t) /\ + Rabs (RiemannInt_SF (wn n)) < un n) -> + sigT (fun l:R => Un_cv (fun N:nat => RiemannInt_SF (vn N)) l). +intros; apply R_complete; unfold Un_cv in H; unfold Cauchy_crit in |- *; + intros; assert (H3 : 0 < eps / 2). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (H _ H3); intros N0 H4; exists N0; intros; unfold R_dist in |- *; + unfold R_dist in H4; elim (H1 n); elim (H1 m); intros; + replace (RiemannInt_SF (vn n) - RiemannInt_SF (vn m)) with + (RiemannInt_SF (vn n) + -1 * RiemannInt_SF (vn m)); + [ idtac | ring ]; rewrite <- StepFun_P30; + apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (vn n) (vn m)))))). +apply StepFun_P34; assumption. +apply Rle_lt_trans with + (RiemannInt_SF (mkStepFun (StepFun_P28 1 (wn n) (wn m)))). +apply StepFun_P37; try assumption. +intros; simpl in |- *; + apply Rle_trans with (Rabs (vn n x - f x) + Rabs (f x - vn m x)). +replace (vn n x + -1 * vn m x) with (vn n x - f x + (f x - vn m x)); + [ apply Rabs_triang | ring ]. +assert (H12 : Rmin a b = a). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +assert (H13 : Rmax a b = b). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +rewrite <- H12 in H11; pattern b at 2 in H11; rewrite <- H13 in H11; + rewrite Rmult_1_l; apply Rplus_le_compat. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H9. +elim H11; intros; split; left; assumption. +apply H7. +elim H11; intros; split; left; assumption. +rewrite StepFun_P30; rewrite Rmult_1_l; apply Rlt_trans with (un n + un m). +apply Rle_lt_trans with + (Rabs (RiemannInt_SF (wn n)) + Rabs (RiemannInt_SF (wn m))). +apply Rplus_le_compat; apply RRle_abs. +apply Rplus_lt_compat; assumption. +apply Rle_lt_trans with (Rabs (un n) + Rabs (un m)). +apply Rplus_le_compat; apply RRle_abs. +replace (pos (un n)) with (un n - 0); [ idtac | ring ]; + replace (pos (un m)) with (un m - 0); [ idtac | ring ]; + rewrite (double_var eps); apply Rplus_lt_compat; apply H4; + assumption. Qed. -Lemma RiemannInt_P3 : (f:R->R;a,b:R;un:nat->posreal;vn,wn:nat->(StepFun a b)) (Un_cv un R0) -> ((n:nat)((t:R)``(Rmin a b)<=t<=(Rmax a b)``->``(Rabsolu ((f t)-(vn n t)))<=(wn n t)``)/\``(Rabsolu (RiemannInt_SF (wn n)))<(un n)``)->(sigTT R ([l:R](Un_cv ([N:nat](RiemannInt_SF (vn N))) l))). -Intros; Case (total_order_Rle a b); Intro. -Apply RiemannInt_P2 with f un wn; Assumption. -Assert H1 : ``b<=a``; Auto with real. -Pose vn' := [n:nat](mkStepFun (StepFun_P6 (pre (vn n)))); Pose wn' := [n:nat](mkStepFun (StepFun_P6 (pre (wn n)))); Assert H2 : (n:nat)((t:R)``(Rmin b a)<=t<=(Rmax b a)``->``(Rabsolu ((f t)-(vn' n t)))<=(wn' n t)``)/\``(Rabsolu (RiemannInt_SF (wn' n)))<(un n)``. -Intro; Elim (H0 n0); Intros; Split. -Intros; Apply (H2 t); Elim H4; Clear H4; Intros; Split; [Apply Rle_trans with (Rmin b a); Try Assumption; Right; Unfold Rmin | Apply Rle_trans with (Rmax b a); Try Assumption; Right; Unfold Rmax]; (Case (total_order_Rle a b); Case (total_order_Rle b a); Intros; Try Reflexivity Orelse Apply Rle_antisym; [Assumption | Assumption | Auto with real | Auto with real]). -Generalize H3; Unfold RiemannInt_SF; Case (total_order_Rle a b); Case (total_order_Rle b a); Unfold wn'; Intros; (Replace (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (wn n0))))) (subdivision (mkStepFun (StepFun_P6 (pre (wn n0)))))) with (Int_SF (subdivision_val (wn n0)) (subdivision (wn n0))); [Idtac | Apply StepFun_P17 with (fe (wn n0)) a b; [Apply StepFun_P1 | Apply StepFun_P2; Apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre (wn n0)))))]]). -Apply H4. -Rewrite Rabsolu_Ropp; Apply H4. -Rewrite Rabsolu_Ropp in H4; Apply H4. -Apply H4. -Assert H3 := (RiemannInt_P2 H H1 H2); Elim H3; Intros; Apply existTT with ``-x``; Unfold Un_cv; Unfold Un_cv in p; Intros; Elim (p ? H4); Intros; Exists x0; Intros; Generalize (H5 ? H6); Unfold R_dist RiemannInt_SF; Case (total_order_Rle b a); Case (total_order_Rle a b); Intros. -Elim n; Assumption. -Unfold vn' in H7; Replace (Int_SF (subdivision_val (vn n0)) (subdivision (vn n0))) with (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (vn n0))))) (subdivision (mkStepFun (StepFun_P6 (pre (vn n0)))))); [Unfold Rminus; Rewrite Ropp_Ropp; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Apply H7 | Symmetry; Apply StepFun_P17 with (fe (vn n0)) a b; [Apply StepFun_P1 | Apply StepFun_P2; Apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre (vn n0)))))]]. -Elim n1; Assumption. -Elim n2; Assumption. +Lemma RiemannInt_P3 : + forall (f:R -> R) (a b:R) (un:nat -> posreal) (vn wn:nat -> StepFun a b), + Un_cv un 0 -> + (forall n:nat, + (forall t:R, Rmin a b <= t <= Rmax a b -> Rabs (f t - vn n t) <= wn n t) /\ + Rabs (RiemannInt_SF (wn n)) < un n) -> + sigT (fun l:R => Un_cv (fun N:nat => RiemannInt_SF (vn N)) l). +intros; case (Rle_dec a b); intro. +apply RiemannInt_P2 with f un wn; assumption. +assert (H1 : b <= a); auto with real. +pose (vn' := fun n:nat => mkStepFun (StepFun_P6 (pre (vn n)))); + pose (wn' := fun n:nat => mkStepFun (StepFun_P6 (pre (wn n)))); + assert + (H2 : + forall n:nat, + (forall t:R, + Rmin b a <= t <= Rmax b a -> Rabs (f t - vn' n t) <= wn' n t) /\ + Rabs (RiemannInt_SF (wn' n)) < un n). +intro; elim (H0 n0); intros; split. +intros; apply (H2 t); elim H4; clear H4; intros; split; + [ apply Rle_trans with (Rmin b a); try assumption; right; + unfold Rmin in |- * + | apply Rle_trans with (Rmax b a); try assumption; right; + unfold Rmax in |- * ]; + (case (Rle_dec a b); case (Rle_dec b a); intros; + try reflexivity || apply Rle_antisym; + [ assumption | assumption | auto with real | auto with real ]). +generalize H3; unfold RiemannInt_SF in |- *; case (Rle_dec a b); + case (Rle_dec b a); unfold wn' in |- *; intros; + (replace + (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (wn n0))))) + (subdivision (mkStepFun (StepFun_P6 (pre (wn n0)))))) with + (Int_SF (subdivision_val (wn n0)) (subdivision (wn n0))); + [ idtac + | apply StepFun_P17 with (fe (wn n0)) a b; + [ apply StepFun_P1 + | apply StepFun_P2; + apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre (wn n0))))) ] ]). +apply H4. +rewrite Rabs_Ropp; apply H4. +rewrite Rabs_Ropp in H4; apply H4. +apply H4. +assert (H3 := RiemannInt_P2 _ _ _ _ H H1 H2); elim H3; intros; + apply existT with (- x); unfold Un_cv in |- *; unfold Un_cv in p; + intros; elim (p _ H4); intros; exists x0; intros; + generalize (H5 _ H6); unfold R_dist, RiemannInt_SF in |- *; + case (Rle_dec b a); case (Rle_dec a b); intros. +elim n; assumption. +unfold vn' in H7; + replace (Int_SF (subdivision_val (vn n0)) (subdivision (vn n0))) with + (Int_SF (subdivision_val (mkStepFun (StepFun_P6 (pre (vn n0))))) + (subdivision (mkStepFun (StepFun_P6 (pre (vn n0)))))); + [ unfold Rminus in |- *; rewrite Ropp_involutive; rewrite <- Rabs_Ropp; + rewrite Ropp_plus_distr; rewrite Ropp_involutive; + apply H7 + | symmetry in |- *; apply StepFun_P17 with (fe (vn n0)) a b; + [ apply StepFun_P1 + | apply StepFun_P2; + apply (StepFun_P1 (mkStepFun (StepFun_P6 (pre (vn n0))))) ] ]. +elim n1; assumption. +elim n2; assumption. Qed. -Lemma RiemannInt_exists : (f:R->R;a,b:R;pr:(Riemann_integrable f a b);un:nat->posreal) (Un_cv un R0) -> (sigTT ? [l:R](Un_cv [N:nat](RiemannInt_SF (phi_sequence un pr N)) l)). -Intros f; Intros; Apply RiemannInt_P3 with f un [n:nat](projT1 ? ? (phi_sequence_prop un pr n)); [Apply H | Intro; Apply (projT2 ? ? (phi_sequence_prop un pr n))]. +Lemma RiemannInt_exists : + forall (f:R -> R) (a b:R) (pr:Riemann_integrable f a b) + (un:nat -> posreal), + Un_cv un 0 -> + sigT + (fun l:R => Un_cv (fun N:nat => RiemannInt_SF (phi_sequence un pr N)) l). +intros f; intros; + apply RiemannInt_P3 with + f un (fun n:nat => projT1 (phi_sequence_prop un pr n)); + [ apply H | intro; apply (projT2 (phi_sequence_prop un pr n)) ]. Qed. -Lemma RiemannInt_P4 : (f:R->R;a,b,l:R;pr1,pr2:(Riemann_integrable f a b);un,vn:nat->posreal) (Un_cv un R0) -> (Un_cv vn R0) -> (Un_cv [N:nat](RiemannInt_SF (phi_sequence un pr1 N)) l) -> (Un_cv [N:nat](RiemannInt_SF (phi_sequence vn pr2 N)) l). -Unfold Un_cv; Unfold R_dist; Intros f; Intros; Assert H3 : ``0<eps/3``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Elim (H ? H3); Clear H; Intros N0 H; Elim (H0 ? H3); Clear H0; Intros N1 H0; Elim (H1 ? H3); Clear H1; Intros N2 H1; Pose N := (max (max N0 N1) N2); Exists N; Intros; Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF [(phi_sequence vn pr2 n)])-(RiemannInt_SF [(phi_sequence un pr1 n)])))+(Rabsolu ((RiemannInt_SF [(phi_sequence un pr1 n)])-l))``. -Replace ``(RiemannInt_SF [(phi_sequence vn pr2 n)])-l`` with ``((RiemannInt_SF [(phi_sequence vn pr2 n)])-(RiemannInt_SF [(phi_sequence un pr1 n)]))+((RiemannInt_SF [(phi_sequence un pr1 n)])-l)``; [Apply Rabsolu_triang | Ring]. -Replace ``eps`` with ``2*eps/3+eps/3``. -Apply Rplus_lt. -Elim (phi_sequence_prop vn pr2 n); Intros psi_vn H5; Elim (phi_sequence_prop un pr1 n); Intros psi_un H6; Replace ``(RiemannInt_SF [(phi_sequence vn pr2 n)])-(RiemannInt_SF [(phi_sequence un pr1 n)])`` with ``(RiemannInt_SF [(phi_sequence vn pr2 n)])+(-1)*(RiemannInt_SF [(phi_sequence un pr1 n)])``; [Idtac | Ring]; Rewrite <- StepFun_P30. -Case (total_order_Rle a b); Intro. -Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 ``-1`` (phi_sequence vn pr2 n) (phi_sequence un pr1 n)))))). -Apply StepFun_P34; Assumption. -Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 R1 psi_un psi_vn))). -Apply StepFun_P37; Try Assumption; Intros; Simpl; Rewrite Rmult_1l; Apply Rle_trans with ``(Rabsolu ([(phi_sequence vn pr2 n x)]-(f x)))+(Rabsolu ((f x)-[(phi_sequence un pr1 n x)]))``. -Replace ``[(phi_sequence vn pr2 n x)]+-1*[(phi_sequence un pr1 n x)]`` with ``([(phi_sequence vn pr2 n x)]-(f x))+((f x)-[(phi_sequence un pr1 n x)])``; [Apply Rabsolu_triang | Ring]. -Assert H10 : (Rmin a b)==a. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Assert H11 : (Rmax a b)==b. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Rewrite (Rplus_sym (psi_un x)); Apply Rplus_le. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Elim H5; Intros; Apply H8. -Rewrite H10; Rewrite H11; Elim H7; Intros; Split; Left; Assumption. -Elim H6; Intros; Apply H8. -Rewrite H10; Rewrite H11; Elim H7; Intros; Split; Left; Assumption. -Rewrite StepFun_P30; Rewrite Rmult_1l; Rewrite double; Apply Rplus_lt. -Apply Rlt_trans with (pos (un n)). -Elim H6; Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF psi_un)). -Apply Rle_Rabsolu. -Assumption. -Replace (pos (un n)) with (Rabsolu ``(un n)-0``); [Apply H; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_trans with (max N0 N1); Apply le_max_l | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (un n))]. -Apply Rlt_trans with (pos (vn n)). -Elim H5; Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF psi_vn)). -Apply Rle_Rabsolu; Assumption. -Assumption. -Replace (pos (vn n)) with (Rabsolu ``(vn n)-0``); [Apply H0; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_trans with (max N0 N1); [Apply le_max_r | Apply le_max_l] | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (vn n))]. -Rewrite StepFun_P39; Rewrite Rabsolu_Ropp; Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 ``-1`` (phi_sequence vn pr2 n) (phi_sequence un pr1 n))))))))). -Apply StepFun_P34; Try Auto with real. -Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 R1 psi_vn psi_un)))))). -Apply StepFun_P37. -Auto with real. -Intros; Simpl; Rewrite Rmult_1l; Apply Rle_trans with ``(Rabsolu ([(phi_sequence vn pr2 n x)]-(f x)))+(Rabsolu ((f x)-[(phi_sequence un pr1 n x)]))``. -Replace ``[(phi_sequence vn pr2 n x)]+-1*[(phi_sequence un pr1 n x)]`` with ``([(phi_sequence vn pr2 n x)]-(f x))+((f x)-[(phi_sequence un pr1 n x)])``; [Apply Rabsolu_triang | Ring]. -Assert H10 : (Rmin a b)==b. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Elim n0; Assumption | Reflexivity]. -Assert H11 : (Rmax a b)==a. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Elim n0; Assumption | Reflexivity]. -Apply Rplus_le. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Elim H5; Intros; Apply H8. -Rewrite H10; Rewrite H11; Elim H7; Intros; Split; Left; Assumption. -Elim H6; Intros; Apply H8. -Rewrite H10; Rewrite H11; Elim H7; Intros; Split; Left; Assumption. -Rewrite <- (Ropp_Ropp (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 R1 psi_vn psi_un))))))); Rewrite <- StepFun_P39; Rewrite StepFun_P30; Rewrite Rmult_1l; Rewrite double; Rewrite Ropp_distr1; Apply Rplus_lt. -Apply Rlt_trans with (pos (vn n)). -Elim H5; Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF psi_vn)). -Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. -Assumption. -Replace (pos (vn n)) with (Rabsolu ``(vn n)-0``); [Apply H0; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_trans with (max N0 N1); [Apply le_max_r | Apply le_max_l] | Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (vn n))]. -Apply Rlt_trans with (pos (un n)). -Elim H6; Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF psi_un)). -Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu; Assumption. -Assumption. -Replace (pos (un n)) with (Rabsolu ``(un n)-0``); [Apply H; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_trans with (max N0 N1); Apply le_max_l | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (un n))]. -Apply H1; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_max_r. -Apply r_Rmult_mult with ``3``; [Unfold Rdiv; Rewrite Rmult_Rplus_distr; Do 2 Rewrite (Rmult_sym ``3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. +Lemma RiemannInt_P4 : + forall (f:R -> R) (a b l:R) (pr1 pr2:Riemann_integrable f a b) + (un vn:nat -> posreal), + Un_cv un 0 -> + Un_cv vn 0 -> + Un_cv (fun N:nat => RiemannInt_SF (phi_sequence un pr1 N)) l -> + Un_cv (fun N:nat => RiemannInt_SF (phi_sequence vn pr2 N)) l. +unfold Un_cv in |- *; unfold R_dist in |- *; intros f; intros; + assert (H3 : 0 < eps / 3). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (H _ H3); clear H; intros N0 H; elim (H0 _ H3); clear H0; intros N1 H0; + elim (H1 _ H3); clear H1; intros N2 H1; pose (N := max (max N0 N1) N2); + exists N; intros; + apply Rle_lt_trans with + (Rabs + (RiemannInt_SF (phi_sequence vn pr2 n) - + RiemannInt_SF (phi_sequence un pr1 n)) + + Rabs (RiemannInt_SF (phi_sequence un pr1 n) - l)). +replace (RiemannInt_SF (phi_sequence vn pr2 n) - l) with + (RiemannInt_SF (phi_sequence vn pr2 n) - + RiemannInt_SF (phi_sequence un pr1 n) + + (RiemannInt_SF (phi_sequence un pr1 n) - l)); [ apply Rabs_triang | ring ]. +replace eps with (2 * (eps / 3) + eps / 3). +apply Rplus_lt_compat. +elim (phi_sequence_prop vn pr2 n); intros psi_vn H5; + elim (phi_sequence_prop un pr1 n); intros psi_un H6; + replace + (RiemannInt_SF (phi_sequence vn pr2 n) - + RiemannInt_SF (phi_sequence un pr1 n)) with + (RiemannInt_SF (phi_sequence vn pr2 n) + + -1 * RiemannInt_SF (phi_sequence un pr1 n)); [ idtac | ring ]; + rewrite <- StepFun_P30. +case (Rle_dec a b); intro. +apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun + (StepFun_P32 + (mkStepFun + (StepFun_P28 (-1) (phi_sequence vn pr2 n) + (phi_sequence un pr1 n)))))). +apply StepFun_P34; assumption. +apply Rle_lt_trans with + (RiemannInt_SF (mkStepFun (StepFun_P28 1 psi_un psi_vn))). +apply StepFun_P37; try assumption; intros; simpl in |- *; rewrite Rmult_1_l; + apply Rle_trans with + (Rabs (phi_sequence vn pr2 n x - f x) + + Rabs (f x - phi_sequence un pr1 n x)). +replace (phi_sequence vn pr2 n x + -1 * phi_sequence un pr1 n x) with + (phi_sequence vn pr2 n x - f x + (f x - phi_sequence un pr1 n x)); + [ apply Rabs_triang | ring ]. +assert (H10 : Rmin a b = a). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +assert (H11 : Rmax a b = b). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +rewrite (Rplus_comm (psi_un x)); apply Rplus_le_compat. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim H5; intros; apply H8. +rewrite H10; rewrite H11; elim H7; intros; split; left; assumption. +elim H6; intros; apply H8. +rewrite H10; rewrite H11; elim H7; intros; split; left; assumption. +rewrite StepFun_P30; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat. +apply Rlt_trans with (pos (un n)). +elim H6; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_un)). +apply RRle_abs. +assumption. +replace (pos (un n)) with (Rabs (un n - 0)); + [ apply H; unfold ge in |- *; apply le_trans with N; try assumption; + unfold N in |- *; apply le_trans with (max N0 N1); + apply le_max_l + | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; + apply Rle_ge; left; apply (cond_pos (un n)) ]. +apply Rlt_trans with (pos (vn n)). +elim H5; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_vn)). +apply RRle_abs; assumption. +assumption. +replace (pos (vn n)) with (Rabs (vn n - 0)); + [ apply H0; unfold ge in |- *; apply le_trans with N; try assumption; + unfold N in |- *; apply le_trans with (max N0 N1); + [ apply le_max_r | apply le_max_l ] + | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; + apply Rle_ge; left; apply (cond_pos (vn n)) ]. +rewrite StepFun_P39; rewrite Rabs_Ropp; + apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun + (StepFun_P32 + (mkStepFun + (StepFun_P6 + (pre + (mkStepFun + (StepFun_P28 (-1) (phi_sequence vn pr2 n) + (phi_sequence un pr1 n))))))))). +apply StepFun_P34; try auto with real. +apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 1 psi_vn psi_un)))))). +apply StepFun_P37. +auto with real. +intros; simpl in |- *; rewrite Rmult_1_l; + apply Rle_trans with + (Rabs (phi_sequence vn pr2 n x - f x) + + Rabs (f x - phi_sequence un pr1 n x)). +replace (phi_sequence vn pr2 n x + -1 * phi_sequence un pr1 n x) with + (phi_sequence vn pr2 n x - f x + (f x - phi_sequence un pr1 n x)); + [ apply Rabs_triang | ring ]. +assert (H10 : Rmin a b = b). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ elim n0; assumption | reflexivity ]. +assert (H11 : Rmax a b = a). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ elim n0; assumption | reflexivity ]. +apply Rplus_le_compat. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim H5; intros; apply H8. +rewrite H10; rewrite H11; elim H7; intros; split; left; assumption. +elim H6; intros; apply H8. +rewrite H10; rewrite H11; elim H7; intros; split; left; assumption. +rewrite <- + (Ropp_involutive + (RiemannInt_SF + (mkStepFun + (StepFun_P6 (pre (mkStepFun (StepFun_P28 1 psi_vn psi_un))))))) + ; rewrite <- StepFun_P39; rewrite StepFun_P30; rewrite Rmult_1_l; + rewrite double; rewrite Ropp_plus_distr; apply Rplus_lt_compat. +apply Rlt_trans with (pos (vn n)). +elim H5; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_vn)). +rewrite <- Rabs_Ropp; apply RRle_abs. +assumption. +replace (pos (vn n)) with (Rabs (vn n - 0)); + [ apply H0; unfold ge in |- *; apply le_trans with N; try assumption; + unfold N in |- *; apply le_trans with (max N0 N1); + [ apply le_max_r | apply le_max_l ] + | unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge; + left; apply (cond_pos (vn n)) ]. +apply Rlt_trans with (pos (un n)). +elim H6; intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF psi_un)). +rewrite <- Rabs_Ropp; apply RRle_abs; assumption. +assumption. +replace (pos (un n)) with (Rabs (un n - 0)); + [ apply H; unfold ge in |- *; apply le_trans with N; try assumption; + unfold N in |- *; apply le_trans with (max N0 N1); + apply le_max_l + | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; + apply Rle_ge; left; apply (cond_pos (un n)) ]. +apply H1; unfold ge in |- *; apply le_trans with N; try assumption; + unfold N in |- *; apply le_max_r. +apply Rmult_eq_reg_l with 3; + [ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l; + do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym; [ ring | discrR ] + | discrR ]. Qed. -Lemma RinvN_pos : (n:nat) ``0</((INR n)+1)``. -Intro; Apply Rlt_Rinv; Apply ge0_plus_gt0_is_gt0; [Apply pos_INR | Apply Rlt_R0_R1]. +Lemma RinvN_pos : forall n:nat, 0 < / (INR n + 1). +intro; apply Rinv_0_lt_compat; apply Rplus_le_lt_0_compat; + [ apply pos_INR | apply Rlt_0_1 ]. Qed. -Definition RinvN : nat->posreal := [N:nat](mkposreal ? (RinvN_pos N)). +Definition RinvN (N:nat) : posreal := mkposreal _ (RinvN_pos N). -Lemma RinvN_cv : (Un_cv RinvN R0). -Unfold Un_cv; Intros; Assert H0 := (archimed ``/eps``); Elim H0; Clear H0; Intros; Assert H2 : `0<=(up (Rinv eps))`. -Apply le_IZR; Left; Apply Rlt_trans with ``/eps``; [Apply Rlt_Rinv; Assumption | Assumption]. -Elim (IZN ? H2); Intros; Exists x; Intros; Unfold R_dist; Simpl; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Assert H5 : ``0<(INR n)+1``. -Apply ge0_plus_gt0_is_gt0; [Apply pos_INR | Apply Rlt_R0_R1]. -Rewrite Rabsolu_right; [Idtac | Left; Change ``0</((INR n)+1)``; Apply Rlt_Rinv; Assumption]; Apply Rle_lt_trans with ``/((INR x)+1)``. -Apply Rle_Rinv. -Apply ge0_plus_gt0_is_gt0; [Apply pos_INR | Apply Rlt_R0_R1]. -Assumption. -Do 2 Rewrite <- (Rplus_sym R1); Apply Rle_compatibility; Apply le_INR; Apply H4. -Rewrite <- (Rinv_Rinv eps). -Apply Rinv_lt. -Apply Rmult_lt_pos. -Apply Rlt_Rinv; Assumption. -Apply ge0_plus_gt0_is_gt0; [Apply pos_INR | Apply Rlt_R0_R1]. -Apply Rlt_trans with (INR x); [Rewrite INR_IZR_INZ; Rewrite <- H3; Apply H0 | Pattern 1 (INR x); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1]. -Red; Intro; Rewrite H6 in H; Elim (Rlt_antirefl ? H). +Lemma RinvN_cv : Un_cv RinvN 0. +unfold Un_cv in |- *; intros; assert (H0 := archimed (/ eps)); elim H0; + clear H0; intros; assert (H2 : (0 <= up (/ eps))%Z). +apply le_IZR; left; apply Rlt_trans with (/ eps); + [ apply Rinv_0_lt_compat; assumption | assumption ]. +elim (IZN _ H2); intros; exists x; intros; unfold R_dist in |- *; + simpl in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; assert (H5 : 0 < INR n + 1). +apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ]. +rewrite Rabs_right; + [ idtac + | left; change (0 < / (INR n + 1)) in |- *; apply Rinv_0_lt_compat; + assumption ]; apply Rle_lt_trans with (/ (INR x + 1)). +apply Rle_Rinv. +apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ]. +assumption. +do 2 rewrite <- (Rplus_comm 1); apply Rplus_le_compat_l; apply le_INR; + apply H4. +rewrite <- (Rinv_involutive eps). +apply Rinv_lt_contravar. +apply Rmult_lt_0_compat. +apply Rinv_0_lt_compat; assumption. +apply Rplus_le_lt_0_compat; [ apply pos_INR | apply Rlt_0_1 ]. +apply Rlt_trans with (INR x); + [ rewrite INR_IZR_INZ; rewrite <- H3; apply H0 + | pattern (INR x) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_lt_compat_l; apply Rlt_0_1 ]. +red in |- *; intro; rewrite H6 in H; elim (Rlt_irrefl _ H). Qed. (**********) -Definition RiemannInt [f:R->R;a,b:R;pr:(Riemann_integrable f a b)] : R := Cases -(RiemannInt_exists pr 5!RinvN RinvN_cv) of (existTT a' b') => a' end. - -Lemma RiemannInt_P5 : (f:R->R;a,b:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable f a b)) (RiemannInt pr1)==(RiemannInt pr2). -Intros; Unfold RiemannInt; Case (RiemannInt_exists pr1 5!RinvN RinvN_cv); Case (RiemannInt_exists pr2 5!RinvN RinvN_cv); Intros; EApply UL_sequence; [Apply u0 | Apply RiemannInt_P4 with pr2 RinvN; Apply RinvN_cv Orelse Assumption]. +Definition RiemannInt (f:R -> R) (a b:R) (pr:Riemann_integrable f a b) : R := + match RiemannInt_exists pr RinvN RinvN_cv with + | existT a' b' => a' + end. + +Lemma RiemannInt_P5 : + forall (f:R -> R) (a b:R) (pr1 pr2:Riemann_integrable f a b), + RiemannInt pr1 = RiemannInt pr2. +intros; unfold RiemannInt in |- *; + case (RiemannInt_exists pr1 RinvN RinvN_cv); + case (RiemannInt_exists pr2 RinvN RinvN_cv); intros; + eapply UL_sequence; + [ apply u0 + | apply RiemannInt_P4 with pr2 RinvN; apply RinvN_cv || assumption ]. Qed. (**************************************) (* C°([a,b]) is included in L1([a,b]) *) (**************************************) -Lemma maxN : (a,b:R;del:posreal) ``a<b`` -> (sigTT ? [n:nat]``a+(INR n)*del<b``/\``b<=a+(INR (S n))*del``). -Intros; Pose I := [n:nat]``a+(INR n)*del < b``; Assert H0 : (EX n:nat | (I n)). -Exists O; Unfold I; Rewrite Rmult_Ol; Rewrite Rplus_Or; Assumption. -Cut (Nbound I). -Intro; Assert H2 := (Nzorn H0 H1); Elim H2; Intros; Exists x; Elim p; Intros; Split. -Apply H3. -Case (total_order_T ``a+(INR (S x))*del`` b); Intro. -Elim s; Intro. -Assert H5 := (H4 (S x) a0); Elim (le_Sn_n ? H5). -Right; Symmetry; Assumption. -Left; Apply r. -Assert H1 : ``0<=(b-a)/del``. -Unfold Rdiv; Apply Rmult_le_pos; [Apply Rle_sym2; Apply Rge_minus; Apply Rle_sym1; Left; Apply H | Left; Apply Rlt_Rinv; Apply (cond_pos del)]. -Elim (archimed ``(b-a)/del``); Intros; Assert H4 : `0<=(up (Rdiv (Rminus b a) del))`. -Apply le_IZR; Simpl; Left; Apply Rle_lt_trans with ``(b-a)/del``; Assumption. -Assert H5 := (IZN ? H4); Elim H5; Clear H5; Intros N H5; Unfold Nbound; Exists N; Intros; Unfold I in H6; Apply INR_le; Rewrite H5 in H2; Rewrite <- INR_IZR_INZ in H2; Left; Apply Rle_lt_trans with ``(b-a)/del``; Try Assumption; Apply Rle_monotony_contra with (pos del); [Apply (cond_pos del) | Unfold Rdiv; Rewrite <- (Rmult_sym ``/del``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite Rmult_sym; Apply Rle_anti_compatibility with a; Replace ``a+(b-a)`` with b; [Left; Assumption | Ring] | Assert H7 := (cond_pos del); Red; Intro; Rewrite H8 in H7; Elim (Rlt_antirefl ? H7)]]. +Lemma maxN : + forall (a b:R) (del:posreal), + a < b -> + sigT (fun n:nat => a + INR n * del < b /\ b <= a + INR (S n) * del). +intros; pose (I := fun n:nat => a + INR n * del < b); + assert (H0 : exists n : nat | I n). +exists 0%nat; unfold I in |- *; rewrite Rmult_0_l; rewrite Rplus_0_r; + assumption. +cut (Nbound I). +intro; assert (H2 := Nzorn H0 H1); elim H2; intros; exists x; elim p; intros; + split. +apply H3. +case (total_order_T (a + INR (S x) * del) b); intro. +elim s; intro. +assert (H5 := H4 (S x) a0); elim (le_Sn_n _ H5). +right; symmetry in |- *; assumption. +left; apply r. +assert (H1 : 0 <= (b - a) / del). +unfold Rdiv in |- *; apply Rmult_le_pos; + [ apply Rge_le; apply Rge_minus; apply Rle_ge; left; apply H + | left; apply Rinv_0_lt_compat; apply (cond_pos del) ]. +elim (archimed ((b - a) / del)); intros; + assert (H4 : (0 <= up ((b - a) / del))%Z). +apply le_IZR; simpl in |- *; left; apply Rle_lt_trans with ((b - a) / del); + assumption. +assert (H5 := IZN _ H4); elim H5; clear H5; intros N H5; + unfold Nbound in |- *; exists N; intros; unfold I in H6; + apply INR_le; rewrite H5 in H2; rewrite <- INR_IZR_INZ in H2; + left; apply Rle_lt_trans with ((b - a) / del); try assumption; + apply Rmult_le_reg_l with (pos del); + [ apply (cond_pos del) + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ del)); + rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite Rmult_comm; apply Rplus_le_reg_l with a; + replace (a + (b - a)) with b; [ left; assumption | ring ] + | assert (H7 := cond_pos del); red in |- *; intro; rewrite H8 in H7; + elim (Rlt_irrefl _ H7) ] ]. Qed. -Fixpoint SubEquiN [N:nat] : R->R->posreal->Rlist := -[x:R][y:R][del:posreal] Cases N of -| O => (cons y nil) -| (S p) => (cons x (SubEquiN p ``x+del`` y del)) -end. - -Definition max_N [a,b:R;del:posreal;h:``a<b``] : nat := Cases (maxN del h) of (existTT N H0) => N end. - -Definition SubEqui [a,b:R;del:posreal;h:``a<b``] :Rlist := (SubEquiN (S (max_N del h)) a b del). - -Lemma Heine_cor1 : (f:R->R;a,b:R) ``a<b`` -> ((x:R)``a<=x<=b``->(continuity_pt f x)) -> (eps:posreal) (sigTT ? [delta:posreal]``delta<=b-a``/\(x,y:R)``a<=x<=b``->``a<=y<=b``->``(Rabsolu (x-y)) < delta``->``(Rabsolu ((f x)-(f y))) < eps``). -Intro f; Intros; Pose E := [l:R]``0<l<=b-a``/\(x,y:R)``a <= x <= b``->``a <= y <= b``->``(Rabsolu (x-y)) < l``->``(Rabsolu ((f x)-(f y))) < eps``; Assert H1 : (bound E). -Unfold bound; Exists ``b-a``; Unfold is_upper_bound; Intros; Unfold E in H1; Elim H1; Clear H1; Intros H1 _; Elim H1; Intros; Assumption. -Assert H2 : (EXT x:R | (E x)). -Assert H2 := (Heine f [x:R]``a<=x<=b`` (compact_P3 a b) H0 eps); Elim H2; Intros; Exists (Rmin x ``b-a``); Unfold E; Split; [Split; [Unfold Rmin; Case (total_order_Rle x ``b-a``); Intro; [Apply (cond_pos x) | Apply Rlt_Rminus; Assumption] | Apply Rmin_r] | Intros; Apply H3; Try Assumption; Apply Rlt_le_trans with (Rmin x ``b-a``); [Assumption | Apply Rmin_l]]. -Assert H3 := (complet E H1 H2); Elim H3; Intros; Cut ``0<x<=b-a``. -Intro; Elim H4; Clear H4; Intros; Apply existTT with (mkposreal ? H4); Split. -Apply H5. -Unfold is_lub in p; Elim p; Intros; Unfold is_upper_bound in H6; Pose D := ``(Rabsolu (x0-y))``; Elim (classic (EXT y:R | ``D<y``/\(E y))); Intro. -Elim H11; Intros; Elim H12; Clear H12; Intros; Unfold E in H13; Elim H13; Intros; Apply H15; Assumption. -Assert H12 := (not_ex_all_not ? [y:R]``D < y``/\(E y) H11); Assert H13 : (is_upper_bound E D). -Unfold is_upper_bound; Intros; Assert H14 := (H12 x1); Elim (not_and_or ``D<x1`` (E x1) H14); Intro. -Case (total_order_Rle x1 D); Intro. -Assumption. -Elim H15; Auto with real. -Elim H15; Assumption. -Assert H14 := (H7 ? H13); Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H14 H10)). -Unfold is_lub in p; Unfold is_upper_bound in p; Elim p; Clear p; Intros; Split. -Elim H2; Intros; Assert H7 := (H4 ? H6); Unfold E in H6; Elim H6; Clear H6; Intros H6 _; Elim H6; Intros; Apply Rlt_le_trans with x0; Assumption. -Apply H5; Intros; Unfold E in H6; Elim H6; Clear H6; Intros H6 _; Elim H6; Intros; Assumption. +Fixpoint SubEquiN (N:nat) (x y:R) (del:posreal) {struct N} : Rlist := + match N with + | O => cons y nil + | S p => cons x (SubEquiN p (x + del) y del) + end. + +Definition max_N (a b:R) (del:posreal) (h:a < b) : nat := + match maxN del h with + | existT N H0 => N + end. + +Definition SubEqui (a b:R) (del:posreal) (h:a < b) : Rlist := + SubEquiN (S (max_N del h)) a b del. + +Lemma Heine_cor1 : + forall (f:R -> R) (a b:R), + a < b -> + (forall x:R, a <= x <= b -> continuity_pt f x) -> + forall eps:posreal, + sigT + (fun delta:posreal => + delta <= b - a /\ + (forall x y:R, + a <= x <= b -> + a <= y <= b -> Rabs (x - y) < delta -> Rabs (f x - f y) < eps)). +intro f; intros; + pose + (E := + fun l:R => + 0 < l <= b - a /\ + (forall x y:R, + a <= x <= b -> + a <= y <= b -> Rabs (x - y) < l -> Rabs (f x - f y) < eps)); + assert (H1 : bound E). +unfold bound in |- *; exists (b - a); unfold is_upper_bound in |- *; intros; + unfold E in H1; elim H1; clear H1; intros H1 _; elim H1; + intros; assumption. +assert (H2 : exists x : R | E x). +assert (H2 := Heine f (fun x:R => a <= x <= b) (compact_P3 a b) H0 eps); + elim H2; intros; exists (Rmin x (b - a)); unfold E in |- *; + split; + [ split; + [ unfold Rmin in |- *; case (Rle_dec x (b - a)); intro; + [ apply (cond_pos x) | apply Rlt_Rminus; assumption ] + | apply Rmin_r ] + | intros; apply H3; try assumption; apply Rlt_le_trans with (Rmin x (b - a)); + [ assumption | apply Rmin_l ] ]. +assert (H3 := completeness E H1 H2); elim H3; intros; cut (0 < x <= b - a). +intro; elim H4; clear H4; intros; apply existT with (mkposreal _ H4); split. +apply H5. +unfold is_lub in p; elim p; intros; unfold is_upper_bound in H6; + pose (D := Rabs (x0 - y)); elim (classic ( exists y : R | D < y /\ E y)); + intro. +elim H11; intros; elim H12; clear H12; intros; unfold E in H13; elim H13; + intros; apply H15; assumption. +assert (H12 := not_ex_all_not _ (fun y:R => D < y /\ E y) H11); + assert (H13 : is_upper_bound E D). +unfold is_upper_bound in |- *; intros; assert (H14 := H12 x1); + elim (not_and_or (D < x1) (E x1) H14); intro. +case (Rle_dec x1 D); intro. +assumption. +elim H15; auto with real. +elim H15; assumption. +assert (H14 := H7 _ H13); elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H14 H10)). +unfold is_lub in p; unfold is_upper_bound in p; elim p; clear p; intros; + split. +elim H2; intros; assert (H7 := H4 _ H6); unfold E in H6; elim H6; clear H6; + intros H6 _; elim H6; intros; apply Rlt_le_trans with x0; + assumption. +apply H5; intros; unfold E in H6; elim H6; clear H6; intros H6 _; elim H6; + intros; assumption. Qed. -Lemma Heine_cor2 : (f:(R->R); a,b:R) ((x:R)``a <= x <= b``->(continuity_pt f x))->(eps:posreal)(sigTT posreal [delta:posreal]((x,y:R)``a <= x <= b``->``a <= y <= b``->``(Rabsolu (x-y)) < delta``->``(Rabsolu ((f x)-(f y))) < eps``)). -Intro f; Intros; Case (total_order_T a b); Intro. -Elim s; Intro. -Assert H0 := (Heine_cor1 a0 H eps); Elim H0; Intros; Apply existTT with x; Elim p; Intros; Apply H2; Assumption. -Apply existTT with (mkposreal ? Rlt_R0_R1); Intros; Assert H3 : x==y; [Elim H0; Elim H1; Intros; Rewrite b0 in H3; Rewrite b0 in H5; Apply Rle_antisym; Apply Rle_trans with b; Assumption | Rewrite H3; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (cond_pos eps)]. -Apply existTT with (mkposreal ? Rlt_R0_R1); Intros; Elim H0; Intros; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? (Rle_trans ? ? ? H3 H4) r)). +Lemma Heine_cor2 : + forall (f:R -> R) (a b:R), + (forall x:R, a <= x <= b -> continuity_pt f x) -> + forall eps:posreal, + sigT + (fun delta:posreal => + forall x y:R, + a <= x <= b -> + a <= y <= b -> Rabs (x - y) < delta -> Rabs (f x - f y) < eps). +intro f; intros; case (total_order_T a b); intro. +elim s; intro. +assert (H0 := Heine_cor1 a0 H eps); elim H0; intros; apply existT with x; + elim p; intros; apply H2; assumption. +apply existT with (mkposreal _ Rlt_0_1); intros; assert (H3 : x = y); + [ elim H0; elim H1; intros; rewrite b0 in H3; rewrite b0 in H5; + apply Rle_antisym; apply Rle_trans with b; assumption + | rewrite H3; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + apply (cond_pos eps) ]. +apply existT with (mkposreal _ Rlt_0_1); intros; elim H0; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (Rle_trans _ _ _ H3 H4) r)). Qed. -Lemma SubEqui_P1 : (a,b:R;del:posreal;h:``a<b``) (pos_Rl (SubEqui del h) O)==a. -Intros; Unfold SubEqui; Case (maxN del h); Intros; Reflexivity. +Lemma SubEqui_P1 : + forall (a b:R) (del:posreal) (h:a < b), pos_Rl (SubEqui del h) 0 = a. +intros; unfold SubEqui in |- *; case (maxN del h); intros; reflexivity. Qed. -Lemma SubEqui_P2 : (a,b:R;del:posreal;h:``a<b``) (pos_Rl (SubEqui del h) (pred (Rlength (SubEqui del h))))==b. -Intros; Unfold SubEqui; Case (maxN del h); Intros; Clear a0; Cut (x:nat)(a:R)(del:posreal)(pos_Rl (SubEquiN (S x) a b del) (pred (Rlength (SubEquiN (S x) a b del)))) == b; [Intro; Apply H | Induction x0; [Intros; Reflexivity | Intros; Change (pos_Rl (SubEquiN (S n) ``a0+del0`` b del0) (pred (Rlength (SubEquiN (S n) ``a0+del0`` b del0))))==b; Apply H]]. +Lemma SubEqui_P2 : + forall (a b:R) (del:posreal) (h:a < b), + pos_Rl (SubEqui del h) (pred (Rlength (SubEqui del h))) = b. +intros; unfold SubEqui in |- *; case (maxN del h); intros; clear a0; + cut + (forall (x:nat) (a:R) (del:posreal), + pos_Rl (SubEquiN (S x) a b del) + (pred (Rlength (SubEquiN (S x) a b del))) = b); + [ intro; apply H + | simple induction x0; + [ intros; reflexivity + | intros; + change + (pos_Rl (SubEquiN (S n) (a0 + del0) b del0) + (pred (Rlength (SubEquiN (S n) (a0 + del0) b del0))) = b) + in |- *; apply H ] ]. Qed. -Lemma SubEqui_P3 : (N:nat;a,b:R;del:posreal) (Rlength (SubEquiN N a b del))=(S N). -Induction N; Intros; [Reflexivity | Simpl; Rewrite H; Reflexivity]. +Lemma SubEqui_P3 : + forall (N:nat) (a b:R) (del:posreal), Rlength (SubEquiN N a b del) = S N. +simple induction N; intros; + [ reflexivity | simpl in |- *; rewrite H; reflexivity ]. Qed. -Lemma SubEqui_P4 : (N:nat;a,b:R;del:posreal;i:nat) (lt i (S N)) -> (pos_Rl (SubEquiN (S N) a b del) i)==``a+(INR i)*del``. -Induction N; [Intros; Inversion H; [Simpl; Ring | Elim (le_Sn_O ? H1)] | Intros; Induction i; [Simpl; Ring | Change (pos_Rl (SubEquiN (S n) ``a+del`` b del) i)==``a+(INR (S i))*del``; Rewrite H; [Rewrite S_INR; Ring | Apply lt_S_n; Apply H0]]]. +Lemma SubEqui_P4 : + forall (N:nat) (a b:R) (del:posreal) (i:nat), + (i < S N)%nat -> pos_Rl (SubEquiN (S N) a b del) i = a + INR i * del. +simple induction N; + [ intros; inversion H; [ simpl in |- *; ring | elim (le_Sn_O _ H1) ] + | intros; induction i as [| i Hreci]; + [ simpl in |- *; ring + | change + (pos_Rl (SubEquiN (S n) (a + del) b del) i = a + INR (S i) * del) + in |- *; rewrite H; [ rewrite S_INR; ring | apply lt_S_n; apply H0 ] ] ]. Qed. -Lemma SubEqui_P5 : (a,b:R;del:posreal;h:``a<b``) (Rlength (SubEqui del h))=(S (S (max_N del h))). -Intros; Unfold SubEqui; Apply SubEqui_P3. +Lemma SubEqui_P5 : + forall (a b:R) (del:posreal) (h:a < b), + Rlength (SubEqui del h) = S (S (max_N del h)). +intros; unfold SubEqui in |- *; apply SubEqui_P3. Qed. -Lemma SubEqui_P6 : (a,b:R;del:posreal;h:``a<b``;i:nat) (lt i (S (max_N del h))) -> (pos_Rl (SubEqui del h) i)==``a+(INR i)*del``. -Intros; Unfold SubEqui; Apply SubEqui_P4; Assumption. +Lemma SubEqui_P6 : + forall (a b:R) (del:posreal) (h:a < b) (i:nat), + (i < S (max_N del h))%nat -> pos_Rl (SubEqui del h) i = a + INR i * del. +intros; unfold SubEqui in |- *; apply SubEqui_P4; assumption. Qed. -Lemma SubEqui_P7 : (a,b:R;del:posreal;h:``a<b``) (ordered_Rlist (SubEqui del h)). -Intros; Unfold ordered_Rlist; Intros; Rewrite SubEqui_P5 in H; Simpl in H; Inversion H. -Rewrite (SubEqui_P6 3!del 4!h 5!(max_N del h)). -Replace (S (max_N del h)) with (pred (Rlength (SubEqui del h))). -Rewrite SubEqui_P2; Unfold max_N; Case (maxN del h); Intros; Left; Elim a0; Intros; Assumption. -Rewrite SubEqui_P5; Reflexivity. -Apply lt_n_Sn. -Repeat Rewrite SubEqui_P6. -3:Assumption. -2:Apply le_lt_n_Sm; Assumption. -Apply Rle_compatibility; Rewrite S_INR; Rewrite Rmult_Rplus_distrl; Pattern 1 ``(INR i)*del``; Rewrite <- Rplus_Or; Apply Rle_compatibility; Rewrite Rmult_1l; Left; Apply (cond_pos del). +Lemma SubEqui_P7 : + forall (a b:R) (del:posreal) (h:a < b), ordered_Rlist (SubEqui del h). +intros; unfold ordered_Rlist in |- *; intros; rewrite SubEqui_P5 in H; + simpl in H; inversion H. +rewrite (SubEqui_P6 del h (i:=(max_N del h))). +replace (S (max_N del h)) with (pred (Rlength (SubEqui del h))). +rewrite SubEqui_P2; unfold max_N in |- *; case (maxN del h); intros; left; + elim a0; intros; assumption. +rewrite SubEqui_P5; reflexivity. +apply lt_n_Sn. +repeat rewrite SubEqui_P6. +3: assumption. +2: apply le_lt_n_Sm; assumption. +apply Rplus_le_compat_l; rewrite S_INR; rewrite Rmult_plus_distr_r; + pattern (INR i * del) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l; rewrite Rmult_1_l; left; + apply (cond_pos del). Qed. -Lemma SubEqui_P8 : (a,b:R;del:posreal;h:``a<b``;i:nat) (lt i (Rlength (SubEqui del h))) -> ``a<=(pos_Rl (SubEqui del h) i)<=b``. -Intros; Split. -Pattern 1 a; Rewrite <- (SubEqui_P1 del h); Apply RList_P5. -Apply SubEqui_P7. -Elim (RList_P3 (SubEqui del h) (pos_Rl (SubEqui del h) i)); Intros; Apply H1; Exists i; Split; [Reflexivity | Assumption]. -Pattern 2 b; Rewrite <- (SubEqui_P2 del h); Apply RList_P7; [Apply SubEqui_P7 | Elim (RList_P3 (SubEqui del h) (pos_Rl (SubEqui del h) i)); Intros; Apply H1; Exists i; Split; [Reflexivity | Assumption]]. +Lemma SubEqui_P8 : + forall (a b:R) (del:posreal) (h:a < b) (i:nat), + (i < Rlength (SubEqui del h))%nat -> a <= pos_Rl (SubEqui del h) i <= b. +intros; split. +pattern a at 1 in |- *; rewrite <- (SubEqui_P1 del h); apply RList_P5. +apply SubEqui_P7. +elim (RList_P3 (SubEqui del h) (pos_Rl (SubEqui del h) i)); intros; apply H1; + exists i; split; [ reflexivity | assumption ]. +pattern b at 2 in |- *; rewrite <- (SubEqui_P2 del h); apply RList_P7; + [ apply SubEqui_P7 + | elim (RList_P3 (SubEqui del h) (pos_Rl (SubEqui del h) i)); intros; + apply H1; exists i; split; [ reflexivity | assumption ] ]. Qed. -Lemma SubEqui_P9 : (a,b:R;del:posreal;f:R->R;h:``a<b``) (sigTT ? [g:(StepFun a b)](g b)==(f b)/\(i:nat)(lt i (pred (Rlength (SubEqui del h))))->(constant_D_eq g (co_interval (pos_Rl (SubEqui del h) i) (pos_Rl (SubEqui del h) (S i))) (f (pos_Rl (SubEqui del h) i)))). -Intros; Apply StepFun_P38; [Apply SubEqui_P7 | Apply SubEqui_P1 | Apply SubEqui_P2]. +Lemma SubEqui_P9 : + forall (a b:R) (del:posreal) (f:R -> R) (h:a < b), + sigT + (fun g:StepFun a b => + g b = f b /\ + (forall i:nat, + (i < pred (Rlength (SubEqui del h)))%nat -> + constant_D_eq g + (co_interval (pos_Rl (SubEqui del h) i) + (pos_Rl (SubEqui del h) (S i))) + (f (pos_Rl (SubEqui del h) i)))). +intros; apply StepFun_P38; + [ apply SubEqui_P7 | apply SubEqui_P1 | apply SubEqui_P2 ]. Qed. -Lemma RiemannInt_P6 : (f:R->R;a,b:R) ``a<b`` -> ((x:R)``a<=x<=b``->(continuity_pt f x)) -> (Riemann_integrable f a b). -Intros; Unfold Riemann_integrable; Intro; Assert H1 : ``0<eps/(2*(b-a))``. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos eps) | Apply Rlt_Rinv; Apply Rmult_lt_pos; [Sup0 | Apply Rlt_Rminus; Assumption]]. -Assert H2 : (Rmin a b)==a. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Left; Assumption]. -Assert H3 : (Rmax a b)==b. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Left; Assumption]. -Elim (Heine_cor2 H0 (mkposreal ? H1)); Intros del H4; Elim (SubEqui_P9 del f H); Intros phi [H5 H6]; Split with phi; Split with (mkStepFun (StepFun_P4 a b ``eps/(2*(b-a))``)); Split. -2:Rewrite StepFun_P18; Unfold Rdiv; Rewrite Rinv_Rmult. -2:Do 2 Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -2:Rewrite Rmult_1r; Rewrite Rabsolu_right. -2:Apply Rlt_monotony_contra with ``2``. -2:Sup0. -2:Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -2:Rewrite Rmult_1l; Pattern 1 (pos eps); Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Apply (cond_pos eps). -2:DiscrR. -2:Apply Rle_sym1; Left; Apply Rmult_lt_pos. -2:Apply (cond_pos eps). -2:Apply Rlt_Rinv; Sup0. -2:Apply Rminus_eq_contra; Red; Intro; Clear H6; Rewrite H7 in H; Elim (Rlt_antirefl ? H). -2:DiscrR. -2:Apply Rminus_eq_contra; Red; Intro; Clear H6; Rewrite H7 in H; Elim (Rlt_antirefl ? H). -Intros; Rewrite H2 in H7; Rewrite H3 in H7; Simpl; Unfold fct_cte; Cut (t:R)``a<=t<=b``->t==b\/(EX i:nat | (lt i (pred (Rlength (SubEqui del H))))/\(co_interval (pos_Rl (SubEqui del H) i) (pos_Rl (SubEqui del H) (S i)) t)). -Intro; Elim (H8 ? H7); Intro. -Rewrite H9; Rewrite H5; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Left; Assumption. -Elim H9; Clear H9; Intros I [H9 H10]; Assert H11 := (H6 I H9 t H10); Rewrite H11; Left; Apply H4. -Assumption. -Apply SubEqui_P8; Apply lt_trans with (pred (Rlength (SubEqui del H))). -Assumption. -Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H12 in H9; Elim (lt_n_O ? H9). -Unfold co_interval in H10; Elim H10; Clear H10; Intros; Rewrite Rabsolu_right. -Rewrite SubEqui_P5 in H9; Simpl in H9; Inversion H9. -Apply Rlt_anti_compatibility with (pos_Rl (SubEqui del H) (max_N del H)). -Replace ``(pos_Rl (SubEqui del H) (max_N del H))+(t-(pos_Rl (SubEqui del H) (max_N del H)))`` with t; [Idtac | Ring]; Apply Rlt_le_trans with b. -Rewrite H14 in H12; Assert H13 : (S (max_N del H))=(pred (Rlength (SubEqui del H))). -Rewrite SubEqui_P5; Reflexivity. -Rewrite H13 in H12; Rewrite SubEqui_P2 in H12; Apply H12. -Rewrite SubEqui_P6. -2:Apply lt_n_Sn. -Unfold max_N; Case (maxN del H); Intros; Elim a0; Clear a0; Intros _ H13; Replace ``a+(INR x)*del+del`` with ``a+(INR (S x))*del``; [Assumption | Rewrite S_INR; Ring]. -Apply Rlt_anti_compatibility with (pos_Rl (SubEqui del H) I); Replace ``(pos_Rl (SubEqui del H) I)+(t-(pos_Rl (SubEqui del H) I))`` with t; [Idtac | Ring]; Replace ``(pos_Rl (SubEqui del H) I)+del`` with (pos_Rl (SubEqui del H) (S I)). -Assumption. -Repeat Rewrite SubEqui_P6. -Rewrite S_INR; Ring. -Assumption. -Apply le_lt_n_Sm; Assumption. -Apply Rge_minus; Apply Rle_sym1; Assumption. -Intros; Clear H0 H1 H4 phi H5 H6 t H7; Case (Req_EM t0 b); Intro. -Left; Assumption. -Right; Pose I := [j:nat]``a+(INR j)*del<=t0``; Assert H1 : (EX n:nat | (I n)). -Exists O; Unfold I; Rewrite Rmult_Ol; Rewrite Rplus_Or; Elim H8; Intros; Assumption. -Assert H4 : (Nbound I). -Unfold Nbound; Exists (S (max_N del H)); Intros; Unfold max_N; Case (maxN del H); Intros; Elim a0; Clear a0; Intros _ H5; Apply INR_le; Apply Rle_monotony_contra with (pos del). -Apply (cond_pos del). -Apply Rle_anti_compatibility with a; Do 2 Rewrite (Rmult_sym del); Apply Rle_trans with t0; Unfold I in H4; Try Assumption; Apply Rle_trans with b; Try Assumption; Elim H8; Intros; Assumption. -Elim (Nzorn H1 H4); Intros N [H5 H6]; Assert H7 : (lt N (S (max_N del H))). -Unfold max_N; Case (maxN del H); Intros; Apply INR_lt; Apply Rlt_monotony_contra with (pos del). -Apply (cond_pos del). -Apply Rlt_anti_compatibility with a; Do 2 Rewrite (Rmult_sym del); Apply Rle_lt_trans with t0; Unfold I in H5; Try Assumption; Elim a0; Intros; Apply Rlt_le_trans with b; Try Assumption; Elim H8; Intros. -Elim H11; Intro. -Assumption. -Elim H0; Assumption. -Exists N; Split. -Rewrite SubEqui_P5; Simpl; Assumption. -Unfold co_interval; Split. -Rewrite SubEqui_P6. -Apply H5. -Assumption. -Inversion H7. -Replace (S (max_N del H)) with (pred (Rlength (SubEqui del H))). -Rewrite (SubEqui_P2 del H); Elim H8; Intros. -Elim H11; Intro. -Assumption. -Elim H0; Assumption. -Rewrite SubEqui_P5; Reflexivity. -Rewrite SubEqui_P6. -Case (total_order_Rle ``a+(INR (S N))*del`` t0); Intro. -Assert H11 := (H6 (S N) r); Elim (le_Sn_n ? H11). -Auto with real. -Apply le_lt_n_Sm; Assumption. +Lemma RiemannInt_P6 : + forall (f:R -> R) (a b:R), + a < b -> + (forall x:R, a <= x <= b -> continuity_pt f x) -> Riemann_integrable f a b. +intros; unfold Riemann_integrable in |- *; intro; + assert (H1 : 0 < eps / (2 * (b - a))). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos eps) + | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat; + [ prove_sup0 | apply Rlt_Rminus; assumption ] ]. +assert (H2 : Rmin a b = a). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; left; assumption ]. +assert (H3 : Rmax a b = b). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; left; assumption ]. +elim (Heine_cor2 H0 (mkposreal _ H1)); intros del H4; + elim (SubEqui_P9 del f H); intros phi [H5 H6]; split with phi; + split with (mkStepFun (StepFun_P4 a b (eps / (2 * (b - a))))); + split. +2: rewrite StepFun_P18; unfold Rdiv in |- *; rewrite Rinv_mult_distr. +2: do 2 rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +2: rewrite Rmult_1_r; rewrite Rabs_right. +2: apply Rmult_lt_reg_l with 2. +2: prove_sup0. +2: rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +2: rewrite Rmult_1_l; pattern (pos eps) at 1 in |- *; rewrite <- Rplus_0_r; + rewrite double; apply Rplus_lt_compat_l; apply (cond_pos eps). +2: discrR. +2: apply Rle_ge; left; apply Rmult_lt_0_compat. +2: apply (cond_pos eps). +2: apply Rinv_0_lt_compat; prove_sup0. +2: apply Rminus_eq_contra; red in |- *; intro; clear H6; rewrite H7 in H; + elim (Rlt_irrefl _ H). +2: discrR. +2: apply Rminus_eq_contra; red in |- *; intro; clear H6; rewrite H7 in H; + elim (Rlt_irrefl _ H). +intros; rewrite H2 in H7; rewrite H3 in H7; simpl in |- *; + unfold fct_cte in |- *; + cut + (forall t:R, + a <= t <= b -> + t = b \/ + ( exists i : nat + | (i < pred (Rlength (SubEqui del H)))%nat /\ + co_interval (pos_Rl (SubEqui del H) i) (pos_Rl (SubEqui del H) (S i)) + t)). +intro; elim (H8 _ H7); intro. +rewrite H9; rewrite H5; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; left; assumption. +elim H9; clear H9; intros I [H9 H10]; assert (H11 := H6 I H9 t H10); + rewrite H11; left; apply H4. +assumption. +apply SubEqui_P8; apply lt_trans with (pred (Rlength (SubEqui del H))). +assumption. +apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H9; + elim (lt_n_O _ H9). +unfold co_interval in H10; elim H10; clear H10; intros; rewrite Rabs_right. +rewrite SubEqui_P5 in H9; simpl in H9; inversion H9. +apply Rplus_lt_reg_r with (pos_Rl (SubEqui del H) (max_N del H)). +replace + (pos_Rl (SubEqui del H) (max_N del H) + + (t - pos_Rl (SubEqui del H) (max_N del H))) with t; + [ idtac | ring ]; apply Rlt_le_trans with b. +rewrite H14 in H12; + assert (H13 : S (max_N del H) = pred (Rlength (SubEqui del H))). +rewrite SubEqui_P5; reflexivity. +rewrite H13 in H12; rewrite SubEqui_P2 in H12; apply H12. +rewrite SubEqui_P6. +2: apply lt_n_Sn. +unfold max_N in |- *; case (maxN del H); intros; elim a0; clear a0; + intros _ H13; replace (a + INR x * del + del) with (a + INR (S x) * del); + [ assumption | rewrite S_INR; ring ]. +apply Rplus_lt_reg_r with (pos_Rl (SubEqui del H) I); + replace (pos_Rl (SubEqui del H) I + (t - pos_Rl (SubEqui del H) I)) with t; + [ idtac | ring ]; + replace (pos_Rl (SubEqui del H) I + del) with (pos_Rl (SubEqui del H) (S I)). +assumption. +repeat rewrite SubEqui_P6. +rewrite S_INR; ring. +assumption. +apply le_lt_n_Sm; assumption. +apply Rge_minus; apply Rle_ge; assumption. +intros; clear H0 H1 H4 phi H5 H6 t H7; case (Req_dec t0 b); intro. +left; assumption. +right; pose (I := fun j:nat => a + INR j * del <= t0); + assert (H1 : exists n : nat | I n). +exists 0%nat; unfold I in |- *; rewrite Rmult_0_l; rewrite Rplus_0_r; elim H8; + intros; assumption. +assert (H4 : Nbound I). +unfold Nbound in |- *; exists (S (max_N del H)); intros; unfold max_N in |- *; + case (maxN del H); intros; elim a0; clear a0; intros _ H5; + apply INR_le; apply Rmult_le_reg_l with (pos del). +apply (cond_pos del). +apply Rplus_le_reg_l with a; do 2 rewrite (Rmult_comm del); + apply Rle_trans with t0; unfold I in H4; try assumption; + apply Rle_trans with b; try assumption; elim H8; intros; + assumption. +elim (Nzorn H1 H4); intros N [H5 H6]; assert (H7 : (N < S (max_N del H))%nat). +unfold max_N in |- *; case (maxN del H); intros; apply INR_lt; + apply Rmult_lt_reg_l with (pos del). +apply (cond_pos del). +apply Rplus_lt_reg_r with a; do 2 rewrite (Rmult_comm del); + apply Rle_lt_trans with t0; unfold I in H5; try assumption; + elim a0; intros; apply Rlt_le_trans with b; try assumption; + elim H8; intros. +elim H11; intro. +assumption. +elim H0; assumption. +exists N; split. +rewrite SubEqui_P5; simpl in |- *; assumption. +unfold co_interval in |- *; split. +rewrite SubEqui_P6. +apply H5. +assumption. +inversion H7. +replace (S (max_N del H)) with (pred (Rlength (SubEqui del H))). +rewrite (SubEqui_P2 del H); elim H8; intros. +elim H11; intro. +assumption. +elim H0; assumption. +rewrite SubEqui_P5; reflexivity. +rewrite SubEqui_P6. +case (Rle_dec (a + INR (S N) * del) t0); intro. +assert (H11 := H6 (S N) r); elim (le_Sn_n _ H11). +auto with real. +apply le_lt_n_Sm; assumption. Qed. -Lemma RiemannInt_P7 : (f:R->R;a:R) (Riemann_integrable f a a). -Unfold Riemann_integrable; Intro f; Intros; Split with (mkStepFun (StepFun_P4 a a (f a))); Split with (mkStepFun (StepFun_P4 a a R0)); Split. -Intros; Simpl; Unfold fct_cte; Replace t with a. -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Right; Reflexivity. -Generalize H; Unfold Rmin Rmax; Case (total_order_Rle a a); Intros; Elim H0; Intros; Apply Rle_antisym; Assumption. -Rewrite StepFun_P18; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Apply (cond_pos eps). +Lemma RiemannInt_P7 : forall (f:R -> R) (a:R), Riemann_integrable f a a. +unfold Riemann_integrable in |- *; intro f; intros; + split with (mkStepFun (StepFun_P4 a a (f a))); + split with (mkStepFun (StepFun_P4 a a 0)); split. +intros; simpl in |- *; unfold fct_cte in |- *; replace t with a. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; right; + reflexivity. +generalize H; unfold Rmin, Rmax in |- *; case (Rle_dec a a); intros; elim H0; + intros; apply Rle_antisym; assumption. +rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0; apply (cond_pos eps). Qed. -Lemma continuity_implies_RiemannInt : (f:R->R;a,b:R) ``a<=b`` -> ((x:R)``a<=x<=b``->(continuity_pt f x)) -> (Riemann_integrable f a b). -Intros; Case (total_order_T a b); Intro; [Elim s; Intro; [Apply RiemannInt_P6; Assumption | Rewrite b0; Apply RiemannInt_P7] | Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r))]. +Lemma continuity_implies_RiemannInt : + forall (f:R -> R) (a b:R), + a <= b -> + (forall x:R, a <= x <= b -> continuity_pt f x) -> Riemann_integrable f a b. +intros; case (total_order_T a b); intro; + [ elim s; intro; + [ apply RiemannInt_P6; assumption | rewrite b0; apply RiemannInt_P7 ] + | elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)) ]. Qed. -Lemma RiemannInt_P8 : (f:R->R;a,b:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable f b a)) ``(RiemannInt pr1)==-(RiemannInt pr2)``. -Intro f; Intros; EApply UL_sequence. -Unfold RiemannInt; Case (RiemannInt_exists pr1 5!RinvN RinvN_cv); Intros; Apply u. -Unfold RiemannInt; Case (RiemannInt_exists pr2 5!RinvN RinvN_cv); Intros; Cut (EXT psi1:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr1 n)] t)))<= (psi1 n t)``)/\``(Rabsolu (RiemannInt_SF (psi1 n))) < (RinvN n)``). -Cut (EXT psi2:nat->(StepFun b a) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr2 n)] t)))<= (psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``). -Intros; Elim H; Clear H; Intros psi2 H; Elim H0; Clear H0; Intros psi1 H0; Assert H1 := RinvN_cv; Unfold Un_cv; Intros; Assert H3 : ``0<eps/3``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Unfold Un_cv in H1; Elim (H1 ? H3); Clear H1; Intros N0 H1; Unfold R_dist in H1; Simpl in H1; Assert H4 : (n:nat)(ge n N0)->``(RinvN n)<eps/3``. -Intros; Assert H5 := (H1 ? H4); Replace (pos (RinvN n)) with ``(Rabsolu (/((INR n)+1)-0))``; [Assumption | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Left; Apply (cond_pos (RinvN n))]. -Clear H1; Unfold Un_cv in u; Elim (u ? H3); Clear u; Intros N1 H1; Exists (max N0 N1); Intros; Unfold R_dist; Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)])))+(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x))``. -Rewrite <- (Rabsolu_Ropp ``(RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x``); Replace ``(RiemannInt_SF [(phi_sequence RinvN pr1 n)])- -x`` with ``((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))+ -((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x)``; [Apply Rabsolu_triang | Ring]. -Replace eps with ``2*eps/3+eps/3``. -Apply Rplus_lt. -Rewrite (StepFun_P39 (phi_sequence RinvN pr2 n)); Replace ``(RiemannInt_SF [(phi_sequence RinvN pr1 n)])+ -(RiemannInt_SF (mkStepFun (StepFun_P6 (pre [(phi_sequence RinvN pr2 n)]))))`` with ``(RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(-1)*(RiemannInt_SF (mkStepFun (StepFun_P6 (pre [(phi_sequence RinvN pr2 n)]))))``; [Idtac | Ring]; Rewrite <- StepFun_P30. -Case (total_order_Rle a b); Intro. -Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 ``-1`` (phi_sequence RinvN pr1 n) (mkStepFun (StepFun_P6 (pre (phi_sequence RinvN pr2 n))))))))). -Apply StepFun_P34; Assumption. -Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 ``1`` (psi1 n) (mkStepFun (StepFun_P6 (pre (psi2 n))))))). -Apply StepFun_P37; Try Assumption. -Intros; Simpl; Rewrite Rmult_1l; Apply Rle_trans with ``(Rabsolu (([(phi_sequence RinvN pr1 n)] x0)-(f x0)))+(Rabsolu ((f x0)-([(phi_sequence RinvN pr2 n)] x0)))``. -Replace ``([(phi_sequence RinvN pr1 n)] x0)+ -1*([(phi_sequence RinvN pr2 n)] x0)`` with ``(([(phi_sequence RinvN pr1 n)] x0)-(f x0))+((f x0)-([(phi_sequence RinvN pr2 n)] x0))``; [Apply Rabsolu_triang | Ring]. -Assert H7 : (Rmin a b)==a. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Assert H8 : (Rmax a b)==b. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Apply Rplus_le. -Elim (H0 n); Intros; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H9; Rewrite H7; Rewrite H8. -Elim H6; Intros; Split; Left; Assumption. -Elim (H n); Intros; Apply H9; Rewrite H7; Rewrite H8. -Elim H6; Intros; Split; Left; Assumption. -Rewrite StepFun_P30; Rewrite Rmult_1l; Rewrite double; Apply Rplus_lt. -Elim (H0 n); Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi1 n))); [Apply Rle_Rabsolu | Apply Rlt_trans with (pos (RinvN n)); [Assumption | Apply H4; Unfold ge; Apply le_trans with (max N0 N1); [Apply le_max_l | Assumption]]]. -Elim (H n); Intros; Rewrite <- (Ropp_Ropp (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (psi2 n)))))); Rewrite <- StepFun_P39; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi2 n))); [Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu | Apply Rlt_trans with (pos (RinvN n)); [Assumption | Apply H4; Unfold ge; Apply le_trans with (max N0 N1); [Apply le_max_l | Assumption]]]. -Assert Hyp : ``b<=a``. -Auto with real. -Rewrite StepFun_P39; Rewrite Rabsolu_Ropp; Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P6 (StepFun_P28 ``-1`` (phi_sequence RinvN pr1 n) (mkStepFun (StepFun_P6 (pre (phi_sequence RinvN pr2 n)))))))))). -Apply StepFun_P34; Assumption. -Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 ``1`` (mkStepFun (StepFun_P6 (pre (psi1 n)))) (psi2 n)))). -Apply StepFun_P37; Try Assumption. -Intros; Simpl; Rewrite Rmult_1l; Apply Rle_trans with ``(Rabsolu (([(phi_sequence RinvN pr1 n)] x0)-(f x0)))+(Rabsolu ((f x0)-([(phi_sequence RinvN pr2 n)] x0)))``. -Replace ``([(phi_sequence RinvN pr1 n)] x0)+ -1*([(phi_sequence RinvN pr2 n)] x0)`` with ``(([(phi_sequence RinvN pr1 n)] x0)-(f x0))+((f x0)-([(phi_sequence RinvN pr2 n)] x0))``; [Apply Rabsolu_triang | Ring]. -Assert H7 : (Rmin a b)==b. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Elim n0; Assumption | Reflexivity]. -Assert H8 : (Rmax a b)==a. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Elim n0; Assumption | Reflexivity]. -Apply Rplus_le. -Elim (H0 n); Intros; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H9; Rewrite H7; Rewrite H8. -Elim H6; Intros; Split; Left; Assumption. -Elim (H n); Intros; Apply H9; Rewrite H7; Rewrite H8; Elim H6; Intros; Split; Left; Assumption. -Rewrite StepFun_P30; Rewrite Rmult_1l; Rewrite double; Apply Rplus_lt. -Elim (H0 n); Intros; Rewrite <- (Ropp_Ropp (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (psi1 n)))))); Rewrite <- StepFun_P39; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi1 n))); [Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu | Apply Rlt_trans with (pos (RinvN n)); [Assumption | Apply H4; Unfold ge; Apply le_trans with (max N0 N1); [Apply le_max_l | Assumption]]]. -Elim (H n); Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi2 n))); [Apply Rle_Rabsolu | Apply Rlt_trans with (pos (RinvN n)); [Assumption | Apply H4; Unfold ge; Apply le_trans with (max N0 N1); [Apply le_max_l | Assumption]]]. -Unfold R_dist in H1; Apply H1; Unfold ge; Apply le_trans with (max N0 N1); [Apply le_max_r | Assumption]. -Apply r_Rmult_mult with ``3``; [Unfold Rdiv; Rewrite Rmult_Rplus_distr; Do 2 Rewrite (Rmult_sym ``3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr2 n)); Intro; Rewrite Rmin_sym; Rewrite RmaxSym; Apply (projT2 ? ? (phi_sequence_prop RinvN pr2 n)). -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr1 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr1 n)). +Lemma RiemannInt_P8 : + forall (f:R -> R) (a b:R) (pr1:Riemann_integrable f a b) + (pr2:Riemann_integrable f b a), RiemannInt pr1 = - RiemannInt pr2. +intro f; intros; eapply UL_sequence. +unfold RiemannInt in |- *; case (RiemannInt_exists pr1 RinvN RinvN_cv); + intros; apply u. +unfold RiemannInt in |- *; case (RiemannInt_exists pr2 RinvN RinvN_cv); + intros; + cut + ( exists psi1 : nat -> StepFun a b + | (forall n:nat, + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> + Rabs (f t - phi_sequence RinvN pr1 n t) <= psi1 n t) /\ + Rabs (RiemannInt_SF (psi1 n)) < RinvN n)). +cut + ( exists psi2 : nat -> StepFun b a + | (forall n:nat, + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> + Rabs (f t - phi_sequence RinvN pr2 n t) <= psi2 n t) /\ + Rabs (RiemannInt_SF (psi2 n)) < RinvN n)). +intros; elim H; clear H; intros psi2 H; elim H0; clear H0; intros psi1 H0; + assert (H1 := RinvN_cv); unfold Un_cv in |- *; intros; + assert (H3 : 0 < eps / 3). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +unfold Un_cv in H1; elim (H1 _ H3); clear H1; intros N0 H1; + unfold R_dist in H1; simpl in H1; + assert (H4 : forall n:nat, (n >= N0)%nat -> RinvN n < eps / 3). +intros; assert (H5 := H1 _ H4); + replace (pos (RinvN n)) with (Rabs (/ (INR n + 1) - 0)); + [ assumption + | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; + left; apply (cond_pos (RinvN n)) ]. +clear H1; unfold Un_cv in u; elim (u _ H3); clear u; intros N1 H1; + exists (max N0 N1); intros; unfold R_dist in |- *; + apply Rle_lt_trans with + (Rabs + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + RiemannInt_SF (phi_sequence RinvN pr2 n)) + + Rabs (RiemannInt_SF (phi_sequence RinvN pr2 n) - x)). +rewrite <- (Rabs_Ropp (RiemannInt_SF (phi_sequence RinvN pr2 n) - x)); + replace (RiemannInt_SF (phi_sequence RinvN pr1 n) - - x) with + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + RiemannInt_SF (phi_sequence RinvN pr2 n) + + - (RiemannInt_SF (phi_sequence RinvN pr2 n) - x)); + [ apply Rabs_triang | ring ]. +replace eps with (2 * (eps / 3) + eps / 3). +apply Rplus_lt_compat. +rewrite (StepFun_P39 (phi_sequence RinvN pr2 n)); + replace + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + - RiemannInt_SF (mkStepFun (StepFun_P6 (pre (phi_sequence RinvN pr2 n))))) + with + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + -1 * + RiemannInt_SF (mkStepFun (StepFun_P6 (pre (phi_sequence RinvN pr2 n))))); + [ idtac | ring ]; rewrite <- StepFun_P30. +case (Rle_dec a b); intro. +apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun + (StepFun_P32 + (mkStepFun + (StepFun_P28 (-1) (phi_sequence RinvN pr1 n) + (mkStepFun (StepFun_P6 (pre (phi_sequence RinvN pr2 n))))))))). +apply StepFun_P34; assumption. +apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun + (StepFun_P28 1 (psi1 n) (mkStepFun (StepFun_P6 (pre (psi2 n))))))). +apply StepFun_P37; try assumption. +intros; simpl in |- *; rewrite Rmult_1_l; + apply Rle_trans with + (Rabs (phi_sequence RinvN pr1 n x0 - f x0) + + Rabs (f x0 - phi_sequence RinvN pr2 n x0)). +replace (phi_sequence RinvN pr1 n x0 + -1 * phi_sequence RinvN pr2 n x0) with + (phi_sequence RinvN pr1 n x0 - f x0 + (f x0 - phi_sequence RinvN pr2 n x0)); + [ apply Rabs_triang | ring ]. +assert (H7 : Rmin a b = a). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +assert (H8 : Rmax a b = b). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +apply Rplus_le_compat. +elim (H0 n); intros; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H9; + rewrite H7; rewrite H8. +elim H6; intros; split; left; assumption. +elim (H n); intros; apply H9; rewrite H7; rewrite H8. +elim H6; intros; split; left; assumption. +rewrite StepFun_P30; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat. +elim (H0 n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n))); + [ apply RRle_abs + | apply Rlt_trans with (pos (RinvN n)); + [ assumption + | apply H4; unfold ge in |- *; apply le_trans with (max N0 N1); + [ apply le_max_l | assumption ] ] ]. +elim (H n); intros; + rewrite <- + (Ropp_involutive (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (psi2 n)))))) + ; rewrite <- StepFun_P39; + apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n))); + [ rewrite <- Rabs_Ropp; apply RRle_abs + | apply Rlt_trans with (pos (RinvN n)); + [ assumption + | apply H4; unfold ge in |- *; apply le_trans with (max N0 N1); + [ apply le_max_l | assumption ] ] ]. +assert (Hyp : b <= a). +auto with real. +rewrite StepFun_P39; rewrite Rabs_Ropp; + apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun + (StepFun_P32 + (mkStepFun + (StepFun_P6 + (StepFun_P28 (-1) (phi_sequence RinvN pr1 n) + (mkStepFun (StepFun_P6 (pre (phi_sequence RinvN pr2 n)))))))))). +apply StepFun_P34; assumption. +apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun + (StepFun_P28 1 (mkStepFun (StepFun_P6 (pre (psi1 n)))) (psi2 n)))). +apply StepFun_P37; try assumption. +intros; simpl in |- *; rewrite Rmult_1_l; + apply Rle_trans with + (Rabs (phi_sequence RinvN pr1 n x0 - f x0) + + Rabs (f x0 - phi_sequence RinvN pr2 n x0)). +replace (phi_sequence RinvN pr1 n x0 + -1 * phi_sequence RinvN pr2 n x0) with + (phi_sequence RinvN pr1 n x0 - f x0 + (f x0 - phi_sequence RinvN pr2 n x0)); + [ apply Rabs_triang | ring ]. +assert (H7 : Rmin a b = b). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ elim n0; assumption | reflexivity ]. +assert (H8 : Rmax a b = a). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ elim n0; assumption | reflexivity ]. +apply Rplus_le_compat. +elim (H0 n); intros; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H9; + rewrite H7; rewrite H8. +elim H6; intros; split; left; assumption. +elim (H n); intros; apply H9; rewrite H7; rewrite H8; elim H6; intros; split; + left; assumption. +rewrite StepFun_P30; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat. +elim (H0 n); intros; + rewrite <- + (Ropp_involutive (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (psi1 n)))))) + ; rewrite <- StepFun_P39; + apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n))); + [ rewrite <- Rabs_Ropp; apply RRle_abs + | apply Rlt_trans with (pos (RinvN n)); + [ assumption + | apply H4; unfold ge in |- *; apply le_trans with (max N0 N1); + [ apply le_max_l | assumption ] ] ]. +elim (H n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n))); + [ apply RRle_abs + | apply Rlt_trans with (pos (RinvN n)); + [ assumption + | apply H4; unfold ge in |- *; apply le_trans with (max N0 N1); + [ apply le_max_l | assumption ] ] ]. +unfold R_dist in H1; apply H1; unfold ge in |- *; + apply le_trans with (max N0 N1); [ apply le_max_r | assumption ]. +apply Rmult_eq_reg_l with 3; + [ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l; + do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym; [ ring | discrR ] + | discrR ]. +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro; + rewrite Rmin_comm; rewrite RmaxSym; + apply (projT2 (phi_sequence_prop RinvN pr2 n)). +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro; + apply (projT2 (phi_sequence_prop RinvN pr1 n)). Qed. -Lemma RiemannInt_P9 : (f:R->R;a:R;pr:(Riemann_integrable f a a)) ``(RiemannInt pr)==0``. -Intros; Assert H := (RiemannInt_P8 pr pr); Apply r_Rmult_mult with ``2``; [Rewrite Rmult_Or; Rewrite double; Pattern 2 (RiemannInt pr); Rewrite H; Apply Rplus_Ropp_r | DiscrR]. +Lemma RiemannInt_P9 : + forall (f:R -> R) (a:R) (pr:Riemann_integrable f a a), RiemannInt pr = 0. +intros; assert (H := RiemannInt_P8 pr pr); apply Rmult_eq_reg_l with 2; + [ rewrite Rmult_0_r; rewrite double; pattern (RiemannInt pr) at 2 in |- *; + rewrite H; apply Rplus_opp_r + | discrR ]. Qed. -Lemma Req_EM_T :(r1,r2:R) (sumboolT (r1==r2) ``r1<>r2``). -Intros; Elim (total_order_T r1 r2);Intros; [Elim a;Intro; [Right; Red; Intro; Rewrite H in a0; Elim (Rlt_antirefl ``r2`` a0) | Left;Assumption] | Right; Red; Intro; Rewrite H in b; Elim (Rlt_antirefl ``r2`` b)]. +Lemma Req_EM_T : forall r1 r2:R, {r1 = r2} + {r1 <> r2}. +intros; elim (total_order_T r1 r2); intros; + [ elim a; intro; + [ right; red in |- *; intro; rewrite H in a0; elim (Rlt_irrefl r2 a0) + | left; assumption ] + | right; red in |- *; intro; rewrite H in b; elim (Rlt_irrefl r2 b) ]. Qed. (* L1([a,b]) is a vectorial space *) -Lemma RiemannInt_P10 : (f,g:R->R;a,b,l:R) (Riemann_integrable f a b) -> (Riemann_integrable g a b) -> (Riemann_integrable [x:R]``(f x)+l*(g x)`` a b). -Unfold Riemann_integrable; Intros f g; Intros; Case (Req_EM_T l R0); Intro. -Elim (X eps); Intros; Split with x; Elim p; Intros; Split with x0; Elim p0; Intros; Split; Try Assumption; Rewrite e; Intros; Rewrite Rmult_Ol; Rewrite Rplus_Or; Apply H; Assumption. -Assert H : ``0<eps/2``. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos eps) | Apply Rlt_Rinv; Sup0]. -Assert H0 : ``0<eps/(2*(Rabsolu l))``. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos eps) | Apply Rlt_Rinv; Apply Rmult_lt_pos; [Sup0 | Apply Rabsolu_pos_lt; Assumption]]. -Elim (X (mkposreal ? H)); Intros; Elim (X0 (mkposreal ? H0)); Intros; Split with (mkStepFun (StepFun_P28 l x x0)); Elim p0; Elim p; Intros; Split with (mkStepFun (StepFun_P28 (Rabsolu l) x1 x2)); Elim p1; Elim p2; Clear p1 p2 p0 p X X0; Intros; Split. -Intros; Simpl; Apply Rle_trans with ``(Rabsolu ((f t)-(x t)))+(Rabsolu (l*((g t)-(x0 t))))``. -Replace ``(f t)+l*(g t)-((x t)+l*(x0 t))`` with ``((f t)-(x t))+ l*((g t)-(x0 t))``; [Apply Rabsolu_triang | Ring]. -Apply Rplus_le; [Apply H3; Assumption | Rewrite Rabsolu_mult; Apply Rle_monotony; [Apply Rabsolu_pos | Apply H1; Assumption]]. -Rewrite StepFun_P30; Apply Rle_lt_trans with ``(Rabsolu (RiemannInt_SF x1))+(Rabsolu ((Rabsolu l)*(RiemannInt_SF x2)))``. -Apply Rabsolu_triang. -Rewrite (double_var eps); Apply Rplus_lt. -Apply H4. -Rewrite Rabsolu_mult; Rewrite Rabsolu_Rabsolu; Apply Rlt_monotony_contra with ``/(Rabsolu l)``. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym; [Rewrite Rmult_1l; Replace ``/(Rabsolu l)*eps/2`` with ``eps/(2*(Rabsolu l))``; [Apply H2 | Unfold Rdiv; Rewrite Rinv_Rmult; [Ring | DiscrR | Apply Rabsolu_no_R0; Assumption]] | Apply Rabsolu_no_R0; Assumption]. +Lemma RiemannInt_P10 : + forall (f g:R -> R) (a b l:R), + Riemann_integrable f a b -> + Riemann_integrable g a b -> + Riemann_integrable (fun x:R => f x + l * g x) a b. +unfold Riemann_integrable in |- *; intros f g; intros; case (Req_EM_T l 0); + intro. +elim (X eps); intros; split with x; elim p; intros; split with x0; elim p0; + intros; split; try assumption; rewrite e; intros; + rewrite Rmult_0_l; rewrite Rplus_0_r; apply H; assumption. +assert (H : 0 < eps / 2). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ]. +assert (H0 : 0 < eps / (2 * Rabs l)). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos eps) + | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat; + [ prove_sup0 | apply Rabs_pos_lt; assumption ] ]. +elim (X (mkposreal _ H)); intros; elim (X0 (mkposreal _ H0)); intros; + split with (mkStepFun (StepFun_P28 l x x0)); elim p0; + elim p; intros; split with (mkStepFun (StepFun_P28 (Rabs l) x1 x2)); + elim p1; elim p2; clear p1 p2 p0 p X X0; intros; split. +intros; simpl in |- *; + apply Rle_trans with (Rabs (f t - x t) + Rabs (l * (g t - x0 t))). +replace (f t + l * g t - (x t + l * x0 t)) with + (f t - x t + l * (g t - x0 t)); [ apply Rabs_triang | ring ]. +apply Rplus_le_compat; + [ apply H3; assumption + | rewrite Rabs_mult; apply Rmult_le_compat_l; + [ apply Rabs_pos | apply H1; assumption ] ]. +rewrite StepFun_P30; + apply Rle_lt_trans with + (Rabs (RiemannInt_SF x1) + Rabs (Rabs l * RiemannInt_SF x2)). +apply Rabs_triang. +rewrite (double_var eps); apply Rplus_lt_compat. +apply H4. +rewrite Rabs_mult; rewrite Rabs_Rabsolu; apply Rmult_lt_reg_l with (/ Rabs l). +apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym; + [ rewrite Rmult_1_l; + replace (/ Rabs l * (eps / 2)) with (eps / (2 * Rabs l)); + [ apply H2 + | unfold Rdiv in |- *; rewrite Rinv_mult_distr; + [ ring | discrR | apply Rabs_no_R0; assumption ] ] + | apply Rabs_no_R0; assumption ]. Qed. -Lemma RiemannInt_P11 : (f:R->R;a,b,l:R;un:nat->posreal;phi1,phi2,psi1,psi2:nat->(StepFun a b)) (Un_cv un R0) -> ((n:nat)((t:R)``(Rmin a b)<=t<=(Rmax a b)``->``(Rabsolu ((f t)-(phi1 n t)))<=(psi1 n t)``)/\``(Rabsolu (RiemannInt_SF (psi1 n)))<(un n)``) -> ((n:nat)((t:R)``(Rmin a b)<=t<=(Rmax a b)``->``(Rabsolu ((f t)-(phi2 n t)))<=(psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n)))<(un n)``) -> (Un_cv [N:nat](RiemannInt_SF (phi1 N)) l) -> (Un_cv [N:nat](RiemannInt_SF (phi2 N)) l). -Unfold Un_cv; Intro f; Intros; Intros. -Case (total_order_Rle a b); Intro Hyp. -Assert H4 : ``0<eps/3``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Elim (H ? H4); Clear H; Intros N0 H. -Elim (H2 ? H4); Clear H2; Intros N1 H2. -Pose N := (max N0 N1); Exists N; Intros; Unfold R_dist. -Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF (phi2 n))-(RiemannInt_SF (phi1 n))))+(Rabsolu ((RiemannInt_SF (phi1 n))-l))``. -Replace ``(RiemannInt_SF (phi2 n))-l`` with ``((RiemannInt_SF (phi2 n))-(RiemannInt_SF (phi1 n)))+((RiemannInt_SF (phi1 n))-l)``; [Apply Rabsolu_triang | Ring]. -Replace ``eps`` with ``2*eps/3+eps/3``. -Apply Rplus_lt. -Replace ``(RiemannInt_SF (phi2 n))-(RiemannInt_SF (phi1 n))`` with ``(RiemannInt_SF (phi2 n))+(-1)*(RiemannInt_SF (phi1 n))``; [Idtac | Ring]. -Rewrite <- StepFun_P30. -Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 ``-1`` (phi2 n) (phi1 n)))))). -Apply StepFun_P34; Assumption. -Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 R1 (psi1 n) (psi2 n)))). -Apply StepFun_P37; Try Assumption; Intros; Simpl; Rewrite Rmult_1l. -Apply Rle_trans with ``(Rabsolu ((phi2 n x)-(f x)))+(Rabsolu ((f x)-(phi1 n x)))``. -Replace ``(phi2 n x)+-1*(phi1 n x)`` with ``((phi2 n x)-(f x))+((f x)-(phi1 n x))``; [Apply Rabsolu_triang | Ring]. -Rewrite (Rplus_sym (psi1 n x)); Apply Rplus_le. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Elim (H1 n); Intros; Apply H7. -Assert H10 : (Rmin a b)==a. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Assert H11 : (Rmax a b)==b. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Rewrite H10; Rewrite H11; Elim H6; Intros; Split; Left; Assumption. -Elim (H0 n); Intros; Apply H7; Assert H10 : (Rmin a b)==a. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Assert H11 : (Rmax a b)==b. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Rewrite H10; Rewrite H11; Elim H6; Intros; Split; Left; Assumption. -Rewrite StepFun_P30; Rewrite Rmult_1l; Rewrite double; Apply Rplus_lt. -Apply Rlt_trans with (pos (un n)). -Elim (H0 n); Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi1 n))). -Apply Rle_Rabsolu. -Assumption. -Replace (pos (un n)) with (R_dist (un n) R0). -Apply H; Unfold ge; Apply le_trans with N; Try Assumption. -Unfold N; Apply le_max_l. -Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right. -Apply Rle_sym1; Left; Apply (cond_pos (un n)). -Apply Rlt_trans with (pos (un n)). -Elim (H1 n); Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi2 n))). -Apply Rle_Rabsolu; Assumption. -Assumption. -Replace (pos (un n)) with (R_dist (un n) R0). -Apply H; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_max_l. -Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (un n)). -Unfold R_dist in H2; Apply H2; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_max_r. -Apply r_Rmult_mult with ``3``; [Unfold Rdiv; Rewrite Rmult_Rplus_distr; Do 2 Rewrite (Rmult_sym ``3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. -Assert H4 : ``0<eps/3``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Elim (H ? H4); Clear H; Intros N0 H. -Elim (H2 ? H4); Clear H2; Intros N1 H2. -Pose N := (max N0 N1); Exists N; Intros; Unfold R_dist. -Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF (phi2 n))-(RiemannInt_SF (phi1 n))))+(Rabsolu ((RiemannInt_SF (phi1 n))-l))``. -Replace ``(RiemannInt_SF (phi2 n))-l`` with ``((RiemannInt_SF (phi2 n))-(RiemannInt_SF (phi1 n)))+((RiemannInt_SF (phi1 n))-l)``; [Apply Rabsolu_triang | Ring]. -Assert Hyp_b : ``b<=a``. -Auto with real. -Replace ``eps`` with ``2*eps/3+eps/3``. -Apply Rplus_lt. -Replace ``(RiemannInt_SF (phi2 n))-(RiemannInt_SF (phi1 n))`` with ``(RiemannInt_SF (phi2 n))+(-1)*(RiemannInt_SF (phi1 n))``; [Idtac | Ring]. -Rewrite <- StepFun_P30. -Rewrite StepFun_P39. -Rewrite Rabsolu_Ropp. -Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 ``-1`` (phi2 n) (phi1 n))))))))). -Apply StepFun_P34; Try Assumption. -Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P6 (pre (mkStepFun (StepFun_P28 R1 (psi1 n) (psi2 n))))))). -Apply StepFun_P37; Try Assumption. -Intros; Simpl; Rewrite Rmult_1l. -Apply Rle_trans with ``(Rabsolu ((phi2 n x)-(f x)))+(Rabsolu ((f x)-(phi1 n x)))``. -Replace ``(phi2 n x)+-1*(phi1 n x)`` with ``((phi2 n x)-(f x))+((f x)-(phi1 n x))``; [Apply Rabsolu_triang | Ring]. -Rewrite (Rplus_sym (psi1 n x)); Apply Rplus_le. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Elim (H1 n); Intros; Apply H7. -Assert H10 : (Rmin a b)==b. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Elim Hyp; Assumption | Reflexivity]. -Assert H11 : (Rmax a b)==a. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Elim Hyp; Assumption | Reflexivity]. -Rewrite H10; Rewrite H11; Elim H6; Intros; Split; Left; Assumption. -Elim (H0 n); Intros; Apply H7; Assert H10 : (Rmin a b)==b. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Elim Hyp; Assumption | Reflexivity]. -Assert H11 : (Rmax a b)==a. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Elim Hyp; Assumption | Reflexivity]. -Rewrite H10; Rewrite H11; Elim H6; Intros; Split; Left; Assumption. -Rewrite <- (Ropp_Ropp (RiemannInt_SF - (mkStepFun - (StepFun_P6 (pre (mkStepFun (StepFun_P28 R1 (psi1 n) (psi2 n)))))))). -Rewrite <- StepFun_P39. -Rewrite StepFun_P30. -Rewrite Rmult_1l; Rewrite double. -Rewrite Ropp_distr1; Apply Rplus_lt. -Apply Rlt_trans with (pos (un n)). -Elim (H0 n); Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi1 n))). -Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. -Assumption. -Replace (pos (un n)) with (R_dist (un n) R0). -Apply H; Unfold ge; Apply le_trans with N; Try Assumption. -Unfold N; Apply le_max_l. -Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right. -Apply Rle_sym1; Left; Apply (cond_pos (un n)). -Apply Rlt_trans with (pos (un n)). -Elim (H1 n); Intros; Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi2 n))). -Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu; Assumption. -Assumption. -Replace (pos (un n)) with (R_dist (un n) R0). -Apply H; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_max_l. -Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (un n)). -Unfold R_dist in H2; Apply H2; Unfold ge; Apply le_trans with N; Try Assumption; Unfold N; Apply le_max_r. -Apply r_Rmult_mult with ``3``; [Unfold Rdiv; Rewrite Rmult_Rplus_distr; Do 2 Rewrite (Rmult_sym ``3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. +Lemma RiemannInt_P11 : + forall (f:R -> R) (a b l:R) (un:nat -> posreal) + (phi1 phi2 psi1 psi2:nat -> StepFun a b), + Un_cv un 0 -> + (forall n:nat, + (forall t:R, + Rmin a b <= t <= Rmax a b -> Rabs (f t - phi1 n t) <= psi1 n t) /\ + Rabs (RiemannInt_SF (psi1 n)) < un n) -> + (forall n:nat, + (forall t:R, + Rmin a b <= t <= Rmax a b -> Rabs (f t - phi2 n t) <= psi2 n t) /\ + Rabs (RiemannInt_SF (psi2 n)) < un n) -> + Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) l -> + Un_cv (fun N:nat => RiemannInt_SF (phi2 N)) l. +unfold Un_cv in |- *; intro f; intros; intros. +case (Rle_dec a b); intro Hyp. +assert (H4 : 0 < eps / 3). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (H _ H4); clear H; intros N0 H. +elim (H2 _ H4); clear H2; intros N1 H2. +pose (N := max N0 N1); exists N; intros; unfold R_dist in |- *. +apply Rle_lt_trans with + (Rabs (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n)) + + Rabs (RiemannInt_SF (phi1 n) - l)). +replace (RiemannInt_SF (phi2 n) - l) with + (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n) + + (RiemannInt_SF (phi1 n) - l)); [ apply Rabs_triang | ring ]. +replace eps with (2 * (eps / 3) + eps / 3). +apply Rplus_lt_compat. +replace (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n)) with + (RiemannInt_SF (phi2 n) + -1 * RiemannInt_SF (phi1 n)); + [ idtac | ring ]. +rewrite <- StepFun_P30. +apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (phi2 n) (phi1 n)))))). +apply StepFun_P34; assumption. +apply Rle_lt_trans with + (RiemannInt_SF (mkStepFun (StepFun_P28 1 (psi1 n) (psi2 n)))). +apply StepFun_P37; try assumption; intros; simpl in |- *; rewrite Rmult_1_l. +apply Rle_trans with (Rabs (phi2 n x - f x) + Rabs (f x - phi1 n x)). +replace (phi2 n x + -1 * phi1 n x) with (phi2 n x - f x + (f x - phi1 n x)); + [ apply Rabs_triang | ring ]. +rewrite (Rplus_comm (psi1 n x)); apply Rplus_le_compat. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim (H1 n); intros; apply H7. +assert (H10 : Rmin a b = a). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +assert (H11 : Rmax a b = b). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +rewrite H10; rewrite H11; elim H6; intros; split; left; assumption. +elim (H0 n); intros; apply H7; assert (H10 : Rmin a b = a). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +assert (H11 : Rmax a b = b). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +rewrite H10; rewrite H11; elim H6; intros; split; left; assumption. +rewrite StepFun_P30; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat. +apply Rlt_trans with (pos (un n)). +elim (H0 n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n))). +apply RRle_abs. +assumption. +replace (pos (un n)) with (R_dist (un n) 0). +apply H; unfold ge in |- *; apply le_trans with N; try assumption. +unfold N in |- *; apply le_max_l. +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; apply Rabs_right. +apply Rle_ge; left; apply (cond_pos (un n)). +apply Rlt_trans with (pos (un n)). +elim (H1 n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n))). +apply RRle_abs; assumption. +assumption. +replace (pos (un n)) with (R_dist (un n) 0). +apply H; unfold ge in |- *; apply le_trans with N; try assumption; + unfold N in |- *; apply le_max_l. +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge; + left; apply (cond_pos (un n)). +unfold R_dist in H2; apply H2; unfold ge in |- *; apply le_trans with N; + try assumption; unfold N in |- *; apply le_max_r. +apply Rmult_eq_reg_l with 3; + [ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l; + do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym; [ ring | discrR ] + | discrR ]. +assert (H4 : 0 < eps / 3). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (H _ H4); clear H; intros N0 H. +elim (H2 _ H4); clear H2; intros N1 H2. +pose (N := max N0 N1); exists N; intros; unfold R_dist in |- *. +apply Rle_lt_trans with + (Rabs (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n)) + + Rabs (RiemannInt_SF (phi1 n) - l)). +replace (RiemannInt_SF (phi2 n) - l) with + (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n) + + (RiemannInt_SF (phi1 n) - l)); [ apply Rabs_triang | ring ]. +assert (Hyp_b : b <= a). +auto with real. +replace eps with (2 * (eps / 3) + eps / 3). +apply Rplus_lt_compat. +replace (RiemannInt_SF (phi2 n) - RiemannInt_SF (phi1 n)) with + (RiemannInt_SF (phi2 n) + -1 * RiemannInt_SF (phi1 n)); + [ idtac | ring ]. +rewrite <- StepFun_P30. +rewrite StepFun_P39. +rewrite Rabs_Ropp. +apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun + (StepFun_P32 + (mkStepFun + (StepFun_P6 + (pre (mkStepFun (StepFun_P28 (-1) (phi2 n) (phi1 n))))))))). +apply StepFun_P34; try assumption. +apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun + (StepFun_P6 (pre (mkStepFun (StepFun_P28 1 (psi1 n) (psi2 n))))))). +apply StepFun_P37; try assumption. +intros; simpl in |- *; rewrite Rmult_1_l. +apply Rle_trans with (Rabs (phi2 n x - f x) + Rabs (f x - phi1 n x)). +replace (phi2 n x + -1 * phi1 n x) with (phi2 n x - f x + (f x - phi1 n x)); + [ apply Rabs_triang | ring ]. +rewrite (Rplus_comm (psi1 n x)); apply Rplus_le_compat. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; elim (H1 n); intros; apply H7. +assert (H10 : Rmin a b = b). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ elim Hyp; assumption | reflexivity ]. +assert (H11 : Rmax a b = a). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ elim Hyp; assumption | reflexivity ]. +rewrite H10; rewrite H11; elim H6; intros; split; left; assumption. +elim (H0 n); intros; apply H7; assert (H10 : Rmin a b = b). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ elim Hyp; assumption | reflexivity ]. +assert (H11 : Rmax a b = a). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ elim Hyp; assumption | reflexivity ]. +rewrite H10; rewrite H11; elim H6; intros; split; left; assumption. +rewrite <- + (Ropp_involutive + (RiemannInt_SF + (mkStepFun + (StepFun_P6 (pre (mkStepFun (StepFun_P28 1 (psi1 n) (psi2 n)))))))) + . +rewrite <- StepFun_P39. +rewrite StepFun_P30. +rewrite Rmult_1_l; rewrite double. +rewrite Ropp_plus_distr; apply Rplus_lt_compat. +apply Rlt_trans with (pos (un n)). +elim (H0 n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n))). +rewrite <- Rabs_Ropp; apply RRle_abs. +assumption. +replace (pos (un n)) with (R_dist (un n) 0). +apply H; unfold ge in |- *; apply le_trans with N; try assumption. +unfold N in |- *; apply le_max_l. +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; apply Rabs_right. +apply Rle_ge; left; apply (cond_pos (un n)). +apply Rlt_trans with (pos (un n)). +elim (H1 n); intros; apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n))). +rewrite <- Rabs_Ropp; apply RRle_abs; assumption. +assumption. +replace (pos (un n)) with (R_dist (un n) 0). +apply H; unfold ge in |- *; apply le_trans with N; try assumption; + unfold N in |- *; apply le_max_l. +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge; + left; apply (cond_pos (un n)). +unfold R_dist in H2; apply H2; unfold ge in |- *; apply le_trans with N; + try assumption; unfold N in |- *; apply le_max_r. +apply Rmult_eq_reg_l with 3; + [ unfold Rdiv in |- *; rewrite Rmult_plus_distr_l; + do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym; [ ring | discrR ] + | discrR ]. Qed. -Lemma RiemannInt_P12 : (f,g:R->R;a,b,l:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable g a b);pr3:(Riemann_integrable [x:R]``(f x)+l*(g x)`` a b)) ``a<=b`` -> ``(RiemannInt pr3)==(RiemannInt pr1)+l*(RiemannInt pr2)``. -Intro f; Intros; Case (Req_EM l R0); Intro. -Pattern 2 l; Rewrite H0; Rewrite Rmult_Ol; Rewrite Rplus_Or; Unfold RiemannInt; Case (RiemannInt_exists pr3 5!RinvN RinvN_cv); Case (RiemannInt_exists pr1 5!RinvN RinvN_cv); Intros; EApply UL_sequence; [Apply u0 | Pose psi1 := [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr1 n)); Pose psi2 := [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr3 n)); Apply RiemannInt_P11 with f RinvN (phi_sequence RinvN pr1) psi1 psi2; [Apply RinvN_cv | Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr1 n)) | Intro; Assert H1 : ((t:R) ``(Rmin a b) <= t``/\``t <= (Rmax a b)`` -> (Rle (Rabsolu (Rminus ``(f t)+l*(g t)`` (phi_sequence RinvN pr3 n t))) (psi2 n t))) /\ ``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``; [Apply (projT2 ? ? (phi_sequence_prop RinvN pr3 n)) | Elim H1; Intros; Split; Try Assumption; Intros; Replace (f t) with ``(f t)+l*(g t)``; [Apply H2; Assumption | Rewrite H0; Ring]] | Assumption]]. -EApply UL_sequence. -Unfold RiemannInt; Case (RiemannInt_exists pr3 5!RinvN RinvN_cv); Intros; Apply u. -Unfold Un_cv; Intros; Unfold RiemannInt; Case (RiemannInt_exists pr1 5!RinvN RinvN_cv); Case (RiemannInt_exists pr2 5!RinvN RinvN_cv); Unfold Un_cv; Intros; Assert H2 : ``0<eps/5``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Elim (u0 ? H2); Clear u0; Intros N0 H3; Assert H4 := RinvN_cv; Unfold Un_cv in H4; Elim (H4 ? H2); Clear H4 H2; Intros N1 H4; Assert H5 : ``0<eps/(5*(Rabsolu l))``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rmult_lt_pos; [Sup0 | Apply Rabsolu_pos_lt; Assumption]]. -Elim (u ? H5); Clear u; Intros N2 H6; Assert H7 := RinvN_cv; Unfold Un_cv in H7; Elim (H7 ? H5); Clear H7 H5; Intros N3 H5; Unfold R_dist in H3 H4 H5 H6; Pose N := (max (max N0 N1) (max N2 N3)). -Assert H7 : (n:nat) (ge n N1)->``(RinvN n)< eps/5``. -Intros; Replace (pos (RinvN n)) with ``(Rabsolu ((RinvN n)-0))``; [Unfold RinvN; Apply H4; Assumption | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Left; Apply (cond_pos (RinvN n))]. -Clear H4; Assert H4 := H7; Clear H7; Assert H7 : (n:nat) (ge n N3)->``(RinvN n)< eps/(5*(Rabsolu l))``. -Intros; Replace (pos (RinvN n)) with ``(Rabsolu ((RinvN n)-0))``; [Unfold RinvN; Apply H5; Assumption | Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Left; Apply (cond_pos (RinvN n))]. -Clear H5; Assert H5 := H7; Clear H7; Exists N; Intros; Unfold R_dist. -Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+l*(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))))+(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x0))+(Rabsolu l)*(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x))``. -Apply Rle_trans with ``(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+l*(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))))+(Rabsolu (((RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x0)+l*((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x)))``. -Replace ``(RiemannInt_SF [(phi_sequence RinvN pr3 n)])-(x0+l*x)`` with ``(((RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+l*(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))))+(((RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x0)+l*((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x))``; [Apply Rabsolu_triang | Ring]. -Rewrite Rplus_assoc; Apply Rle_compatibility; Rewrite <- Rabsolu_mult; Replace ``(RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x0+l*((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x)`` with ``((RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x0)+(l*((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x))``; [Apply Rabsolu_triang | Ring]. -Replace eps with ``3*eps/5+eps/5+eps/5``. -Repeat Apply Rplus_lt. -Assert H7 : (EXT psi1:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr1 n)] t)))<= (psi1 n t)``)/\``(Rabsolu (RiemannInt_SF (psi1 n))) < (RinvN n)``). -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr1 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr1 n0)). -Assert H8 : (EXT psi2:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((g t)-([(phi_sequence RinvN pr2 n)] t)))<= (psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``). -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr2 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr2 n0)). -Assert H9 : (EXT psi3:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu (((f t)+l*(g t))-([(phi_sequence RinvN pr3 n)] t)))<= (psi3 n t)``)/\``(Rabsolu (RiemannInt_SF (psi3 n))) < (RinvN n)``). -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr3 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr3 n0)). -Elim H7; Clear H7; Intros psi1 H7; Elim H8; Clear H8; Intros psi2 H8; Elim H9; Clear H9; Intros psi3 H9; Replace ``(RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+l*(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))`` with ``(RiemannInt_SF [(phi_sequence RinvN pr3 n)])+(-1)*((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+l*(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))``; [Idtac | Ring]; Do 2 Rewrite <- StepFun_P30; Assert H10 : (Rmin a b)==a. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Assert H11 : (Rmax a b)==b. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Rewrite H10 in H7; Rewrite H10 in H8; Rewrite H10 in H9; Rewrite H11 in H7; Rewrite H11 in H8; Rewrite H11 in H9; Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 ``-1`` (phi_sequence RinvN pr3 n) (mkStepFun (StepFun_P28 l (phi_sequence RinvN pr1 n) (phi_sequence RinvN pr2 n)))))))). -Apply StepFun_P34; Assumption. -Apply Rle_lt_trans with (RiemannInt_SF (mkStepFun (StepFun_P28 R1 (psi3 n) (mkStepFun (StepFun_P28 (Rabsolu l) (psi1 n) (psi2 n)))))). -Apply StepFun_P37; Try Assumption. -Intros; Simpl; Rewrite Rmult_1l. -Apply Rle_trans with ``(Rabsolu (([(phi_sequence RinvN pr3 n)] x1)-((f x1)+l*(g x1))))+(Rabsolu (((f x1)+l*(g x1))+ -1*(([(phi_sequence RinvN pr1 n)] x1)+l*([(phi_sequence RinvN pr2 n)] x1))))``. -Replace ``([(phi_sequence RinvN pr3 n)] x1)+ -1*(([(phi_sequence RinvN pr1 n)] x1)+l*([(phi_sequence RinvN pr2 n)] x1))`` with ``(([(phi_sequence RinvN pr3 n)] x1)-((f x1)+l*(g x1)))+(((f x1)+l*(g x1))+ -1*(([(phi_sequence RinvN pr1 n)] x1)+l*([(phi_sequence RinvN pr2 n)] x1)))``; [Apply Rabsolu_triang | Ring]. -Rewrite Rplus_assoc; Apply Rplus_le. -Elim (H9 n); Intros; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H13. -Elim H12; Intros; Split; Left; Assumption. -Apply Rle_trans with ``(Rabsolu ((f x1)-([(phi_sequence RinvN pr1 n)] x1)))+(Rabsolu l)*(Rabsolu ((g x1)-([(phi_sequence RinvN pr2 n)] x1)))``. -Rewrite <- Rabsolu_mult; Replace ``((f x1)+(l*(g x1)+ -1*(([(phi_sequence RinvN pr1 n)] x1)+l*([(phi_sequence RinvN pr2 n)] x1))))`` with ``((f x1)-([(phi_sequence RinvN pr1 n)] x1))+l*((g x1)-([(phi_sequence RinvN pr2 n)] x1))``; [Apply Rabsolu_triang | Ring]. -Apply Rplus_le. -Elim (H7 n); Intros; Apply H13. -Elim H12; Intros; Split; Left; Assumption. -Apply Rle_monotony; [Apply Rabsolu_pos | Elim (H8 n); Intros; Apply H13; Elim H12; Intros; Split; Left; Assumption]. -Do 2 Rewrite StepFun_P30; Rewrite Rmult_1l; Replace ``3*eps/5`` with ``eps/5+(eps/5+eps/5)``; [Repeat Apply Rplus_lt | Ring]. -Apply Rlt_trans with (pos (RinvN n)); [Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi3 n))); [Apply Rle_Rabsolu | Elim (H9 n); Intros; Assumption] | Apply H4; Unfold ge; Apply le_trans with N; [Apply le_trans with (max N0 N1); [Apply le_max_r | Unfold N; Apply le_max_l] | Assumption]]. -Apply Rlt_trans with (pos (RinvN n)); [Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi1 n))); [Apply Rle_Rabsolu | Elim (H7 n); Intros; Assumption] | Apply H4; Unfold ge; Apply le_trans with N; [Apply le_trans with (max N0 N1); [Apply le_max_r | Unfold N; Apply le_max_l] | Assumption]]. -Apply Rlt_monotony_contra with ``/(Rabsolu l)``. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Replace ``/(Rabsolu l)*eps/5`` with ``eps/(5*(Rabsolu l))``. -Apply Rlt_trans with (pos (RinvN n)); [Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi2 n))); [Apply Rle_Rabsolu | Elim (H8 n); Intros; Assumption] | Apply H5; Unfold ge; Apply le_trans with N; [Apply le_trans with (max N2 N3); [Apply le_max_r | Unfold N; Apply le_max_r] | Assumption]]. -Unfold Rdiv; Rewrite Rinv_Rmult; [Ring | DiscrR | Apply Rabsolu_no_R0; Assumption]. -Apply Rabsolu_no_R0; Assumption. -Apply H3; Unfold ge; Apply le_trans with (max N0 N1); [Apply le_max_l | Apply le_trans with N; [Unfold N; Apply le_max_l | Assumption]]. -Apply Rlt_monotony_contra with ``/(Rabsolu l)``. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Replace ``/(Rabsolu l)*eps/5`` with ``eps/(5*(Rabsolu l))``. -Apply H6; Unfold ge; Apply le_trans with (max N2 N3); [Apply le_max_l | Apply le_trans with N; [Unfold N; Apply le_max_r | Assumption]]. -Unfold Rdiv; Rewrite Rinv_Rmult; [Ring | DiscrR | Apply Rabsolu_no_R0; Assumption]. -Apply Rabsolu_no_R0; Assumption. -Apply r_Rmult_mult with ``5``; [Unfold Rdiv; Do 2 Rewrite Rmult_Rplus_distr; Do 3 Rewrite (Rmult_sym ``5``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. +Lemma RiemannInt_P12 : + forall (f g:R -> R) (a b l:R) (pr1:Riemann_integrable f a b) + (pr2:Riemann_integrable g a b) + (pr3:Riemann_integrable (fun x:R => f x + l * g x) a b), + a <= b -> RiemannInt pr3 = RiemannInt pr1 + l * RiemannInt pr2. +intro f; intros; case (Req_dec l 0); intro. +pattern l at 2 in |- *; rewrite H0; rewrite Rmult_0_l; rewrite Rplus_0_r; + unfold RiemannInt in |- *; case (RiemannInt_exists pr3 RinvN RinvN_cv); + case (RiemannInt_exists pr1 RinvN RinvN_cv); intros; + eapply UL_sequence; + [ apply u0 + | pose (psi1 := fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); + pose (psi2 := fun n:nat => projT1 (phi_sequence_prop RinvN pr3 n)); + apply RiemannInt_P11 with f RinvN (phi_sequence RinvN pr1) psi1 psi2; + [ apply RinvN_cv + | intro; apply (projT2 (phi_sequence_prop RinvN pr1 n)) + | intro; + assert + (H1 : + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> + Rabs (f t + l * g t - phi_sequence RinvN pr3 n t) <= psi2 n t) /\ + Rabs (RiemannInt_SF (psi2 n)) < RinvN n); + [ apply (projT2 (phi_sequence_prop RinvN pr3 n)) + | elim H1; intros; split; try assumption; intros; + replace (f t) with (f t + l * g t); + [ apply H2; assumption | rewrite H0; ring ] ] + | assumption ] ]. +eapply UL_sequence. +unfold RiemannInt in |- *; case (RiemannInt_exists pr3 RinvN RinvN_cv); + intros; apply u. +unfold Un_cv in |- *; intros; unfold RiemannInt in |- *; + case (RiemannInt_exists pr1 RinvN RinvN_cv); + case (RiemannInt_exists pr2 RinvN RinvN_cv); unfold Un_cv in |- *; + intros; assert (H2 : 0 < eps / 5). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (u0 _ H2); clear u0; intros N0 H3; assert (H4 := RinvN_cv); + unfold Un_cv in H4; elim (H4 _ H2); clear H4 H2; intros N1 H4; + assert (H5 : 0 < eps / (5 * Rabs l)). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption + | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat; + [ prove_sup0 | apply Rabs_pos_lt; assumption ] ]. +elim (u _ H5); clear u; intros N2 H6; assert (H7 := RinvN_cv); + unfold Un_cv in H7; elim (H7 _ H5); clear H7 H5; intros N3 H5; + unfold R_dist in H3, H4, H5, H6; pose (N := max (max N0 N1) (max N2 N3)). +assert (H7 : forall n:nat, (n >= N1)%nat -> RinvN n < eps / 5). +intros; replace (pos (RinvN n)) with (Rabs (RinvN n - 0)); + [ unfold RinvN in |- *; apply H4; assumption + | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; + left; apply (cond_pos (RinvN n)) ]. +clear H4; assert (H4 := H7); clear H7; + assert (H7 : forall n:nat, (n >= N3)%nat -> RinvN n < eps / (5 * Rabs l)). +intros; replace (pos (RinvN n)) with (Rabs (RinvN n - 0)); + [ unfold RinvN in |- *; apply H5; assumption + | unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply Rabs_right; + left; apply (cond_pos (RinvN n)) ]. +clear H5; assert (H5 := H7); clear H7; exists N; intros; + unfold R_dist in |- *. +apply Rle_lt_trans with + (Rabs + (RiemannInt_SF (phi_sequence RinvN pr3 n) - + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + l * RiemannInt_SF (phi_sequence RinvN pr2 n))) + + Rabs (RiemannInt_SF (phi_sequence RinvN pr1 n) - x0) + + Rabs l * Rabs (RiemannInt_SF (phi_sequence RinvN pr2 n) - x)). +apply Rle_trans with + (Rabs + (RiemannInt_SF (phi_sequence RinvN pr3 n) - + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + l * RiemannInt_SF (phi_sequence RinvN pr2 n))) + + Rabs + (RiemannInt_SF (phi_sequence RinvN pr1 n) - x0 + + l * (RiemannInt_SF (phi_sequence RinvN pr2 n) - x))). +replace (RiemannInt_SF (phi_sequence RinvN pr3 n) - (x0 + l * x)) with + (RiemannInt_SF (phi_sequence RinvN pr3 n) - + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + l * RiemannInt_SF (phi_sequence RinvN pr2 n)) + + (RiemannInt_SF (phi_sequence RinvN pr1 n) - x0 + + l * (RiemannInt_SF (phi_sequence RinvN pr2 n) - x))); + [ apply Rabs_triang | ring ]. +rewrite Rplus_assoc; apply Rplus_le_compat_l; rewrite <- Rabs_mult; + replace + (RiemannInt_SF (phi_sequence RinvN pr1 n) - x0 + + l * (RiemannInt_SF (phi_sequence RinvN pr2 n) - x)) with + (RiemannInt_SF (phi_sequence RinvN pr1 n) - x0 + + l * (RiemannInt_SF (phi_sequence RinvN pr2 n) - x)); + [ apply Rabs_triang | ring ]. +replace eps with (3 * (eps / 5) + eps / 5 + eps / 5). +repeat apply Rplus_lt_compat. +assert + (H7 : + exists psi1 : nat -> StepFun a b + | (forall n:nat, + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> + Rabs (f t - phi_sequence RinvN pr1 n t) <= psi1 n t) /\ + Rabs (RiemannInt_SF (psi1 n)) < RinvN n)). +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro; + apply (projT2 (phi_sequence_prop RinvN pr1 n0)). +assert + (H8 : + exists psi2 : nat -> StepFun a b + | (forall n:nat, + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> + Rabs (g t - phi_sequence RinvN pr2 n t) <= psi2 n t) /\ + Rabs (RiemannInt_SF (psi2 n)) < RinvN n)). +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro; + apply (projT2 (phi_sequence_prop RinvN pr2 n0)). +assert + (H9 : + exists psi3 : nat -> StepFun a b + | (forall n:nat, + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> + Rabs (f t + l * g t - phi_sequence RinvN pr3 n t) <= psi3 n t) /\ + Rabs (RiemannInt_SF (psi3 n)) < RinvN n)). +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr3 n)); intro; + apply (projT2 (phi_sequence_prop RinvN pr3 n0)). +elim H7; clear H7; intros psi1 H7; elim H8; clear H8; intros psi2 H8; elim H9; + clear H9; intros psi3 H9; + replace + (RiemannInt_SF (phi_sequence RinvN pr3 n) - + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + l * RiemannInt_SF (phi_sequence RinvN pr2 n))) with + (RiemannInt_SF (phi_sequence RinvN pr3 n) + + -1 * + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + l * RiemannInt_SF (phi_sequence RinvN pr2 n))); + [ idtac | ring ]; do 2 rewrite <- StepFun_P30; assert (H10 : Rmin a b = a). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +assert (H11 : Rmax a b = b). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +rewrite H10 in H7; rewrite H10 in H8; rewrite H10 in H9; rewrite H11 in H7; + rewrite H11 in H8; rewrite H11 in H9; + apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun + (StepFun_P32 + (mkStepFun + (StepFun_P28 (-1) (phi_sequence RinvN pr3 n) + (mkStepFun + (StepFun_P28 l (phi_sequence RinvN pr1 n) + (phi_sequence RinvN pr2 n)))))))). +apply StepFun_P34; assumption. +apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun + (StepFun_P28 1 (psi3 n) + (mkStepFun (StepFun_P28 (Rabs l) (psi1 n) (psi2 n)))))). +apply StepFun_P37; try assumption. +intros; simpl in |- *; rewrite Rmult_1_l. +apply Rle_trans with + (Rabs (phi_sequence RinvN pr3 n x1 - (f x1 + l * g x1)) + + Rabs + (f x1 + l * g x1 + + -1 * (phi_sequence RinvN pr1 n x1 + l * phi_sequence RinvN pr2 n x1))). +replace + (phi_sequence RinvN pr3 n x1 + + -1 * (phi_sequence RinvN pr1 n x1 + l * phi_sequence RinvN pr2 n x1)) with + (phi_sequence RinvN pr3 n x1 - (f x1 + l * g x1) + + (f x1 + l * g x1 + + -1 * (phi_sequence RinvN pr1 n x1 + l * phi_sequence RinvN pr2 n x1))); + [ apply Rabs_triang | ring ]. +rewrite Rplus_assoc; apply Rplus_le_compat. +elim (H9 n); intros; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; + apply H13. +elim H12; intros; split; left; assumption. +apply Rle_trans with + (Rabs (f x1 - phi_sequence RinvN pr1 n x1) + + Rabs l * Rabs (g x1 - phi_sequence RinvN pr2 n x1)). +rewrite <- Rabs_mult; + replace + (f x1 + + (l * g x1 + + -1 * (phi_sequence RinvN pr1 n x1 + l * phi_sequence RinvN pr2 n x1))) + with + (f x1 - phi_sequence RinvN pr1 n x1 + + l * (g x1 - phi_sequence RinvN pr2 n x1)); [ apply Rabs_triang | ring ]. +apply Rplus_le_compat. +elim (H7 n); intros; apply H13. +elim H12; intros; split; left; assumption. +apply Rmult_le_compat_l; + [ apply Rabs_pos + | elim (H8 n); intros; apply H13; elim H12; intros; split; left; assumption ]. +do 2 rewrite StepFun_P30; rewrite Rmult_1_l; + replace (3 * (eps / 5)) with (eps / 5 + (eps / 5 + eps / 5)); + [ repeat apply Rplus_lt_compat | ring ]. +apply Rlt_trans with (pos (RinvN n)); + [ apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi3 n))); + [ apply RRle_abs | elim (H9 n); intros; assumption ] + | apply H4; unfold ge in |- *; apply le_trans with N; + [ apply le_trans with (max N0 N1); + [ apply le_max_r | unfold N in |- *; apply le_max_l ] + | assumption ] ]. +apply Rlt_trans with (pos (RinvN n)); + [ apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n))); + [ apply RRle_abs | elim (H7 n); intros; assumption ] + | apply H4; unfold ge in |- *; apply le_trans with N; + [ apply le_trans with (max N0 N1); + [ apply le_max_r | unfold N in |- *; apply le_max_l ] + | assumption ] ]. +apply Rmult_lt_reg_l with (/ Rabs l). +apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; replace (/ Rabs l * (eps / 5)) with (eps / (5 * Rabs l)). +apply Rlt_trans with (pos (RinvN n)); + [ apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n))); + [ apply RRle_abs | elim (H8 n); intros; assumption ] + | apply H5; unfold ge in |- *; apply le_trans with N; + [ apply le_trans with (max N2 N3); + [ apply le_max_r | unfold N in |- *; apply le_max_r ] + | assumption ] ]. +unfold Rdiv in |- *; rewrite Rinv_mult_distr; + [ ring | discrR | apply Rabs_no_R0; assumption ]. +apply Rabs_no_R0; assumption. +apply H3; unfold ge in |- *; apply le_trans with (max N0 N1); + [ apply le_max_l + | apply le_trans with N; [ unfold N in |- *; apply le_max_l | assumption ] ]. +apply Rmult_lt_reg_l with (/ Rabs l). +apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; replace (/ Rabs l * (eps / 5)) with (eps / (5 * Rabs l)). +apply H6; unfold ge in |- *; apply le_trans with (max N2 N3); + [ apply le_max_l + | apply le_trans with N; [ unfold N in |- *; apply le_max_r | assumption ] ]. +unfold Rdiv in |- *; rewrite Rinv_mult_distr; + [ ring | discrR | apply Rabs_no_R0; assumption ]. +apply Rabs_no_R0; assumption. +apply Rmult_eq_reg_l with 5; + [ unfold Rdiv in |- *; do 2 rewrite Rmult_plus_distr_l; + do 3 rewrite (Rmult_comm 5); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym; [ ring | discrR ] + | discrR ]. Qed. -Lemma RiemannInt_P13 : (f,g:R->R;a,b,l:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable g a b);pr3:(Riemann_integrable [x:R]``(f x)+l*(g x)`` a b)) ``(RiemannInt pr3)==(RiemannInt pr1)+l*(RiemannInt pr2)``. -Intros; Case (total_order_Rle a b); Intro; [Apply RiemannInt_P12; Assumption | Assert H : ``b<=a``; [Auto with real | Replace (RiemannInt pr3) with (Ropp (RiemannInt (RiemannInt_P1 pr3))); [Idtac | Symmetry; Apply RiemannInt_P8]; Replace (RiemannInt pr2) with (Ropp (RiemannInt (RiemannInt_P1 pr2))); [Idtac | Symmetry; Apply RiemannInt_P8]; Replace (RiemannInt pr1) with (Ropp (RiemannInt (RiemannInt_P1 pr1))); [Idtac | Symmetry; Apply RiemannInt_P8]; Rewrite (RiemannInt_P12 (RiemannInt_P1 pr1) (RiemannInt_P1 pr2) (RiemannInt_P1 pr3) H); Ring]]. +Lemma RiemannInt_P13 : + forall (f g:R -> R) (a b l:R) (pr1:Riemann_integrable f a b) + (pr2:Riemann_integrable g a b) + (pr3:Riemann_integrable (fun x:R => f x + l * g x) a b), + RiemannInt pr3 = RiemannInt pr1 + l * RiemannInt pr2. +intros; case (Rle_dec a b); intro; + [ apply RiemannInt_P12; assumption + | assert (H : b <= a); + [ auto with real + | replace (RiemannInt pr3) with (- RiemannInt (RiemannInt_P1 pr3)); + [ idtac | symmetry in |- *; apply RiemannInt_P8 ]; + replace (RiemannInt pr2) with (- RiemannInt (RiemannInt_P1 pr2)); + [ idtac | symmetry in |- *; apply RiemannInt_P8 ]; + replace (RiemannInt pr1) with (- RiemannInt (RiemannInt_P1 pr1)); + [ idtac | symmetry in |- *; apply RiemannInt_P8 ]; + rewrite + (RiemannInt_P12 (RiemannInt_P1 pr1) (RiemannInt_P1 pr2) + (RiemannInt_P1 pr3) H); ring ] ]. Qed. -Lemma RiemannInt_P14 : (a,b,c:R) (Riemann_integrable (fct_cte c) a b). -Unfold Riemann_integrable; Intros; Split with (mkStepFun (StepFun_P4 a b c)); Split with (mkStepFun (StepFun_P4 a b R0)); Split; [Intros; Simpl; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Unfold fct_cte; Right; Reflexivity | Rewrite StepFun_P18; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Apply (cond_pos eps)]. +Lemma RiemannInt_P14 : forall a b c:R, Riemann_integrable (fct_cte c) a b. +unfold Riemann_integrable in |- *; intros; + split with (mkStepFun (StepFun_P4 a b c)); + split with (mkStepFun (StepFun_P4 a b 0)); split; + [ intros; simpl in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; unfold fct_cte in |- *; right; + reflexivity + | rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0; + apply (cond_pos eps) ]. Qed. -Lemma RiemannInt_P15 : (a,b,c:R;pr:(Riemann_integrable (fct_cte c) a b)) ``(RiemannInt pr)==c*(b-a)``. -Intros; Unfold RiemannInt; Case (RiemannInt_exists 1!(fct_cte c) 2!a 3!b pr 5!RinvN RinvN_cv); Intros; EApply UL_sequence. -Apply u. -Pose phi1 := [N:nat](phi_sequence RinvN 2!(fct_cte c) 3!a 4!b pr N); Change (Un_cv [N:nat](RiemannInt_SF (phi1 N)) ``c*(b-a)``); Pose f := (fct_cte c); Assert H1 : (EXT psi1:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr n)] t)))<= (psi1 n t)``)/\``(Rabsolu (RiemannInt_SF (psi1 n))) < (RinvN n)``). -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr n)). -Elim H1; Clear H1; Intros psi1 H1; Pose phi2 := [n:nat](mkStepFun (StepFun_P4 a b c)); Pose psi2 := [n:nat](mkStepFun (StepFun_P4 a b R0)); Apply RiemannInt_P11 with f RinvN phi2 psi2 psi1; Try Assumption. -Apply RinvN_cv. -Intro; Split. -Intros; Unfold f; Simpl; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Unfold fct_cte; Right; Reflexivity. -Unfold psi2; Rewrite StepFun_P18; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Apply (cond_pos (RinvN n)). -Unfold Un_cv; Intros; Split with O; Intros; Unfold R_dist; Unfold phi2; Rewrite StepFun_P18; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply H. +Lemma RiemannInt_P15 : + forall (a b c:R) (pr:Riemann_integrable (fct_cte c) a b), + RiemannInt pr = c * (b - a). +intros; unfold RiemannInt in |- *; case (RiemannInt_exists pr RinvN RinvN_cv); + intros; eapply UL_sequence. +apply u. +pose (phi1 := fun N:nat => phi_sequence RinvN pr N); + change (Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) (c * (b - a))) in |- *; + pose (f := fct_cte c); + assert + (H1 : + exists psi1 : nat -> StepFun a b + | (forall n:nat, + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> + Rabs (f t - phi_sequence RinvN pr n t) <= psi1 n t) /\ + Rabs (RiemannInt_SF (psi1 n)) < RinvN n)). +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr n)); intro; + apply (projT2 (phi_sequence_prop RinvN pr n)). +elim H1; clear H1; intros psi1 H1; + pose (phi2 := fun n:nat => mkStepFun (StepFun_P4 a b c)); + pose (psi2 := fun n:nat => mkStepFun (StepFun_P4 a b 0)); + apply RiemannInt_P11 with f RinvN phi2 psi2 psi1; + try assumption. +apply RinvN_cv. +intro; split. +intros; unfold f in |- *; simpl in |- *; unfold Rminus in |- *; + rewrite Rplus_opp_r; rewrite Rabs_R0; unfold fct_cte in |- *; + right; reflexivity. +unfold psi2 in |- *; rewrite StepFun_P18; rewrite Rmult_0_l; rewrite Rabs_R0; + apply (cond_pos (RinvN n)). +unfold Un_cv in |- *; intros; split with 0%nat; intros; unfold R_dist in |- *; + unfold phi2 in |- *; rewrite StepFun_P18; unfold Rminus in |- *; + rewrite Rplus_opp_r; rewrite Rabs_R0; apply H. Qed. -Lemma RiemannInt_P16 : (f:R->R;a,b:R) (Riemann_integrable f a b) -> (Riemann_integrable [x:R](Rabsolu (f x)) a b). -Unfold Riemann_integrable; Intro f; Intros; Elim (X eps); Clear X; Intros phi [psi [H H0]]; Split with (mkStepFun (StepFun_P32 phi)); Split with psi; Split; Try Assumption; Intros; Simpl; Apply Rle_trans with ``(Rabsolu ((f t)-(phi t)))``; [Apply Rabsolu_triang_inv2 | Apply H; Assumption]. +Lemma RiemannInt_P16 : + forall (f:R -> R) (a b:R), + Riemann_integrable f a b -> Riemann_integrable (fun x:R => Rabs (f x)) a b. +unfold Riemann_integrable in |- *; intro f; intros; elim (X eps); clear X; + intros phi [psi [H H0]]; split with (mkStepFun (StepFun_P32 phi)); + split with psi; split; try assumption; intros; simpl in |- *; + apply Rle_trans with (Rabs (f t - phi t)); + [ apply Rabs_triang_inv2 | apply H; assumption ]. Qed. -Lemma Rle_cv_lim : (Un,Vn:nat->R;l1,l2:R) ((n:nat)``(Un n)<=(Vn n)``) -> (Un_cv Un l1) -> (Un_cv Vn l2) -> ``l1<=l2``. -Intros; Case (total_order_Rle l1 l2); Intro. -Assumption. -Assert H2 : ``l2<l1``. -Auto with real. -Clear n; Assert H3 : ``0<(l1-l2)/2``. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply Rlt_Rminus; Assumption | Apply Rlt_Rinv; Sup0]. -Elim (H1 ? H3); Elim (H0 ? H3); Clear H0 H1; Unfold R_dist; Intros; Pose N := (max x x0); Cut ``(Vn N)<(Un N)``. -Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? (H N) H4)). -Apply Rlt_trans with ``(l1+l2)/2``. -Apply Rlt_anti_compatibility with ``-l2``; Replace ``-l2+(l1+l2)/2`` with ``(l1-l2)/2``. -Rewrite Rplus_sym; Apply Rle_lt_trans with ``(Rabsolu ((Vn N)-l2))``. -Apply Rle_Rabsolu. -Apply H1; Unfold ge; Unfold N; Apply le_max_r. -Apply r_Rmult_mult with ``2``; [Unfold Rdiv; Do 2 Rewrite -> (Rmult_sym ``2``); Rewrite (Rmult_Rplus_distrl ``-l2`` ``(l1+l2)*/2`` ``2``); Repeat Rewrite -> Rmult_assoc; Rewrite <- Rinv_l_sym; [ Ring | DiscrR ] | DiscrR]. -Apply Ropp_Rlt; Apply Rlt_anti_compatibility with l1; Replace ``l1+ -((l1+l2)/2)`` with ``(l1-l2)/2``. -Apply Rle_lt_trans with ``(Rabsolu ((Un N)-l1))``. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply Rle_Rabsolu. -Apply H0; Unfold ge; Unfold N; Apply le_max_l. -Apply r_Rmult_mult with ``2``; [Unfold Rdiv; Do 2 Rewrite -> (Rmult_sym ``2``); Rewrite (Rmult_Rplus_distrl ``l1`` ``-((l1+l2)*/2)`` ``2``); Rewrite <- Ropp_mul1; Repeat Rewrite -> Rmult_assoc; Rewrite <- Rinv_l_sym; [ Ring | DiscrR ] | DiscrR]. +Lemma Rle_cv_lim : + forall (Un Vn:nat -> R) (l1 l2:R), + (forall n:nat, Un n <= Vn n) -> Un_cv Un l1 -> Un_cv Vn l2 -> l1 <= l2. +intros; case (Rle_dec l1 l2); intro. +assumption. +assert (H2 : l2 < l1). +auto with real. +clear n; assert (H3 : 0 < (l1 - l2) / 2). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply Rlt_Rminus; assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (H1 _ H3); elim (H0 _ H3); clear H0 H1; unfold R_dist in |- *; intros; + pose (N := max x x0); cut (Vn N < Un N). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (H N) H4)). +apply Rlt_trans with ((l1 + l2) / 2). +apply Rplus_lt_reg_r with (- l2); + replace (- l2 + (l1 + l2) / 2) with ((l1 - l2) / 2). +rewrite Rplus_comm; apply Rle_lt_trans with (Rabs (Vn N - l2)). +apply RRle_abs. +apply H1; unfold ge in |- *; unfold N in |- *; apply le_max_r. +apply Rmult_eq_reg_l with 2; + [ unfold Rdiv in |- *; do 2 rewrite (Rmult_comm 2); + rewrite (Rmult_plus_distr_r (- l2) ((l1 + l2) * / 2) 2); + repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; + [ ring | discrR ] + | discrR ]. +apply Ropp_lt_cancel; apply Rplus_lt_reg_r with l1; + replace (l1 + - ((l1 + l2) / 2)) with ((l1 - l2) / 2). +apply Rle_lt_trans with (Rabs (Un N - l1)). +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply RRle_abs. +apply H0; unfold ge in |- *; unfold N in |- *; apply le_max_l. +apply Rmult_eq_reg_l with 2; + [ unfold Rdiv in |- *; do 2 rewrite (Rmult_comm 2); + rewrite (Rmult_plus_distr_r l1 (- ((l1 + l2) * / 2)) 2); + rewrite <- Ropp_mult_distr_l_reverse; repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym; [ ring | discrR ] + | discrR ]. Qed. -Lemma RiemannInt_P17 : (f:R->R;a,b:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable [x:R](Rabsolu (f x)) a b)) ``a<=b`` -> ``(Rabsolu (RiemannInt pr1))<=(RiemannInt pr2)``. -Intro f; Intros; Unfold RiemannInt; Case (RiemannInt_exists 1!f 2!a 3!b pr1 5!RinvN RinvN_cv); Case (RiemannInt_exists 1!([x0:R](Rabsolu (f x0))) 2!a 3!b pr2 5!RinvN RinvN_cv); Intros; Pose phi1 := (phi_sequence RinvN pr1); Pose phi2 := [N:nat](mkStepFun (StepFun_P32 (phi1 N))); Apply Rle_cv_lim with [N:nat](Rabsolu (RiemannInt_SF (phi1 N))) [N:nat](RiemannInt_SF (phi2 N)). -Intro; Unfold phi2; Apply StepFun_P34; Assumption. -Fold phi1 in u0; Apply (continuity_seq Rabsolu [N:nat](RiemannInt_SF (phi1 N)) x0); Try Assumption. -Apply continuity_Rabsolu. -Pose phi3 := (phi_sequence RinvN pr2); Assert H0 : (EXT psi3:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((Rabsolu (f t))-((phi3 n) t)))<= (psi3 n t)``)/\``(Rabsolu (RiemannInt_SF (psi3 n))) < (RinvN n)``). -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr2 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr2 n)). -Assert H1 : (EXT psi2:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((Rabsolu (f t))-((phi2 n) t)))<= (psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``). -Assert H1 : (EXT psi2:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-((phi1 n) t)))<= (psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``). -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr1 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr1 n)). -Elim H1; Clear H1; Intros psi2 H1; Split with psi2; Intros; Elim (H1 n); Clear H1; Intros; Split; Try Assumption. -Intros; Unfold phi2; Simpl; Apply Rle_trans with ``(Rabsolu ((f t)-((phi1 n) t)))``. -Apply Rabsolu_triang_inv2. -Apply H1; Assumption. -Elim H0; Clear H0; Intros psi3 H0; Elim H1; Clear H1; Intros psi2 H1; Apply RiemannInt_P11 with [x:R](Rabsolu (f x)) RinvN phi3 psi3 psi2; Try Assumption; Apply RinvN_cv. +Lemma RiemannInt_P17 : + forall (f:R -> R) (a b:R) (pr1:Riemann_integrable f a b) + (pr2:Riemann_integrable (fun x:R => Rabs (f x)) a b), + a <= b -> Rabs (RiemannInt pr1) <= RiemannInt pr2. +intro f; intros; unfold RiemannInt in |- *; + case (RiemannInt_exists pr1 RinvN RinvN_cv); + case (RiemannInt_exists pr2 RinvN RinvN_cv); intros; + pose (phi1 := phi_sequence RinvN pr1); + pose (phi2 := fun N:nat => mkStepFun (StepFun_P32 (phi1 N))); + apply Rle_cv_lim with + (fun N:nat => Rabs (RiemannInt_SF (phi1 N))) + (fun N:nat => RiemannInt_SF (phi2 N)). +intro; unfold phi2 in |- *; apply StepFun_P34; assumption. +fold phi1 in u0; + apply (continuity_seq Rabs (fun N:nat => RiemannInt_SF (phi1 N)) x0); + try assumption. +apply Rcontinuity_abs. +pose (phi3 := phi_sequence RinvN pr2); + assert + (H0 : + exists psi3 : nat -> StepFun a b + | (forall n:nat, + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> + Rabs (Rabs (f t) - phi3 n t) <= psi3 n t) /\ + Rabs (RiemannInt_SF (psi3 n)) < RinvN n)). +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro; + apply (projT2 (phi_sequence_prop RinvN pr2 n)). +assert + (H1 : + exists psi2 : nat -> StepFun a b + | (forall n:nat, + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> + Rabs (Rabs (f t) - phi2 n t) <= psi2 n t) /\ + Rabs (RiemannInt_SF (psi2 n)) < RinvN n)). +assert + (H1 : + exists psi2 : nat -> StepFun a b + | (forall n:nat, + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> Rabs (f t - phi1 n t) <= psi2 n t) /\ + Rabs (RiemannInt_SF (psi2 n)) < RinvN n)). +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro; + apply (projT2 (phi_sequence_prop RinvN pr1 n)). +elim H1; clear H1; intros psi2 H1; split with psi2; intros; elim (H1 n); + clear H1; intros; split; try assumption. +intros; unfold phi2 in |- *; simpl in |- *; + apply Rle_trans with (Rabs (f t - phi1 n t)). +apply Rabs_triang_inv2. +apply H1; assumption. +elim H0; clear H0; intros psi3 H0; elim H1; clear H1; intros psi2 H1; + apply RiemannInt_P11 with (fun x:R => Rabs (f x)) RinvN phi3 psi3 psi2; + try assumption; apply RinvN_cv. Qed. -Lemma RiemannInt_P18 : (f,g:R->R;a,b:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable g a b)) ``a<=b`` -> ((x:R)``a<x<b``->``(f x)==(g x)``) -> ``(RiemannInt pr1)==(RiemannInt pr2)``. -Intro f; Intros; Unfold RiemannInt; Case (RiemannInt_exists 1!f 2!a 3!b pr1 5!RinvN RinvN_cv); Case (RiemannInt_exists 1!g 2!a 3!b pr2 5!RinvN RinvN_cv); Intros; EApply UL_sequence. -Apply u0. -Pose phi1 := [N:nat](phi_sequence RinvN 2!f 3!a 4!b pr1 N); Change (Un_cv [N:nat](RiemannInt_SF (phi1 N)) x); Assert H1 : (EXT psi1:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-((phi1 n) t)))<= (psi1 n t)``)/\``(Rabsolu (RiemannInt_SF (psi1 n))) < (RinvN n)``). -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr1 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr1 n)). -Elim H1; Clear H1; Intros psi1 H1; Pose phi2 := [N:nat](phi_sequence RinvN 2!g 3!a 4!b pr2 N). -Pose phi2_aux := [N:nat][x:R](Cases (Req_EM_T x a) of - | (leftT _) => (f a) - | (rightT _) => (Cases (Req_EM_T x b) of - | (leftT _) => (f b) - | (rightT _) => (phi2 N x) end) end). -Cut (N:nat)(IsStepFun (phi2_aux N) a b). -Intro; Pose phi2_m := [N:nat](mkStepFun (X N)). -Assert H2 : (EXT psi2:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((g t)-((phi2 n) t)))<= (psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``). -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr2 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr2 n)). -Elim H2; Clear H2; Intros psi2 H2; Apply RiemannInt_P11 with f RinvN phi2_m psi2 psi1; Try Assumption. -Apply RinvN_cv. -Intro; Elim (H2 n); Intros; Split; Try Assumption. -Intros; Unfold phi2_m; Simpl; Unfold phi2_aux; Case (Req_EM_T t a); Case (Req_EM_T t b); Intros. -Rewrite e0; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rle_trans with ``(Rabsolu ((g t)-((phi2 n) t)))``. -Apply Rabsolu_pos. -Pattern 3 a; Rewrite <- e0; Apply H3; Assumption. -Rewrite e; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rle_trans with ``(Rabsolu ((g t)-((phi2 n) t)))``. -Apply Rabsolu_pos. -Pattern 3 a; Rewrite <- e; Apply H3; Assumption. -Rewrite e; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply Rle_trans with ``(Rabsolu ((g t)-((phi2 n) t)))``. -Apply Rabsolu_pos. -Pattern 3 b; Rewrite <- e; Apply H3; Assumption. -Replace (f t) with (g t). -Apply H3; Assumption. -Symmetry; Apply H0; Elim H5; Clear H5; Intros. -Assert H7 : (Rmin a b)==a. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n2; Assumption]. -Assert H8 : (Rmax a b)==b. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n2; Assumption]. -Rewrite H7 in H5; Rewrite H8 in H6; Split. -Elim H5; Intro; [Assumption | Elim n1; Symmetry; Assumption]. -Elim H6; Intro; [Assumption | Elim n0; Assumption]. -Cut (N:nat)(RiemannInt_SF (phi2_m N))==(RiemannInt_SF (phi2 N)). -Intro; Unfold Un_cv; Intros; Elim (u ? H4); Intros; Exists x1; Intros; Rewrite (H3 n); Apply H5; Assumption. -Intro; Apply Rle_antisym. -Apply StepFun_P37; Try Assumption. -Intros; Unfold phi2_m; Simpl; Unfold phi2_aux; Case (Req_EM_T x1 a); Case (Req_EM_T x1 b); Intros. -Elim H3; Intros; Rewrite e0 in H4; Elim (Rlt_antirefl ? H4). -Elim H3; Intros; Rewrite e in H4; Elim (Rlt_antirefl ? H4). -Elim H3; Intros; Rewrite e in H5; Elim (Rlt_antirefl ? H5). -Right; Reflexivity. -Apply StepFun_P37; Try Assumption. -Intros; Unfold phi2_m; Simpl; Unfold phi2_aux; Case (Req_EM_T x1 a); Case (Req_EM_T x1 b); Intros. -Elim H3; Intros; Rewrite e0 in H4; Elim (Rlt_antirefl ? H4). -Elim H3; Intros; Rewrite e in H4; Elim (Rlt_antirefl ? H4). -Elim H3; Intros; Rewrite e in H5; Elim (Rlt_antirefl ? H5). -Right; Reflexivity. -Intro; Assert H2 := (pre (phi2 N)); Unfold IsStepFun in H2; Unfold is_subdivision in H2; Elim H2; Clear H2; Intros l [lf H2]; Split with l; Split with lf; Unfold adapted_couple in H2; Decompose [and] H2; Clear H2; Unfold adapted_couple; Repeat Split; Try Assumption. -Intros; Assert H9 := (H8 i H2); Unfold constant_D_eq open_interval in H9; Unfold constant_D_eq open_interval; Intros; Rewrite <- (H9 x1 H7); Assert H10 : ``a<=(pos_Rl l i)``. -Replace a with (Rmin a b). -Rewrite <- H5; Elim (RList_P6 l); Intros; Apply H10. -Assumption. -Apply le_O_n. -Apply lt_trans with (pred (Rlength l)); [Assumption | Apply lt_pred_n_n]. -Apply neq_O_lt; Intro; Rewrite <- H12 in H6; Discriminate. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Assert H11 : ``(pos_Rl l (S i))<=b``. -Replace b with (Rmax a b). -Rewrite <- H4; Elim (RList_P6 l); Intros; Apply H11. -Assumption. -Apply lt_le_S; Assumption. -Apply lt_pred_n_n; Apply neq_O_lt; Intro; Rewrite <- H13 in H6; Discriminate. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Elim H7; Clear H7; Intros; Unfold phi2_aux; Case (Req_EM_T x1 a); Case (Req_EM_T x1 b); Intros. -Rewrite e in H12; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H11 H12)). -Rewrite e in H7; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H10 H7)). -Rewrite e in H12; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H11 H12)). -Reflexivity. +Lemma RiemannInt_P18 : + forall (f g:R -> R) (a b:R) (pr1:Riemann_integrable f a b) + (pr2:Riemann_integrable g a b), + a <= b -> + (forall x:R, a < x < b -> f x = g x) -> RiemannInt pr1 = RiemannInt pr2. +intro f; intros; unfold RiemannInt in |- *; + case (RiemannInt_exists pr1 RinvN RinvN_cv); + case (RiemannInt_exists pr2 RinvN RinvN_cv); intros; + eapply UL_sequence. +apply u0. +pose (phi1 := fun N:nat => phi_sequence RinvN pr1 N); + change (Un_cv (fun N:nat => RiemannInt_SF (phi1 N)) x) in |- *; + assert + (H1 : + exists psi1 : nat -> StepFun a b + | (forall n:nat, + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> + Rabs (f t - phi1 n t) <= psi1 n t) /\ + Rabs (RiemannInt_SF (psi1 n)) < RinvN n)). +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro; + apply (projT2 (phi_sequence_prop RinvN pr1 n)). +elim H1; clear H1; intros psi1 H1; + pose (phi2 := fun N:nat => phi_sequence RinvN pr2 N). +pose + (phi2_aux := + fun (N:nat) (x:R) => + match Req_EM_T x a with + | left _ => f a + | right _ => + match Req_EM_T x b with + | left _ => f b + | right _ => phi2 N x + end + end). +cut (forall N:nat, IsStepFun (phi2_aux N) a b). +intro; pose (phi2_m := fun N:nat => mkStepFun (X N)). +assert + (H2 : + exists psi2 : nat -> StepFun a b + | (forall n:nat, + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> Rabs (g t - phi2 n t) <= psi2 n t) /\ + Rabs (RiemannInt_SF (psi2 n)) < RinvN n)). +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro; + apply (projT2 (phi_sequence_prop RinvN pr2 n)). +elim H2; clear H2; intros psi2 H2; + apply RiemannInt_P11 with f RinvN phi2_m psi2 psi1; + try assumption. +apply RinvN_cv. +intro; elim (H2 n); intros; split; try assumption. +intros; unfold phi2_m in |- *; simpl in |- *; unfold phi2_aux in |- *; + case (Req_EM_T t a); case (Req_EM_T t b); intros. +rewrite e0; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + apply Rle_trans with (Rabs (g t - phi2 n t)). +apply Rabs_pos. +pattern a at 3 in |- *; rewrite <- e0; apply H3; assumption. +rewrite e; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + apply Rle_trans with (Rabs (g t - phi2 n t)). +apply Rabs_pos. +pattern a at 3 in |- *; rewrite <- e; apply H3; assumption. +rewrite e; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + apply Rle_trans with (Rabs (g t - phi2 n t)). +apply Rabs_pos. +pattern b at 3 in |- *; rewrite <- e; apply H3; assumption. +replace (f t) with (g t). +apply H3; assumption. +symmetry in |- *; apply H0; elim H5; clear H5; intros. +assert (H7 : Rmin a b = a). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n2; assumption ]. +assert (H8 : Rmax a b = b). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n2; assumption ]. +rewrite H7 in H5; rewrite H8 in H6; split. +elim H5; intro; [ assumption | elim n1; symmetry in |- *; assumption ]. +elim H6; intro; [ assumption | elim n0; assumption ]. +cut (forall N:nat, RiemannInt_SF (phi2_m N) = RiemannInt_SF (phi2 N)). +intro; unfold Un_cv in |- *; intros; elim (u _ H4); intros; exists x1; intros; + rewrite (H3 n); apply H5; assumption. +intro; apply Rle_antisym. +apply StepFun_P37; try assumption. +intros; unfold phi2_m in |- *; simpl in |- *; unfold phi2_aux in |- *; + case (Req_EM_T x1 a); case (Req_EM_T x1 b); intros. +elim H3; intros; rewrite e0 in H4; elim (Rlt_irrefl _ H4). +elim H3; intros; rewrite e in H4; elim (Rlt_irrefl _ H4). +elim H3; intros; rewrite e in H5; elim (Rlt_irrefl _ H5). +right; reflexivity. +apply StepFun_P37; try assumption. +intros; unfold phi2_m in |- *; simpl in |- *; unfold phi2_aux in |- *; + case (Req_EM_T x1 a); case (Req_EM_T x1 b); intros. +elim H3; intros; rewrite e0 in H4; elim (Rlt_irrefl _ H4). +elim H3; intros; rewrite e in H4; elim (Rlt_irrefl _ H4). +elim H3; intros; rewrite e in H5; elim (Rlt_irrefl _ H5). +right; reflexivity. +intro; assert (H2 := pre (phi2 N)); unfold IsStepFun in H2; + unfold is_subdivision in H2; elim H2; clear H2; intros l [lf H2]; + split with l; split with lf; unfold adapted_couple in H2; + decompose [and] H2; clear H2; unfold adapted_couple in |- *; + repeat split; try assumption. +intros; assert (H9 := H8 i H2); unfold constant_D_eq, open_interval in H9; + unfold constant_D_eq, open_interval in |- *; intros; + rewrite <- (H9 x1 H7); assert (H10 : a <= pos_Rl l i). +replace a with (Rmin a b). +rewrite <- H5; elim (RList_P6 l); intros; apply H10. +assumption. +apply le_O_n. +apply lt_trans with (pred (Rlength l)); [ assumption | apply lt_pred_n_n ]. +apply neq_O_lt; intro; rewrite <- H12 in H6; discriminate. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +assert (H11 : pos_Rl l (S i) <= b). +replace b with (Rmax a b). +rewrite <- H4; elim (RList_P6 l); intros; apply H11. +assumption. +apply lt_le_S; assumption. +apply lt_pred_n_n; apply neq_O_lt; intro; rewrite <- H13 in H6; discriminate. +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +elim H7; clear H7; intros; unfold phi2_aux in |- *; case (Req_EM_T x1 a); + case (Req_EM_T x1 b); intros. +rewrite e in H12; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H11 H12)). +rewrite e in H7; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H10 H7)). +rewrite e in H12; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H11 H12)). +reflexivity. Qed. -Lemma RiemannInt_P19 : (f,g:R->R;a,b:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable g a b)) ``a<=b`` -> ((x:R)``a<x<b``->``(f x)<=(g x)``) -> ``(RiemannInt pr1)<=(RiemannInt pr2)``. -Intro f; Intros; Apply Rle_anti_compatibility with ``-(RiemannInt pr1)``; Rewrite Rplus_Ropp_l; Rewrite Rplus_sym; Apply Rle_trans with (Rabsolu (RiemannInt (RiemannInt_P10 ``-1`` pr2 pr1))). -Apply Rabsolu_pos. -Replace ``(RiemannInt pr2)+ -(RiemannInt pr1)`` with (RiemannInt (RiemannInt_P16 (RiemannInt_P10 ``-1`` pr2 pr1))). -Apply (RiemannInt_P17 (RiemannInt_P10 ``-1`` pr2 pr1) (RiemannInt_P16 (RiemannInt_P10 ``-1`` pr2 pr1))); Assumption. -Replace ``(RiemannInt pr2)+-(RiemannInt pr1)`` with (RiemannInt (RiemannInt_P10 ``-1`` pr2 pr1)). -Apply RiemannInt_P18; Try Assumption. -Intros; Apply Rabsolu_right. -Apply Rle_sym1; Apply Rle_anti_compatibility with (f x); Rewrite Rplus_Or; Replace ``(f x)+((g x)+ -1*(f x))`` with (g x); [Apply H0; Assumption | Ring]. -Rewrite (RiemannInt_P12 pr2 pr1 (RiemannInt_P10 ``-1`` pr2 pr1)); [Ring | Assumption]. +Lemma RiemannInt_P19 : + forall (f g:R -> R) (a b:R) (pr1:Riemann_integrable f a b) + (pr2:Riemann_integrable g a b), + a <= b -> + (forall x:R, a < x < b -> f x <= g x) -> RiemannInt pr1 <= RiemannInt pr2. +intro f; intros; apply Rplus_le_reg_l with (- RiemannInt pr1); + rewrite Rplus_opp_l; rewrite Rplus_comm; + apply Rle_trans with (Rabs (RiemannInt (RiemannInt_P10 (-1) pr2 pr1))). +apply Rabs_pos. +replace (RiemannInt pr2 + - RiemannInt pr1) with + (RiemannInt (RiemannInt_P16 (RiemannInt_P10 (-1) pr2 pr1))). +apply + (RiemannInt_P17 (RiemannInt_P10 (-1) pr2 pr1) + (RiemannInt_P16 (RiemannInt_P10 (-1) pr2 pr1))); + assumption. +replace (RiemannInt pr2 + - RiemannInt pr1) with + (RiemannInt (RiemannInt_P10 (-1) pr2 pr1)). +apply RiemannInt_P18; try assumption. +intros; apply Rabs_right. +apply Rle_ge; apply Rplus_le_reg_l with (f x); rewrite Rplus_0_r; + replace (f x + (g x + -1 * f x)) with (g x); [ apply H0; assumption | ring ]. +rewrite (RiemannInt_P12 pr2 pr1 (RiemannInt_P10 (-1) pr2 pr1)); + [ ring | assumption ]. Qed. -Lemma FTC_P1 : (f:R->R;a,b:R) ``a<=b`` -> ((x:R)``a<=x<=b``->(continuity_pt f x)) -> ((x:R)``a<=x``->``x<=b``->(Riemann_integrable f a x)). -Intros; Apply continuity_implies_RiemannInt; [Assumption | Intros; Apply H0; Elim H3; Intros; Split; Assumption Orelse Apply Rle_trans with x; Assumption]. +Lemma FTC_P1 : + forall (f:R -> R) (a b:R), + a <= b -> + (forall x:R, a <= x <= b -> continuity_pt f x) -> + forall x:R, a <= x -> x <= b -> Riemann_integrable f a x. +intros; apply continuity_implies_RiemannInt; + [ assumption + | intros; apply H0; elim H3; intros; split; + assumption || apply Rle_trans with x; assumption ]. Qed. -V7only [Notation FTC_P2 := Rle_refl.]. - -Definition primitive [f:R->R;a,b:R;h:``a<=b``;pr:((x:R)``a<=x``->``x<=b``->(Riemann_integrable f a x))] : R->R := [x:R] Cases (total_order_Rle a x) of - | (leftT r) => Cases (total_order_Rle x b) of - | (leftT r0) => (RiemannInt (pr x r r0)) - | (rightT _) => ``(f b)*(x-b)+(RiemannInt (pr b h (FTC_P2 b)))`` end - | (rightT _) => ``(f a)*(x-a)`` end. - -Lemma RiemannInt_P20 : (f:R->R;a,b:R;h:``a<=b``;pr:((x:R)``a<=x``->``x<=b``->(Riemann_integrable f a x));pr0:(Riemann_integrable f a b)) ``(RiemannInt pr0)==(primitive h pr b)-(primitive h pr a)``. -Intros; Replace (primitive h pr a) with R0. -Replace (RiemannInt pr0) with (primitive h pr b). -Ring. -Unfold primitive; Case (total_order_Rle a b); Case (total_order_Rle b b); Intros; [Apply RiemannInt_P5 | Elim n; Right; Reflexivity | Elim n; Assumption | Elim n0; Assumption]. -Symmetry; Unfold primitive; Case (total_order_Rle a a); Case (total_order_Rle a b); Intros; [Apply RiemannInt_P9 | Elim n; Assumption | Elim n; Right; Reflexivity | Elim n0; Right; Reflexivity]. + +Definition primitive (f:R -> R) (a b:R) (h:a <= b) + (pr:forall x:R, a <= x -> x <= b -> Riemann_integrable f a x) + (x:R) : R := + match Rle_dec a x with + | left r => + match Rle_dec x b with + | left r0 => RiemannInt (pr x r r0) + | right _ => f b * (x - b) + RiemannInt (pr b h (Rle_refl b)) + end + | right _ => f a * (x - a) + end. + +Lemma RiemannInt_P20 : + forall (f:R -> R) (a b:R) (h:a <= b) + (pr:forall x:R, a <= x -> x <= b -> Riemann_integrable f a x) + (pr0:Riemann_integrable f a b), + RiemannInt pr0 = primitive h pr b - primitive h pr a. +intros; replace (primitive h pr a) with 0. +replace (RiemannInt pr0) with (primitive h pr b). +ring. +unfold primitive in |- *; case (Rle_dec a b); case (Rle_dec b b); intros; + [ apply RiemannInt_P5 + | elim n; right; reflexivity + | elim n; assumption + | elim n0; assumption ]. +symmetry in |- *; unfold primitive in |- *; case (Rle_dec a a); + case (Rle_dec a b); intros; + [ apply RiemannInt_P9 + | elim n; assumption + | elim n; right; reflexivity + | elim n0; right; reflexivity ]. Qed. -Lemma RiemannInt_P21 : (f:R->R;a,b,c:R) ``a<=b``-> ``b<=c`` -> (Riemann_integrable f a b) -> (Riemann_integrable f b c) -> (Riemann_integrable f a c). -Unfold Riemann_integrable; Intros f a b c Hyp1 Hyp2 X X0 eps. -Assert H : ``0<eps/2``. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos eps) | Apply Rlt_Rinv; Sup0]. -Elim (X (mkposreal ? H)); Clear X; Intros phi1 [psi1 H1]; Elim (X0 (mkposreal ? H)); Clear X0; Intros phi2 [psi2 H2]. -Pose phi3 := [x:R] Cases (total_order_Rle a x) of - | (leftT _) => Cases (total_order_Rle x b) of - | (leftT _) => (phi1 x) - | (rightT _) => (phi2 x) end - | (rightT _) => R0 end. -Pose psi3 := [x:R] Cases (total_order_Rle a x) of - | (leftT _) => Cases (total_order_Rle x b) of - | (leftT _) => (psi1 x) - | (rightT _) => (psi2 x) end - | (rightT _) => R0 end. -Cut (IsStepFun phi3 a c). -Intro; Cut (IsStepFun psi3 a b). -Intro; Cut (IsStepFun psi3 b c). -Intro; Cut (IsStepFun psi3 a c). -Intro; Split with (mkStepFun X); Split with (mkStepFun X2); Simpl; Split. -Intros; Unfold phi3 psi3; Case (total_order_Rle t b); Case (total_order_Rle a t); Intros. -Elim H1; Intros; Apply H3. -Replace (Rmin a b) with a. -Replace (Rmax a b) with b. -Split; Assumption. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Elim n; Replace a with (Rmin a c). -Elim H0; Intros; Assumption. -Unfold Rmin; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n0; Apply Rle_trans with b; Assumption]. -Elim H2; Intros; Apply H3. -Replace (Rmax b c) with (Rmax a c). -Elim H0; Intros; Split; Try Assumption. -Replace (Rmin b c) with b. -Auto with real. -Unfold Rmin; Case (total_order_Rle b c); Intro; [Reflexivity | Elim n0; Assumption]. -Unfold Rmax; Case (total_order_Rle a c); Case (total_order_Rle b c); Intros; Try (Elim n0; Assumption Orelse Elim n0; Apply Rle_trans with b; Assumption). -Reflexivity. -Elim n; Replace a with (Rmin a c). -Elim H0; Intros; Assumption. -Unfold Rmin; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n1; Apply Rle_trans with b; Assumption]. -Rewrite <- (StepFun_P43 X0 X1 X2). -Apply Rle_lt_trans with ``(Rabsolu (RiemannInt_SF (mkStepFun X0)))+(Rabsolu (RiemannInt_SF (mkStepFun X1)))``. -Apply Rabsolu_triang. -Rewrite (double_var eps); Replace (RiemannInt_SF (mkStepFun X0)) with (RiemannInt_SF psi1). -Replace (RiemannInt_SF (mkStepFun X1)) with (RiemannInt_SF psi2). -Apply Rplus_lt. -Elim H1; Intros; Assumption. -Elim H2; Intros; Assumption. -Apply Rle_antisym. -Apply StepFun_P37; Try Assumption. -Simpl; Intros; Unfold psi3; Elim H0; Clear H0; Intros; Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; [Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H0)) | Right; Reflexivity | Elim n; Apply Rle_trans with b; [Assumption | Left; Assumption] | Elim n0; Apply Rle_trans with b; [Assumption | Left; Assumption]]. -Apply StepFun_P37; Try Assumption. -Simpl; Intros; Unfold psi3; Elim H0; Clear H0; Intros; Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; [Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H0)) | Right; Reflexivity | Elim n; Apply Rle_trans with b; [Assumption | Left; Assumption] | Elim n0; Apply Rle_trans with b; [Assumption | Left; Assumption]]. -Apply Rle_antisym. -Apply StepFun_P37; Try Assumption. -Simpl; Intros; Unfold psi3; Elim H0; Clear H0; Intros; Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; [Right; Reflexivity | Elim n; Left; Assumption | Elim n; Left; Assumption | Elim n0; Left; Assumption]. -Apply StepFun_P37; Try Assumption. -Simpl; Intros; Unfold psi3; Elim H0; Clear H0; Intros; Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; [Right; Reflexivity | Elim n; Left; Assumption | Elim n; Left; Assumption | Elim n0; Left; Assumption]. -Apply StepFun_P46 with b; Assumption. -Assert H3 := (pre psi2); Unfold IsStepFun in H3; Unfold is_subdivision in H3; Elim H3; Clear H3; Intros l1 [lf1 H3]; Split with l1; Split with lf1; Unfold adapted_couple in H3; Decompose [and] H3; Clear H3; Unfold adapted_couple; Repeat Split; Try Assumption. -Intros; Assert H9 := (H8 i H3); Unfold constant_D_eq open_interval; Unfold constant_D_eq open_interval in H9; Intros; Rewrite <- (H9 x H7); Unfold psi3; Assert H10 : ``b<x``. -Apply Rle_lt_trans with (pos_Rl l1 i). -Replace b with (Rmin b c). -Rewrite <- H5; Elim (RList_P6 l1); Intros; Apply H10; Try Assumption. -Apply le_O_n. -Apply lt_trans with (pred (Rlength l1)); Try Assumption; Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H12 in H6; Discriminate. -Unfold Rmin; Case (total_order_Rle b c); Intro; [Reflexivity | Elim n; Assumption]. -Elim H7; Intros; Assumption. -Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; [Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H10)) | Reflexivity | Elim n; Apply Rle_trans with b; [Assumption | Left; Assumption] | Elim n0; Apply Rle_trans with b; [Assumption | Left; Assumption]]. -Assert H3 := (pre psi1); Unfold IsStepFun in H3; Unfold is_subdivision in H3; Elim H3; Clear H3; Intros l1 [lf1 H3]; Split with l1; Split with lf1; Unfold adapted_couple in H3; Decompose [and] H3; Clear H3; Unfold adapted_couple; Repeat Split; Try Assumption. -Intros; Assert H9 := (H8 i H3); Unfold constant_D_eq open_interval; Unfold constant_D_eq open_interval in H9; Intros; Rewrite <- (H9 x H7); Unfold psi3; Assert H10 : ``x<=b``. -Apply Rle_trans with (pos_Rl l1 (S i)). -Elim H7; Intros; Left; Assumption. -Replace b with (Rmax a b). -Rewrite <- H4; Elim (RList_P6 l1); Intros; Apply H10; Try Assumption. -Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H12 in H6; Discriminate. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Assert H11 : ``a<=x``. -Apply Rle_trans with (pos_Rl l1 i). -Replace a with (Rmin a b). -Rewrite <- H5; Elim (RList_P6 l1); Intros; Apply H11; Try Assumption. -Apply le_O_n. -Apply lt_trans with (pred (Rlength l1)); Try Assumption; Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H6; Discriminate. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Left; Elim H7; Intros; Assumption. -Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; Reflexivity Orelse Elim n; Assumption. -Apply StepFun_P46 with b. -Assert H3 := (pre phi1); Unfold IsStepFun in H3; Unfold is_subdivision in H3; Elim H3; Clear H3; Intros l1 [lf1 H3]; Split with l1; Split with lf1; Unfold adapted_couple in H3; Decompose [and] H3; Clear H3; Unfold adapted_couple; Repeat Split; Try Assumption. -Intros; Assert H9 := (H8 i H3); Unfold constant_D_eq open_interval; Unfold constant_D_eq open_interval in H9; Intros; Rewrite <- (H9 x H7); Unfold psi3; Assert H10 : ``x<=b``. -Apply Rle_trans with (pos_Rl l1 (S i)). -Elim H7; Intros; Left; Assumption. -Replace b with (Rmax a b). -Rewrite <- H4; Elim (RList_P6 l1); Intros; Apply H10; Try Assumption. -Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H12 in H6; Discriminate. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Assert H11 : ``a<=x``. -Apply Rle_trans with (pos_Rl l1 i). -Replace a with (Rmin a b). -Rewrite <- H5; Elim (RList_P6 l1); Intros; Apply H11; Try Assumption. -Apply le_O_n. -Apply lt_trans with (pred (Rlength l1)); Try Assumption; Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H6; Discriminate. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Left; Elim H7; Intros; Assumption. -Unfold phi3; Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; Reflexivity Orelse Elim n; Assumption. -Assert H3 := (pre phi2); Unfold IsStepFun in H3; Unfold is_subdivision in H3; Elim H3; Clear H3; Intros l1 [lf1 H3]; Split with l1; Split with lf1; Unfold adapted_couple in H3; Decompose [and] H3; Clear H3; Unfold adapted_couple; Repeat Split; Try Assumption. -Intros; Assert H9 := (H8 i H3); Unfold constant_D_eq open_interval; Unfold constant_D_eq open_interval in H9; Intros; Rewrite <- (H9 x H7); Unfold psi3; Assert H10 : ``b<x``. -Apply Rle_lt_trans with (pos_Rl l1 i). -Replace b with (Rmin b c). -Rewrite <- H5; Elim (RList_P6 l1); Intros; Apply H10; Try Assumption. -Apply le_O_n. -Apply lt_trans with (pred (Rlength l1)); Try Assumption; Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H12 in H6; Discriminate. -Unfold Rmin; Case (total_order_Rle b c); Intro; [Reflexivity | Elim n; Assumption]. -Elim H7; Intros; Assumption. -Unfold phi3; Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; [Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H10)) | Reflexivity | Elim n; Apply Rle_trans with b; [Assumption | Left; Assumption] | Elim n0; Apply Rle_trans with b; [Assumption | Left; Assumption]]. +Lemma RiemannInt_P21 : + forall (f:R -> R) (a b c:R), + a <= b -> + b <= c -> + Riemann_integrable f a b -> + Riemann_integrable f b c -> Riemann_integrable f a c. +unfold Riemann_integrable in |- *; intros f a b c Hyp1 Hyp2 X X0 eps. +assert (H : 0 < eps / 2). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (X (mkposreal _ H)); clear X; intros phi1 [psi1 H1]; + elim (X0 (mkposreal _ H)); clear X0; intros phi2 [psi2 H2]. +pose + (phi3 := + fun x:R => + match Rle_dec a x with + | left _ => + match Rle_dec x b with + | left _ => phi1 x + | right _ => phi2 x + end + | right _ => 0 + end). +pose + (psi3 := + fun x:R => + match Rle_dec a x with + | left _ => + match Rle_dec x b with + | left _ => psi1 x + | right _ => psi2 x + end + | right _ => 0 + end). +cut (IsStepFun phi3 a c). +intro; cut (IsStepFun psi3 a b). +intro; cut (IsStepFun psi3 b c). +intro; cut (IsStepFun psi3 a c). +intro; split with (mkStepFun X); split with (mkStepFun X2); simpl in |- *; + split. +intros; unfold phi3, psi3 in |- *; case (Rle_dec t b); case (Rle_dec a t); + intros. +elim H1; intros; apply H3. +replace (Rmin a b) with a. +replace (Rmax a b) with b. +split; assumption. +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +elim n; replace a with (Rmin a c). +elim H0; intros; assumption. +unfold Rmin in |- *; case (Rle_dec a c); intro; + [ reflexivity | elim n0; apply Rle_trans with b; assumption ]. +elim H2; intros; apply H3. +replace (Rmax b c) with (Rmax a c). +elim H0; intros; split; try assumption. +replace (Rmin b c) with b. +auto with real. +unfold Rmin in |- *; case (Rle_dec b c); intro; + [ reflexivity | elim n0; assumption ]. +unfold Rmax in |- *; case (Rle_dec a c); case (Rle_dec b c); intros; + try (elim n0; assumption || elim n0; apply Rle_trans with b; assumption). +reflexivity. +elim n; replace a with (Rmin a c). +elim H0; intros; assumption. +unfold Rmin in |- *; case (Rle_dec a c); intro; + [ reflexivity | elim n1; apply Rle_trans with b; assumption ]. +rewrite <- (StepFun_P43 X0 X1 X2). +apply Rle_lt_trans with + (Rabs (RiemannInt_SF (mkStepFun X0)) + Rabs (RiemannInt_SF (mkStepFun X1))). +apply Rabs_triang. +rewrite (double_var eps); + replace (RiemannInt_SF (mkStepFun X0)) with (RiemannInt_SF psi1). +replace (RiemannInt_SF (mkStepFun X1)) with (RiemannInt_SF psi2). +apply Rplus_lt_compat. +elim H1; intros; assumption. +elim H2; intros; assumption. +apply Rle_antisym. +apply StepFun_P37; try assumption. +simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros; + case (Rle_dec a x); case (Rle_dec x b); intros; + [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H0)) + | right; reflexivity + | elim n; apply Rle_trans with b; [ assumption | left; assumption ] + | elim n0; apply Rle_trans with b; [ assumption | left; assumption ] ]. +apply StepFun_P37; try assumption. +simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros; + case (Rle_dec a x); case (Rle_dec x b); intros; + [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H0)) + | right; reflexivity + | elim n; apply Rle_trans with b; [ assumption | left; assumption ] + | elim n0; apply Rle_trans with b; [ assumption | left; assumption ] ]. +apply Rle_antisym. +apply StepFun_P37; try assumption. +simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros; + case (Rle_dec a x); case (Rle_dec x b); intros; + [ right; reflexivity + | elim n; left; assumption + | elim n; left; assumption + | elim n0; left; assumption ]. +apply StepFun_P37; try assumption. +simpl in |- *; intros; unfold psi3 in |- *; elim H0; clear H0; intros; + case (Rle_dec a x); case (Rle_dec x b); intros; + [ right; reflexivity + | elim n; left; assumption + | elim n; left; assumption + | elim n0; left; assumption ]. +apply StepFun_P46 with b; assumption. +assert (H3 := pre psi2); unfold IsStepFun in H3; unfold is_subdivision in H3; + elim H3; clear H3; intros l1 [lf1 H3]; split with l1; + split with lf1; unfold adapted_couple in H3; decompose [and] H3; + clear H3; unfold adapted_couple in |- *; repeat split; + try assumption. +intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *; + unfold constant_D_eq, open_interval in H9; intros; + rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : b < x). +apply Rle_lt_trans with (pos_Rl l1 i). +replace b with (Rmin b c). +rewrite <- H5; elim (RList_P6 l1); intros; apply H10; try assumption. +apply le_O_n. +apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n; + apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6; + discriminate. +unfold Rmin in |- *; case (Rle_dec b c); intro; + [ reflexivity | elim n; assumption ]. +elim H7; intros; assumption. +case (Rle_dec a x); case (Rle_dec x b); intros; + [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H10)) + | reflexivity + | elim n; apply Rle_trans with b; [ assumption | left; assumption ] + | elim n0; apply Rle_trans with b; [ assumption | left; assumption ] ]. +assert (H3 := pre psi1); unfold IsStepFun in H3; unfold is_subdivision in H3; + elim H3; clear H3; intros l1 [lf1 H3]; split with l1; + split with lf1; unfold adapted_couple in H3; decompose [and] H3; + clear H3; unfold adapted_couple in |- *; repeat split; + try assumption. +intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *; + unfold constant_D_eq, open_interval in H9; intros; + rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : x <= b). +apply Rle_trans with (pos_Rl l1 (S i)). +elim H7; intros; left; assumption. +replace b with (Rmax a b). +rewrite <- H4; elim (RList_P6 l1); intros; apply H10; try assumption. +apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6; + discriminate. +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +assert (H11 : a <= x). +apply Rle_trans with (pos_Rl l1 i). +replace a with (Rmin a b). +rewrite <- H5; elim (RList_P6 l1); intros; apply H11; try assumption. +apply le_O_n. +apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n; + apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H6; + discriminate. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +left; elim H7; intros; assumption. +case (Rle_dec a x); case (Rle_dec x b); intros; reflexivity || elim n; + assumption. +apply StepFun_P46 with b. +assert (H3 := pre phi1); unfold IsStepFun in H3; unfold is_subdivision in H3; + elim H3; clear H3; intros l1 [lf1 H3]; split with l1; + split with lf1; unfold adapted_couple in H3; decompose [and] H3; + clear H3; unfold adapted_couple in |- *; repeat split; + try assumption. +intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *; + unfold constant_D_eq, open_interval in H9; intros; + rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : x <= b). +apply Rle_trans with (pos_Rl l1 (S i)). +elim H7; intros; left; assumption. +replace b with (Rmax a b). +rewrite <- H4; elim (RList_P6 l1); intros; apply H10; try assumption. +apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6; + discriminate. +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +assert (H11 : a <= x). +apply Rle_trans with (pos_Rl l1 i). +replace a with (Rmin a b). +rewrite <- H5; elim (RList_P6 l1); intros; apply H11; try assumption. +apply le_O_n. +apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n; + apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H6; + discriminate. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +left; elim H7; intros; assumption. +unfold phi3 in |- *; case (Rle_dec a x); case (Rle_dec x b); intros; + reflexivity || elim n; assumption. +assert (H3 := pre phi2); unfold IsStepFun in H3; unfold is_subdivision in H3; + elim H3; clear H3; intros l1 [lf1 H3]; split with l1; + split with lf1; unfold adapted_couple in H3; decompose [and] H3; + clear H3; unfold adapted_couple in |- *; repeat split; + try assumption. +intros; assert (H9 := H8 i H3); unfold constant_D_eq, open_interval in |- *; + unfold constant_D_eq, open_interval in H9; intros; + rewrite <- (H9 x H7); unfold psi3 in |- *; assert (H10 : b < x). +apply Rle_lt_trans with (pos_Rl l1 i). +replace b with (Rmin b c). +rewrite <- H5; elim (RList_P6 l1); intros; apply H10; try assumption. +apply le_O_n. +apply lt_trans with (pred (Rlength l1)); try assumption; apply lt_pred_n_n; + apply neq_O_lt; red in |- *; intro; rewrite <- H12 in H6; + discriminate. +unfold Rmin in |- *; case (Rle_dec b c); intro; + [ reflexivity | elim n; assumption ]. +elim H7; intros; assumption. +unfold phi3 in |- *; case (Rle_dec a x); case (Rle_dec x b); intros; + [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H10)) + | reflexivity + | elim n; apply Rle_trans with b; [ assumption | left; assumption ] + | elim n0; apply Rle_trans with b; [ assumption | left; assumption ] ]. Qed. -Lemma RiemannInt_P22 : (f:R->R;a,b,c:R) (Riemann_integrable f a b) -> ``a<=c<=b`` -> (Riemann_integrable f a c). -Unfold Riemann_integrable; Intros; Elim (X eps); Clear X; Intros phi [psi H0]; Elim H; Elim H0; Clear H H0; Intros; Assert H3 : (IsStepFun phi a c). -Apply StepFun_P44 with b. -Apply (pre phi). -Split; Assumption. -Assert H4 : (IsStepFun psi a c). -Apply StepFun_P44 with b. -Apply (pre psi). -Split; Assumption. -Split with (mkStepFun H3); Split with (mkStepFun H4); Split. -Simpl; Intros; Apply H. -Replace (Rmin a b) with (Rmin a c). -Elim H5; Intros; Split; Try Assumption. -Apply Rle_trans with (Rmax a c); Try Assumption. -Replace (Rmax a b) with b. -Replace (Rmax a c) with c. -Assumption. -Unfold Rmax; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n; Assumption]. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. -Unfold Rmin; Case (total_order_Rle a c); Case (total_order_Rle a b); Intros; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption | Elim n; Assumption | Elim n0; Assumption]. -Rewrite Rabsolu_right. -Assert H5 : (IsStepFun psi c b). -Apply StepFun_P46 with a. -Apply StepFun_P6; Assumption. -Apply (pre psi). -Replace (RiemannInt_SF (mkStepFun H4)) with ``(RiemannInt_SF psi)-(RiemannInt_SF (mkStepFun H5))``. -Apply Rle_lt_trans with (RiemannInt_SF psi). -Unfold Rminus; Pattern 2 (RiemannInt_SF psi); Rewrite <- Rplus_Or; Apply Rle_compatibility; Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Replace R0 with (RiemannInt_SF (mkStepFun (StepFun_P4 c b R0))). -Apply StepFun_P37; Try Assumption. -Intros; Simpl; Unfold fct_cte; Apply Rle_trans with ``(Rabsolu ((f x)-(phi x)))``. -Apply Rabsolu_pos. -Apply H. -Replace (Rmin a b) with a. -Replace (Rmax a b) with b. -Elim H6; Intros; Split; Left. -Apply Rle_lt_trans with c; Assumption. -Assumption. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. -Rewrite StepFun_P18; Ring. -Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF psi)). -Apply Rle_Rabsolu. -Assumption. -Assert H6 : (IsStepFun psi a b). -Apply (pre psi). -Replace (RiemannInt_SF psi) with (RiemannInt_SF (mkStepFun H6)). -Rewrite <- (StepFun_P43 H4 H5 H6); Ring. -Unfold RiemannInt_SF; Case (total_order_Rle a b); Intro. -EApply StepFun_P17. -Apply StepFun_P1. -Simpl; Apply StepFun_P1. -Apply eq_Ropp; EApply StepFun_P17. -Apply StepFun_P1. -Simpl; Apply StepFun_P1. -Apply Rle_sym1; Replace R0 with (RiemannInt_SF (mkStepFun (StepFun_P4 a c R0))). -Apply StepFun_P37; Try Assumption. -Intros; Simpl; Unfold fct_cte; Apply Rle_trans with ``(Rabsolu ((f x)-(phi x)))``. -Apply Rabsolu_pos. -Apply H. -Replace (Rmin a b) with a. -Replace (Rmax a b) with b. -Elim H5; Intros; Split; Left. -Assumption. -Apply Rlt_le_trans with c; Assumption. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. -Rewrite StepFun_P18; Ring. +Lemma RiemannInt_P22 : + forall (f:R -> R) (a b c:R), + Riemann_integrable f a b -> a <= c <= b -> Riemann_integrable f a c. +unfold Riemann_integrable in |- *; intros; elim (X eps); clear X; + intros phi [psi H0]; elim H; elim H0; clear H H0; + intros; assert (H3 : IsStepFun phi a c). +apply StepFun_P44 with b. +apply (pre phi). +split; assumption. +assert (H4 : IsStepFun psi a c). +apply StepFun_P44 with b. +apply (pre psi). +split; assumption. +split with (mkStepFun H3); split with (mkStepFun H4); split. +simpl in |- *; intros; apply H. +replace (Rmin a b) with (Rmin a c). +elim H5; intros; split; try assumption. +apply Rle_trans with (Rmax a c); try assumption. +replace (Rmax a b) with b. +replace (Rmax a c) with c. +assumption. +unfold Rmax in |- *; case (Rle_dec a c); intro; + [ reflexivity | elim n; assumption ]. +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; apply Rle_trans with c; assumption ]. +unfold Rmin in |- *; case (Rle_dec a c); case (Rle_dec a b); intros; + [ reflexivity + | elim n; apply Rle_trans with c; assumption + | elim n; assumption + | elim n0; assumption ]. +rewrite Rabs_right. +assert (H5 : IsStepFun psi c b). +apply StepFun_P46 with a. +apply StepFun_P6; assumption. +apply (pre psi). +replace (RiemannInt_SF (mkStepFun H4)) with + (RiemannInt_SF psi - RiemannInt_SF (mkStepFun H5)). +apply Rle_lt_trans with (RiemannInt_SF psi). +unfold Rminus in |- *; pattern (RiemannInt_SF psi) at 2 in |- *; + rewrite <- Rplus_0_r; apply Rplus_le_compat_l; rewrite <- Ropp_0; + apply Ropp_ge_le_contravar; apply Rle_ge; + replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 c b 0))). +apply StepFun_P37; try assumption. +intros; simpl in |- *; unfold fct_cte in |- *; + apply Rle_trans with (Rabs (f x - phi x)). +apply Rabs_pos. +apply H. +replace (Rmin a b) with a. +replace (Rmax a b) with b. +elim H6; intros; split; left. +apply Rle_lt_trans with c; assumption. +assumption. +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; apply Rle_trans with c; assumption ]. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; apply Rle_trans with c; assumption ]. +rewrite StepFun_P18; ring. +apply Rle_lt_trans with (Rabs (RiemannInt_SF psi)). +apply RRle_abs. +assumption. +assert (H6 : IsStepFun psi a b). +apply (pre psi). +replace (RiemannInt_SF psi) with (RiemannInt_SF (mkStepFun H6)). +rewrite <- (StepFun_P43 H4 H5 H6); ring. +unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro. +eapply StepFun_P17. +apply StepFun_P1. +simpl in |- *; apply StepFun_P1. +apply Ropp_eq_compat; eapply StepFun_P17. +apply StepFun_P1. +simpl in |- *; apply StepFun_P1. +apply Rle_ge; replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 a c 0))). +apply StepFun_P37; try assumption. +intros; simpl in |- *; unfold fct_cte in |- *; + apply Rle_trans with (Rabs (f x - phi x)). +apply Rabs_pos. +apply H. +replace (Rmin a b) with a. +replace (Rmax a b) with b. +elim H5; intros; split; left. +assumption. +apply Rlt_le_trans with c; assumption. +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; apply Rle_trans with c; assumption ]. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; apply Rle_trans with c; assumption ]. +rewrite StepFun_P18; ring. Qed. -Lemma RiemannInt_P23 : (f:R->R;a,b,c:R) (Riemann_integrable f a b) -> ``a<=c<=b`` -> (Riemann_integrable f c b). -Unfold Riemann_integrable; Intros; Elim (X eps); Clear X; Intros phi [psi H0]; Elim H; Elim H0; Clear H H0; Intros; Assert H3 : (IsStepFun phi c b). -Apply StepFun_P45 with a. -Apply (pre phi). -Split; Assumption. -Assert H4 : (IsStepFun psi c b). -Apply StepFun_P45 with a. -Apply (pre psi). -Split; Assumption. -Split with (mkStepFun H3); Split with (mkStepFun H4); Split. -Simpl; Intros; Apply H. -Replace (Rmax a b) with (Rmax c b). -Elim H5; Intros; Split; Try Assumption. -Apply Rle_trans with (Rmin c b); Try Assumption. -Replace (Rmin a b) with a. -Replace (Rmin c b) with c. -Assumption. -Unfold Rmin; Case (total_order_Rle c b); Intro; [Reflexivity | Elim n; Assumption]. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. -Unfold Rmax; Case (total_order_Rle c b); Case (total_order_Rle a b); Intros; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption | Elim n; Assumption | Elim n0; Assumption]. -Rewrite Rabsolu_right. -Assert H5 : (IsStepFun psi a c). -Apply StepFun_P46 with b. -Apply (pre psi). -Apply StepFun_P6; Assumption. -Replace (RiemannInt_SF (mkStepFun H4)) with ``(RiemannInt_SF psi)-(RiemannInt_SF (mkStepFun H5))``. -Apply Rle_lt_trans with (RiemannInt_SF psi). -Unfold Rminus; Pattern 2 (RiemannInt_SF psi); Rewrite <- Rplus_Or; Apply Rle_compatibility; Rewrite <- Ropp_O; Apply Rge_Ropp; Apply Rle_sym1; Replace R0 with (RiemannInt_SF (mkStepFun (StepFun_P4 a c R0))). -Apply StepFun_P37; Try Assumption. -Intros; Simpl; Unfold fct_cte; Apply Rle_trans with ``(Rabsolu ((f x)-(phi x)))``. -Apply Rabsolu_pos. -Apply H. -Replace (Rmin a b) with a. -Replace (Rmax a b) with b. -Elim H6; Intros; Split; Left. -Assumption. -Apply Rlt_le_trans with c; Assumption. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. -Rewrite StepFun_P18; Ring. -Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF psi)). -Apply Rle_Rabsolu. -Assumption. -Assert H6 : (IsStepFun psi a b). -Apply (pre psi). -Replace (RiemannInt_SF psi) with (RiemannInt_SF (mkStepFun H6)). -Rewrite <- (StepFun_P43 H5 H4 H6); Ring. -Unfold RiemannInt_SF; Case (total_order_Rle a b); Intro. -EApply StepFun_P17. -Apply StepFun_P1. -Simpl; Apply StepFun_P1. -Apply eq_Ropp; EApply StepFun_P17. -Apply StepFun_P1. -Simpl; Apply StepFun_P1. -Apply Rle_sym1; Replace R0 with (RiemannInt_SF (mkStepFun (StepFun_P4 c b R0))). -Apply StepFun_P37; Try Assumption. -Intros; Simpl; Unfold fct_cte; Apply Rle_trans with ``(Rabsolu ((f x)-(phi x)))``. -Apply Rabsolu_pos. -Apply H. -Replace (Rmin a b) with a. -Replace (Rmax a b) with b. -Elim H5; Intros; Split; Left. -Apply Rle_lt_trans with c; Assumption. -Assumption. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Apply Rle_trans with c; Assumption]. -Rewrite StepFun_P18; Ring. +Lemma RiemannInt_P23 : + forall (f:R -> R) (a b c:R), + Riemann_integrable f a b -> a <= c <= b -> Riemann_integrable f c b. +unfold Riemann_integrable in |- *; intros; elim (X eps); clear X; + intros phi [psi H0]; elim H; elim H0; clear H H0; + intros; assert (H3 : IsStepFun phi c b). +apply StepFun_P45 with a. +apply (pre phi). +split; assumption. +assert (H4 : IsStepFun psi c b). +apply StepFun_P45 with a. +apply (pre psi). +split; assumption. +split with (mkStepFun H3); split with (mkStepFun H4); split. +simpl in |- *; intros; apply H. +replace (Rmax a b) with (Rmax c b). +elim H5; intros; split; try assumption. +apply Rle_trans with (Rmin c b); try assumption. +replace (Rmin a b) with a. +replace (Rmin c b) with c. +assumption. +unfold Rmin in |- *; case (Rle_dec c b); intro; + [ reflexivity | elim n; assumption ]. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; apply Rle_trans with c; assumption ]. +unfold Rmax in |- *; case (Rle_dec c b); case (Rle_dec a b); intros; + [ reflexivity + | elim n; apply Rle_trans with c; assumption + | elim n; assumption + | elim n0; assumption ]. +rewrite Rabs_right. +assert (H5 : IsStepFun psi a c). +apply StepFun_P46 with b. +apply (pre psi). +apply StepFun_P6; assumption. +replace (RiemannInt_SF (mkStepFun H4)) with + (RiemannInt_SF psi - RiemannInt_SF (mkStepFun H5)). +apply Rle_lt_trans with (RiemannInt_SF psi). +unfold Rminus in |- *; pattern (RiemannInt_SF psi) at 2 in |- *; + rewrite <- Rplus_0_r; apply Rplus_le_compat_l; rewrite <- Ropp_0; + apply Ropp_ge_le_contravar; apply Rle_ge; + replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 a c 0))). +apply StepFun_P37; try assumption. +intros; simpl in |- *; unfold fct_cte in |- *; + apply Rle_trans with (Rabs (f x - phi x)). +apply Rabs_pos. +apply H. +replace (Rmin a b) with a. +replace (Rmax a b) with b. +elim H6; intros; split; left. +assumption. +apply Rlt_le_trans with c; assumption. +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; apply Rle_trans with c; assumption ]. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; apply Rle_trans with c; assumption ]. +rewrite StepFun_P18; ring. +apply Rle_lt_trans with (Rabs (RiemannInt_SF psi)). +apply RRle_abs. +assumption. +assert (H6 : IsStepFun psi a b). +apply (pre psi). +replace (RiemannInt_SF psi) with (RiemannInt_SF (mkStepFun H6)). +rewrite <- (StepFun_P43 H5 H4 H6); ring. +unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro. +eapply StepFun_P17. +apply StepFun_P1. +simpl in |- *; apply StepFun_P1. +apply Ropp_eq_compat; eapply StepFun_P17. +apply StepFun_P1. +simpl in |- *; apply StepFun_P1. +apply Rle_ge; replace 0 with (RiemannInt_SF (mkStepFun (StepFun_P4 c b 0))). +apply StepFun_P37; try assumption. +intros; simpl in |- *; unfold fct_cte in |- *; + apply Rle_trans with (Rabs (f x - phi x)). +apply Rabs_pos. +apply H. +replace (Rmin a b) with a. +replace (Rmax a b) with b. +elim H5; intros; split; left. +apply Rle_lt_trans with c; assumption. +assumption. +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; apply Rle_trans with c; assumption ]. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; apply Rle_trans with c; assumption ]. +rewrite StepFun_P18; ring. Qed. -Lemma RiemannInt_P24 : (f:R->R;a,b,c:R) (Riemann_integrable f a b) -> (Riemann_integrable f b c) -> (Riemann_integrable f a c). -Intros; Case (total_order_Rle a b); Case (total_order_Rle b c); Intros. -Apply RiemannInt_P21 with b; Assumption. -Case (total_order_Rle a c); Intro. -Apply RiemannInt_P22 with b; Try Assumption. -Split; [Assumption | Auto with real]. -Apply RiemannInt_P1; Apply RiemannInt_P22 with b. -Apply RiemannInt_P1; Assumption. -Split; Auto with real. -Case (total_order_Rle a c); Intro. -Apply RiemannInt_P23 with b; Try Assumption. -Split; Auto with real. -Apply RiemannInt_P1; Apply RiemannInt_P23 with b. -Apply RiemannInt_P1; Assumption. -Split; [Assumption | Auto with real]. -Apply RiemannInt_P1; Apply RiemannInt_P21 with b; Auto with real Orelse Apply RiemannInt_P1; Assumption. +Lemma RiemannInt_P24 : + forall (f:R -> R) (a b c:R), + Riemann_integrable f a b -> + Riemann_integrable f b c -> Riemann_integrable f a c. +intros; case (Rle_dec a b); case (Rle_dec b c); intros. +apply RiemannInt_P21 with b; assumption. +case (Rle_dec a c); intro. +apply RiemannInt_P22 with b; try assumption. +split; [ assumption | auto with real ]. +apply RiemannInt_P1; apply RiemannInt_P22 with b. +apply RiemannInt_P1; assumption. +split; auto with real. +case (Rle_dec a c); intro. +apply RiemannInt_P23 with b; try assumption. +split; auto with real. +apply RiemannInt_P1; apply RiemannInt_P23 with b. +apply RiemannInt_P1; assumption. +split; [ assumption | auto with real ]. +apply RiemannInt_P1; apply RiemannInt_P21 with b; + auto with real || apply RiemannInt_P1; assumption. Qed. -Lemma RiemannInt_P25 : (f:R->R;a,b,c:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable f b c);pr3:(Riemann_integrable f a c)) ``a<=b``->``b<=c``->``(RiemannInt pr1)+(RiemannInt pr2)==(RiemannInt pr3)``. -Intros f a b c pr1 pr2 pr3 Hyp1 Hyp2; Unfold RiemannInt; Case (RiemannInt_exists 1!f 2!a 3!b pr1 5!RinvN RinvN_cv); Case (RiemannInt_exists 1!f 2!b 3!c pr2 5!RinvN RinvN_cv); Case (RiemannInt_exists 1!f 2!a 3!c pr3 5!RinvN RinvN_cv); Intros; Symmetry; EApply UL_sequence. -Apply u. -Unfold Un_cv; Intros; Assert H0 : ``0<eps/3``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Elim (u1 ? H0); Clear u1; Intros N1 H1; Elim (u0 ? H0); Clear u0; Intros N2 H2; Cut (Un_cv [n:nat]``(RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))`` R0). -Intro; Elim (H3 ? H0); Clear H3; Intros N3 H3; Pose N0 := (max (max N1 N2) N3); Exists N0; Intros; Unfold R_dist; Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))))+(Rabsolu (((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))-(x1+x0)))``. -Replace ``(RiemannInt_SF [(phi_sequence RinvN pr3 n)])-(x1+x0)`` with ``((RiemannInt_SF [(phi_sequence RinvN pr3 n)])-((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)])))+(((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))-(x1+x0))``; [Apply Rabsolu_triang | Ring]. -Replace eps with ``eps/3+eps/3+eps/3``. -Rewrite Rplus_assoc; Apply Rplus_lt. -Unfold R_dist in H3; Cut (ge n N3). -Intro; Assert H6 := (H3 ? H5); Unfold Rminus in H6; Rewrite Ropp_O in H6; Rewrite Rplus_Or in H6; Apply H6. -Unfold ge; Apply le_trans with N0; [Unfold N0; Apply le_max_r | Assumption]. -Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x1))+(Rabsolu ((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x0))``. -Replace ``((RiemannInt_SF [(phi_sequence RinvN pr1 n)])+(RiemannInt_SF [(phi_sequence RinvN pr2 n)]))-(x1+x0)`` with ``((RiemannInt_SF [(phi_sequence RinvN pr1 n)])-x1)+((RiemannInt_SF [(phi_sequence RinvN pr2 n)])-x0)``; [Apply Rabsolu_triang | Ring]. -Apply Rplus_lt. -Unfold R_dist in H1; Apply H1. -Unfold ge; Apply le_trans with N0; [Apply le_trans with (max N1 N2); [Apply le_max_l | Unfold N0; Apply le_max_l] | Assumption]. -Unfold R_dist in H2; Apply H2. -Unfold ge; Apply le_trans with N0; [Apply le_trans with (max N1 N2); [Apply le_max_r | Unfold N0; Apply le_max_l] | Assumption]. -Apply r_Rmult_mult with ``3``; [Unfold Rdiv; Repeat Rewrite Rmult_Rplus_distr; Do 2 Rewrite (Rmult_sym ``3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. -Clear x u x0 x1 eps H H0 N1 H1 N2 H2; Assert H1 : (EXT psi1:nat->(StepFun a b) | (n:nat) ((t:R)``(Rmin a b) <= t``/\``t <= (Rmax a b)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr1 n)] t)))<= (psi1 n t)``)/\``(Rabsolu (RiemannInt_SF (psi1 n))) < (RinvN n)``). -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr1 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr1 n)). -Assert H2 : (EXT psi2:nat->(StepFun b c) | (n:nat) ((t:R)``(Rmin b c) <= t``/\``t <= (Rmax b c)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr2 n)] t)))<= (psi2 n t)``)/\``(Rabsolu (RiemannInt_SF (psi2 n))) < (RinvN n)``). -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr2 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr2 n)). -Assert H3 : (EXT psi3:nat->(StepFun a c) | (n:nat) ((t:R)``(Rmin a c) <= t``/\``t <= (Rmax a c)``->``(Rabsolu ((f t)-([(phi_sequence RinvN pr3 n)] t)))<= (psi3 n t)``)/\``(Rabsolu (RiemannInt_SF (psi3 n))) < (RinvN n)``). -Split with [n:nat](projT1 ? ? (phi_sequence_prop RinvN pr3 n)); Intro; Apply (projT2 ? ? (phi_sequence_prop RinvN pr3 n)). -Elim H1; Clear H1; Intros psi1 H1; Elim H2; Clear H2; Intros psi2 H2; Elim H3; Clear H3; Intros psi3 H3; Assert H := RinvN_cv; Unfold Un_cv; Intros; Assert H4 : ``0<eps/3``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Elim (H ? H4); Clear H; Intros N0 H; Assert H5 : (n:nat)(ge n N0)->``(RinvN n)<eps/3``. -Intros; Replace (pos (RinvN n)) with ``(R_dist (mkposreal (/((INR n)+1)) (RinvN_pos n)) 0)``. -Apply H; Assumption. -Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rabsolu_right; Apply Rle_sym1; Left; Apply (cond_pos (RinvN n)). -Exists N0; Intros; Elim (H1 n); Elim (H2 n); Elim (H3 n); Clear H1 H2 H3; Intros; Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Pose phi1 := (phi_sequence RinvN pr1 n); Fold phi1 in H8; Pose phi2 := (phi_sequence RinvN pr2 n); Fold phi2 in H3; Pose phi3 := (phi_sequence RinvN pr3 n); Fold phi2 in H1; Assert H10 : (IsStepFun phi3 a b). -Apply StepFun_P44 with c. -Apply (pre phi3). -Split; Assumption. -Assert H11 : (IsStepFun (psi3 n) a b). -Apply StepFun_P44 with c. -Apply (pre (psi3 n)). -Split; Assumption. -Assert H12 : (IsStepFun phi3 b c). -Apply StepFun_P45 with a. -Apply (pre phi3). -Split; Assumption. -Assert H13 : (IsStepFun (psi3 n) b c). -Apply StepFun_P45 with a. -Apply (pre (psi3 n)). -Split; Assumption. -Replace (RiemannInt_SF phi3) with ``(RiemannInt_SF (mkStepFun H10))+(RiemannInt_SF (mkStepFun H12))``. -Apply Rle_lt_trans with ``(Rabsolu ((RiemannInt_SF (mkStepFun H10))-(RiemannInt_SF phi1)))+(Rabsolu ((RiemannInt_SF (mkStepFun H12))-(RiemannInt_SF phi2)))``. -Replace ``(RiemannInt_SF (mkStepFun H10))+(RiemannInt_SF (mkStepFun H12))+ -((RiemannInt_SF phi1)+(RiemannInt_SF phi2))`` with ``((RiemannInt_SF (mkStepFun H10))-(RiemannInt_SF phi1))+((RiemannInt_SF (mkStepFun H12))-(RiemannInt_SF phi2))``; [Apply Rabsolu_triang | Ring]. -Replace ``(RiemannInt_SF (mkStepFun H10))-(RiemannInt_SF phi1)`` with (RiemannInt_SF (mkStepFun (StepFun_P28 ``-1`` (mkStepFun H10) phi1))). -Replace ``(RiemannInt_SF (mkStepFun H12))-(RiemannInt_SF phi2)`` with (RiemannInt_SF (mkStepFun (StepFun_P28 ``-1`` (mkStepFun H12) phi2))). -Apply Rle_lt_trans with ``(RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1)))))+(RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H12) phi2)))))``. -Apply Rle_trans with ``(Rabsolu (RiemannInt_SF (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1))))+(RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H12) phi2)))))``. -Apply Rle_compatibility. -Apply StepFun_P34; Try Assumption. -Do 2 Rewrite <- (Rplus_sym (RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 ``-1`` (mkStepFun H12) phi2)))))); Apply Rle_compatibility; Apply StepFun_P34; Try Assumption. -Apply Rle_lt_trans with ``(RiemannInt_SF (mkStepFun (StepFun_P28 R1 (mkStepFun H11) (psi1 n))))+(RiemannInt_SF (mkStepFun (StepFun_P28 R1 (mkStepFun H13) (psi2 n))))``. -Apply Rle_trans with ``(RiemannInt_SF (mkStepFun (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1)))))+(RiemannInt_SF (mkStepFun (StepFun_P28 R1 (mkStepFun H13) (psi2 n))))``. -Apply Rle_compatibility; Apply StepFun_P37; Try Assumption. -Intros; Simpl; Rewrite Rmult_1l; Apply Rle_trans with ``(Rabsolu ((f x)-(phi3 x)))+(Rabsolu ((f x)-(phi2 x)))``. -Rewrite <- (Rabsolu_Ropp ``(f x)-(phi3 x)``); Rewrite Ropp_distr2; Replace ``(phi3 x)+ -1*(phi2 x)`` with ``((phi3 x)-(f x))+((f x)-(phi2 x))``; [Apply Rabsolu_triang | Ring]. -Apply Rplus_le. -Fold phi3 in H1; Apply H1. -Elim H14; Intros; Split. -Replace (Rmin a c) with a. -Apply Rle_trans with b; Try Assumption. -Left; Assumption. -Unfold Rmin; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n0; Apply Rle_trans with b; Assumption]. -Replace (Rmax a c) with c. -Left; Assumption. -Unfold Rmax; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n0; Apply Rle_trans with b; Assumption]. -Apply H3. -Elim H14; Intros; Split. -Replace (Rmin b c) with b. -Left; Assumption. -Unfold Rmin; Case (total_order_Rle b c); Intro; [Reflexivity | Elim n0; Assumption]. -Replace (Rmax b c) with c. -Left; Assumption. -Unfold Rmax; Case (total_order_Rle b c); Intro; [Reflexivity | Elim n0; Assumption]. -Do 2 Rewrite <- (Rplus_sym ``(RiemannInt_SF (mkStepFun (StepFun_P28 R1 (mkStepFun H13) (psi2 n))))``); Apply Rle_compatibility; Apply StepFun_P37; Try Assumption. -Intros; Simpl; Rewrite Rmult_1l; Apply Rle_trans with ``(Rabsolu ((f x)-(phi3 x)))+(Rabsolu ((f x)-(phi1 x)))``. -Rewrite <- (Rabsolu_Ropp ``(f x)-(phi3 x)``); Rewrite Ropp_distr2; Replace ``(phi3 x)+ -1*(phi1 x)`` with ``((phi3 x)-(f x))+((f x)-(phi1 x))``; [Apply Rabsolu_triang | Ring]. -Apply Rplus_le. -Apply H1. -Elim H14; Intros; Split. -Replace (Rmin a c) with a. -Left; Assumption. -Unfold Rmin; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n0; Apply Rle_trans with b; Assumption]. -Replace (Rmax a c) with c. -Apply Rle_trans with b. -Left; Assumption. -Assumption. -Unfold Rmax; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n0; Apply Rle_trans with b; Assumption]. -Apply H8. -Elim H14; Intros; Split. -Replace (Rmin a b) with a. -Left; Assumption. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Replace (Rmax a b) with b. -Left; Assumption. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n0; Assumption]. -Do 2 Rewrite StepFun_P30. -Do 2 Rewrite Rmult_1l; Replace ``(RiemannInt_SF (mkStepFun H11))+(RiemannInt_SF (psi1 n))+((RiemannInt_SF (mkStepFun H13))+(RiemannInt_SF (psi2 n)))`` with ``(RiemannInt_SF (psi3 n))+(RiemannInt_SF (psi1 n))+(RiemannInt_SF (psi2 n))``. -Replace eps with ``eps/3+eps/3+eps/3``. -Repeat Rewrite Rplus_assoc; Repeat Apply Rplus_lt. -Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi3 n))). -Apply Rle_Rabsolu. -Apply Rlt_trans with (pos (RinvN n)). -Assumption. -Apply H5; Assumption. -Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi1 n))). -Apply Rle_Rabsolu. -Apply Rlt_trans with (pos (RinvN n)). -Assumption. -Apply H5; Assumption. -Apply Rle_lt_trans with (Rabsolu (RiemannInt_SF (psi2 n))). -Apply Rle_Rabsolu. -Apply Rlt_trans with (pos (RinvN n)). -Assumption. -Apply H5; Assumption. -Apply r_Rmult_mult with ``3``; [Unfold Rdiv; Repeat Rewrite Rmult_Rplus_distr; Do 2 Rewrite (Rmult_sym ``3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | DiscrR] | DiscrR]. -Replace (RiemannInt_SF (psi3 n)) with (RiemannInt_SF (mkStepFun (pre (psi3 n)))). -Rewrite <- (StepFun_P43 H11 H13 (pre (psi3 n))); Ring. -Reflexivity. -Rewrite StepFun_P30; Ring. -Rewrite StepFun_P30; Ring. -Apply (StepFun_P43 H10 H12 (pre phi3)). +Lemma RiemannInt_P25 : + forall (f:R -> R) (a b c:R) (pr1:Riemann_integrable f a b) + (pr2:Riemann_integrable f b c) (pr3:Riemann_integrable f a c), + a <= b -> b <= c -> RiemannInt pr1 + RiemannInt pr2 = RiemannInt pr3. +intros f a b c pr1 pr2 pr3 Hyp1 Hyp2; unfold RiemannInt in |- *; + case (RiemannInt_exists pr1 RinvN RinvN_cv); + case (RiemannInt_exists pr2 RinvN RinvN_cv); + case (RiemannInt_exists pr3 RinvN RinvN_cv); intros; + symmetry in |- *; eapply UL_sequence. +apply u. +unfold Un_cv in |- *; intros; assert (H0 : 0 < eps / 3). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (u1 _ H0); clear u1; intros N1 H1; elim (u0 _ H0); clear u0; + intros N2 H2; + cut + (Un_cv + (fun n:nat => + RiemannInt_SF (phi_sequence RinvN pr3 n) - + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + RiemannInt_SF (phi_sequence RinvN pr2 n))) 0). +intro; elim (H3 _ H0); clear H3; intros N3 H3; + pose (N0 := max (max N1 N2) N3); exists N0; intros; + unfold R_dist in |- *; + apply Rle_lt_trans with + (Rabs + (RiemannInt_SF (phi_sequence RinvN pr3 n) - + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + RiemannInt_SF (phi_sequence RinvN pr2 n))) + + Rabs + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + RiemannInt_SF (phi_sequence RinvN pr2 n) - (x1 + x0))). +replace (RiemannInt_SF (phi_sequence RinvN pr3 n) - (x1 + x0)) with + (RiemannInt_SF (phi_sequence RinvN pr3 n) - + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + RiemannInt_SF (phi_sequence RinvN pr2 n)) + + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + RiemannInt_SF (phi_sequence RinvN pr2 n) - (x1 + x0))); + [ apply Rabs_triang | ring ]. +replace eps with (eps / 3 + eps / 3 + eps / 3). +rewrite Rplus_assoc; apply Rplus_lt_compat. +unfold R_dist in H3; cut (n >= N3)%nat. +intro; assert (H6 := H3 _ H5); unfold Rminus in H6; rewrite Ropp_0 in H6; + rewrite Rplus_0_r in H6; apply H6. +unfold ge in |- *; apply le_trans with N0; + [ unfold N0 in |- *; apply le_max_r | assumption ]. +apply Rle_lt_trans with + (Rabs (RiemannInt_SF (phi_sequence RinvN pr1 n) - x1) + + Rabs (RiemannInt_SF (phi_sequence RinvN pr2 n) - x0)). +replace + (RiemannInt_SF (phi_sequence RinvN pr1 n) + + RiemannInt_SF (phi_sequence RinvN pr2 n) - (x1 + x0)) with + (RiemannInt_SF (phi_sequence RinvN pr1 n) - x1 + + (RiemannInt_SF (phi_sequence RinvN pr2 n) - x0)); + [ apply Rabs_triang | ring ]. +apply Rplus_lt_compat. +unfold R_dist in H1; apply H1. +unfold ge in |- *; apply le_trans with N0; + [ apply le_trans with (max N1 N2); + [ apply le_max_l | unfold N0 in |- *; apply le_max_l ] + | assumption ]. +unfold R_dist in H2; apply H2. +unfold ge in |- *; apply le_trans with N0; + [ apply le_trans with (max N1 N2); + [ apply le_max_r | unfold N0 in |- *; apply le_max_l ] + | assumption ]. +apply Rmult_eq_reg_l with 3; + [ unfold Rdiv in |- *; repeat rewrite Rmult_plus_distr_l; + do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym; [ ring | discrR ] + | discrR ]. +clear x u x0 x1 eps H H0 N1 H1 N2 H2; + assert + (H1 : + exists psi1 : nat -> StepFun a b + | (forall n:nat, + (forall t:R, + Rmin a b <= t /\ t <= Rmax a b -> + Rabs (f t - phi_sequence RinvN pr1 n t) <= psi1 n t) /\ + Rabs (RiemannInt_SF (psi1 n)) < RinvN n)). +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr1 n)); intro; + apply (projT2 (phi_sequence_prop RinvN pr1 n)). +assert + (H2 : + exists psi2 : nat -> StepFun b c + | (forall n:nat, + (forall t:R, + Rmin b c <= t /\ t <= Rmax b c -> + Rabs (f t - phi_sequence RinvN pr2 n t) <= psi2 n t) /\ + Rabs (RiemannInt_SF (psi2 n)) < RinvN n)). +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr2 n)); intro; + apply (projT2 (phi_sequence_prop RinvN pr2 n)). +assert + (H3 : + exists psi3 : nat -> StepFun a c + | (forall n:nat, + (forall t:R, + Rmin a c <= t /\ t <= Rmax a c -> + Rabs (f t - phi_sequence RinvN pr3 n t) <= psi3 n t) /\ + Rabs (RiemannInt_SF (psi3 n)) < RinvN n)). +split with (fun n:nat => projT1 (phi_sequence_prop RinvN pr3 n)); intro; + apply (projT2 (phi_sequence_prop RinvN pr3 n)). +elim H1; clear H1; intros psi1 H1; elim H2; clear H2; intros psi2 H2; elim H3; + clear H3; intros psi3 H3; assert (H := RinvN_cv); + unfold Un_cv in |- *; intros; assert (H4 : 0 < eps / 3). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (H _ H4); clear H; intros N0 H; + assert (H5 : forall n:nat, (n >= N0)%nat -> RinvN n < eps / 3). +intros; + replace (pos (RinvN n)) with + (R_dist (mkposreal (/ (INR n + 1)) (RinvN_pos n)) 0). +apply H; assumption. +unfold R_dist in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; apply Rabs_right; apply Rle_ge; + left; apply (cond_pos (RinvN n)). +exists N0; intros; elim (H1 n); elim (H2 n); elim (H3 n); clear H1 H2 H3; + intros; unfold R_dist in |- *; unfold Rminus in |- *; + rewrite Ropp_0; rewrite Rplus_0_r; pose (phi1 := phi_sequence RinvN pr1 n); + fold phi1 in H8; pose (phi2 := phi_sequence RinvN pr2 n); + fold phi2 in H3; pose (phi3 := phi_sequence RinvN pr3 n); + fold phi2 in H1; assert (H10 : IsStepFun phi3 a b). +apply StepFun_P44 with c. +apply (pre phi3). +split; assumption. +assert (H11 : IsStepFun (psi3 n) a b). +apply StepFun_P44 with c. +apply (pre (psi3 n)). +split; assumption. +assert (H12 : IsStepFun phi3 b c). +apply StepFun_P45 with a. +apply (pre phi3). +split; assumption. +assert (H13 : IsStepFun (psi3 n) b c). +apply StepFun_P45 with a. +apply (pre (psi3 n)). +split; assumption. +replace (RiemannInt_SF phi3) with + (RiemannInt_SF (mkStepFun H10) + RiemannInt_SF (mkStepFun H12)). +apply Rle_lt_trans with + (Rabs (RiemannInt_SF (mkStepFun H10) - RiemannInt_SF phi1) + + Rabs (RiemannInt_SF (mkStepFun H12) - RiemannInt_SF phi2)). +replace + (RiemannInt_SF (mkStepFun H10) + RiemannInt_SF (mkStepFun H12) + + - (RiemannInt_SF phi1 + RiemannInt_SF phi2)) with + (RiemannInt_SF (mkStepFun H10) - RiemannInt_SF phi1 + + (RiemannInt_SF (mkStepFun H12) - RiemannInt_SF phi2)); + [ apply Rabs_triang | ring ]. +replace (RiemannInt_SF (mkStepFun H10) - RiemannInt_SF phi1) with + (RiemannInt_SF (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1))). +replace (RiemannInt_SF (mkStepFun H12) - RiemannInt_SF phi2) with + (RiemannInt_SF (mkStepFun (StepFun_P28 (-1) (mkStepFun H12) phi2))). +apply Rle_lt_trans with + (RiemannInt_SF + (mkStepFun + (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1)))) + + RiemannInt_SF + (mkStepFun + (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H12) phi2))))). +apply Rle_trans with + (Rabs (RiemannInt_SF (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1))) + + RiemannInt_SF + (mkStepFun + (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H12) phi2))))). +apply Rplus_le_compat_l. +apply StepFun_P34; try assumption. +do 2 + rewrite <- + (Rplus_comm + (RiemannInt_SF + (mkStepFun + (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H12) phi2)))))) + ; apply Rplus_le_compat_l; apply StepFun_P34; try assumption. +apply Rle_lt_trans with + (RiemannInt_SF (mkStepFun (StepFun_P28 1 (mkStepFun H11) (psi1 n))) + + RiemannInt_SF (mkStepFun (StepFun_P28 1 (mkStepFun H13) (psi2 n)))). +apply Rle_trans with + (RiemannInt_SF + (mkStepFun + (StepFun_P32 (mkStepFun (StepFun_P28 (-1) (mkStepFun H10) phi1)))) + + RiemannInt_SF (mkStepFun (StepFun_P28 1 (mkStepFun H13) (psi2 n)))). +apply Rplus_le_compat_l; apply StepFun_P37; try assumption. +intros; simpl in |- *; rewrite Rmult_1_l; + apply Rle_trans with (Rabs (f x - phi3 x) + Rabs (f x - phi2 x)). +rewrite <- (Rabs_Ropp (f x - phi3 x)); rewrite Ropp_minus_distr; + replace (phi3 x + -1 * phi2 x) with (phi3 x - f x + (f x - phi2 x)); + [ apply Rabs_triang | ring ]. +apply Rplus_le_compat. +fold phi3 in H1; apply H1. +elim H14; intros; split. +replace (Rmin a c) with a. +apply Rle_trans with b; try assumption. +left; assumption. +unfold Rmin in |- *; case (Rle_dec a c); intro; + [ reflexivity | elim n0; apply Rle_trans with b; assumption ]. +replace (Rmax a c) with c. +left; assumption. +unfold Rmax in |- *; case (Rle_dec a c); intro; + [ reflexivity | elim n0; apply Rle_trans with b; assumption ]. +apply H3. +elim H14; intros; split. +replace (Rmin b c) with b. +left; assumption. +unfold Rmin in |- *; case (Rle_dec b c); intro; + [ reflexivity | elim n0; assumption ]. +replace (Rmax b c) with c. +left; assumption. +unfold Rmax in |- *; case (Rle_dec b c); intro; + [ reflexivity | elim n0; assumption ]. +do 2 + rewrite <- + (Rplus_comm + (RiemannInt_SF (mkStepFun (StepFun_P28 1 (mkStepFun H13) (psi2 n))))) + ; apply Rplus_le_compat_l; apply StepFun_P37; try assumption. +intros; simpl in |- *; rewrite Rmult_1_l; + apply Rle_trans with (Rabs (f x - phi3 x) + Rabs (f x - phi1 x)). +rewrite <- (Rabs_Ropp (f x - phi3 x)); rewrite Ropp_minus_distr; + replace (phi3 x + -1 * phi1 x) with (phi3 x - f x + (f x - phi1 x)); + [ apply Rabs_triang | ring ]. +apply Rplus_le_compat. +apply H1. +elim H14; intros; split. +replace (Rmin a c) with a. +left; assumption. +unfold Rmin in |- *; case (Rle_dec a c); intro; + [ reflexivity | elim n0; apply Rle_trans with b; assumption ]. +replace (Rmax a c) with c. +apply Rle_trans with b. +left; assumption. +assumption. +unfold Rmax in |- *; case (Rle_dec a c); intro; + [ reflexivity | elim n0; apply Rle_trans with b; assumption ]. +apply H8. +elim H14; intros; split. +replace (Rmin a b) with a. +left; assumption. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +replace (Rmax a b) with b. +left; assumption. +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n0; assumption ]. +do 2 rewrite StepFun_P30. +do 2 rewrite Rmult_1_l; + replace + (RiemannInt_SF (mkStepFun H11) + RiemannInt_SF (psi1 n) + + (RiemannInt_SF (mkStepFun H13) + RiemannInt_SF (psi2 n))) with + (RiemannInt_SF (psi3 n) + RiemannInt_SF (psi1 n) + RiemannInt_SF (psi2 n)). +replace eps with (eps / 3 + eps / 3 + eps / 3). +repeat rewrite Rplus_assoc; repeat apply Rplus_lt_compat. +apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi3 n))). +apply RRle_abs. +apply Rlt_trans with (pos (RinvN n)). +assumption. +apply H5; assumption. +apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi1 n))). +apply RRle_abs. +apply Rlt_trans with (pos (RinvN n)). +assumption. +apply H5; assumption. +apply Rle_lt_trans with (Rabs (RiemannInt_SF (psi2 n))). +apply RRle_abs. +apply Rlt_trans with (pos (RinvN n)). +assumption. +apply H5; assumption. +apply Rmult_eq_reg_l with 3; + [ unfold Rdiv in |- *; repeat rewrite Rmult_plus_distr_l; + do 2 rewrite (Rmult_comm 3); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym; [ ring | discrR ] + | discrR ]. +replace (RiemannInt_SF (psi3 n)) with + (RiemannInt_SF (mkStepFun (pre (psi3 n)))). +rewrite <- (StepFun_P43 H11 H13 (pre (psi3 n))); ring. +reflexivity. +rewrite StepFun_P30; ring. +rewrite StepFun_P30; ring. +apply (StepFun_P43 H10 H12 (pre phi3)). Qed. -Lemma RiemannInt_P26 : (f:R->R;a,b,c:R;pr1:(Riemann_integrable f a b);pr2:(Riemann_integrable f b c);pr3:(Riemann_integrable f a c)) ``(RiemannInt pr1)+(RiemannInt pr2)==(RiemannInt pr3)``. -Intros; Case (total_order_Rle a b); Case (total_order_Rle b c); Intros. -Apply RiemannInt_P25; Assumption. -Case (total_order_Rle a c); Intro. -Assert H : ``c<=b``. -Auto with real. -Rewrite <- (RiemannInt_P25 pr3 (RiemannInt_P1 pr2) pr1 r0 H); Rewrite (RiemannInt_P8 pr2 (RiemannInt_P1 pr2)); Ring. -Assert H : ``c<=a``. -Auto with real. -Rewrite (RiemannInt_P8 pr2 (RiemannInt_P1 pr2)); Rewrite <- (RiemannInt_P25 (RiemannInt_P1 pr3) pr1 (RiemannInt_P1 pr2) H r); Rewrite (RiemannInt_P8 pr3 (RiemannInt_P1 pr3)); Ring. -Assert H : ``b<=a``. -Auto with real. -Case (total_order_Rle a c); Intro. -Rewrite <- (RiemannInt_P25 (RiemannInt_P1 pr1) pr3 pr2 H r0); Rewrite (RiemannInt_P8 pr1 (RiemannInt_P1 pr1)); Ring. -Assert H0 : ``c<=a``. -Auto with real. -Rewrite (RiemannInt_P8 pr1 (RiemannInt_P1 pr1)); Rewrite <- (RiemannInt_P25 pr2 (RiemannInt_P1 pr3) (RiemannInt_P1 pr1) r H0); Rewrite (RiemannInt_P8 pr3 (RiemannInt_P1 pr3)); Ring. -Rewrite (RiemannInt_P8 pr1 (RiemannInt_P1 pr1)); Rewrite (RiemannInt_P8 pr2 (RiemannInt_P1 pr2)); Rewrite (RiemannInt_P8 pr3 (RiemannInt_P1 pr3)); Rewrite <- (RiemannInt_P25 (RiemannInt_P1 pr2) (RiemannInt_P1 pr1) (RiemannInt_P1 pr3)); [Ring | Auto with real | Auto with real]. +Lemma RiemannInt_P26 : + forall (f:R -> R) (a b c:R) (pr1:Riemann_integrable f a b) + (pr2:Riemann_integrable f b c) (pr3:Riemann_integrable f a c), + RiemannInt pr1 + RiemannInt pr2 = RiemannInt pr3. +intros; case (Rle_dec a b); case (Rle_dec b c); intros. +apply RiemannInt_P25; assumption. +case (Rle_dec a c); intro. +assert (H : c <= b). +auto with real. +rewrite <- (RiemannInt_P25 pr3 (RiemannInt_P1 pr2) pr1 r0 H); + rewrite (RiemannInt_P8 pr2 (RiemannInt_P1 pr2)); ring. +assert (H : c <= a). +auto with real. +rewrite (RiemannInt_P8 pr2 (RiemannInt_P1 pr2)); + rewrite <- (RiemannInt_P25 (RiemannInt_P1 pr3) pr1 (RiemannInt_P1 pr2) H r); + rewrite (RiemannInt_P8 pr3 (RiemannInt_P1 pr3)); ring. +assert (H : b <= a). +auto with real. +case (Rle_dec a c); intro. +rewrite <- (RiemannInt_P25 (RiemannInt_P1 pr1) pr3 pr2 H r0); + rewrite (RiemannInt_P8 pr1 (RiemannInt_P1 pr1)); ring. +assert (H0 : c <= a). +auto with real. +rewrite (RiemannInt_P8 pr1 (RiemannInt_P1 pr1)); + rewrite <- (RiemannInt_P25 pr2 (RiemannInt_P1 pr3) (RiemannInt_P1 pr1) r H0); + rewrite (RiemannInt_P8 pr3 (RiemannInt_P1 pr3)); ring. +rewrite (RiemannInt_P8 pr1 (RiemannInt_P1 pr1)); + rewrite (RiemannInt_P8 pr2 (RiemannInt_P1 pr2)); + rewrite (RiemannInt_P8 pr3 (RiemannInt_P1 pr3)); + rewrite <- + (RiemannInt_P25 (RiemannInt_P1 pr2) (RiemannInt_P1 pr1) (RiemannInt_P1 pr3)) + ; [ ring | auto with real | auto with real ]. Qed. -Lemma RiemannInt_P27 : (f:R->R;a,b,x:R;h:``a<=b``;C0:((x:R)``a<=x<=b``->(continuity_pt f x))) ``a<x<b`` -> (derivable_pt_lim (primitive h (FTC_P1 h C0)) x (f x)). -Intro f; Intros; Elim H; Clear H; Intros; Assert H1 : (continuity_pt f x). -Apply C0; Split; Left; Assumption. -Unfold derivable_pt_lim; Intros; Assert Hyp : ``0<eps/2``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Elim (H1 ? Hyp); Unfold dist D_x no_cond; Simpl; Unfold R_dist; Intros; Pose del := (Rmin x0 (Rmin ``b-x`` ``x-a``)); Assert H4 : ``0<del``. -Unfold del; Unfold Rmin; Case (total_order_Rle ``b-x`` ``x-a``); Intro. -Case (total_order_Rle x0 ``b-x``); Intro; [Elim H3; Intros; Assumption | Apply Rlt_Rminus; Assumption]. -Case (total_order_Rle x0 ``x-a``); Intro; [Elim H3; Intros; Assumption | Apply Rlt_Rminus; Assumption]. -Split with (mkposreal ? H4); Intros; Assert H7 : (Riemann_integrable f x ``x+h0``). -Case (total_order_Rle x ``x+h0``); Intro. -Apply continuity_implies_RiemannInt; Try Assumption. -Intros; Apply C0; Elim H7; Intros; Split. -Apply Rle_trans with x; [Left; Assumption | Assumption]. -Apply Rle_trans with ``x+h0``. -Assumption. -Left; Apply Rlt_le_trans with ``x+del``. -Apply Rlt_compatibility; Apply Rle_lt_trans with (Rabsolu h0); [Apply Rle_Rabsolu | Apply H6]. -Unfold del; Apply Rle_trans with ``x+(Rmin (b-x) (x-a))``. -Apply Rle_compatibility; Apply Rmin_r. -Pattern 2 b; Replace b with ``x+(b-x)``; [Apply Rle_compatibility; Apply Rmin_l | Ring]. -Apply RiemannInt_P1; Apply continuity_implies_RiemannInt; Auto with real. -Intros; Apply C0; Elim H7; Intros; Split. -Apply Rle_trans with ``x+h0``. -Left; Apply Rle_lt_trans with ``x-del``. -Unfold del; Apply Rle_trans with ``x-(Rmin (b-x) (x-a))``. -Pattern 1 a; Replace a with ``x+(a-x)``; [Idtac | Ring]. -Unfold Rminus; Apply Rle_compatibility; Apply Ropp_Rle. -Rewrite Ropp_Ropp; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Rewrite (Rplus_sym x); Apply Rmin_r. -Unfold Rminus; Apply Rle_compatibility; Apply Ropp_Rle. -Do 2 Rewrite Ropp_Ropp; Apply Rmin_r. -Unfold Rminus; Apply Rlt_compatibility; Apply Ropp_Rlt. -Rewrite Ropp_Ropp; Apply Rle_lt_trans with (Rabsolu h0); [Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu | Apply H6]. -Assumption. -Apply Rle_trans with x; [Assumption | Left; Assumption]. -Replace ``(primitive h (FTC_P1 h C0) (x+h0))-(primitive h (FTC_P1 h C0) x)`` with (RiemannInt H7). -Replace (f x) with ``(RiemannInt (RiemannInt_P14 x (x+h0) (f x)))/h0``. -Replace ``(RiemannInt H7)/h0-(RiemannInt (RiemannInt_P14 x (x+h0) (f x)))/h0`` with ``((RiemannInt H7)-(RiemannInt (RiemannInt_P14 x (x+h0) (f x))))/h0``. -Replace ``(RiemannInt H7)-(RiemannInt (RiemannInt_P14 x (x+h0) (f x)))`` with (RiemannInt (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x)))). -Unfold Rdiv; Rewrite Rabsolu_mult; Case (total_order_Rle x ``x+h0``); Intro. -Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P16 (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x+h0) (f x)))))*(Rabsolu (/h0))``. -Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. -Apply Rabsolu_pos. -Apply (RiemannInt_P17 (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x))) (RiemannInt_P16 (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x))))); Assumption. -Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P14 x (x+h0) (eps/2)))*(Rabsolu (/h0))``. -Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. -Apply Rabsolu_pos. -Apply RiemannInt_P19; Try Assumption. -Intros; Replace ``(f x1)+ -1*(fct_cte (f x) x1)`` with ``(f x1)-(f x)``. -Unfold fct_cte; Case (Req_EM x x1); Intro. -Rewrite H9; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Left; Assumption. -Elim H3; Intros; Left; Apply H11. -Repeat Split. -Assumption. -Rewrite Rabsolu_right. -Apply Rlt_anti_compatibility with x; Replace ``x+(x1-x)`` with x1; [Idtac | Ring]. -Apply Rlt_le_trans with ``x+h0``. -Elim H8; Intros; Assumption. -Apply Rle_compatibility; Apply Rle_trans with del. -Left; Apply Rle_lt_trans with (Rabsolu h0); [Apply Rle_Rabsolu | Assumption]. -Unfold del; Apply Rmin_l. -Apply Rge_minus; Apply Rle_sym1; Left; Elim H8; Intros; Assumption. -Unfold fct_cte; Ring. -Rewrite RiemannInt_P15. -Rewrite Rmult_assoc; Replace ``(x+h0-x)*(Rabsolu (/h0))`` with R1. -Rewrite Rmult_1r; Unfold Rdiv; Apply Rlt_monotony_contra with ``2``; [Sup0 | Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Pattern 1 eps; Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Rewrite Rabsolu_right. -Replace ``x+h0-x`` with h0; [Idtac | Ring]. -Apply Rinv_r_sym. -Assumption. -Apply Rle_sym1; Left; Apply Rlt_Rinv. -Elim r; Intro. -Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or; Assumption. -Elim H5; Symmetry; Apply r_Rplus_plus with x; Rewrite Rplus_Or; Assumption. -Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P16 (RiemannInt_P1 (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x+h0) (f x))))))*(Rabsolu (/h0))``. -Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. -Apply Rabsolu_pos. -Replace (RiemannInt (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x)))) with ``-(RiemannInt (RiemannInt_P1 (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x+h0) (f x)))))``. -Rewrite Rabsolu_Ropp; Apply (RiemannInt_P17 (RiemannInt_P1 (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x)))) (RiemannInt_P16 (RiemannInt_P1 (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x)))))); Auto with real. -Symmetry; Apply RiemannInt_P8. -Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P14 (x+h0) x (eps/2)))*(Rabsolu (/h0))``. -Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. -Apply Rabsolu_pos. -Apply RiemannInt_P19. -Auto with real. -Intros; Replace ``(f x1)+ -1*(fct_cte (f x) x1)`` with ``(f x1)-(f x)``. -Unfold fct_cte; Case (Req_EM x x1); Intro. -Rewrite H9; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Left; Assumption. -Elim H3; Intros; Left; Apply H11. -Repeat Split. -Assumption. -Rewrite Rabsolu_left. -Apply Rlt_anti_compatibility with ``x1-x0``; Replace ``x1-x0+x0`` with x1; [Idtac | Ring]. -Replace ``x1-x0+ -(x1-x)`` with ``x-x0``; [Idtac | Ring]. -Apply Rle_lt_trans with ``x+h0``. -Unfold Rminus; Apply Rle_compatibility; Apply Ropp_Rle. -Rewrite Ropp_Ropp; Apply Rle_trans with (Rabsolu h0). -Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. -Apply Rle_trans with del; [Left; Assumption | Unfold del; Apply Rmin_l]. -Elim H8; Intros; Assumption. -Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or; Replace ``x+(x1-x)`` with x1; [Elim H8; Intros; Assumption | Ring]. -Unfold fct_cte; Ring. -Rewrite RiemannInt_P15. -Rewrite Rmult_assoc; Replace ``(x-(x+h0))*(Rabsolu (/h0))`` with R1. -Rewrite Rmult_1r; Unfold Rdiv; Apply Rlt_monotony_contra with ``2``; [Sup0 | Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Pattern 1 eps; Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Rewrite Rabsolu_left. -Replace ``x-(x+h0)`` with ``-h0``; [Idtac | Ring]. -Rewrite Ropp_mul1; Rewrite Ropp_mul3; Rewrite Ropp_Ropp; Apply Rinv_r_sym. -Assumption. -Apply Rlt_Rinv2. -Assert H8 : ``x+h0<x``. -Auto with real. -Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or; Assumption. -Rewrite (RiemannInt_P13 H7 (RiemannInt_P14 x ``x+h0`` (f x)) (RiemannInt_P10 ``-1`` H7 (RiemannInt_P14 x ``x+h0`` (f x)))). -Ring. -Unfold Rdiv Rminus; Rewrite Rmult_Rplus_distrl; Ring. -Rewrite RiemannInt_P15; Apply r_Rmult_mult with h0; [Unfold Rdiv; Rewrite -> (Rmult_sym h0); Repeat Rewrite -> Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | Assumption] | Assumption]. -Cut ``a<=x+h0``. -Cut ``x+h0<=b``. -Intros; Unfold primitive. -Case (total_order_Rle a ``x+h0``); Case (total_order_Rle ``x+h0`` b); Case (total_order_Rle a x); Case (total_order_Rle x b); Intros; Try (Elim n; Assumption Orelse Left; Assumption). -Rewrite <- (RiemannInt_P26 (FTC_P1 h C0 r0 r) H7 (FTC_P1 h C0 r2 r1)); Ring. -Apply Rle_anti_compatibility with ``-x``; Replace ``-x+(x+h0)`` with h0; [Idtac | Ring]. -Rewrite Rplus_sym; Apply Rle_trans with (Rabsolu h0). -Apply Rle_Rabsolu. -Apply Rle_trans with del; [Left; Assumption | Unfold del; Apply Rle_trans with ``(Rmin (b-x) (x-a))``; [Apply Rmin_r | Apply Rmin_l]]. -Apply Ropp_Rle; Apply Rle_anti_compatibility with ``x``; Replace ``x+-(x+h0)`` with ``-h0``; [Idtac | Ring]. -Apply Rle_trans with (Rabsolu h0); [Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu | Apply Rle_trans with del; [Left; Assumption | Unfold del; Apply Rle_trans with ``(Rmin (b-x) (x-a))``; Apply Rmin_r]]. +Lemma RiemannInt_P27 : + forall (f:R -> R) (a b x:R) (h:a <= b) + (C0:forall x:R, a <= x <= b -> continuity_pt f x), + a < x < b -> derivable_pt_lim (primitive h (FTC_P1 h C0)) x (f x). +intro f; intros; elim H; clear H; intros; assert (H1 : continuity_pt f x). +apply C0; split; left; assumption. +unfold derivable_pt_lim in |- *; intros; assert (Hyp : 0 < eps / 2). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (H1 _ Hyp); unfold dist, D_x, no_cond in |- *; simpl in |- *; + unfold R_dist in |- *; intros; pose (del := Rmin x0 (Rmin (b - x) (x - a))); + assert (H4 : 0 < del). +unfold del in |- *; unfold Rmin in |- *; case (Rle_dec (b - x) (x - a)); + intro. +case (Rle_dec x0 (b - x)); intro; + [ elim H3; intros; assumption | apply Rlt_Rminus; assumption ]. +case (Rle_dec x0 (x - a)); intro; + [ elim H3; intros; assumption | apply Rlt_Rminus; assumption ]. +split with (mkposreal _ H4); intros; + assert (H7 : Riemann_integrable f x (x + h0)). +case (Rle_dec x (x + h0)); intro. +apply continuity_implies_RiemannInt; try assumption. +intros; apply C0; elim H7; intros; split. +apply Rle_trans with x; [ left; assumption | assumption ]. +apply Rle_trans with (x + h0). +assumption. +left; apply Rlt_le_trans with (x + del). +apply Rplus_lt_compat_l; apply Rle_lt_trans with (Rabs h0); + [ apply RRle_abs | apply H6 ]. +unfold del in |- *; apply Rle_trans with (x + Rmin (b - x) (x - a)). +apply Rplus_le_compat_l; apply Rmin_r. +pattern b at 2 in |- *; replace b with (x + (b - x)); + [ apply Rplus_le_compat_l; apply Rmin_l | ring ]. +apply RiemannInt_P1; apply continuity_implies_RiemannInt; auto with real. +intros; apply C0; elim H7; intros; split. +apply Rle_trans with (x + h0). +left; apply Rle_lt_trans with (x - del). +unfold del in |- *; apply Rle_trans with (x - Rmin (b - x) (x - a)). +pattern a at 1 in |- *; replace a with (x + (a - x)); [ idtac | ring ]. +unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_le_cancel. +rewrite Ropp_involutive; rewrite Ropp_plus_distr; rewrite Ropp_involutive; + rewrite (Rplus_comm x); apply Rmin_r. +unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_le_cancel. +do 2 rewrite Ropp_involutive; apply Rmin_r. +unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_cancel. +rewrite Ropp_involutive; apply Rle_lt_trans with (Rabs h0); + [ rewrite <- Rabs_Ropp; apply RRle_abs | apply H6 ]. +assumption. +apply Rle_trans with x; [ assumption | left; assumption ]. +replace (primitive h (FTC_P1 h C0) (x + h0) - primitive h (FTC_P1 h C0) x) + with (RiemannInt H7). +replace (f x) with (RiemannInt (RiemannInt_P14 x (x + h0) (f x)) / h0). +replace + (RiemannInt H7 / h0 - RiemannInt (RiemannInt_P14 x (x + h0) (f x)) / h0) + with ((RiemannInt H7 - RiemannInt (RiemannInt_P14 x (x + h0) (f x))) / h0). +replace (RiemannInt H7 - RiemannInt (RiemannInt_P14 x (x + h0) (f x))) with + (RiemannInt (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))). +unfold Rdiv in |- *; rewrite Rabs_mult; case (Rle_dec x (x + h0)); intro. +apply Rle_lt_trans with + (RiemannInt + (RiemannInt_P16 + (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))) * + Rabs (/ h0)). +do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l. +apply Rabs_pos. +apply + (RiemannInt_P17 (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x))) + (RiemannInt_P16 + (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x))))); + assumption. +apply Rle_lt_trans with + (RiemannInt (RiemannInt_P14 x (x + h0) (eps / 2)) * Rabs (/ h0)). +do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l. +apply Rabs_pos. +apply RiemannInt_P19; try assumption. +intros; replace (f x1 + -1 * fct_cte (f x) x1) with (f x1 - f x). +unfold fct_cte in |- *; case (Req_dec x x1); intro. +rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; left; + assumption. +elim H3; intros; left; apply H11. +repeat split. +assumption. +rewrite Rabs_right. +apply Rplus_lt_reg_r with x; replace (x + (x1 - x)) with x1; [ idtac | ring ]. +apply Rlt_le_trans with (x + h0). +elim H8; intros; assumption. +apply Rplus_le_compat_l; apply Rle_trans with del. +left; apply Rle_lt_trans with (Rabs h0); [ apply RRle_abs | assumption ]. +unfold del in |- *; apply Rmin_l. +apply Rge_minus; apply Rle_ge; left; elim H8; intros; assumption. +unfold fct_cte in |- *; ring. +rewrite RiemannInt_P15. +rewrite Rmult_assoc; replace ((x + h0 - x) * Rabs (/ h0)) with 1. +rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r; + rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +rewrite Rabs_right. +replace (x + h0 - x) with h0; [ idtac | ring ]. +apply Rinv_r_sym. +assumption. +apply Rle_ge; left; apply Rinv_0_lt_compat. +elim r; intro. +apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; assumption. +elim H5; symmetry in |- *; apply Rplus_eq_reg_l with x; rewrite Rplus_0_r; + assumption. +apply Rle_lt_trans with + (RiemannInt + (RiemannInt_P16 + (RiemannInt_P1 + (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x))))) * + Rabs (/ h0)). +do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l. +apply Rabs_pos. +replace + (RiemannInt (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))) with + (- + RiemannInt + (RiemannInt_P1 (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x))))). +rewrite Rabs_Ropp; + apply + (RiemannInt_P17 + (RiemannInt_P1 + (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))) + (RiemannInt_P16 + (RiemannInt_P1 + (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))))); + auto with real. +symmetry in |- *; apply RiemannInt_P8. +apply Rle_lt_trans with + (RiemannInt (RiemannInt_P14 (x + h0) x (eps / 2)) * Rabs (/ h0)). +do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l. +apply Rabs_pos. +apply RiemannInt_P19. +auto with real. +intros; replace (f x1 + -1 * fct_cte (f x) x1) with (f x1 - f x). +unfold fct_cte in |- *; case (Req_dec x x1); intro. +rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; left; + assumption. +elim H3; intros; left; apply H11. +repeat split. +assumption. +rewrite Rabs_left. +apply Rplus_lt_reg_r with (x1 - x0); replace (x1 - x0 + x0) with x1; + [ idtac | ring ]. +replace (x1 - x0 + - (x1 - x)) with (x - x0); [ idtac | ring ]. +apply Rle_lt_trans with (x + h0). +unfold Rminus in |- *; apply Rplus_le_compat_l; apply Ropp_le_cancel. +rewrite Ropp_involutive; apply Rle_trans with (Rabs h0). +rewrite <- Rabs_Ropp; apply RRle_abs. +apply Rle_trans with del; + [ left; assumption | unfold del in |- *; apply Rmin_l ]. +elim H8; intros; assumption. +apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; + replace (x + (x1 - x)) with x1; [ elim H8; intros; assumption | ring ]. +unfold fct_cte in |- *; ring. +rewrite RiemannInt_P15. +rewrite Rmult_assoc; replace ((x - (x + h0)) * Rabs (/ h0)) with 1. +rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r; + rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +rewrite Rabs_left. +replace (x - (x + h0)) with (- h0); [ idtac | ring ]. +rewrite Ropp_mult_distr_l_reverse; rewrite Ropp_mult_distr_r_reverse; + rewrite Ropp_involutive; apply Rinv_r_sym. +assumption. +apply Rinv_lt_0_compat. +assert (H8 : x + h0 < x). +auto with real. +apply Rplus_lt_reg_r with x; rewrite Rplus_0_r; assumption. +rewrite + (RiemannInt_P13 H7 (RiemannInt_P14 x (x + h0) (f x)) + (RiemannInt_P10 (-1) H7 (RiemannInt_P14 x (x + h0) (f x)))) + . +ring. +unfold Rdiv, Rminus in |- *; rewrite Rmult_plus_distr_r; ring. +rewrite RiemannInt_P15; apply Rmult_eq_reg_l with h0; + [ unfold Rdiv in |- *; rewrite (Rmult_comm h0); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym; [ ring | assumption ] + | assumption ]. +cut (a <= x + h0). +cut (x + h0 <= b). +intros; unfold primitive in |- *. +case (Rle_dec a (x + h0)); case (Rle_dec (x + h0) b); case (Rle_dec a x); + case (Rle_dec x b); intros; try (elim n; assumption || left; assumption). +rewrite <- (RiemannInt_P26 (FTC_P1 h C0 r0 r) H7 (FTC_P1 h C0 r2 r1)); ring. +apply Rplus_le_reg_l with (- x); replace (- x + (x + h0)) with h0; + [ idtac | ring ]. +rewrite Rplus_comm; apply Rle_trans with (Rabs h0). +apply RRle_abs. +apply Rle_trans with del; + [ left; assumption + | unfold del in |- *; apply Rle_trans with (Rmin (b - x) (x - a)); + [ apply Rmin_r | apply Rmin_l ] ]. +apply Ropp_le_cancel; apply Rplus_le_reg_l with x; + replace (x + - (x + h0)) with (- h0); [ idtac | ring ]. +apply Rle_trans with (Rabs h0); + [ rewrite <- Rabs_Ropp; apply RRle_abs + | apply Rle_trans with del; + [ left; assumption + | unfold del in |- *; apply Rle_trans with (Rmin (b - x) (x - a)); + apply Rmin_r ] ]. Qed. -Lemma RiemannInt_P28 : (f:R->R;a,b,x:R;h:``a<=b``;C0:((x:R)``a<=x<=b``->(continuity_pt f x))) ``a<=x<=b`` -> (derivable_pt_lim (primitive h (FTC_P1 h C0)) x (f x)). -Intro f; Intros; Elim h; Intro. -Elim H; Clear H; Intros; Elim H; Intro. -Elim H1; Intro. -Apply RiemannInt_P27; Split; Assumption. -Pose f_b := [x:R]``(f b)*(x-b)+(RiemannInt [(FTC_P1 h C0 h (FTC_P2 b))])``; Rewrite H3. -Assert H4 : (derivable_pt_lim f_b b (f b)). -Unfold f_b; Pattern 2 (f b); Replace (f b) with ``(f b)+0``. -Change (derivable_pt_lim (plus_fct (mult_fct (fct_cte (f b)) (minus_fct id (fct_cte b))) (fct_cte (RiemannInt (FTC_P1 h C0 h (FTC_P2 b))))) b ``(f b)+0``). -Apply derivable_pt_lim_plus. -Pattern 2 (f b); Replace (f b) with ``0*((minus_fct id (fct_cte b)) b)+((fct_cte (f b)) b)*1``. -Apply derivable_pt_lim_mult. -Apply derivable_pt_lim_const. -Replace R1 with ``1-0``; [Idtac | Ring]. -Apply derivable_pt_lim_minus. -Apply derivable_pt_lim_id. -Apply derivable_pt_lim_const. -Unfold fct_cte; Ring. -Apply derivable_pt_lim_const. -Ring. -Unfold derivable_pt_lim; Intros; Elim (H4 ? H5); Intros; Assert H7 : (continuity_pt f b). -Apply C0; Split; [Left; Assumption | Right; Reflexivity]. -Assert H8 : ``0<eps/2``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Elim (H7 ? H8); Unfold D_x no_cond dist; Simpl; Unfold R_dist; Intros; Pose del := (Rmin x0 (Rmin x1 ``b-a``)); Assert H10 : ``0<del``. -Unfold del; Unfold Rmin; Case (total_order_Rle x1 ``b-a``); Intros. -Case (total_order_Rle x0 x1); Intro; [Apply (cond_pos x0) | Elim H9; Intros; Assumption]. -Case (total_order_Rle x0 ``b-a``); Intro; [Apply (cond_pos x0) | Apply Rlt_Rminus; Assumption]. -Split with (mkposreal ? H10); Intros; Case (case_Rabsolu h0); Intro. -Assert H14 : ``b+h0<b``. -Pattern 2 b; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Assumption. -Assert H13 : (Riemann_integrable f ``b+h0`` b). -Apply continuity_implies_RiemannInt. -Left; Assumption. -Intros; Apply C0; Elim H13; Intros; Split; Try Assumption. -Apply Rle_trans with ``b+h0``; Try Assumption. -Apply Rle_anti_compatibility with ``-a-h0``. -Replace ``-a-h0+a`` with ``-h0``; [Idtac | Ring]. -Replace ``-a-h0+(b+h0)`` with ``b-a``; [Idtac | Ring]. -Apply Rle_trans with del. -Apply Rle_trans with (Rabsolu h0). -Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. -Left; Assumption. -Unfold del; Apply Rle_trans with (Rmin x1 ``b-a``); Apply Rmin_r. -Replace ``[(primitive h (FTC_P1 h C0) (b+h0))]-[(primitive h (FTC_P1 h C0) b)]`` with ``-(RiemannInt H13)``. -Replace (f b) with ``-[(RiemannInt (RiemannInt_P14 (b+h0) b (f b)))]/h0``. -Rewrite <- Rabsolu_Ropp; Unfold Rminus; Unfold Rdiv; Rewrite Ropp_mul1; Rewrite Ropp_distr1; Repeat Rewrite Ropp_Ropp; Replace ``(RiemannInt H13)*/h0+ -(RiemannInt (RiemannInt_P14 (b+h0) b (f b)))*/h0`` with ``((RiemannInt H13)-(RiemannInt (RiemannInt_P14 (b+h0) b (f b))))/h0``. -Replace ``(RiemannInt H13)-(RiemannInt (RiemannInt_P14 (b+h0) b (f b)))`` with (RiemannInt (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 ``b+h0`` b (f b)))). -Unfold Rdiv; Rewrite Rabsolu_mult; Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P16 (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b+h0) b (f b)))))*(Rabsolu (/h0))``. -Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. -Apply Rabsolu_pos. -Apply (RiemannInt_P17 (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 ``b+h0`` b (f b))) (RiemannInt_P16 (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 ``b+h0`` b (f b))))); Left; Assumption. -Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P14 (b+h0) b (eps/2)))*(Rabsolu (/h0))``. -Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. -Apply Rabsolu_pos. -Apply RiemannInt_P19. -Left; Assumption. -Intros; Replace ``(f x2)+ -1*(fct_cte (f b) x2)`` with ``(f x2)-(f b)``. -Unfold fct_cte; Case (Req_EM b x2); Intro. -Rewrite H16; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Left; Assumption. -Elim H9; Intros; Left; Apply H18. -Repeat Split. -Assumption. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Rewrite Rabsolu_right. -Apply Rlt_anti_compatibility with ``x2-x1``; Replace ``x2-x1+(b-x2)`` with ``b-x1``; [Idtac | Ring]. -Replace ``x2-x1+x1`` with x2; [Idtac | Ring]. -Apply Rlt_le_trans with ``b+h0``. -2:Elim H15; Intros; Left; Assumption. -Unfold Rminus; Apply Rlt_compatibility; Apply Ropp_Rlt; Rewrite Ropp_Ropp; Apply Rle_lt_trans with (Rabsolu h0). -Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. -Apply Rlt_le_trans with del; [Assumption | Unfold del; Apply Rle_trans with (Rmin x1 ``b-a``); [Apply Rmin_r | Apply Rmin_l]]. -Apply Rle_sym1; Left; Apply Rlt_Rminus; Elim H15; Intros; Assumption. -Unfold fct_cte; Ring. -Rewrite RiemannInt_P15. -Rewrite Rmult_assoc; Replace ``(b-(b+h0))*(Rabsolu (/h0))`` with R1. -Rewrite Rmult_1r; Unfold Rdiv; Apply Rlt_monotony_contra with ``2``; [Sup0 | Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Pattern 1 eps; Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Rewrite Rabsolu_left. -Apply r_Rmult_mult with h0; [Do 2 Rewrite (Rmult_sym h0); Rewrite Rmult_assoc; Rewrite Ropp_mul1; Rewrite <- Rinv_l_sym; [ Ring | Assumption ] | Assumption]. -Apply Rlt_Rinv2; Assumption. -Rewrite (RiemannInt_P13 H13 (RiemannInt_P14 ``b+h0`` b (f b)) (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 ``b+h0`` b (f b)))); Ring. -Unfold Rdiv Rminus; Rewrite Rmult_Rplus_distrl; Ring. -Rewrite RiemannInt_P15. -Rewrite <- Ropp_mul1; Apply r_Rmult_mult with h0; [Repeat Rewrite (Rmult_sym h0); Unfold Rdiv; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Ring | Assumption] | Assumption]. -Cut ``a<=b+h0``. -Cut ``b+h0<=b``. -Intros; Unfold primitive; Case (total_order_Rle a ``b+h0``); Case (total_order_Rle ``b+h0`` b); Case (total_order_Rle a b); Case (total_order_Rle b b); Intros; Try (Elim n; Right; Reflexivity) Orelse (Elim n; Left; Assumption). -Rewrite <- (RiemannInt_P26 (FTC_P1 h C0 r3 r2) H13 (FTC_P1 h C0 r1 r0)); Ring. -Elim n; Assumption. -Left; Assumption. -Apply Rle_anti_compatibility with ``-a-h0``. -Replace ``-a-h0+a`` with ``-h0``; [Idtac | Ring]. -Replace ``-a-h0+(b+h0)`` with ``b-a``; [Idtac | Ring]. -Apply Rle_trans with del. -Apply Rle_trans with (Rabsolu h0). -Rewrite <- Rabsolu_Ropp; Apply Rle_Rabsolu. -Left; Assumption. -Unfold del; Apply Rle_trans with (Rmin x1 ``b-a``); Apply Rmin_r. -Cut (primitive h (FTC_P1 h C0) b)==(f_b b). -Intro; Cut (primitive h (FTC_P1 h C0) ``b+h0``)==(f_b ``b+h0``). -Intro; Rewrite H13; Rewrite H14; Apply H6. -Assumption. -Apply Rlt_le_trans with del; [Assumption | Unfold del; Apply Rmin_l]. -Assert H14 : ``b<b+h0``. -Pattern 1 b; Rewrite <- Rplus_Or; Apply Rlt_compatibility. -Assert H14 := (Rle_sym2 ? ? r); Elim H14; Intro. -Assumption. -Elim H11; Symmetry; Assumption. -Unfold primitive; Case (total_order_Rle a ``b+h0``); Case (total_order_Rle ``b+h0`` b); Intros; [Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 H14)) | Unfold f_b; Reflexivity | Elim n; Left; Apply Rlt_trans with b; Assumption | Elim n0; Left; Apply Rlt_trans with b; Assumption]. -Unfold f_b; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rmult_Or; Rewrite Rplus_Ol; Unfold primitive; Case (total_order_Rle a b); Case (total_order_Rle b b); Intros; [Apply RiemannInt_P5 | Elim n; Right; Reflexivity | Elim n; Left; Assumption | Elim n; Right; Reflexivity]. +Lemma RiemannInt_P28 : + forall (f:R -> R) (a b x:R) (h:a <= b) + (C0:forall x:R, a <= x <= b -> continuity_pt f x), + a <= x <= b -> derivable_pt_lim (primitive h (FTC_P1 h C0)) x (f x). +intro f; intros; elim h; intro. +elim H; clear H; intros; elim H; intro. +elim H1; intro. +apply RiemannInt_P27; split; assumption. +pose + (f_b := fun x:R => f b * (x - b) + RiemannInt (FTC_P1 h C0 h (Rle_refl b))); + rewrite H3. +assert (H4 : derivable_pt_lim f_b b (f b)). +unfold f_b in |- *; pattern (f b) at 2 in |- *; replace (f b) with (f b + 0). +change + (derivable_pt_lim + ((fct_cte (f b) * (id - fct_cte b))%F + + fct_cte (RiemannInt (FTC_P1 h C0 h (Rle_refl b)))) b ( + f b + 0)) in |- *. +apply derivable_pt_lim_plus. +pattern (f b) at 2 in |- *; + replace (f b) with (0 * (id - fct_cte b)%F b + fct_cte (f b) b * 1). +apply derivable_pt_lim_mult. +apply derivable_pt_lim_const. +replace 1 with (1 - 0); [ idtac | ring ]. +apply derivable_pt_lim_minus. +apply derivable_pt_lim_id. +apply derivable_pt_lim_const. +unfold fct_cte in |- *; ring. +apply derivable_pt_lim_const. +ring. +unfold derivable_pt_lim in |- *; intros; elim (H4 _ H5); intros; + assert (H7 : continuity_pt f b). +apply C0; split; [ left; assumption | right; reflexivity ]. +assert (H8 : 0 < eps / 2). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (H7 _ H8); unfold D_x, no_cond, dist in |- *; simpl in |- *; + unfold R_dist in |- *; intros; pose (del := Rmin x0 (Rmin x1 (b - a))); + assert (H10 : 0 < del). +unfold del in |- *; unfold Rmin in |- *; case (Rle_dec x1 (b - a)); intros. +case (Rle_dec x0 x1); intro; + [ apply (cond_pos x0) | elim H9; intros; assumption ]. +case (Rle_dec x0 (b - a)); intro; + [ apply (cond_pos x0) | apply Rlt_Rminus; assumption ]. +split with (mkposreal _ H10); intros; case (Rcase_abs h0); intro. +assert (H14 : b + h0 < b). +pattern b at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + assumption. +assert (H13 : Riemann_integrable f (b + h0) b). +apply continuity_implies_RiemannInt. +left; assumption. +intros; apply C0; elim H13; intros; split; try assumption. +apply Rle_trans with (b + h0); try assumption. +apply Rplus_le_reg_l with (- a - h0). +replace (- a - h0 + a) with (- h0); [ idtac | ring ]. +replace (- a - h0 + (b + h0)) with (b - a); [ idtac | ring ]. +apply Rle_trans with del. +apply Rle_trans with (Rabs h0). +rewrite <- Rabs_Ropp; apply RRle_abs. +left; assumption. +unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r. +replace (primitive h (FTC_P1 h C0) (b + h0) - primitive h (FTC_P1 h C0) b) + with (- RiemannInt H13). +replace (f b) with (- RiemannInt (RiemannInt_P14 (b + h0) b (f b)) / h0). +rewrite <- Rabs_Ropp; unfold Rminus in |- *; unfold Rdiv in |- *; + rewrite Ropp_mult_distr_l_reverse; rewrite Ropp_plus_distr; + repeat rewrite Ropp_involutive; + replace + (RiemannInt H13 * / h0 + + - RiemannInt (RiemannInt_P14 (b + h0) b (f b)) * / h0) with + ((RiemannInt H13 - RiemannInt (RiemannInt_P14 (b + h0) b (f b))) / h0). +replace (RiemannInt H13 - RiemannInt (RiemannInt_P14 (b + h0) b (f b))) with + (RiemannInt (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b)))). +unfold Rdiv in |- *; rewrite Rabs_mult; + apply Rle_lt_trans with + (RiemannInt + (RiemannInt_P16 + (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b)))) * + Rabs (/ h0)). +do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l. +apply Rabs_pos. +apply + (RiemannInt_P17 (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b))) + (RiemannInt_P16 + (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b))))); + left; assumption. +apply Rle_lt_trans with + (RiemannInt (RiemannInt_P14 (b + h0) b (eps / 2)) * Rabs (/ h0)). +do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l. +apply Rabs_pos. +apply RiemannInt_P19. +left; assumption. +intros; replace (f x2 + -1 * fct_cte (f b) x2) with (f x2 - f b). +unfold fct_cte in |- *; case (Req_dec b x2); intro. +rewrite H16; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + left; assumption. +elim H9; intros; left; apply H18. +repeat split. +assumption. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; rewrite Rabs_right. +apply Rplus_lt_reg_r with (x2 - x1); + replace (x2 - x1 + (b - x2)) with (b - x1); [ idtac | ring ]. +replace (x2 - x1 + x1) with x2; [ idtac | ring ]. +apply Rlt_le_trans with (b + h0). +2: elim H15; intros; left; assumption. +unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_cancel; + rewrite Ropp_involutive; apply Rle_lt_trans with (Rabs h0). +rewrite <- Rabs_Ropp; apply RRle_abs. +apply Rlt_le_trans with del; + [ assumption + | unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); + [ apply Rmin_r | apply Rmin_l ] ]. +apply Rle_ge; left; apply Rlt_Rminus; elim H15; intros; assumption. +unfold fct_cte in |- *; ring. +rewrite RiemannInt_P15. +rewrite Rmult_assoc; replace ((b - (b + h0)) * Rabs (/ h0)) with 1. +rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r; + rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +rewrite Rabs_left. +apply Rmult_eq_reg_l with h0; + [ do 2 rewrite (Rmult_comm h0); rewrite Rmult_assoc; + rewrite Ropp_mult_distr_l_reverse; rewrite <- Rinv_l_sym; + [ ring | assumption ] + | assumption ]. +apply Rinv_lt_0_compat; assumption. +rewrite + (RiemannInt_P13 H13 (RiemannInt_P14 (b + h0) b (f b)) + (RiemannInt_P10 (-1) H13 (RiemannInt_P14 (b + h0) b (f b)))) + ; ring. +unfold Rdiv, Rminus in |- *; rewrite Rmult_plus_distr_r; ring. +rewrite RiemannInt_P15. +rewrite <- Ropp_mult_distr_l_reverse; apply Rmult_eq_reg_l with h0; + [ repeat rewrite (Rmult_comm h0); unfold Rdiv in |- *; + repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym; + [ ring | assumption ] + | assumption ]. +cut (a <= b + h0). +cut (b + h0 <= b). +intros; unfold primitive in |- *; case (Rle_dec a (b + h0)); + case (Rle_dec (b + h0) b); case (Rle_dec a b); case (Rle_dec b b); + intros; try (elim n; right; reflexivity) || (elim n; left; assumption). +rewrite <- (RiemannInt_P26 (FTC_P1 h C0 r3 r2) H13 (FTC_P1 h C0 r1 r0)); ring. +elim n; assumption. +left; assumption. +apply Rplus_le_reg_l with (- a - h0). +replace (- a - h0 + a) with (- h0); [ idtac | ring ]. +replace (- a - h0 + (b + h0)) with (b - a); [ idtac | ring ]. +apply Rle_trans with del. +apply Rle_trans with (Rabs h0). +rewrite <- Rabs_Ropp; apply RRle_abs. +left; assumption. +unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r. +cut (primitive h (FTC_P1 h C0) b = f_b b). +intro; cut (primitive h (FTC_P1 h C0) (b + h0) = f_b (b + h0)). +intro; rewrite H13; rewrite H14; apply H6. +assumption. +apply Rlt_le_trans with del; + [ assumption | unfold del in |- *; apply Rmin_l ]. +assert (H14 : b < b + h0). +pattern b at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. +assert (H14 := Rge_le _ _ r); elim H14; intro. +assumption. +elim H11; symmetry in |- *; assumption. +unfold primitive in |- *; case (Rle_dec a (b + h0)); + case (Rle_dec (b + h0) b); intros; + [ elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 H14)) + | unfold f_b in |- *; reflexivity + | elim n; left; apply Rlt_trans with b; assumption + | elim n0; left; apply Rlt_trans with b; assumption ]. +unfold f_b in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rmult_0_r; rewrite Rplus_0_l; unfold primitive in |- *; + case (Rle_dec a b); case (Rle_dec b b); intros; + [ apply RiemannInt_P5 + | elim n; right; reflexivity + | elim n; left; assumption + | elim n; right; reflexivity ]. (*****) -Pose f_a := [x:R]``(f a)*(x-a)``; Rewrite <- H2; Assert H3 : (derivable_pt_lim f_a a (f a)). -Unfold f_a; Change (derivable_pt_lim (mult_fct (fct_cte (f a)) (minus_fct id (fct_cte a))) a (f a)); Pattern 2 (f a); Replace (f a) with ``0*((minus_fct id (fct_cte a)) a)+((fct_cte (f a)) a)*1``. -Apply derivable_pt_lim_mult. -Apply derivable_pt_lim_const. -Replace R1 with ``1-0``; [Idtac | Ring]. -Apply derivable_pt_lim_minus. -Apply derivable_pt_lim_id. -Apply derivable_pt_lim_const. -Unfold fct_cte; Ring. -Unfold derivable_pt_lim; Intros; Elim (H3 ? H4); Intros. -Assert H6 : (continuity_pt f a). -Apply C0; Split; [Right; Reflexivity | Left; Assumption]. -Assert H7 : ``0<eps/2``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Elim (H6 ? H7); Unfold D_x no_cond dist; Simpl; Unfold R_dist; Intros. -Pose del := (Rmin x0 (Rmin x1 ``b-a``)). -Assert H9 : ``0<del``. -Unfold del; Unfold Rmin. -Case (total_order_Rle x1 ``b-a``); Intros. -Case (total_order_Rle x0 x1); Intro. -Apply (cond_pos x0). -Elim H8; Intros; Assumption. -Case (total_order_Rle x0 ``b-a``); Intro. -Apply (cond_pos x0). -Apply Rlt_Rminus; Assumption. -Split with (mkposreal ? H9). -Intros; Case (case_Rabsolu h0); Intro. -Assert H12 : ``a+h0<a``. -Pattern 2 a; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Assumption. -Unfold primitive. -Case (total_order_Rle a ``a+h0``); Case (total_order_Rle ``a+h0`` b); Case (total_order_Rle a a); Case (total_order_Rle a b); Intros; Try (Elim n; Left; Assumption) Orelse (Elim n; Right; Reflexivity). -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r3 H12)). -Elim n; Left; Apply Rlt_trans with a; Assumption. -Rewrite RiemannInt_P9; Replace R0 with (f_a a). -Replace ``(f a)*(a+h0-a)`` with (f_a ``a+h0``). -Apply H5; Try Assumption. -Apply Rlt_le_trans with del; [Assumption | Unfold del; Apply Rmin_l]. -Unfold f_a; Ring. -Unfold f_a; Ring. -Elim n; Left; Apply Rlt_trans with a; Assumption. -Assert H12 : ``a<a+h0``. -Pattern 1 a; Rewrite <- Rplus_Or; Apply Rlt_compatibility. -Assert H12 := (Rle_sym2 ? ? r); Elim H12; Intro. -Assumption. -Elim H10; Symmetry; Assumption. -Assert H13 : (Riemann_integrable f a ``a+h0``). -Apply continuity_implies_RiemannInt. -Left; Assumption. -Intros; Apply C0; Elim H13; Intros; Split; Try Assumption. -Apply Rle_trans with ``a+h0``; Try Assumption. -Apply Rle_anti_compatibility with ``-b-h0``. -Replace ``-b-h0+b`` with ``-h0``; [Idtac | Ring]. -Replace ``-b-h0+(a+h0)`` with ``a-b``; [Idtac | Ring]. -Apply Ropp_Rle; Rewrite Ropp_Ropp; Rewrite Ropp_distr2; Apply Rle_trans with del. -Apply Rle_trans with (Rabsolu h0); [Apply Rle_Rabsolu | Left; Assumption]. -Unfold del; Apply Rle_trans with (Rmin x1 ``b-a``); Apply Rmin_r. -Replace ``(primitive h (FTC_P1 h C0) (a+h0))-(primitive h (FTC_P1 h C0) a)`` with ``(RiemannInt H13)``. -Replace (f a) with ``(RiemannInt (RiemannInt_P14 a (a+h0) (f a)))/h0``. -Replace ``(RiemannInt H13)/h0-(RiemannInt (RiemannInt_P14 a (a+h0) (f a)))/h0`` with ``((RiemannInt H13)-(RiemannInt (RiemannInt_P14 a (a+h0) (f a))))/h0``. -Replace ``(RiemannInt H13)-(RiemannInt (RiemannInt_P14 a (a+h0) (f a)))`` with (RiemannInt (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 a ``a+h0`` (f a)))). -Unfold Rdiv; Rewrite Rabsolu_mult; Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P16 (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a+h0) (f a)))))*(Rabsolu (/h0))``. -Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. -Apply Rabsolu_pos. -Apply (RiemannInt_P17 (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 a ``a+h0`` (f a))) (RiemannInt_P16 (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 a ``a+h0`` (f a))))); Left; Assumption. -Apply Rle_lt_trans with ``(RiemannInt (RiemannInt_P14 a (a+h0) (eps/2)))*(Rabsolu (/h0))``. -Do 2 Rewrite <- (Rmult_sym ``(Rabsolu (/h0))``); Apply Rle_monotony. -Apply Rabsolu_pos. -Apply RiemannInt_P19. -Left; Assumption. -Intros; Replace ``(f x2)+ -1*(fct_cte (f a) x2)`` with ``(f x2)-(f a)``. -Unfold fct_cte; Case (Req_EM a x2); Intro. -Rewrite H15; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Left; Assumption. -Elim H8; Intros; Left; Apply H17; Repeat Split. -Assumption. -Rewrite Rabsolu_right. -Apply Rlt_anti_compatibility with a; Replace ``a+(x2-a)`` with x2; [Idtac | Ring]. -Apply Rlt_le_trans with ``a+h0``. -Elim H14; Intros; Assumption. -Apply Rle_compatibility; Left; Apply Rle_lt_trans with (Rabsolu h0). -Apply Rle_Rabsolu. -Apply Rlt_le_trans with del; [Assumption | Unfold del; Apply Rle_trans with (Rmin x1 ``b-a``); [Apply Rmin_r | Apply Rmin_l]]. -Apply Rle_sym1; Left; Apply Rlt_Rminus; Elim H14; Intros; Assumption. -Unfold fct_cte; Ring. -Rewrite RiemannInt_P15. -Rewrite Rmult_assoc; Replace ``((a+h0)-a)*(Rabsolu (/h0))`` with R1. -Rewrite Rmult_1r; Unfold Rdiv; Apply Rlt_monotony_contra with ``2``; [Sup0 | Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Pattern 1 eps; Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Rewrite Rabsolu_right. -Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Rewrite <- Rinv_r_sym; [ Reflexivity | Assumption ]. -Apply Rle_sym1; Left; Apply Rlt_Rinv; Assert H14 := (Rle_sym2 ? ? r); Elim H14; Intro. -Assumption. -Elim H10; Symmetry; Assumption. -Rewrite (RiemannInt_P13 H13 (RiemannInt_P14 a ``a+h0`` (f a)) (RiemannInt_P10 ``-1`` H13 (RiemannInt_P14 a ``a+h0`` (f a)))); Ring. -Unfold Rdiv Rminus; Rewrite Rmult_Rplus_distrl; Ring. -Rewrite RiemannInt_P15. -Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Unfold Rdiv; Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym; [ Ring | Assumption ]. -Cut ``a<=a+h0``. -Cut ``a+h0<=b``. -Intros; Unfold primitive; Case (total_order_Rle a ``a+h0``); Case (total_order_Rle ``a+h0`` b); Case (total_order_Rle a a); Case (total_order_Rle a b); Intros; Try (Elim n; Right; Reflexivity) Orelse (Elim n; Left; Assumption). -Rewrite RiemannInt_P9; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply RiemannInt_P5. -Elim n; Assumption. -Elim n; Assumption. -2:Left; Assumption. -Apply Rle_anti_compatibility with ``-a``; Replace ``-a+(a+h0)`` with h0; [Idtac | Ring]. -Rewrite Rplus_sym; Apply Rle_trans with del; [Apply Rle_trans with (Rabsolu h0); [Apply Rle_Rabsolu | Left; Assumption] | Unfold del; Apply Rle_trans with (Rmin x1 ``b-a``); Apply Rmin_r]. +pose (f_a := fun x:R => f a * (x - a)); rewrite <- H2; + assert (H3 : derivable_pt_lim f_a a (f a)). +unfold f_a in |- *; + change (derivable_pt_lim (fct_cte (f a) * (id - fct_cte a)%F) a (f a)) + in |- *; pattern (f a) at 2 in |- *; + replace (f a) with (0 * (id - fct_cte a)%F a + fct_cte (f a) a * 1). +apply derivable_pt_lim_mult. +apply derivable_pt_lim_const. +replace 1 with (1 - 0); [ idtac | ring ]. +apply derivable_pt_lim_minus. +apply derivable_pt_lim_id. +apply derivable_pt_lim_const. +unfold fct_cte in |- *; ring. +unfold derivable_pt_lim in |- *; intros; elim (H3 _ H4); intros. +assert (H6 : continuity_pt f a). +apply C0; split; [ right; reflexivity | left; assumption ]. +assert (H7 : 0 < eps / 2). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +elim (H6 _ H7); unfold D_x, no_cond, dist in |- *; simpl in |- *; + unfold R_dist in |- *; intros. +pose (del := Rmin x0 (Rmin x1 (b - a))). +assert (H9 : 0 < del). +unfold del in |- *; unfold Rmin in |- *. +case (Rle_dec x1 (b - a)); intros. +case (Rle_dec x0 x1); intro. +apply (cond_pos x0). +elim H8; intros; assumption. +case (Rle_dec x0 (b - a)); intro. +apply (cond_pos x0). +apply Rlt_Rminus; assumption. +split with (mkposreal _ H9). +intros; case (Rcase_abs h0); intro. +assert (H12 : a + h0 < a). +pattern a at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + assumption. +unfold primitive in |- *. +case (Rle_dec a (a + h0)); case (Rle_dec (a + h0) b); case (Rle_dec a a); + case (Rle_dec a b); intros; + try (elim n; left; assumption) || (elim n; right; reflexivity). +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H12)). +elim n; left; apply Rlt_trans with a; assumption. +rewrite RiemannInt_P9; replace 0 with (f_a a). +replace (f a * (a + h0 - a)) with (f_a (a + h0)). +apply H5; try assumption. +apply Rlt_le_trans with del; + [ assumption | unfold del in |- *; apply Rmin_l ]. +unfold f_a in |- *; ring. +unfold f_a in |- *; ring. +elim n; left; apply Rlt_trans with a; assumption. +assert (H12 : a < a + h0). +pattern a at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. +assert (H12 := Rge_le _ _ r); elim H12; intro. +assumption. +elim H10; symmetry in |- *; assumption. +assert (H13 : Riemann_integrable f a (a + h0)). +apply continuity_implies_RiemannInt. +left; assumption. +intros; apply C0; elim H13; intros; split; try assumption. +apply Rle_trans with (a + h0); try assumption. +apply Rplus_le_reg_l with (- b - h0). +replace (- b - h0 + b) with (- h0); [ idtac | ring ]. +replace (- b - h0 + (a + h0)) with (a - b); [ idtac | ring ]. +apply Ropp_le_cancel; rewrite Ropp_involutive; rewrite Ropp_minus_distr; + apply Rle_trans with del. +apply Rle_trans with (Rabs h0); [ apply RRle_abs | left; assumption ]. +unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r. +replace (primitive h (FTC_P1 h C0) (a + h0) - primitive h (FTC_P1 h C0) a) + with (RiemannInt H13). +replace (f a) with (RiemannInt (RiemannInt_P14 a (a + h0) (f a)) / h0). +replace + (RiemannInt H13 / h0 - RiemannInt (RiemannInt_P14 a (a + h0) (f a)) / h0) + with ((RiemannInt H13 - RiemannInt (RiemannInt_P14 a (a + h0) (f a))) / h0). +replace (RiemannInt H13 - RiemannInt (RiemannInt_P14 a (a + h0) (f a))) with + (RiemannInt (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a)))). +unfold Rdiv in |- *; rewrite Rabs_mult; + apply Rle_lt_trans with + (RiemannInt + (RiemannInt_P16 + (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a)))) * + Rabs (/ h0)). +do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l. +apply Rabs_pos. +apply + (RiemannInt_P17 (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a))) + (RiemannInt_P16 + (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a))))); + left; assumption. +apply Rle_lt_trans with + (RiemannInt (RiemannInt_P14 a (a + h0) (eps / 2)) * Rabs (/ h0)). +do 2 rewrite <- (Rmult_comm (Rabs (/ h0))); apply Rmult_le_compat_l. +apply Rabs_pos. +apply RiemannInt_P19. +left; assumption. +intros; replace (f x2 + -1 * fct_cte (f a) x2) with (f x2 - f a). +unfold fct_cte in |- *; case (Req_dec a x2); intro. +rewrite H15; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + left; assumption. +elim H8; intros; left; apply H17; repeat split. +assumption. +rewrite Rabs_right. +apply Rplus_lt_reg_r with a; replace (a + (x2 - a)) with x2; [ idtac | ring ]. +apply Rlt_le_trans with (a + h0). +elim H14; intros; assumption. +apply Rplus_le_compat_l; left; apply Rle_lt_trans with (Rabs h0). +apply RRle_abs. +apply Rlt_le_trans with del; + [ assumption + | unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); + [ apply Rmin_r | apply Rmin_l ] ]. +apply Rle_ge; left; apply Rlt_Rminus; elim H14; intros; assumption. +unfold fct_cte in |- *; ring. +rewrite RiemannInt_P15. +rewrite Rmult_assoc; replace ((a + h0 - a) * Rabs (/ h0)) with 1. +rewrite Rmult_1_r; unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; pattern eps at 1 in |- *; rewrite <- Rplus_0_r; + rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +rewrite Rabs_right. +rewrite Rplus_comm; unfold Rminus in |- *; rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; rewrite <- Rinv_r_sym; + [ reflexivity | assumption ]. +apply Rle_ge; left; apply Rinv_0_lt_compat; assert (H14 := Rge_le _ _ r); + elim H14; intro. +assumption. +elim H10; symmetry in |- *; assumption. +rewrite + (RiemannInt_P13 H13 (RiemannInt_P14 a (a + h0) (f a)) + (RiemannInt_P10 (-1) H13 (RiemannInt_P14 a (a + h0) (f a)))) + ; ring. +unfold Rdiv, Rminus in |- *; rewrite Rmult_plus_distr_r; ring. +rewrite RiemannInt_P15. +rewrite Rplus_comm; unfold Rminus in |- *; rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; unfold Rdiv in |- *; + rewrite Rmult_assoc; rewrite <- Rinv_r_sym; [ ring | assumption ]. +cut (a <= a + h0). +cut (a + h0 <= b). +intros; unfold primitive in |- *; case (Rle_dec a (a + h0)); + case (Rle_dec (a + h0) b); case (Rle_dec a a); case (Rle_dec a b); + intros; try (elim n; right; reflexivity) || (elim n; left; assumption). +rewrite RiemannInt_P9; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; apply RiemannInt_P5. +elim n; assumption. +elim n; assumption. +2: left; assumption. +apply Rplus_le_reg_l with (- a); replace (- a + (a + h0)) with h0; + [ idtac | ring ]. +rewrite Rplus_comm; apply Rle_trans with del; + [ apply Rle_trans with (Rabs h0); [ apply RRle_abs | left; assumption ] + | unfold del in |- *; apply Rle_trans with (Rmin x1 (b - a)); apply Rmin_r ]. (*****) -Assert H1 : x==a. -Rewrite <- H0 in H; Elim H; Intros; Apply Rle_antisym; Assumption. -Pose f_a := [x:R]``(f a)*(x-a)``. -Assert H2 : (derivable_pt_lim f_a a (f a)). -Unfold f_a; Change (derivable_pt_lim (mult_fct (fct_cte (f a)) (minus_fct id (fct_cte a))) a (f a)); Pattern 2 (f a); Replace (f a) with ``0*((minus_fct id (fct_cte a)) a)+((fct_cte (f a)) a)*1``. -Apply derivable_pt_lim_mult. -Apply derivable_pt_lim_const. -Replace R1 with ``1-0``; [Idtac | Ring]. -Apply derivable_pt_lim_minus. -Apply derivable_pt_lim_id. -Apply derivable_pt_lim_const. -Unfold fct_cte; Ring. -Pose f_b := [x:R]``(f b)*(x-b)+(RiemannInt (FTC_P1 h C0 b h (FTC_P2 b)))``. -Assert H3 : (derivable_pt_lim f_b b (f b)). -Unfold f_b; Pattern 2 (f b); Replace (f b) with ``(f b)+0``. -Change (derivable_pt_lim (plus_fct (mult_fct (fct_cte (f b)) (minus_fct id (fct_cte b))) (fct_cte (RiemannInt (FTC_P1 h C0 h (FTC_P2 b))))) b ``(f b)+0``). -Apply derivable_pt_lim_plus. -Pattern 2 (f b); Replace (f b) with ``0*((minus_fct id (fct_cte b)) b)+((fct_cte (f b)) b)*1``. -Apply derivable_pt_lim_mult. -Apply derivable_pt_lim_const. -Replace R1 with ``1-0``; [Idtac | Ring]. -Apply derivable_pt_lim_minus. -Apply derivable_pt_lim_id. -Apply derivable_pt_lim_const. -Unfold fct_cte; Ring. -Apply derivable_pt_lim_const. -Ring. -Unfold derivable_pt_lim; Intros; Elim (H2 ? H4); Intros; Elim (H3 ? H4); Intros; Pose del := (Rmin x0 x1). -Assert H7 : ``0<del``. -Unfold del; Unfold Rmin; Case (total_order_Rle x0 x1); Intro. -Apply (cond_pos x0). -Apply (cond_pos x1). -Split with (mkposreal ? H7); Intros; Case (case_Rabsolu h0); Intro. -Assert H10 : ``a+h0<a``. -Pattern 2 a; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Assumption. -Rewrite H1; Unfold primitive; Case (total_order_Rle a ``a+h0``); Case (total_order_Rle ``a+h0`` b); Case (total_order_Rle a a); Case (total_order_Rle a b); Intros; Try (Elim n; Right; Assumption Orelse Reflexivity). -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r3 H10)). -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r2 H10)). -Rewrite RiemannInt_P9; Replace R0 with (f_a a). -Replace ``(f a)*(a+h0-a)`` with (f_a ``a+h0``). -Apply H5; Try Assumption. -Apply Rlt_le_trans with del; Try Assumption. -Unfold del; Apply Rmin_l. -Unfold f_a; Ring. -Unfold f_a; Ring. -Elim n; Rewrite <- H0; Left; Assumption. -Assert H10 : ``a<a+h0``. -Pattern 1 a; Rewrite <- Rplus_Or; Apply Rlt_compatibility. -Assert H10 := (Rle_sym2 ? ? r); Elim H10; Intro. -Assumption. -Elim H8; Symmetry; Assumption. -Rewrite H0 in H1; Rewrite H1; Unfold primitive; Case (total_order_Rle a ``b+h0``); Case (total_order_Rle ``b+h0`` b); Case (total_order_Rle a b); Case (total_order_Rle b b); Intros; Try (Elim n; Right; Assumption Orelse Reflexivity). -Rewrite H0 in H10; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r2 H10)). -Repeat Rewrite RiemannInt_P9. -Replace (RiemannInt (FTC_P1 h C0 r1 r0)) with (f_b b). -Fold (f_b ``b+h0``). -Apply H6; Try Assumption. -Apply Rlt_le_trans with del; Try Assumption. -Unfold del; Apply Rmin_r. -Unfold f_b; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rmult_Or; Rewrite Rplus_Ol; Apply RiemannInt_P5. -Elim n; Rewrite <- H0; Left; Assumption. -Elim n0; Rewrite <- H0; Left; Assumption. +assert (H1 : x = a). +rewrite <- H0 in H; elim H; intros; apply Rle_antisym; assumption. +pose (f_a := fun x:R => f a * (x - a)). +assert (H2 : derivable_pt_lim f_a a (f a)). +unfold f_a in |- *; + change (derivable_pt_lim (fct_cte (f a) * (id - fct_cte a)%F) a (f a)) + in |- *; pattern (f a) at 2 in |- *; + replace (f a) with (0 * (id - fct_cte a)%F a + fct_cte (f a) a * 1). +apply derivable_pt_lim_mult. +apply derivable_pt_lim_const. +replace 1 with (1 - 0); [ idtac | ring ]. +apply derivable_pt_lim_minus. +apply derivable_pt_lim_id. +apply derivable_pt_lim_const. +unfold fct_cte in |- *; ring. +pose + (f_b := fun x:R => f b * (x - b) + RiemannInt (FTC_P1 h C0 h (Rle_refl b))). +assert (H3 : derivable_pt_lim f_b b (f b)). +unfold f_b in |- *; pattern (f b) at 2 in |- *; replace (f b) with (f b + 0). +change + (derivable_pt_lim + ((fct_cte (f b) * (id - fct_cte b))%F + + fct_cte (RiemannInt (FTC_P1 h C0 h (Rle_refl b)))) b ( + f b + 0)) in |- *. +apply derivable_pt_lim_plus. +pattern (f b) at 2 in |- *; + replace (f b) with (0 * (id - fct_cte b)%F b + fct_cte (f b) b * 1). +apply derivable_pt_lim_mult. +apply derivable_pt_lim_const. +replace 1 with (1 - 0); [ idtac | ring ]. +apply derivable_pt_lim_minus. +apply derivable_pt_lim_id. +apply derivable_pt_lim_const. +unfold fct_cte in |- *; ring. +apply derivable_pt_lim_const. +ring. +unfold derivable_pt_lim in |- *; intros; elim (H2 _ H4); intros; + elim (H3 _ H4); intros; pose (del := Rmin x0 x1). +assert (H7 : 0 < del). +unfold del in |- *; unfold Rmin in |- *; case (Rle_dec x0 x1); intro. +apply (cond_pos x0). +apply (cond_pos x1). +split with (mkposreal _ H7); intros; case (Rcase_abs h0); intro. +assert (H10 : a + h0 < a). +pattern a at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + assumption. +rewrite H1; unfold primitive in |- *; case (Rle_dec a (a + h0)); + case (Rle_dec (a + h0) b); case (Rle_dec a a); case (Rle_dec a b); + intros; try (elim n; right; assumption || reflexivity). +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H10)). +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r2 H10)). +rewrite RiemannInt_P9; replace 0 with (f_a a). +replace (f a * (a + h0 - a)) with (f_a (a + h0)). +apply H5; try assumption. +apply Rlt_le_trans with del; try assumption. +unfold del in |- *; apply Rmin_l. +unfold f_a in |- *; ring. +unfold f_a in |- *; ring. +elim n; rewrite <- H0; left; assumption. +assert (H10 : a < a + h0). +pattern a at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. +assert (H10 := Rge_le _ _ r); elim H10; intro. +assumption. +elim H8; symmetry in |- *; assumption. +rewrite H0 in H1; rewrite H1; unfold primitive in |- *; + case (Rle_dec a (b + h0)); case (Rle_dec (b + h0) b); + case (Rle_dec a b); case (Rle_dec b b); intros; + try (elim n; right; assumption || reflexivity). +rewrite H0 in H10; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r2 H10)). +repeat rewrite RiemannInt_P9. +replace (RiemannInt (FTC_P1 h C0 r1 r0)) with (f_b b). +fold (f_b (b + h0)) in |- *. +apply H6; try assumption. +apply Rlt_le_trans with del; try assumption. +unfold del in |- *; apply Rmin_r. +unfold f_b in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rmult_0_r; rewrite Rplus_0_l; apply RiemannInt_P5. +elim n; rewrite <- H0; left; assumption. +elim n0; rewrite <- H0; left; assumption. Qed. -Lemma RiemannInt_P29 : (f:R->R;a,b;h:``a<=b``;C0:((x:R)``a<=x<=b``->(continuity_pt f x))) (antiderivative f (primitive h (FTC_P1 h C0)) a b). -Intro f; Intros; Unfold antiderivative; Split; Try Assumption; Intros; Assert H0 := (RiemannInt_P28 h C0 H); Assert H1 : (derivable_pt (primitive h (FTC_P1 h C0)) x); [Unfold derivable_pt; Split with (f x); Apply H0 | Split with H1; Symmetry; Apply derive_pt_eq_0; Apply H0]. +Lemma RiemannInt_P29 : + forall (f:R -> R) a b (h:a <= b) + (C0:forall x:R, a <= x <= b -> continuity_pt f x), + antiderivative f (primitive h (FTC_P1 h C0)) a b. +intro f; intros; unfold antiderivative in |- *; split; try assumption; intros; + assert (H0 := RiemannInt_P28 h C0 H); + assert (H1 : derivable_pt (primitive h (FTC_P1 h C0)) x); + [ unfold derivable_pt in |- *; split with (f x); apply H0 + | split with H1; symmetry in |- *; apply derive_pt_eq_0; apply H0 ]. Qed. -Lemma RiemannInt_P30 : (f:R->R;a,b:R) ``a<=b`` -> ((x:R)``a<=x<=b``->(continuity_pt f x)) -> (sigTT ? [g:R->R](antiderivative f g a b)). -Intros; Split with (primitive H (FTC_P1 H H0)); Apply RiemannInt_P29. +Lemma RiemannInt_P30 : + forall (f:R -> R) (a b:R), + a <= b -> + (forall x:R, a <= x <= b -> continuity_pt f x) -> + sigT (fun g:R -> R => antiderivative f g a b). +intros; split with (primitive H (FTC_P1 H H0)); apply RiemannInt_P29. Qed. -Record C1_fun : Type := mkC1 { -c1 :> R->R; -diff0 : (derivable c1); -cont1 : (continuity (derive c1 diff0)) }. +Record C1_fun : Type := mkC1 + {c1 :> R -> R; diff0 : derivable c1; cont1 : continuity (derive c1 diff0)}. -Lemma RiemannInt_P31 : (f:C1_fun;a,b:R) ``a<=b`` -> (antiderivative (derive f (diff0 f)) f a b). -Intro f; Intros; Unfold antiderivative; Split; Try Assumption; Intros; Split with (diff0 f x); Reflexivity. +Lemma RiemannInt_P31 : + forall (f:C1_fun) (a b:R), + a <= b -> antiderivative (derive f (diff0 f)) f a b. +intro f; intros; unfold antiderivative in |- *; split; try assumption; intros; + split with (diff0 f x); reflexivity. Qed. -Lemma RiemannInt_P32 : (f:C1_fun;a,b:R) (Riemann_integrable (derive f (diff0 f)) a b). -Intro f; Intros; Case (total_order_Rle a b); Intro; [Apply continuity_implies_RiemannInt; Try Assumption; Intros; Apply (cont1 f) | Assert H : ``b<=a``; [Auto with real | Apply RiemannInt_P1; Apply continuity_implies_RiemannInt; Try Assumption; Intros; Apply (cont1 f)]]. +Lemma RiemannInt_P32 : + forall (f:C1_fun) (a b:R), Riemann_integrable (derive f (diff0 f)) a b. +intro f; intros; case (Rle_dec a b); intro; + [ apply continuity_implies_RiemannInt; try assumption; intros; + apply (cont1 f) + | assert (H : b <= a); + [ auto with real + | apply RiemannInt_P1; apply continuity_implies_RiemannInt; + try assumption; intros; apply (cont1 f) ] ]. Qed. -Lemma RiemannInt_P33 : (f:C1_fun;a,b:R;pr:(Riemann_integrable (derive f (diff0 f)) a b)) ``a<=b`` -> (RiemannInt pr)==``(f b)-(f a)``. -Intro f; Intros; Assert H0 : (x:R)``a<=x<=b``->(continuity_pt (derive f (diff0 f)) x). -Intros; Apply (cont1 f). -Rewrite (RiemannInt_P20 H (FTC_P1 H H0) pr); Assert H1 := (RiemannInt_P29 H H0); Assert H2 := (RiemannInt_P31 f H); Elim (antiderivative_Ucte (derive f (diff0 f)) ? ? ? ? H1 H2); Intros C H3; Repeat Rewrite H3; [Ring | Split; [Right; Reflexivity | Assumption] | Split; [Assumption | Right; Reflexivity]]. +Lemma RiemannInt_P33 : + forall (f:C1_fun) (a b:R) (pr:Riemann_integrable (derive f (diff0 f)) a b), + a <= b -> RiemannInt pr = f b - f a. +intro f; intros; + assert + (H0 : forall x:R, a <= x <= b -> continuity_pt (derive f (diff0 f)) x). +intros; apply (cont1 f). +rewrite (RiemannInt_P20 H (FTC_P1 H H0) pr); + assert (H1 := RiemannInt_P29 H H0); assert (H2 := RiemannInt_P31 f H); + elim (antiderivative_Ucte (derive f (diff0 f)) _ _ _ _ H1 H2); + intros C H3; repeat rewrite H3; + [ ring + | split; [ right; reflexivity | assumption ] + | split; [ assumption | right; reflexivity ] ]. Qed. -Lemma FTC_Riemann : (f:C1_fun;a,b:R;pr:(Riemann_integrable (derive f (diff0 f)) a b)) (RiemannInt pr)==``(f b)-(f a)``. -Intro f; Intros; Case (total_order_Rle a b); Intro; [Apply RiemannInt_P33; Assumption | Assert H : ``b<=a``; [Auto with real | Assert H0 := (RiemannInt_P1 pr); Rewrite (RiemannInt_P8 pr H0); Rewrite (RiemannInt_P33 H0 H); Ring]]. -Qed. +Lemma FTC_Riemann : + forall (f:C1_fun) (a b:R) (pr:Riemann_integrable (derive f (diff0 f)) a b), + RiemannInt pr = f b - f a. +intro f; intros; case (Rle_dec a b); intro; + [ apply RiemannInt_P33; assumption + | assert (H : b <= a); + [ auto with real + | assert (H0 := RiemannInt_P1 pr); rewrite (RiemannInt_P8 pr H0); + rewrite (RiemannInt_P33 _ H0 H); ring ] ]. +Qed.
\ No newline at end of file diff --git a/theories/Reals/RiemannInt_SF.v b/theories/Reals/RiemannInt_SF.v index f81c57997..5f47466ac 100644 --- a/theories/Reals/RiemannInt_SF.v +++ b/theories/Reals/RiemannInt_SF.v @@ -8,1393 +8,2625 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Ranalysis. -Require Classical_Prop. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import Ranalysis. +Require Import Classical_Prop. Open Local Scope R_scope. -Implicit Arguments On. +Set Implicit Arguments. (**************************************************) (* Each bounded subset of N has a maximal element *) (**************************************************) -Definition Nbound [I:nat->Prop] : Prop := (EX n:nat | (i:nat)(I i)->(le i n)). +Definition Nbound (I:nat -> Prop) : Prop := + exists n : nat | (forall i:nat, I i -> (i <= n)%nat). -Lemma IZN_var:(z:Z)(`0<=z`)->{ n:nat | z=(INZ n)}. -Intros; Apply inject_nat_complete_inf; Assumption. +Lemma IZN_var : forall z:Z, (0 <= z)%Z -> {n : nat | z = Z_of_nat n}. +intros; apply Z_of_nat_complete_inf; assumption. Qed. -Lemma Nzorn : (I:nat->Prop) (EX n:nat | (I n)) -> (Nbound I) -> (sigTT ? [n:nat](I n)/\(i:nat)(I i)->(le i n)). -Intros I H H0; Pose E := [x:R](EX i:nat | (I i)/\(INR i)==x); Assert H1 : (bound E). -Unfold Nbound in H0; Elim H0; Intros N H1; Unfold bound; Exists (INR N); Unfold is_upper_bound; Intros; Unfold E in H2; Elim H2; Intros; Elim H3; Intros; Rewrite <- H5; Apply le_INR; Apply H1; Assumption. -Assert H2 : (EXT x:R | (E x)). -Elim H; Intros; Exists (INR x); Unfold E; Exists x; Split; [Assumption | Reflexivity]. -Assert H3 := (complet E H1 H2); Elim H3; Intros; Unfold is_lub in p; Elim p; Clear p; Intros; Unfold is_upper_bound in H4 H5; Assert H6 : ``0<=x``. -Elim H2; Intros; Unfold E in H6; Elim H6; Intros; Elim H7; Intros; Apply Rle_trans with x0; [Rewrite <- H9; Change ``(INR O)<=(INR x1)``; Apply le_INR; Apply le_O_n | Apply H4; Assumption]. -Assert H7 := (archimed x); Elim H7; Clear H7; Intros; Assert H9 : ``x<=(IZR (up x))-1``. -Apply H5; Intros; Assert H10 := (H4 ? H9); Unfold E in H9; Elim H9; Intros; Elim H11; Intros; Rewrite <- H13; Apply Rle_anti_compatibility with R1; Replace ``1+((IZR (up x))-1)`` with (IZR (up x)); [Idtac | Ring]; Replace ``1+(INR x1)`` with (INR (S x1)); [Idtac | Rewrite S_INR; Ring]. -Assert H14 : `0<=(up x)`. -Apply le_IZR; Apply Rle_trans with x; [Apply H6 | Left; Assumption]. -Assert H15 := (IZN ? H14); Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- INR_IZR_INZ; Apply le_INR; Apply lt_le_S; Apply INR_lt; Rewrite H13; Apply Rle_lt_trans with x; [Assumption | Rewrite INR_IZR_INZ; Rewrite <- H15; Assumption]. -Assert H10 : ``x==(IZR (up x))-1``. -Apply Rle_antisym; [Assumption | Apply Rle_anti_compatibility with ``-x+1``; Replace `` -x+1+((IZR (up x))-1)`` with ``(IZR (up x))-x``; [Idtac | Ring]; Replace ``-x+1+x`` with R1; [Assumption | Ring]]. -Assert H11 : `0<=(up x)`. -Apply le_IZR; Apply Rle_trans with x; [Apply H6 | Left; Assumption]. -Assert H12 := (IZN_var H11); Elim H12; Clear H12; Intros; Assert H13 : (E x). -Elim (classic (E x)); Intro; Try Assumption. -Cut ((y:R)(E y)->``y<=x-1``). -Intro; Assert H14 := (H5 ? H13); Cut ``x-1<x``. -Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H14 H15)). -Apply Rminus_lt; Replace ``x-1-x`` with ``-1``; [Idtac | Ring]; Rewrite <- Ropp_O; Apply Rlt_Ropp; Apply Rlt_R0_R1. -Intros; Assert H14 := (H4 ? H13); Elim H14; Intro; Unfold E in H13; Elim H13; Intros; Elim H16; Intros; Apply Rle_anti_compatibility with R1. -Replace ``1+(x-1)`` with x; [Idtac | Ring]; Rewrite <- H18; Replace ``1+(INR x1)`` with (INR (S x1)); [Idtac | Rewrite S_INR; Ring]. -Cut x==(INR (pred x0)). -Intro; Rewrite H19; Apply le_INR; Apply lt_le_S; Apply INR_lt; Rewrite H18; Rewrite <- H19; Assumption. -Rewrite H10; Rewrite p; Rewrite <- INR_IZR_INZ; Replace R1 with (INR (S O)); [Idtac | Reflexivity]; Rewrite <- minus_INR. -Replace (minus x0 (S O)) with (pred x0); [Reflexivity | Case x0; [Reflexivity | Intro; Simpl; Apply minus_n_O]]. -Induction x0; [Rewrite p in H7; Rewrite <- INR_IZR_INZ in H7; Simpl in H7; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H6 H7)) | Apply le_n_S; Apply le_O_n]. -Rewrite H15 in H13; Elim H12; Assumption. -Split with (pred x0); Unfold E in H13; Elim H13; Intros; Elim H12; Intros; Rewrite H10 in H15; Rewrite p in H15; Rewrite <- INR_IZR_INZ in H15; Assert H16 : ``(INR x0)==(INR x1)+1``. -Rewrite H15; Ring. -Rewrite <- S_INR in H16; Assert H17 := (INR_eq ? ? H16); Rewrite H17; Simpl; Split. -Assumption. -Intros; Apply INR_le; Rewrite H15; Rewrite <- H15; Elim H12; Intros; Rewrite H20; Apply H4; Unfold E; Exists i; Split; [Assumption | Reflexivity]. +Lemma Nzorn : + forall I:nat -> Prop, + ( exists n : nat | I n) -> + Nbound I -> sigT (fun n:nat => I n /\ (forall i:nat, I i -> (i <= n)%nat)). +intros I H H0; pose (E := fun x:R => exists i : nat | I i /\ INR i = x); + assert (H1 : bound E). +unfold Nbound in H0; elim H0; intros N H1; unfold bound in |- *; + exists (INR N); unfold is_upper_bound in |- *; intros; + unfold E in H2; elim H2; intros; elim H3; intros; + rewrite <- H5; apply le_INR; apply H1; assumption. +assert (H2 : exists x : R | E x). +elim H; intros; exists (INR x); unfold E in |- *; exists x; split; + [ assumption | reflexivity ]. +assert (H3 := completeness E H1 H2); elim H3; intros; unfold is_lub in p; + elim p; clear p; intros; unfold is_upper_bound in H4, H5; + assert (H6 : 0 <= x). +elim H2; intros; unfold E in H6; elim H6; intros; elim H7; intros; + apply Rle_trans with x0; + [ rewrite <- H9; change (INR 0 <= INR x1) in |- *; apply le_INR; + apply le_O_n + | apply H4; assumption ]. +assert (H7 := archimed x); elim H7; clear H7; intros; + assert (H9 : x <= IZR (up x) - 1). +apply H5; intros; assert (H10 := H4 _ H9); unfold E in H9; elim H9; intros; + elim H11; intros; rewrite <- H13; apply Rplus_le_reg_l with 1; + replace (1 + (IZR (up x) - 1)) with (IZR (up x)); + [ idtac | ring ]; replace (1 + INR x1) with (INR (S x1)); + [ idtac | rewrite S_INR; ring ]. +assert (H14 : (0 <= up x)%Z). +apply le_IZR; apply Rle_trans with x; [ apply H6 | left; assumption ]. +assert (H15 := IZN _ H14); elim H15; clear H15; intros; rewrite H15; + rewrite <- INR_IZR_INZ; apply le_INR; apply lt_le_S; + apply INR_lt; rewrite H13; apply Rle_lt_trans with x; + [ assumption | rewrite INR_IZR_INZ; rewrite <- H15; assumption ]. +assert (H10 : x = IZR (up x) - 1). +apply Rle_antisym; + [ assumption + | apply Rplus_le_reg_l with (- x + 1); + replace (- x + 1 + (IZR (up x) - 1)) with (IZR (up x) - x); + [ idtac | ring ]; replace (- x + 1 + x) with 1; + [ assumption | ring ] ]. +assert (H11 : (0 <= up x)%Z). +apply le_IZR; apply Rle_trans with x; [ apply H6 | left; assumption ]. +assert (H12 := IZN_var H11); elim H12; clear H12; intros; assert (H13 : E x). +elim (classic (E x)); intro; try assumption. +cut (forall y:R, E y -> y <= x - 1). +intro; assert (H14 := H5 _ H13); cut (x - 1 < x). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H14 H15)). +apply Rminus_lt; replace (x - 1 - x) with (-1); [ idtac | ring ]; + rewrite <- Ropp_0; apply Ropp_lt_gt_contravar; apply Rlt_0_1. +intros; assert (H14 := H4 _ H13); elim H14; intro; unfold E in H13; elim H13; + intros; elim H16; intros; apply Rplus_le_reg_l with 1. +replace (1 + (x - 1)) with x; [ idtac | ring ]; rewrite <- H18; + replace (1 + INR x1) with (INR (S x1)); [ idtac | rewrite S_INR; ring ]. +cut (x = INR (pred x0)). +intro; rewrite H19; apply le_INR; apply lt_le_S; apply INR_lt; rewrite H18; + rewrite <- H19; assumption. +rewrite H10; rewrite p; rewrite <- INR_IZR_INZ; replace 1 with (INR 1); + [ idtac | reflexivity ]; rewrite <- minus_INR. +replace (x0 - 1)%nat with (pred x0); + [ reflexivity + | case x0; [ reflexivity | intro; simpl in |- *; apply minus_n_O ] ]. +induction x0 as [| x0 Hrecx0]; + [ rewrite p in H7; rewrite <- INR_IZR_INZ in H7; simpl in H7; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H6 H7)) + | apply le_n_S; apply le_O_n ]. +rewrite H15 in H13; elim H12; assumption. +split with (pred x0); unfold E in H13; elim H13; intros; elim H12; intros; + rewrite H10 in H15; rewrite p in H15; rewrite <- INR_IZR_INZ in H15; + assert (H16 : INR x0 = INR x1 + 1). +rewrite H15; ring. +rewrite <- S_INR in H16; assert (H17 := INR_eq _ _ H16); rewrite H17; + simpl in |- *; split. +assumption. +intros; apply INR_le; rewrite H15; rewrite <- H15; elim H12; intros; + rewrite H20; apply H4; unfold E in |- *; exists i; + split; [ assumption | reflexivity ]. Qed. (*******************************************) (* Step functions *) (*******************************************) -Definition open_interval [a,b:R] : R->Prop := [x:R]``a<x<b``. -Definition co_interval [a,b:R] : R->Prop := [x:R]``a<=x<b``. +Definition open_interval (a b x:R) : Prop := a < x < b. +Definition co_interval (a b x:R) : Prop := a <= x < b. -Definition adapted_couple [f:R->R;a,b:R;l,lf:Rlist] : Prop := (ordered_Rlist l)/\``(pos_Rl l O)==(Rmin a b)``/\``(pos_Rl l (pred (Rlength l)))==(Rmax a b)``/\(Rlength l)=(S (Rlength lf))/\(i:nat)(lt i (pred (Rlength l)))->(constant_D_eq f (open_interval (pos_Rl l i) (pos_Rl l (S i))) (pos_Rl lf i)). +Definition adapted_couple (f:R -> R) (a b:R) (l lf:Rlist) : Prop := + ordered_Rlist l /\ + pos_Rl l 0 = Rmin a b /\ + pos_Rl l (pred (Rlength l)) = Rmax a b /\ + Rlength l = S (Rlength lf) /\ + (forall i:nat, + (i < pred (Rlength l))%nat -> + constant_D_eq f (open_interval (pos_Rl l i) (pos_Rl l (S i))) + (pos_Rl lf i)). -Definition adapted_couple_opt [f:R->R;a,b:R;l,lf:Rlist] := (adapted_couple f a b l lf)/\((i:nat)(lt i (pred (Rlength lf)))->(``(pos_Rl lf i)<>(pos_Rl lf (S i))``\/``(f (pos_Rl l (S i)))<>(pos_Rl lf i)``))/\((i:nat)(lt i (pred (Rlength l)))->``(pos_Rl l i)<>(pos_Rl l (S i))``). +Definition adapted_couple_opt (f:R -> R) (a b:R) (l lf:Rlist) := + adapted_couple f a b l lf /\ + (forall i:nat, + (i < pred (Rlength lf))%nat -> + pos_Rl lf i <> pos_Rl lf (S i) \/ f (pos_Rl l (S i)) <> pos_Rl lf i) /\ + (forall i:nat, (i < pred (Rlength l))%nat -> pos_Rl l i <> pos_Rl l (S i)). -Definition is_subdivision [f:R->R;a,b:R;l:Rlist] : Type := (sigTT ? [l0:Rlist](adapted_couple f a b l l0)). +Definition is_subdivision (f:R -> R) (a b:R) (l:Rlist) : Type := + sigT (fun l0:Rlist => adapted_couple f a b l l0). -Definition IsStepFun [f:R->R;a,b:R] : Type := (SigT ? [l:Rlist](is_subdivision f a b l)). +Definition IsStepFun (f:R -> R) (a b:R) : Type := + sigT (fun l:Rlist => is_subdivision f a b l). (* Class of step functions *) -Record StepFun [a,b:R] : Type := mkStepFun { - fe:> R->R; - pre:(IsStepFun fe a b)}. - -Definition subdivision [a,b:R;f:(StepFun a b)] : Rlist := (projT1 ? ? (pre f)). - -Definition subdivision_val [a,b:R;f:(StepFun a b)] : Rlist := Cases (projT2 ? ? (pre f)) of (existTT a b) => a end. - -Fixpoint Int_SF [l:Rlist] : Rlist -> R := -[k:Rlist] Cases l of -| nil => R0 -| (cons a l') => Cases k of - | nil => R0 - | (cons x nil) => R0 - | (cons x (cons y k')) => ``a*(y-x)+(Int_SF l' (cons y k'))`` - end -end. +Record StepFun (a b:R) : Type := mkStepFun + {fe :> R -> R; pre : IsStepFun fe a b}. + +Definition subdivision (a b:R) (f:StepFun a b) : Rlist := projT1 (pre f). + +Definition subdivision_val (a b:R) (f:StepFun a b) : Rlist := + match projT2 (pre f) with + | existT a b => a + end. + +Fixpoint Int_SF (l k:Rlist) {struct l} : R := + match l with + | nil => 0 + | cons a l' => + match k with + | nil => 0 + | cons x nil => 0 + | cons x (cons y k') => a * (y - x) + Int_SF l' (cons y k') + end + end. (* Integral of step functions *) -Definition RiemannInt_SF [a,b:R;f:(StepFun a b)] : R := -Cases (total_order_Rle a b) of - (leftT _) => (Int_SF (subdivision_val f) (subdivision f)) -| (rightT _) => ``-(Int_SF (subdivision_val f) (subdivision f))`` -end. +Definition RiemannInt_SF (a b:R) (f:StepFun a b) : R := + match Rle_dec a b with + | left _ => Int_SF (subdivision_val f) (subdivision f) + | right _ => - Int_SF (subdivision_val f) (subdivision f) + end. (********************************) (* Properties of step functions *) (********************************) -Lemma StepFun_P1 : (a,b:R;f:(StepFun a b)) (adapted_couple f a b (subdivision f) (subdivision_val f)). -Intros a b f; Unfold subdivision_val; Case (projT2 Rlist ([l:Rlist](is_subdivision f a b l)) (pre f)); Intros; Apply a0. +Lemma StepFun_P1 : + forall (a b:R) (f:StepFun a b), + adapted_couple f a b (subdivision f) (subdivision_val f). +intros a b f; unfold subdivision_val in |- *; case (projT2 (pre f)); intros; + apply a0. Qed. -Lemma StepFun_P2 : (a,b:R;f:R->R;l,lf:Rlist) (adapted_couple f a b l lf) -> (adapted_couple f b a l lf). -Unfold adapted_couple; Intros; Decompose [and] H; Clear H; Repeat Split; Try Assumption. -Rewrite H2; Unfold Rmin; Case (total_order_Rle a b); Intro; Case (total_order_Rle b a); Intro; Try Reflexivity. -Apply Rle_antisym; Assumption. -Apply Rle_antisym; Auto with real. -Rewrite H1; Unfold Rmax; Case (total_order_Rle a b); Intro; Case (total_order_Rle b a); Intro; Try Reflexivity. -Apply Rle_antisym; Assumption. -Apply Rle_antisym; Auto with real. +Lemma StepFun_P2 : + forall (a b:R) (f:R -> R) (l lf:Rlist), + adapted_couple f a b l lf -> adapted_couple f b a l lf. +unfold adapted_couple in |- *; intros; decompose [and] H; clear H; + repeat split; try assumption. +rewrite H2; unfold Rmin in |- *; case (Rle_dec a b); intro; + case (Rle_dec b a); intro; try reflexivity. +apply Rle_antisym; assumption. +apply Rle_antisym; auto with real. +rewrite H1; unfold Rmax in |- *; case (Rle_dec a b); intro; + case (Rle_dec b a); intro; try reflexivity. +apply Rle_antisym; assumption. +apply Rle_antisym; auto with real. Qed. -Lemma StepFun_P3 : (a,b,c:R) ``a<=b`` -> (adapted_couple (fct_cte c) a b (cons a (cons b nil)) (cons c nil)). -Intros; Unfold adapted_couple; Repeat Split. -Unfold ordered_Rlist; Intros; Simpl in H0; Inversion H0; [Simpl; Assumption | Elim (le_Sn_O ? H2)]. -Simpl; Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Simpl; Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Unfold constant_D_eq open_interval; Intros; Simpl in H0; Inversion H0; [Reflexivity | Elim (le_Sn_O ? H3)]. +Lemma StepFun_P3 : + forall a b c:R, + a <= b -> + adapted_couple (fct_cte c) a b (cons a (cons b nil)) (cons c nil). +intros; unfold adapted_couple in |- *; repeat split. +unfold ordered_Rlist in |- *; intros; simpl in H0; inversion H0; + [ simpl in |- *; assumption | elim (le_Sn_O _ H2) ]. +simpl in |- *; unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +simpl in |- *; unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +unfold constant_D_eq, open_interval in |- *; intros; simpl in H0; + inversion H0; [ reflexivity | elim (le_Sn_O _ H3) ]. Qed. -Lemma StepFun_P4 : (a,b,c:R) (IsStepFun (fct_cte c) a b). -Intros; Unfold IsStepFun; Case (total_order_Rle a b); Intro. -Apply Specif.existT with (cons a (cons b nil)); Unfold is_subdivision; Apply existTT with (cons c nil); Apply (StepFun_P3 c r). -Apply Specif.existT with (cons b (cons a nil)); Unfold is_subdivision; Apply existTT with (cons c nil); Apply StepFun_P2; Apply StepFun_P3; Auto with real. +Lemma StepFun_P4 : forall a b c:R, IsStepFun (fct_cte c) a b. +intros; unfold IsStepFun in |- *; case (Rle_dec a b); intro. +apply existT with (cons a (cons b nil)); unfold is_subdivision in |- *; + apply existT with (cons c nil); apply (StepFun_P3 c r). +apply existT with (cons b (cons a nil)); unfold is_subdivision in |- *; + apply existT with (cons c nil); apply StepFun_P2; + apply StepFun_P3; auto with real. Qed. -Lemma StepFun_P5 : (a,b:R;f:R->R;l:Rlist) (is_subdivision f a b l) -> (is_subdivision f b a l). -Unfold is_subdivision; Intros; Elim X; Intros; Exists x; Unfold adapted_couple in p; Decompose [and] p; Clear p; Unfold adapted_couple; Repeat Split; Try Assumption. -Rewrite H1; Unfold Rmin; Case (total_order_Rle a b); Intro; Case (total_order_Rle b a); Intro; Try Reflexivity. -Apply Rle_antisym; Assumption. -Apply Rle_antisym; Auto with real. -Rewrite H0; Unfold Rmax; Case (total_order_Rle a b); Intro; Case (total_order_Rle b a); Intro; Try Reflexivity. -Apply Rle_antisym; Assumption. -Apply Rle_antisym; Auto with real. +Lemma StepFun_P5 : + forall (a b:R) (f:R -> R) (l:Rlist), + is_subdivision f a b l -> is_subdivision f b a l. +unfold is_subdivision in |- *; intros; elim X; intros; exists x; + unfold adapted_couple in p; decompose [and] p; clear p; + unfold adapted_couple in |- *; repeat split; try assumption. +rewrite H1; unfold Rmin in |- *; case (Rle_dec a b); intro; + case (Rle_dec b a); intro; try reflexivity. +apply Rle_antisym; assumption. +apply Rle_antisym; auto with real. +rewrite H0; unfold Rmax in |- *; case (Rle_dec a b); intro; + case (Rle_dec b a); intro; try reflexivity. +apply Rle_antisym; assumption. +apply Rle_antisym; auto with real. Qed. -Lemma StepFun_P6 : (f:R->R;a,b:R) (IsStepFun f a b) -> (IsStepFun f b a). -Unfold IsStepFun; Intros; Elim X; Intros; Apply Specif.existT with x; Apply StepFun_P5; Assumption. +Lemma StepFun_P6 : + forall (f:R -> R) (a b:R), IsStepFun f a b -> IsStepFun f b a. +unfold IsStepFun in |- *; intros; elim X; intros; apply existT with x; + apply StepFun_P5; assumption. Qed. -Lemma StepFun_P7 : (a,b,r1,r2,r3:R;f:R->R;l,lf:Rlist) ``a<=b`` -> (adapted_couple f a b (cons r1 (cons r2 l)) (cons r3 lf)) -> (adapted_couple f r2 b (cons r2 l) lf). -Unfold adapted_couple; Intros; Decompose [and] H0; Clear H0; Assert H5 : (Rmax a b)==b. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Assert H7 : ``r2<=b``. -Rewrite H5 in H2; Rewrite <- H2; Apply RList_P7; [Assumption | Simpl; Right; Left; Reflexivity]. -Repeat Split. -Apply RList_P4 with r1; Assumption. -Rewrite H5 in H2; Unfold Rmin; Case (total_order_Rle r2 b); Intro; [Reflexivity | Elim n; Assumption]. -Unfold Rmax; Case (total_order_Rle r2 b); Intro; [Rewrite H5 in H2; Rewrite <- H2; Reflexivity | Elim n; Assumption]. -Simpl in H4; Simpl; Apply INR_eq; Apply r_Rplus_plus with R1; Do 2 Rewrite (Rplus_sym R1); Do 2 Rewrite <- S_INR; Rewrite H4; Reflexivity. -Intros; Unfold constant_D_eq open_interval; Intros; Unfold constant_D_eq open_interval in H6; Assert H9 : (lt (S i) (pred (Rlength (cons r1 (cons r2 l))))). -Simpl; Simpl in H0; Apply lt_n_S; Assumption. -Assert H10 := (H6 ? H9); Apply H10; Assumption. +Lemma StepFun_P7 : + forall (a b r1 r2 r3:R) (f:R -> R) (l lf:Rlist), + a <= b -> + adapted_couple f a b (cons r1 (cons r2 l)) (cons r3 lf) -> + adapted_couple f r2 b (cons r2 l) lf. +unfold adapted_couple in |- *; intros; decompose [and] H0; clear H0; + assert (H5 : Rmax a b = b). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +assert (H7 : r2 <= b). +rewrite H5 in H2; rewrite <- H2; apply RList_P7; + [ assumption | simpl in |- *; right; left; reflexivity ]. +repeat split. +apply RList_P4 with r1; assumption. +rewrite H5 in H2; unfold Rmin in |- *; case (Rle_dec r2 b); intro; + [ reflexivity | elim n; assumption ]. +unfold Rmax in |- *; case (Rle_dec r2 b); intro; + [ rewrite H5 in H2; rewrite <- H2; reflexivity | elim n; assumption ]. +simpl in H4; simpl in |- *; apply INR_eq; apply Rplus_eq_reg_l with 1; + do 2 rewrite (Rplus_comm 1); do 2 rewrite <- S_INR; + rewrite H4; reflexivity. +intros; unfold constant_D_eq, open_interval in |- *; intros; + unfold constant_D_eq, open_interval in H6; + assert (H9 : (S i < pred (Rlength (cons r1 (cons r2 l))))%nat). +simpl in |- *; simpl in H0; apply lt_n_S; assumption. +assert (H10 := H6 _ H9); apply H10; assumption. Qed. -Lemma StepFun_P8 : (f:R->R;l1,lf1:Rlist;a,b:R) (adapted_couple f a b l1 lf1) -> a==b -> (Int_SF lf1 l1)==R0. -Induction l1. -Intros; Induction lf1; Reflexivity. -Induction r0. -Intros; Induction lf1. -Reflexivity. -Unfold adapted_couple in H0; Decompose [and] H0; Clear H0; Simpl in H5; Discriminate. -Intros; Induction lf1. -Reflexivity. -Simpl; Cut r==r1. -Intro; Rewrite H3; Rewrite (H0 lf1 r b). -Ring. -Rewrite H3; Apply StepFun_P7 with a r r3; [Right; Assumption | Assumption]. -Clear H H0 Hreclf1 r0; Unfold adapted_couple in H1; Decompose [and] H1; Intros; Simpl in H4; Rewrite H4; Unfold Rmin; Case (total_order_Rle a b); Intro; [Assumption | Reflexivity]. -Unfold adapted_couple in H1; Decompose [and] H1; Intros; Apply Rle_antisym. -Apply (H3 O); Simpl; Apply lt_O_Sn. -Simpl in H5; Rewrite H2 in H5; Rewrite H5; Replace (Rmin b b) with (Rmax a b); [Rewrite <- H4; Apply RList_P7; [Assumption | Simpl; Right; Left; Reflexivity] | Unfold Rmin Rmax; Case (total_order_Rle b b); Case (total_order_Rle a b); Intros; Try Assumption Orelse Reflexivity]. +Lemma StepFun_P8 : + forall (f:R -> R) (l1 lf1:Rlist) (a b:R), + adapted_couple f a b l1 lf1 -> a = b -> Int_SF lf1 l1 = 0. +simple induction l1. +intros; induction lf1 as [| r lf1 Hreclf1]; reflexivity. +simple induction r0. +intros; induction lf1 as [| r1 lf1 Hreclf1]. +reflexivity. +unfold adapted_couple in H0; decompose [and] H0; clear H0; simpl in H5; + discriminate. +intros; induction lf1 as [| r3 lf1 Hreclf1]. +reflexivity. +simpl in |- *; cut (r = r1). +intro; rewrite H3; rewrite (H0 lf1 r b). +ring. +rewrite H3; apply StepFun_P7 with a r r3; [ right; assumption | assumption ]. +clear H H0 Hreclf1 r0; unfold adapted_couple in H1; decompose [and] H1; + intros; simpl in H4; rewrite H4; unfold Rmin in |- *; + case (Rle_dec a b); intro; [ assumption | reflexivity ]. +unfold adapted_couple in H1; decompose [and] H1; intros; apply Rle_antisym. +apply (H3 0%nat); simpl in |- *; apply lt_O_Sn. +simpl in H5; rewrite H2 in H5; rewrite H5; replace (Rmin b b) with (Rmax a b); + [ rewrite <- H4; apply RList_P7; + [ assumption | simpl in |- *; right; left; reflexivity ] + | unfold Rmin, Rmax in |- *; case (Rle_dec b b); case (Rle_dec a b); intros; + try assumption || reflexivity ]. Qed. -Lemma StepFun_P9 : (a,b:R;f:R->R;l,lf:Rlist) (adapted_couple f a b l lf) -> ``a<>b`` -> (le (2) (Rlength l)). -Intros; Unfold adapted_couple in H; Decompose [and] H; Clear H; Induction l; [Simpl in H4; Discriminate | Induction l; [Simpl in H3; Simpl in H2; Generalize H3; Generalize H2; Unfold Rmin Rmax; Case (total_order_Rle a b); Intros; Elim H0; Rewrite <- H5; Rewrite <- H7; Reflexivity | Simpl; Do 2 Apply le_n_S; Apply le_O_n]]. +Lemma StepFun_P9 : + forall (a b:R) (f:R -> R) (l lf:Rlist), + adapted_couple f a b l lf -> a <> b -> (2 <= Rlength l)%nat. +intros; unfold adapted_couple in H; decompose [and] H; clear H; + induction l as [| r l Hrecl]; + [ simpl in H4; discriminate + | induction l as [| r0 l Hrecl0]; + [ simpl in H3; simpl in H2; generalize H3; generalize H2; + unfold Rmin, Rmax in |- *; case (Rle_dec a b); + intros; elim H0; rewrite <- H5; rewrite <- H7; + reflexivity + | simpl in |- *; do 2 apply le_n_S; apply le_O_n ] ]. Qed. -Lemma StepFun_P10 : (f:R->R;l,lf:Rlist;a,b:R) ``a<=b`` -> (adapted_couple f a b l lf) -> (EXT l':Rlist | (EXT lf':Rlist | (adapted_couple_opt f a b l' lf'))). -Induction l. -Intros; Unfold adapted_couple in H0; Decompose [and] H0; Simpl in H4; Discriminate. -Intros; Case (Req_EM a b); Intro. -Exists (cons a nil); Exists nil; Unfold adapted_couple_opt; Unfold adapted_couple; Unfold ordered_Rlist; Repeat Split; Try (Intros; Simpl in H3; Elim (lt_n_O ? H3)). -Simpl; Rewrite <- H2; Unfold Rmin; Case (total_order_Rle a a); Intro; Reflexivity. -Simpl; Rewrite <- H2; Unfold Rmax; Case (total_order_Rle a a); Intro; Reflexivity. -Elim (RList_P20 ? (StepFun_P9 H1 H2)); Intros t1 [t2 [t3 H3]]; Induction lf. -Unfold adapted_couple in H1; Decompose [and] H1; Rewrite H3 in H7; Simpl in H7; Discriminate. -Clear Hreclf; Assert H4 : (adapted_couple f t2 b r0 lf). -Rewrite H3 in H1; Assert H4 := (RList_P21 ? ? H3); Simpl in H4; Rewrite H4; EApply StepFun_P7; [Apply H0 | Apply H1]. -Cut ``t2<=b``. -Intro; Assert H6 := (H ? ? ? H5 H4); Case (Req_EM t1 t2); Intro Hyp_eq. -Replace a with t2. -Apply H6. -Rewrite <- Hyp_eq; Rewrite H3 in H1; Unfold adapted_couple in H1; Decompose [and] H1; Clear H1; Simpl in H9; Rewrite H9; Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Elim H6; Clear H6; Intros l' [lf' H6]; Case (Req_EM t2 b); Intro. -Exists (cons a (cons b nil)); Exists (cons r1 nil); Unfold adapted_couple_opt; Unfold adapted_couple; Repeat Split. -Unfold ordered_Rlist; Intros; Simpl in H8; Inversion H8; [Simpl; Assumption | Elim (le_Sn_O ? H10)]. -Simpl; Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Simpl; Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Intros; Simpl in H8; Inversion H8. -Unfold constant_D_eq open_interval; Intros; Simpl; Simpl in H9; Rewrite H3 in H1; Unfold adapted_couple in H1; Decompose [and] H1; Apply (H16 O). -Simpl; Apply lt_O_Sn. -Unfold open_interval; Simpl; Rewrite H7; Simpl in H13; Rewrite H13; Unfold Rmin; Case (total_order_Rle a b); Intro; [Assumption | Elim n; Assumption]. -Elim (le_Sn_O ? H10). -Intros; Simpl in H8; Elim (lt_n_O ? H8). -Intros; Simpl in H8; Inversion H8; [Simpl; Assumption | Elim (le_Sn_O ? H10)]. -Assert Hyp_min : (Rmin t2 b)==t2. -Unfold Rmin; Case (total_order_Rle t2 b); Intro; [Reflexivity | Elim n; Assumption]. -Unfold adapted_couple in H6; Elim H6; Clear H6; Intros; Elim (RList_P20 ? (StepFun_P9 H6 H7)); Intros s1 [s2 [s3 H9]]; Induction lf'. -Unfold adapted_couple in H6; Decompose [and] H6; Rewrite H9 in H13; Simpl in H13; Discriminate. -Clear Hreclf'; Case (Req_EM r1 r2); Intro. -Case (Req_EM (f t2) r1); Intro. -Exists (cons t1 (cons s2 s3)); Exists (cons r1 lf'); Rewrite H3 in H1; Rewrite H9 in H6; Unfold adapted_couple in H6 H1; Decompose [and] H1; Decompose [and] H6; Clear H1 H6; Unfold adapted_couple_opt; Unfold adapted_couple; Repeat Split. -Unfold ordered_Rlist; Intros; Simpl in H1; Induction i. -Simpl; Apply Rle_trans with s1. -Replace s1 with t2. -Apply (H12 O). -Simpl; Apply lt_O_Sn. -Simpl in H19; Rewrite H19; Symmetry; Apply Hyp_min. -Apply (H16 O); Simpl; Apply lt_O_Sn. -Change ``(pos_Rl (cons s2 s3) i)<=(pos_Rl (cons s2 s3) (S i))``; Apply (H16 (S i)); Simpl; Assumption. -Simpl; Simpl in H14; Rewrite H14; Reflexivity. -Simpl; Simpl in H18; Rewrite H18; Unfold Rmax; Case (total_order_Rle a b); Case (total_order_Rle t2 b); Intros; Reflexivity Orelse Elim n; Assumption. -Simpl; Simpl in H20; Apply H20. -Intros; Simpl in H1; Unfold constant_D_eq open_interval; Intros; Induction i. -Simpl; Simpl in H6; Case (total_order_T x t2); Intro. -Elim s; Intro. -Apply (H17 O); [Simpl; Apply lt_O_Sn | Unfold open_interval; Simpl; Elim H6; Intros; Split; Assumption]. -Rewrite b0; Assumption. -Rewrite H10; Apply (H22 O); [Simpl; Apply lt_O_Sn | Unfold open_interval; Simpl; Replace s1 with t2; [Elim H6; Intros; Split; Assumption | Simpl in H19; Rewrite H19; Rewrite Hyp_min; Reflexivity]]. -Simpl; Simpl in H6; Apply (H22 (S i)); [Simpl; Assumption | Unfold open_interval; Simpl; Apply H6]. -Intros; Simpl in H1; Rewrite H10; Change ``(pos_Rl (cons r2 lf') i)<>(pos_Rl (cons r2 lf') (S i))``\/``(f (pos_Rl (cons s1 (cons s2 s3)) (S i)))<>(pos_Rl (cons r2 lf') i)``; Rewrite <- H9; Elim H8; Intros; Apply H6; Simpl; Apply H1. -Intros; Induction i. -Simpl; Red; Intro; Elim Hyp_eq; Apply Rle_antisym. -Apply (H12 O); Simpl; Apply lt_O_Sn. -Rewrite <- Hyp_min; Rewrite H6; Simpl in H19; Rewrite <- H19; Apply (H16 O); Simpl; Apply lt_O_Sn. -Elim H8; Intros; Rewrite H9 in H21; Apply (H21 (S i)); Simpl; Simpl in H1; Apply H1. -Exists (cons t1 l'); Exists (cons r1 (cons r2 lf')); Rewrite H9 in H6; Rewrite H3 in H1; Unfold adapted_couple in H1 H6; Decompose [and] H6; Decompose [and] H1; Clear H6 H1; Unfold adapted_couple_opt; Unfold adapted_couple; Repeat Split. -Rewrite H9; Unfold ordered_Rlist; Intros; Simpl in H1; Induction i. -Simpl; Replace s1 with t2. -Apply (H16 O); Simpl; Apply lt_O_Sn. -Simpl in H14; Rewrite H14; Rewrite Hyp_min; Reflexivity. -Change ``(pos_Rl (cons s1 (cons s2 s3)) i)<=(pos_Rl (cons s1 (cons s2 s3)) (S i))``; Apply (H12 i); Simpl; Apply lt_S_n; Assumption. -Simpl; Simpl in H19; Apply H19. -Rewrite H9; Simpl; Simpl in H13; Rewrite H13; Unfold Rmax; Case (total_order_Rle t2 b); Case (total_order_Rle a b); Intros; Reflexivity Orelse Elim n; Assumption. -Rewrite H9; Simpl; Simpl in H15; Rewrite H15; Reflexivity. -Intros; Simpl in H1; Unfold constant_D_eq open_interval; Intros; Induction i. -Simpl; Rewrite H9 in H6; Simpl in H6; Apply (H22 O). -Simpl; Apply lt_O_Sn. -Unfold open_interval; Simpl. -Replace t2 with s1. -Assumption. -Simpl in H14; Rewrite H14; Rewrite Hyp_min; Reflexivity. -Change (f x)==(pos_Rl (cons r2 lf') i); Clear Hreci; Apply (H17 i). -Simpl; Rewrite H9 in H1; Simpl in H1; Apply lt_S_n; Apply H1. -Rewrite H9 in H6; Unfold open_interval; Apply H6. -Intros; Simpl in H1; Induction i. -Simpl; Rewrite H9; Right; Simpl; Replace s1 with t2. -Assumption. -Simpl in H14; Rewrite H14; Rewrite Hyp_min; Reflexivity. -Elim H8; Intros; Apply (H6 i). -Simpl; Apply lt_S_n; Apply H1. -Intros; Rewrite H9; Induction i. -Simpl; Red; Intro; Elim Hyp_eq; Apply Rle_antisym. -Apply (H16 O); Simpl; Apply lt_O_Sn. -Rewrite <- Hyp_min; Rewrite H6; Simpl in H14; Rewrite <- H14; Right; Reflexivity. -Elim H8; Intros; Rewrite <- H9; Apply (H21 i); Rewrite H9; Rewrite H9 in H1; Simpl; Simpl in H1; Apply lt_S_n; Apply H1. -Exists (cons t1 l'); Exists (cons r1 (cons r2 lf')); Rewrite H9 in H6; Rewrite H3 in H1; Unfold adapted_couple in H1 H6; Decompose [and] H6; Decompose [and] H1; Clear H6 H1; Unfold adapted_couple_opt; Unfold adapted_couple; Repeat Split. -Rewrite H9; Unfold ordered_Rlist; Intros; Simpl in H1; Induction i. -Simpl; Replace s1 with t2. -Apply (H15 O); Simpl; Apply lt_O_Sn. -Simpl in H13; Rewrite H13; Rewrite Hyp_min; Reflexivity. -Change ``(pos_Rl (cons s1 (cons s2 s3)) i)<=(pos_Rl (cons s1 (cons s2 s3)) (S i))``; Apply (H11 i); Simpl; Apply lt_S_n; Assumption. -Simpl; Simpl in H18; Apply H18. -Rewrite H9; Simpl; Simpl in H12; Rewrite H12; Unfold Rmax; Case (total_order_Rle t2 b); Case (total_order_Rle a b); Intros; Reflexivity Orelse Elim n; Assumption. -Rewrite H9; Simpl; Simpl in H14; Rewrite H14; Reflexivity. -Intros; Simpl in H1; Unfold constant_D_eq open_interval; Intros; Induction i. -Simpl; Rewrite H9 in H6; Simpl in H6; Apply (H21 O). -Simpl; Apply lt_O_Sn. -Unfold open_interval; Simpl; Replace t2 with s1. -Assumption. -Simpl in H13; Rewrite H13; Rewrite Hyp_min; Reflexivity. -Change (f x)==(pos_Rl (cons r2 lf') i); Clear Hreci; Apply (H16 i). -Simpl; Rewrite H9 in H1; Simpl in H1; Apply lt_S_n; Apply H1. -Rewrite H9 in H6; Unfold open_interval; Apply H6. -Intros; Simpl in H1; Induction i. -Simpl; Left; Assumption. -Elim H8; Intros; Apply (H6 i). -Simpl; Apply lt_S_n; Apply H1. -Intros; Rewrite H9; Induction i. -Simpl; Red; Intro; Elim Hyp_eq; Apply Rle_antisym. -Apply (H15 O); Simpl; Apply lt_O_Sn. -Rewrite <- Hyp_min; Rewrite H6; Simpl in H13; Rewrite <- H13; Right; Reflexivity. -Elim H8; Intros; Rewrite <- H9; Apply (H20 i); Rewrite H9; Rewrite H9 in H1; Simpl; Simpl in H1; Apply lt_S_n; Apply H1. -Rewrite H3 in H1; Clear H4; Unfold adapted_couple in H1; Decompose [and] H1; Clear H1; Clear H H7 H9; Cut (Rmax a b)==b; [Intro; Rewrite H in H5; Rewrite <- H5; Apply RList_P7; [Assumption | Simpl; Right; Left; Reflexivity] | Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]]. +Lemma StepFun_P10 : + forall (f:R -> R) (l lf:Rlist) (a b:R), + a <= b -> + adapted_couple f a b l lf -> + exists l' : Rlist + | ( exists lf' : Rlist | adapted_couple_opt f a b l' lf'). +simple induction l. +intros; unfold adapted_couple in H0; decompose [and] H0; simpl in H4; + discriminate. +intros; case (Req_dec a b); intro. +exists (cons a nil); exists nil; unfold adapted_couple_opt in |- *; + unfold adapted_couple in |- *; unfold ordered_Rlist in |- *; + repeat split; try (intros; simpl in H3; elim (lt_n_O _ H3)). +simpl in |- *; rewrite <- H2; unfold Rmin in |- *; case (Rle_dec a a); intro; + reflexivity. +simpl in |- *; rewrite <- H2; unfold Rmax in |- *; case (Rle_dec a a); intro; + reflexivity. +elim (RList_P20 _ (StepFun_P9 H1 H2)); intros t1 [t2 [t3 H3]]; + induction lf as [| r1 lf Hreclf]. +unfold adapted_couple in H1; decompose [and] H1; rewrite H3 in H7; + simpl in H7; discriminate. +clear Hreclf; assert (H4 : adapted_couple f t2 b r0 lf). +rewrite H3 in H1; assert (H4 := RList_P21 _ _ H3); simpl in H4; rewrite H4; + eapply StepFun_P7; [ apply H0 | apply H1 ]. +cut (t2 <= b). +intro; assert (H6 := H _ _ _ H5 H4); case (Req_dec t1 t2); intro Hyp_eq. +replace a with t2. +apply H6. +rewrite <- Hyp_eq; rewrite H3 in H1; unfold adapted_couple in H1; + decompose [and] H1; clear H1; simpl in H9; rewrite H9; + unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +elim H6; clear H6; intros l' [lf' H6]; case (Req_dec t2 b); intro. +exists (cons a (cons b nil)); exists (cons r1 nil); + unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *; + repeat split. +unfold ordered_Rlist in |- *; intros; simpl in H8; inversion H8; + [ simpl in |- *; assumption | elim (le_Sn_O _ H10) ]. +simpl in |- *; unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +simpl in |- *; unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +intros; simpl in H8; inversion H8. +unfold constant_D_eq, open_interval in |- *; intros; simpl in |- *; + simpl in H9; rewrite H3 in H1; unfold adapted_couple in H1; + decompose [and] H1; apply (H16 0%nat). +simpl in |- *; apply lt_O_Sn. +unfold open_interval in |- *; simpl in |- *; rewrite H7; simpl in H13; + rewrite H13; unfold Rmin in |- *; case (Rle_dec a b); + intro; [ assumption | elim n; assumption ]. +elim (le_Sn_O _ H10). +intros; simpl in H8; elim (lt_n_O _ H8). +intros; simpl in H8; inversion H8; + [ simpl in |- *; assumption | elim (le_Sn_O _ H10) ]. +assert (Hyp_min : Rmin t2 b = t2). +unfold Rmin in |- *; case (Rle_dec t2 b); intro; + [ reflexivity | elim n; assumption ]. +unfold adapted_couple in H6; elim H6; clear H6; intros; + elim (RList_P20 _ (StepFun_P9 H6 H7)); intros s1 [s2 [s3 H9]]; + induction lf' as [| r2 lf' Hreclf']. +unfold adapted_couple in H6; decompose [and] H6; rewrite H9 in H13; + simpl in H13; discriminate. +clear Hreclf'; case (Req_dec r1 r2); intro. +case (Req_dec (f t2) r1); intro. +exists (cons t1 (cons s2 s3)); exists (cons r1 lf'); rewrite H3 in H1; + rewrite H9 in H6; unfold adapted_couple in H6, H1; + decompose [and] H1; decompose [and] H6; clear H1 H6; + unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *; + repeat split. +unfold ordered_Rlist in |- *; intros; simpl in H1; + induction i as [| i Hreci]. +simpl in |- *; apply Rle_trans with s1. +replace s1 with t2. +apply (H12 0%nat). +simpl in |- *; apply lt_O_Sn. +simpl in H19; rewrite H19; symmetry in |- *; apply Hyp_min. +apply (H16 0%nat); simpl in |- *; apply lt_O_Sn. +change (pos_Rl (cons s2 s3) i <= pos_Rl (cons s2 s3) (S i)) in |- *; + apply (H16 (S i)); simpl in |- *; assumption. +simpl in |- *; simpl in H14; rewrite H14; reflexivity. +simpl in |- *; simpl in H18; rewrite H18; unfold Rmax in |- *; + case (Rle_dec a b); case (Rle_dec t2 b); intros; reflexivity || elim n; + assumption. +simpl in |- *; simpl in H20; apply H20. +intros; simpl in H1; unfold constant_D_eq, open_interval in |- *; intros; + induction i as [| i Hreci]. +simpl in |- *; simpl in H6; case (total_order_T x t2); intro. +elim s; intro. +apply (H17 0%nat); + [ simpl in |- *; apply lt_O_Sn + | unfold open_interval in |- *; simpl in |- *; elim H6; intros; split; + assumption ]. +rewrite b0; assumption. +rewrite H10; apply (H22 0%nat); + [ simpl in |- *; apply lt_O_Sn + | unfold open_interval in |- *; simpl in |- *; replace s1 with t2; + [ elim H6; intros; split; assumption + | simpl in H19; rewrite H19; rewrite Hyp_min; reflexivity ] ]. +simpl in |- *; simpl in H6; apply (H22 (S i)); + [ simpl in |- *; assumption + | unfold open_interval in |- *; simpl in |- *; apply H6 ]. +intros; simpl in H1; rewrite H10; + change + (pos_Rl (cons r2 lf') i <> pos_Rl (cons r2 lf') (S i) \/ + f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <> pos_Rl (cons r2 lf') i) + in |- *; rewrite <- H9; elim H8; intros; apply H6; + simpl in |- *; apply H1. +intros; induction i as [| i Hreci]. +simpl in |- *; red in |- *; intro; elim Hyp_eq; apply Rle_antisym. +apply (H12 0%nat); simpl in |- *; apply lt_O_Sn. +rewrite <- Hyp_min; rewrite H6; simpl in H19; rewrite <- H19; + apply (H16 0%nat); simpl in |- *; apply lt_O_Sn. +elim H8; intros; rewrite H9 in H21; apply (H21 (S i)); simpl in |- *; + simpl in H1; apply H1. +exists (cons t1 l'); exists (cons r1 (cons r2 lf')); rewrite H9 in H6; + rewrite H3 in H1; unfold adapted_couple in H1, H6; + decompose [and] H6; decompose [and] H1; clear H6 H1; + unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *; + repeat split. +rewrite H9; unfold ordered_Rlist in |- *; intros; simpl in H1; + induction i as [| i Hreci]. +simpl in |- *; replace s1 with t2. +apply (H16 0%nat); simpl in |- *; apply lt_O_Sn. +simpl in H14; rewrite H14; rewrite Hyp_min; reflexivity. +change + (pos_Rl (cons s1 (cons s2 s3)) i <= pos_Rl (cons s1 (cons s2 s3)) (S i)) + in |- *; apply (H12 i); simpl in |- *; apply lt_S_n; + assumption. +simpl in |- *; simpl in H19; apply H19. +rewrite H9; simpl in |- *; simpl in H13; rewrite H13; unfold Rmax in |- *; + case (Rle_dec t2 b); case (Rle_dec a b); intros; reflexivity || elim n; + assumption. +rewrite H9; simpl in |- *; simpl in H15; rewrite H15; reflexivity. +intros; simpl in H1; unfold constant_D_eq, open_interval in |- *; intros; + induction i as [| i Hreci]. +simpl in |- *; rewrite H9 in H6; simpl in H6; apply (H22 0%nat). +simpl in |- *; apply lt_O_Sn. +unfold open_interval in |- *; simpl in |- *. +replace t2 with s1. +assumption. +simpl in H14; rewrite H14; rewrite Hyp_min; reflexivity. +change (f x = pos_Rl (cons r2 lf') i) in |- *; clear Hreci; apply (H17 i). +simpl in |- *; rewrite H9 in H1; simpl in H1; apply lt_S_n; apply H1. +rewrite H9 in H6; unfold open_interval in |- *; apply H6. +intros; simpl in H1; induction i as [| i Hreci]. +simpl in |- *; rewrite H9; right; simpl in |- *; replace s1 with t2. +assumption. +simpl in H14; rewrite H14; rewrite Hyp_min; reflexivity. +elim H8; intros; apply (H6 i). +simpl in |- *; apply lt_S_n; apply H1. +intros; rewrite H9; induction i as [| i Hreci]. +simpl in |- *; red in |- *; intro; elim Hyp_eq; apply Rle_antisym. +apply (H16 0%nat); simpl in |- *; apply lt_O_Sn. +rewrite <- Hyp_min; rewrite H6; simpl in H14; rewrite <- H14; right; + reflexivity. +elim H8; intros; rewrite <- H9; apply (H21 i); rewrite H9; rewrite H9 in H1; + simpl in |- *; simpl in H1; apply lt_S_n; apply H1. +exists (cons t1 l'); exists (cons r1 (cons r2 lf')); rewrite H9 in H6; + rewrite H3 in H1; unfold adapted_couple in H1, H6; + decompose [and] H6; decompose [and] H1; clear H6 H1; + unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *; + repeat split. +rewrite H9; unfold ordered_Rlist in |- *; intros; simpl in H1; + induction i as [| i Hreci]. +simpl in |- *; replace s1 with t2. +apply (H15 0%nat); simpl in |- *; apply lt_O_Sn. +simpl in H13; rewrite H13; rewrite Hyp_min; reflexivity. +change + (pos_Rl (cons s1 (cons s2 s3)) i <= pos_Rl (cons s1 (cons s2 s3)) (S i)) + in |- *; apply (H11 i); simpl in |- *; apply lt_S_n; + assumption. +simpl in |- *; simpl in H18; apply H18. +rewrite H9; simpl in |- *; simpl in H12; rewrite H12; unfold Rmax in |- *; + case (Rle_dec t2 b); case (Rle_dec a b); intros; reflexivity || elim n; + assumption. +rewrite H9; simpl in |- *; simpl in H14; rewrite H14; reflexivity. +intros; simpl in H1; unfold constant_D_eq, open_interval in |- *; intros; + induction i as [| i Hreci]. +simpl in |- *; rewrite H9 in H6; simpl in H6; apply (H21 0%nat). +simpl in |- *; apply lt_O_Sn. +unfold open_interval in |- *; simpl in |- *; replace t2 with s1. +assumption. +simpl in H13; rewrite H13; rewrite Hyp_min; reflexivity. +change (f x = pos_Rl (cons r2 lf') i) in |- *; clear Hreci; apply (H16 i). +simpl in |- *; rewrite H9 in H1; simpl in H1; apply lt_S_n; apply H1. +rewrite H9 in H6; unfold open_interval in |- *; apply H6. +intros; simpl in H1; induction i as [| i Hreci]. +simpl in |- *; left; assumption. +elim H8; intros; apply (H6 i). +simpl in |- *; apply lt_S_n; apply H1. +intros; rewrite H9; induction i as [| i Hreci]. +simpl in |- *; red in |- *; intro; elim Hyp_eq; apply Rle_antisym. +apply (H15 0%nat); simpl in |- *; apply lt_O_Sn. +rewrite <- Hyp_min; rewrite H6; simpl in H13; rewrite <- H13; right; + reflexivity. +elim H8; intros; rewrite <- H9; apply (H20 i); rewrite H9; rewrite H9 in H1; + simpl in |- *; simpl in H1; apply lt_S_n; apply H1. +rewrite H3 in H1; clear H4; unfold adapted_couple in H1; decompose [and] H1; + clear H1; clear H H7 H9; cut (Rmax a b = b); + [ intro; rewrite H in H5; rewrite <- H5; apply RList_P7; + [ assumption | simpl in |- *; right; left; reflexivity ] + | unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ] ]. Qed. -Lemma StepFun_P11 : (a,b,r,r1,r3,s1,s2,r4:R;r2,lf1,s3,lf2:Rlist;f:R->R) ``a<b`` -> (adapted_couple f a b (cons r (cons r1 r2)) (cons r3 lf1)) -> (adapted_couple_opt f a b (cons s1 (cons s2 s3)) (cons r4 lf2)) -> ``r1<=s2``. -Intros; Unfold adapted_couple_opt in H1; Elim H1; Clear H1; Intros; Unfold adapted_couple in H0 H1; Decompose [and] H0; Decompose [and] H1; Clear H0 H1; Assert H12 : r==s1. -Simpl in H10; Simpl in H5; Rewrite H10; Rewrite H5; Reflexivity. -Assert H14 := (H3 O (lt_O_Sn ?)); Simpl in H14; Elim H14; Intro. -Assert H15 := (H7 O (lt_O_Sn ?)); Simpl in H15; Elim H15; Intro. -Rewrite <- H12 in H1; Case (total_order_Rle r1 s2); Intro; Try Assumption. -Assert H16 : ``s2<r1``; Auto with real. -Induction s3. -Simpl in H9; Rewrite H9 in H16; Cut ``r1<=(Rmax a b)``. -Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H17 H16)). -Rewrite <- H4; Apply RList_P7; [Assumption | Simpl; Right; Left; Reflexivity]. -Clear Hrecs3; Induction lf2. -Simpl in H11; Discriminate. -Clear Hreclf2; Assert H17 : r3==r4. -Pose x := ``(r+s2)/2``; Assert H17 := (H8 O (lt_O_Sn ?)); Assert H18 := (H13 O (lt_O_Sn ?)); Unfold constant_D_eq open_interval in H17 H18; Simpl in H17; Simpl in H18; Rewrite <- (H17 x). -Rewrite <- (H18 x). -Reflexivity. -Rewrite <- H12; Unfold x; Split. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite (Rplus_sym r); Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Unfold x; Split. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Apply Rlt_trans with s2; [Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite (Rplus_sym r); Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]] | Assumption]. -Assert H18 : (f s2)==r3. -Apply (H8 O); [Simpl; Apply lt_O_Sn | Unfold open_interval; Simpl; Split; Assumption]. -Assert H19 : r3 == r5. -Assert H19 := (H7 (S O)); Simpl in H19; Assert H20 := (H19 (lt_n_S ? ? (lt_O_Sn ?))); Elim H20; Intro. -Pose x := ``(s2+(Rmin r1 r0))/2``; Assert H22 := (H8 O); Assert H23 := (H13 (S O)); Simpl in H22; Simpl in H23; Rewrite <- (H22 (lt_O_Sn ?) x). -Rewrite <- (H23 (lt_n_S ? ? (lt_O_Sn ?)) x). -Reflexivity. -Unfold open_interval; Simpl; Unfold x; Split. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Unfold Rmin; Case (total_order_Rle r1 r0); Intro; Assumption | DiscrR]]. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_le_trans with ``r0+(Rmin r1 r0)``; [Do 2 Rewrite <- (Rplus_sym (Rmin r1 r0)); Apply Rlt_compatibility; Assumption | Apply Rle_compatibility; Apply Rmin_r] | DiscrR]]. -Unfold open_interval; Simpl; Unfold x; Split. -Apply Rlt_trans with s2; [Assumption | Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Unfold Rmin; Case (total_order_Rle r1 r0); Intro; Assumption | DiscrR]]]. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_le_trans with ``r1+(Rmin r1 r0)``; [Do 2 Rewrite <- (Rplus_sym (Rmin r1 r0)); Apply Rlt_compatibility; Assumption | Apply Rle_compatibility; Apply Rmin_l] | DiscrR]]. -Elim H2; Clear H2; Intros; Assert H23 := (H22 (S O)); Simpl in H23; Assert H24 := (H23 (lt_n_S ? ? (lt_O_Sn ?))); Elim H24; Assumption. -Elim H2; Intros; Assert H22 := (H20 O); Simpl in H22; Assert H23 := (H22 (lt_O_Sn ?)); Elim H23; Intro; [Elim H24; Rewrite <- H17; Rewrite <- H19; Reflexivity | Elim H24; Rewrite <- H17; Assumption]. -Elim H2; Clear H2; Intros; Assert H17 := (H16 O); Simpl in H17; Elim (H17 (lt_O_Sn ?)); Assumption. -Rewrite <- H0; Rewrite H12; Apply (H7 O); Simpl; Apply lt_O_Sn. +Lemma StepFun_P11 : + forall (a b r r1 r3 s1 s2 r4:R) (r2 lf1 s3 lf2:Rlist) + (f:R -> R), + a < b -> + adapted_couple f a b (cons r (cons r1 r2)) (cons r3 lf1) -> + adapted_couple_opt f a b (cons s1 (cons s2 s3)) (cons r4 lf2) -> r1 <= s2. +intros; unfold adapted_couple_opt in H1; elim H1; clear H1; intros; + unfold adapted_couple in H0, H1; decompose [and] H0; + decompose [and] H1; clear H0 H1; assert (H12 : r = s1). +simpl in H10; simpl in H5; rewrite H10; rewrite H5; reflexivity. +assert (H14 := H3 0%nat (lt_O_Sn _)); simpl in H14; elim H14; intro. +assert (H15 := H7 0%nat (lt_O_Sn _)); simpl in H15; elim H15; intro. +rewrite <- H12 in H1; case (Rle_dec r1 s2); intro; try assumption. +assert (H16 : s2 < r1); auto with real. +induction s3 as [| r0 s3 Hrecs3]. +simpl in H9; rewrite H9 in H16; cut (r1 <= Rmax a b). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H17 H16)). +rewrite <- H4; apply RList_P7; + [ assumption | simpl in |- *; right; left; reflexivity ]. +clear Hrecs3; induction lf2 as [| r5 lf2 Hreclf2]. +simpl in H11; discriminate. +clear Hreclf2; assert (H17 : r3 = r4). +pose (x := (r + s2) / 2); assert (H17 := H8 0%nat (lt_O_Sn _)); + assert (H18 := H13 0%nat (lt_O_Sn _)); + unfold constant_D_eq, open_interval in H17, H18; simpl in H17; + simpl in H18; rewrite <- (H17 x). +rewrite <- (H18 x). +reflexivity. +rewrite <- H12; unfold x in |- *; split. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite (Rplus_comm r); rewrite double; + apply Rplus_lt_compat_l; assumption + | discrR ] ]. +unfold x in |- *; split. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +apply Rlt_trans with s2; + [ apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite (Rplus_comm r); rewrite double; + apply Rplus_lt_compat_l; assumption + | discrR ] ] + | assumption ]. +assert (H18 : f s2 = r3). +apply (H8 0%nat); + [ simpl in |- *; apply lt_O_Sn + | unfold open_interval in |- *; simpl in |- *; split; assumption ]. +assert (H19 : r3 = r5). +assert (H19 := H7 1%nat); simpl in H19; + assert (H20 := H19 (lt_n_S _ _ (lt_O_Sn _))); elim H20; + intro. +pose (x := (s2 + Rmin r1 r0) / 2); assert (H22 := H8 0%nat); + assert (H23 := H13 1%nat); simpl in H22; simpl in H23; + rewrite <- (H22 (lt_O_Sn _) x). +rewrite <- (H23 (lt_n_S _ _ (lt_O_Sn _)) x). +reflexivity. +unfold open_interval in |- *; simpl in |- *; unfold x in |- *; split. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; + unfold Rmin in |- *; case (Rle_dec r1 r0); intro; + assumption + | discrR ] ]. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; + apply Rlt_le_trans with (r0 + Rmin r1 r0); + [ do 2 rewrite <- (Rplus_comm (Rmin r1 r0)); apply Rplus_lt_compat_l; + assumption + | apply Rplus_le_compat_l; apply Rmin_r ] + | discrR ] ]. +unfold open_interval in |- *; simpl in |- *; unfold x in |- *; split. +apply Rlt_trans with s2; + [ assumption + | apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; + unfold Rmin in |- *; case (Rle_dec r1 r0); + intro; assumption + | discrR ] ] ]. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; + apply Rlt_le_trans with (r1 + Rmin r1 r0); + [ do 2 rewrite <- (Rplus_comm (Rmin r1 r0)); apply Rplus_lt_compat_l; + assumption + | apply Rplus_le_compat_l; apply Rmin_l ] + | discrR ] ]. +elim H2; clear H2; intros; assert (H23 := H22 1%nat); simpl in H23; + assert (H24 := H23 (lt_n_S _ _ (lt_O_Sn _))); elim H24; + assumption. +elim H2; intros; assert (H22 := H20 0%nat); simpl in H22; + assert (H23 := H22 (lt_O_Sn _)); elim H23; intro; + [ elim H24; rewrite <- H17; rewrite <- H19; reflexivity + | elim H24; rewrite <- H17; assumption ]. +elim H2; clear H2; intros; assert (H17 := H16 0%nat); simpl in H17; + elim (H17 (lt_O_Sn _)); assumption. +rewrite <- H0; rewrite H12; apply (H7 0%nat); simpl in |- *; apply lt_O_Sn. Qed. -Lemma StepFun_P12 : (a,b:R;f:R->R;l,lf:Rlist) (adapted_couple_opt f a b l lf) -> (adapted_couple_opt f b a l lf). -Unfold adapted_couple_opt; Unfold adapted_couple; Intros; Decompose [and] H; Clear H; Repeat Split; Try Assumption. -Rewrite H0; Unfold Rmin; Case (total_order_Rle a b); Intro; Case (total_order_Rle b a); Intro; Try Reflexivity. -Apply Rle_antisym; Assumption. -Apply Rle_antisym; Auto with real. -Rewrite H3; Unfold Rmax; Case (total_order_Rle a b); Intro; Case (total_order_Rle b a); Intro; Try Reflexivity. -Apply Rle_antisym; Assumption. -Apply Rle_antisym; Auto with real. +Lemma StepFun_P12 : + forall (a b:R) (f:R -> R) (l lf:Rlist), + adapted_couple_opt f a b l lf -> adapted_couple_opt f b a l lf. +unfold adapted_couple_opt in |- *; unfold adapted_couple in |- *; intros; + decompose [and] H; clear H; repeat split; try assumption. +rewrite H0; unfold Rmin in |- *; case (Rle_dec a b); intro; + case (Rle_dec b a); intro; try reflexivity. +apply Rle_antisym; assumption. +apply Rle_antisym; auto with real. +rewrite H3; unfold Rmax in |- *; case (Rle_dec a b); intro; + case (Rle_dec b a); intro; try reflexivity. +apply Rle_antisym; assumption. +apply Rle_antisym; auto with real. Qed. -Lemma StepFun_P13 : (a,b,r,r1,r3,s1,s2,r4:R;r2,lf1,s3,lf2:Rlist;f:R->R) ``a<>b`` -> (adapted_couple f a b (cons r (cons r1 r2)) (cons r3 lf1)) -> (adapted_couple_opt f a b (cons s1 (cons s2 s3)) (cons r4 lf2)) -> ``r1<=s2``. -Intros; Case (total_order_T a b); Intro. -Elim s; Intro. -EApply StepFun_P11; [Apply a0 | Apply H0 | Apply H1]. -Elim H; Assumption. -EApply StepFun_P11; [Apply r0 | Apply StepFun_P2; Apply H0 | Apply StepFun_P12; Apply H1]. +Lemma StepFun_P13 : + forall (a b r r1 r3 s1 s2 r4:R) (r2 lf1 s3 lf2:Rlist) + (f:R -> R), + a <> b -> + adapted_couple f a b (cons r (cons r1 r2)) (cons r3 lf1) -> + adapted_couple_opt f a b (cons s1 (cons s2 s3)) (cons r4 lf2) -> r1 <= s2. +intros; case (total_order_T a b); intro. +elim s; intro. +eapply StepFun_P11; [ apply a0 | apply H0 | apply H1 ]. +elim H; assumption. +eapply StepFun_P11; + [ apply r0 | apply StepFun_P2; apply H0 | apply StepFun_P12; apply H1 ]. Qed. -Lemma StepFun_P14 : (f:R->R;l1,l2,lf1,lf2:Rlist;a,b:R) ``a<=b`` -> (adapted_couple f a b l1 lf1) -> (adapted_couple_opt f a b l2 lf2) -> (Int_SF lf1 l1)==(Int_SF lf2 l2). -Induction l1. -Intros l2 lf1 lf2 a b Hyp H H0; Unfold adapted_couple in H; Decompose [and] H; Clear H H0 H2 H3 H1 H6; Simpl in H4; Discriminate. -Induction r0. -Intros; Case (Req_EM a b); Intro. -Unfold adapted_couple_opt in H2; Elim H2; Intros; Rewrite (StepFun_P8 H4 H3); Rewrite (StepFun_P8 H1 H3); Reflexivity. -Assert H4 := (StepFun_P9 H1 H3); Simpl in H4; Elim (le_Sn_O ? (le_S_n ? ? H4)). -Intros; Clear H; Unfold adapted_couple_opt in H3; Elim H3; Clear H3; Intros; Case (Req_EM a b); Intro. -Rewrite (StepFun_P8 H2 H4); Rewrite (StepFun_P8 H H4); Reflexivity. -Assert Hyp_min : (Rmin a b)==a. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Assert Hyp_max : (Rmax a b)==b. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Elim (RList_P20 ? (StepFun_P9 H H4)); Intros s1 [s2 [s3 H5]]; Rewrite H5 in H; Rewrite H5; Induction lf1. -Unfold adapted_couple in H2; Decompose [and] H2; Clear H H2 H4 H5 H3 H6 H8 H7 H11; Simpl in H9; Discriminate. -Clear Hreclf1; Induction lf2. -Unfold adapted_couple in H; Decompose [and] H; Clear H H2 H4 H5 H3 H6 H8 H7 H11; Simpl in H9; Discriminate. -Clear Hreclf2; Assert H6 : r==s1. -Unfold adapted_couple in H H2; Decompose [and] H; Decompose [and] H2; Clear H H2; Simpl in H13; Simpl in H8; Rewrite H13; Rewrite H8; Reflexivity. -Assert H7 : r3==r4\/r==r1. -Case (Req_EM r r1); Intro. -Right; Assumption. -Left; Cut ``r1<=s2``. -Intro; Unfold adapted_couple in H2 H; Decompose [and] H; Decompose [and] H2; Clear H H2; Pose x := ``(r+r1)/2``; Assert H18 := (H14 O); Assert H20 := (H19 O); Unfold constant_D_eq open_interval in H18 H20; Simpl in H18; Simpl in H20; Rewrite <- (H18 (lt_O_Sn ?) x). -Rewrite <- (H20 (lt_O_Sn ?) x). -Reflexivity. -Assert H21 := (H13 O (lt_O_Sn ?)); Simpl in H21; Elim H21; Intro; [Idtac | Elim H7; Assumption]; Unfold x; Split. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Apply H | DiscrR]]. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite <- (Rplus_sym r1); Rewrite double; Apply Rlt_compatibility; Apply H | DiscrR]]. -Rewrite <- H6; Assert H21 := (H13 O (lt_O_Sn ?)); Simpl in H21; Elim H21; Intro; [Idtac | Elim H7; Assumption]; Unfold x; Split. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Apply H | DiscrR]]. -Apply Rlt_le_trans with r1; [Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite <- (Rplus_sym r1); Rewrite double; Apply Rlt_compatibility; Apply H | DiscrR]] | Assumption]. -EApply StepFun_P13. -Apply H4. -Apply H2. -Unfold adapted_couple_opt; Split. -Apply H. -Rewrite H5 in H3; Apply H3. -Assert H8 : ``r1<=s2``. -EApply StepFun_P13. -Apply H4. -Apply H2. -Unfold adapted_couple_opt; Split. -Apply H. -Rewrite H5 in H3; Apply H3. -Elim H7; Intro. -Simpl; Elim H8; Intro. -Replace ``r4*(s2-s1)`` with ``r3*(r1-r)+r3*(s2-r1)``; [Idtac | Rewrite H9; Rewrite H6; Ring]. -Rewrite Rplus_assoc; Apply Rplus_plus_r; Change (Int_SF lf1 (cons r1 r2))==(Int_SF (cons r3 lf2) (cons r1 (cons s2 s3))); Apply H0 with r1 b. -Unfold adapted_couple in H2; Decompose [and] H2; Clear H2; Replace b with (Rmax a b). -Rewrite <- H12; Apply RList_P7; [Assumption | Simpl; Right; Left; Reflexivity]. -EApply StepFun_P7. -Apply H1. -Apply H2. -Unfold adapted_couple_opt; Split. -Apply StepFun_P7 with a a r3. -Apply H1. -Unfold adapted_couple in H2 H; Decompose [and] H2; Decompose [and] H; Clear H H2; Assert H20 : r==a. -Simpl in H13; Rewrite H13; Apply Hyp_min. -Unfold adapted_couple; Repeat Split. -Unfold ordered_Rlist; Intros; Simpl in H; Induction i. -Simpl; Rewrite <- H20; Apply (H11 O). -Simpl; Apply lt_O_Sn. -Induction i. -Simpl; Assumption. -Change ``(pos_Rl (cons s2 s3) i)<=(pos_Rl (cons s2 s3) (S i))``; Apply (H15 (S i)); Simpl; Apply lt_S_n; Assumption. -Simpl; Symmetry; Apply Hyp_min. -Rewrite <- H17; Reflexivity. -Simpl in H19; Simpl; Rewrite H19; Reflexivity. -Intros; Simpl in H; Unfold constant_D_eq open_interval; Intros; Induction i. -Simpl; Apply (H16 O). -Simpl; Apply lt_O_Sn. -Simpl in H2; Rewrite <- H20 in H2; Unfold open_interval; Simpl; Apply H2. -Clear Hreci; Induction i. -Simpl; Simpl in H2; Rewrite H9; Apply (H21 O). -Simpl; Apply lt_O_Sn. -Unfold open_interval; Simpl; Elim H2; Intros; Split. -Apply Rle_lt_trans with r1; Try Assumption; Rewrite <- H6; Apply (H11 O); Simpl; Apply lt_O_Sn. -Assumption. -Clear Hreci; Simpl; Apply (H21 (S i)). -Simpl; Apply lt_S_n; Assumption. -Unfold open_interval; Apply H2. -Elim H3; Clear H3; Intros; Split. -Rewrite H9; Change (i:nat) (lt i (pred (Rlength (cons r4 lf2)))) ->``(pos_Rl (cons r4 lf2) i)<>(pos_Rl (cons r4 lf2) (S i))``\/``(f (pos_Rl (cons s1 (cons s2 s3)) (S i)))<>(pos_Rl (cons r4 lf2) i)``; Rewrite <- H5; Apply H3. -Rewrite H5 in H11; Intros; Simpl in H12; Induction i. -Simpl; Red; Intro; Rewrite H13 in H10; Elim (Rlt_antirefl ? H10). -Clear Hreci; Apply (H11 (S i)); Simpl; Apply H12. -Rewrite H9; Rewrite H10; Rewrite H6; Apply Rplus_plus_r; Rewrite <- H10; Apply H0 with r1 b. -Unfold adapted_couple in H2; Decompose [and] H2; Clear H2; Replace b with (Rmax a b). -Rewrite <- H12; Apply RList_P7; [Assumption | Simpl; Right; Left; Reflexivity]. -EApply StepFun_P7. -Apply H1. -Apply H2. -Unfold adapted_couple_opt; Split. -Apply StepFun_P7 with a a r3. -Apply H1. -Unfold adapted_couple in H2 H; Decompose [and] H2; Decompose [and] H; Clear H H2; Assert H20 : r==a. -Simpl in H13; Rewrite H13; Apply Hyp_min. -Unfold adapted_couple; Repeat Split. -Unfold ordered_Rlist; Intros; Simpl in H; Induction i. -Simpl; Rewrite <- H20; Apply (H11 O); Simpl; Apply lt_O_Sn. -Rewrite H10; Apply (H15 (S i)); Simpl; Assumption. -Simpl; Symmetry; Apply Hyp_min. -Rewrite <- H17; Rewrite H10; Reflexivity. -Simpl in H19; Simpl; Apply H19. -Intros; Simpl in H; Unfold constant_D_eq open_interval; Intros; Induction i. -Simpl; Apply (H16 O). -Simpl; Apply lt_O_Sn. -Simpl in H2; Rewrite <- H20 in H2; Unfold open_interval; Simpl; Apply H2. -Clear Hreci; Simpl; Apply (H21 (S i)). -Simpl; Assumption. -Rewrite <- H10; Unfold open_interval; Apply H2. -Elim H3; Clear H3; Intros; Split. -Rewrite H5 in H3; Intros; Apply (H3 (S i)). -Simpl; Replace (Rlength lf2) with (S (pred (Rlength lf2))). -Apply lt_n_S; Apply H12. -Symmetry; Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H12; Elim (lt_n_O ? H12). -Intros; Simpl in H12; Rewrite H10; Rewrite H5 in H11; Apply (H11 (S i)); Simpl; Apply lt_n_S; Apply H12. -Simpl; Rewrite H9; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rmult_Or; Rewrite Rplus_Ol; Change (Int_SF lf1 (cons r1 r2))==(Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))); EApply H0. -Apply H1. -2: Rewrite H5 in H3; Unfold adapted_couple_opt; Split; Assumption. -Assert H10 : r==a. -Unfold adapted_couple in H2; Decompose [and] H2; Clear H2; Simpl in H12; Rewrite H12; Apply Hyp_min. -Rewrite <- H9; Rewrite H10; Apply StepFun_P7 with a r r3; [Apply H1 | Pattern 2 a; Rewrite <- H10; Pattern 2 r; Rewrite H9; Apply H2]. +Lemma StepFun_P14 : + forall (f:R -> R) (l1 l2 lf1 lf2:Rlist) (a b:R), + a <= b -> + adapted_couple f a b l1 lf1 -> + adapted_couple_opt f a b l2 lf2 -> Int_SF lf1 l1 = Int_SF lf2 l2. +simple induction l1. +intros l2 lf1 lf2 a b Hyp H H0; unfold adapted_couple in H; decompose [and] H; + clear H H0 H2 H3 H1 H6; simpl in H4; discriminate. +simple induction r0. +intros; case (Req_dec a b); intro. +unfold adapted_couple_opt in H2; elim H2; intros; rewrite (StepFun_P8 H4 H3); + rewrite (StepFun_P8 H1 H3); reflexivity. +assert (H4 := StepFun_P9 H1 H3); simpl in H4; + elim (le_Sn_O _ (le_S_n _ _ H4)). +intros; clear H; unfold adapted_couple_opt in H3; elim H3; clear H3; intros; + case (Req_dec a b); intro. +rewrite (StepFun_P8 H2 H4); rewrite (StepFun_P8 H H4); reflexivity. +assert (Hyp_min : Rmin a b = a). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +assert (Hyp_max : Rmax a b = b). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +elim (RList_P20 _ (StepFun_P9 H H4)); intros s1 [s2 [s3 H5]]; rewrite H5 in H; + rewrite H5; induction lf1 as [| r3 lf1 Hreclf1]. +unfold adapted_couple in H2; decompose [and] H2; + clear H H2 H4 H5 H3 H6 H8 H7 H11; simpl in H9; discriminate. +clear Hreclf1; induction lf2 as [| r4 lf2 Hreclf2]. +unfold adapted_couple in H; decompose [and] H; + clear H H2 H4 H5 H3 H6 H8 H7 H11; simpl in H9; discriminate. +clear Hreclf2; assert (H6 : r = s1). +unfold adapted_couple in H, H2; decompose [and] H; decompose [and] H2; + clear H H2; simpl in H13; simpl in H8; rewrite H13; + rewrite H8; reflexivity. +assert (H7 : r3 = r4 \/ r = r1). +case (Req_dec r r1); intro. +right; assumption. +left; cut (r1 <= s2). +intro; unfold adapted_couple in H2, H; decompose [and] H; decompose [and] H2; + clear H H2; pose (x := (r + r1) / 2); assert (H18 := H14 0%nat); + assert (H20 := H19 0%nat); unfold constant_D_eq, open_interval in H18, H20; + simpl in H18; simpl in H20; rewrite <- (H18 (lt_O_Sn _) x). +rewrite <- (H20 (lt_O_Sn _) x). +reflexivity. +assert (H21 := H13 0%nat (lt_O_Sn _)); simpl in H21; elim H21; intro; + [ idtac | elim H7; assumption ]; unfold x in |- *; + split. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H + | discrR ] ]. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite <- (Rplus_comm r1); rewrite double; + apply Rplus_lt_compat_l; apply H + | discrR ] ]. +rewrite <- H6; assert (H21 := H13 0%nat (lt_O_Sn _)); simpl in H21; elim H21; + intro; [ idtac | elim H7; assumption ]; unfold x in |- *; + split. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; apply H + | discrR ] ]. +apply Rlt_le_trans with r1; + [ apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite <- (Rplus_comm r1); rewrite double; + apply Rplus_lt_compat_l; apply H + | discrR ] ] + | assumption ]. +eapply StepFun_P13. +apply H4. +apply H2. +unfold adapted_couple_opt in |- *; split. +apply H. +rewrite H5 in H3; apply H3. +assert (H8 : r1 <= s2). +eapply StepFun_P13. +apply H4. +apply H2. +unfold adapted_couple_opt in |- *; split. +apply H. +rewrite H5 in H3; apply H3. +elim H7; intro. +simpl in |- *; elim H8; intro. +replace (r4 * (s2 - s1)) with (r3 * (r1 - r) + r3 * (s2 - r1)); + [ idtac | rewrite H9; rewrite H6; ring ]. +rewrite Rplus_assoc; apply Rplus_eq_compat_l; + change + (Int_SF lf1 (cons r1 r2) = Int_SF (cons r3 lf2) (cons r1 (cons s2 s3))) + in |- *; apply H0 with r1 b. +unfold adapted_couple in H2; decompose [and] H2; clear H2; + replace b with (Rmax a b). +rewrite <- H12; apply RList_P7; + [ assumption | simpl in |- *; right; left; reflexivity ]. +eapply StepFun_P7. +apply H1. +apply H2. +unfold adapted_couple_opt in |- *; split. +apply StepFun_P7 with a a r3. +apply H1. +unfold adapted_couple in H2, H; decompose [and] H2; decompose [and] H; + clear H H2; assert (H20 : r = a). +simpl in H13; rewrite H13; apply Hyp_min. +unfold adapted_couple in |- *; repeat split. +unfold ordered_Rlist in |- *; intros; simpl in H; induction i as [| i Hreci]. +simpl in |- *; rewrite <- H20; apply (H11 0%nat). +simpl in |- *; apply lt_O_Sn. +induction i as [| i Hreci0]. +simpl in |- *; assumption. +change (pos_Rl (cons s2 s3) i <= pos_Rl (cons s2 s3) (S i)) in |- *; + apply (H15 (S i)); simpl in |- *; apply lt_S_n; assumption. +simpl in |- *; symmetry in |- *; apply Hyp_min. +rewrite <- H17; reflexivity. +simpl in H19; simpl in |- *; rewrite H19; reflexivity. +intros; simpl in H; unfold constant_D_eq, open_interval in |- *; intros; + induction i as [| i Hreci]. +simpl in |- *; apply (H16 0%nat). +simpl in |- *; apply lt_O_Sn. +simpl in H2; rewrite <- H20 in H2; unfold open_interval in |- *; + simpl in |- *; apply H2. +clear Hreci; induction i as [| i Hreci]. +simpl in |- *; simpl in H2; rewrite H9; apply (H21 0%nat). +simpl in |- *; apply lt_O_Sn. +unfold open_interval in |- *; simpl in |- *; elim H2; intros; split. +apply Rle_lt_trans with r1; try assumption; rewrite <- H6; apply (H11 0%nat); + simpl in |- *; apply lt_O_Sn. +assumption. +clear Hreci; simpl in |- *; apply (H21 (S i)). +simpl in |- *; apply lt_S_n; assumption. +unfold open_interval in |- *; apply H2. +elim H3; clear H3; intros; split. +rewrite H9; + change + (forall i:nat, + (i < pred (Rlength (cons r4 lf2)))%nat -> + pos_Rl (cons r4 lf2) i <> pos_Rl (cons r4 lf2) (S i) \/ + f (pos_Rl (cons s1 (cons s2 s3)) (S i)) <> pos_Rl (cons r4 lf2) i) + in |- *; rewrite <- H5; apply H3. +rewrite H5 in H11; intros; simpl in H12; induction i as [| i Hreci]. +simpl in |- *; red in |- *; intro; rewrite H13 in H10; + elim (Rlt_irrefl _ H10). +clear Hreci; apply (H11 (S i)); simpl in |- *; apply H12. +rewrite H9; rewrite H10; rewrite H6; apply Rplus_eq_compat_l; rewrite <- H10; + apply H0 with r1 b. +unfold adapted_couple in H2; decompose [and] H2; clear H2; + replace b with (Rmax a b). +rewrite <- H12; apply RList_P7; + [ assumption | simpl in |- *; right; left; reflexivity ]. +eapply StepFun_P7. +apply H1. +apply H2. +unfold adapted_couple_opt in |- *; split. +apply StepFun_P7 with a a r3. +apply H1. +unfold adapted_couple in H2, H; decompose [and] H2; decompose [and] H; + clear H H2; assert (H20 : r = a). +simpl in H13; rewrite H13; apply Hyp_min. +unfold adapted_couple in |- *; repeat split. +unfold ordered_Rlist in |- *; intros; simpl in H; induction i as [| i Hreci]. +simpl in |- *; rewrite <- H20; apply (H11 0%nat); simpl in |- *; + apply lt_O_Sn. +rewrite H10; apply (H15 (S i)); simpl in |- *; assumption. +simpl in |- *; symmetry in |- *; apply Hyp_min. +rewrite <- H17; rewrite H10; reflexivity. +simpl in H19; simpl in |- *; apply H19. +intros; simpl in H; unfold constant_D_eq, open_interval in |- *; intros; + induction i as [| i Hreci]. +simpl in |- *; apply (H16 0%nat). +simpl in |- *; apply lt_O_Sn. +simpl in H2; rewrite <- H20 in H2; unfold open_interval in |- *; + simpl in |- *; apply H2. +clear Hreci; simpl in |- *; apply (H21 (S i)). +simpl in |- *; assumption. +rewrite <- H10; unfold open_interval in |- *; apply H2. +elim H3; clear H3; intros; split. +rewrite H5 in H3; intros; apply (H3 (S i)). +simpl in |- *; replace (Rlength lf2) with (S (pred (Rlength lf2))). +apply lt_n_S; apply H12. +symmetry in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *; + intro; rewrite <- H13 in H12; elim (lt_n_O _ H12). +intros; simpl in H12; rewrite H10; rewrite H5 in H11; apply (H11 (S i)); + simpl in |- *; apply lt_n_S; apply H12. +simpl in |- *; rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rmult_0_r; rewrite Rplus_0_l; + change + (Int_SF lf1 (cons r1 r2) = Int_SF (cons r4 lf2) (cons s1 (cons s2 s3))) + in |- *; eapply H0. +apply H1. +2: rewrite H5 in H3; unfold adapted_couple_opt in |- *; split; assumption. +assert (H10 : r = a). +unfold adapted_couple in H2; decompose [and] H2; clear H2; simpl in H12; + rewrite H12; apply Hyp_min. +rewrite <- H9; rewrite H10; apply StepFun_P7 with a r r3; + [ apply H1 + | pattern a at 2 in |- *; rewrite <- H10; pattern r at 2 in |- *; rewrite H9; + apply H2 ]. Qed. -Lemma StepFun_P15 : (f:R->R;l1,l2,lf1,lf2:Rlist;a,b:R) (adapted_couple f a b l1 lf1) -> (adapted_couple_opt f a b l2 lf2) -> (Int_SF lf1 l1)==(Int_SF lf2 l2). -Intros; Case (total_order_Rle a b); Intro; [Apply (StepFun_P14 r H H0) | Assert H1 : ``b<=a``; [Auto with real | EApply StepFun_P14; [Apply H1 | Apply StepFun_P2; Apply H | Apply StepFun_P12; Apply H0]]]. +Lemma StepFun_P15 : + forall (f:R -> R) (l1 l2 lf1 lf2:Rlist) (a b:R), + adapted_couple f a b l1 lf1 -> + adapted_couple_opt f a b l2 lf2 -> Int_SF lf1 l1 = Int_SF lf2 l2. +intros; case (Rle_dec a b); intro; + [ apply (StepFun_P14 r H H0) + | assert (H1 : b <= a); + [ auto with real + | eapply StepFun_P14; + [ apply H1 | apply StepFun_P2; apply H | apply StepFun_P12; apply H0 ] ] ]. Qed. -Lemma StepFun_P16 : (f:R->R;l,lf:Rlist;a,b:R) (adapted_couple f a b l lf) -> (EXT l':Rlist | (EXT lf':Rlist | (adapted_couple_opt f a b l' lf'))). -Intros; Case (total_order_Rle a b); Intro; [Apply (StepFun_P10 r H) | Assert H1 : ``b<=a``; [Auto with real | Assert H2 := (StepFun_P10 H1 (StepFun_P2 H)); Elim H2; Intros l' [lf' H3]; Exists l'; Exists lf'; Apply StepFun_P12; Assumption]]. +Lemma StepFun_P16 : + forall (f:R -> R) (l lf:Rlist) (a b:R), + adapted_couple f a b l lf -> + exists l' : Rlist + | ( exists lf' : Rlist | adapted_couple_opt f a b l' lf'). +intros; case (Rle_dec a b); intro; + [ apply (StepFun_P10 r H) + | assert (H1 : b <= a); + [ auto with real + | assert (H2 := StepFun_P10 H1 (StepFun_P2 H)); elim H2; + intros l' [lf' H3]; exists l'; exists lf'; apply StepFun_P12; + assumption ] ]. Qed. -Lemma StepFun_P17 : (f:R->R;l1,l2,lf1,lf2:Rlist;a,b:R) (adapted_couple f a b l1 lf1) -> (adapted_couple f a b l2 lf2) -> (Int_SF lf1 l1)==(Int_SF lf2 l2). -Intros; Elim (StepFun_P16 H); Intros l' [lf' H1]; Rewrite (StepFun_P15 H H1); Rewrite (StepFun_P15 H0 H1); Reflexivity. +Lemma StepFun_P17 : + forall (f:R -> R) (l1 l2 lf1 lf2:Rlist) (a b:R), + adapted_couple f a b l1 lf1 -> + adapted_couple f a b l2 lf2 -> Int_SF lf1 l1 = Int_SF lf2 l2. +intros; elim (StepFun_P16 H); intros l' [lf' H1]; rewrite (StepFun_P15 H H1); + rewrite (StepFun_P15 H0 H1); reflexivity. Qed. -Lemma StepFun_P18 : (a,b,c:R) (RiemannInt_SF (mkStepFun (StepFun_P4 a b c)))==``c*(b-a)``. -Intros; Unfold RiemannInt_SF; Case (total_order_Rle a b); Intro. -Replace (Int_SF (subdivision_val (mkStepFun (StepFun_P4 a b c))) (subdivision (mkStepFun (StepFun_P4 a b c)))) with (Int_SF (cons c nil) (cons a (cons b nil))); [Simpl; Ring | Apply StepFun_P17 with (fct_cte c) a b; [Apply StepFun_P3; Assumption | Apply (StepFun_P1 (mkStepFun (StepFun_P4 a b c)))]]. -Replace (Int_SF (subdivision_val (mkStepFun (StepFun_P4 a b c))) (subdivision (mkStepFun (StepFun_P4 a b c)))) with (Int_SF (cons c nil) (cons b (cons a nil))); [Simpl; Ring | Apply StepFun_P17 with (fct_cte c) a b; [Apply StepFun_P2; Apply StepFun_P3; Auto with real | Apply (StepFun_P1 (mkStepFun (StepFun_P4 a b c)))]]. +Lemma StepFun_P18 : + forall a b c:R, RiemannInt_SF (mkStepFun (StepFun_P4 a b c)) = c * (b - a). +intros; unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro. +replace + (Int_SF (subdivision_val (mkStepFun (StepFun_P4 a b c))) + (subdivision (mkStepFun (StepFun_P4 a b c)))) with + (Int_SF (cons c nil) (cons a (cons b nil))); + [ simpl in |- *; ring + | apply StepFun_P17 with (fct_cte c) a b; + [ apply StepFun_P3; assumption + | apply (StepFun_P1 (mkStepFun (StepFun_P4 a b c))) ] ]. +replace + (Int_SF (subdivision_val (mkStepFun (StepFun_P4 a b c))) + (subdivision (mkStepFun (StepFun_P4 a b c)))) with + (Int_SF (cons c nil) (cons b (cons a nil))); + [ simpl in |- *; ring + | apply StepFun_P17 with (fct_cte c) a b; + [ apply StepFun_P2; apply StepFun_P3; auto with real + | apply (StepFun_P1 (mkStepFun (StepFun_P4 a b c))) ] ]. Qed. -Lemma StepFun_P19 : (l1:Rlist;f,g:R->R;l:R) (Int_SF (FF l1 [x:R]``(f x)+l*(g x)``) l1)==``(Int_SF (FF l1 f) l1)+l*(Int_SF (FF l1 g) l1)``. -Intros; Induction l1; [Simpl; Ring | Induction l1; Simpl; [Ring | Simpl in Hrecl1; Rewrite Hrecl1; Ring]]. +Lemma StepFun_P19 : + forall (l1:Rlist) (f g:R -> R) (l:R), + Int_SF (FF l1 (fun x:R => f x + l * g x)) l1 = + Int_SF (FF l1 f) l1 + l * Int_SF (FF l1 g) l1. +intros; induction l1 as [| r l1 Hrecl1]; + [ simpl in |- *; ring + | induction l1 as [| r0 l1 Hrecl0]; simpl in |- *; + [ ring | simpl in Hrecl1; rewrite Hrecl1; ring ] ]. Qed. -Lemma StepFun_P20 : (l:Rlist;f:R->R) (lt O (Rlength l)) -> (Rlength l)=(S (Rlength (FF l f))). -Intros l f H; NewInduction l; [Elim (lt_n_n ? H) | Simpl; Rewrite RList_P18; Rewrite RList_P14; Reflexivity]. +Lemma StepFun_P20 : + forall (l:Rlist) (f:R -> R), + (0 < Rlength l)%nat -> Rlength l = S (Rlength (FF l f)). +intros l f H; induction l; + [ elim (lt_irrefl _ H) + | simpl in |- *; rewrite RList_P18; rewrite RList_P14; reflexivity ]. Qed. -Lemma StepFun_P21 : (a,b:R;f:R->R;l:Rlist) (is_subdivision f a b l) -> (adapted_couple f a b l (FF l f)). -Intros; Unfold adapted_couple; Unfold is_subdivision in X; Unfold adapted_couple in X; Elim X; Clear X; Intros; Decompose [and] p; Clear p; Repeat Split; Try Assumption. -Apply StepFun_P20; Rewrite H2; Apply lt_O_Sn. -Intros; Assert H5 := (H4 ? H3); Unfold constant_D_eq open_interval in H5; Unfold constant_D_eq open_interval; Intros; Induction l. -Discriminate. -Unfold FF; Rewrite RList_P12. -Simpl; Change (f x0)==(f (pos_Rl (mid_Rlist (cons r l) r) (S i))); Rewrite RList_P13; Try Assumption; Rewrite (H5 x0 H6); Rewrite H5. -Reflexivity. -Split. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Elim H6; Intros; Apply Rlt_trans with x0; Assumption | DiscrR]]. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Rewrite (Rplus_sym (pos_Rl (cons r l) i)); Apply Rlt_compatibility; Elim H6; Intros; Apply Rlt_trans with x0; Assumption | DiscrR]]. -Rewrite RList_P14; Simpl in H3; Apply H3. +Lemma StepFun_P21 : + forall (a b:R) (f:R -> R) (l:Rlist), + is_subdivision f a b l -> adapted_couple f a b l (FF l f). +intros; unfold adapted_couple in |- *; unfold is_subdivision in X; + unfold adapted_couple in X; elim X; clear X; intros; + decompose [and] p; clear p; repeat split; try assumption. +apply StepFun_P20; rewrite H2; apply lt_O_Sn. +intros; assert (H5 := H4 _ H3); unfold constant_D_eq, open_interval in H5; + unfold constant_D_eq, open_interval in |- *; intros; + induction l as [| r l Hrecl]. +discriminate. +unfold FF in |- *; rewrite RList_P12. +simpl in |- *; + change (f x0 = f (pos_Rl (mid_Rlist (cons r l) r) (S i))) in |- *; + rewrite RList_P13; try assumption; rewrite (H5 x0 H6); + rewrite H5. +reflexivity. +split. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; elim H6; + intros; apply Rlt_trans with x0; assumption + | discrR ] ]. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; + rewrite (Rplus_comm (pos_Rl (cons r l) i)); + apply Rplus_lt_compat_l; elim H6; intros; apply Rlt_trans with x0; + assumption + | discrR ] ]. +rewrite RList_P14; simpl in H3; apply H3. Qed. -Lemma StepFun_P22 : (a,b:R;f,g:R->R;lf,lg:Rlist) ``a<=b`` -> (is_subdivision f a b lf) -> (is_subdivision g a b lg) -> (is_subdivision f a b (cons_ORlist lf lg)). -Unfold is_subdivision; Intros a b f g lf lg Hyp X X0; Elim X; Elim X0; Clear X X0; Intros lg0 p lf0 p0; Assert Hyp_min : (Rmin a b)==a. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Assert Hyp_max : (Rmax a b)==b. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Apply existTT with (FF (cons_ORlist lf lg) f); Unfold adapted_couple in p p0; Decompose [and] p; Decompose [and] p0; Clear p p0; Rewrite Hyp_min in H6; Rewrite Hyp_min in H1; Rewrite Hyp_max in H0; Rewrite Hyp_max in H5; Unfold adapted_couple; Repeat Split. -Apply RList_P2; Assumption. -Rewrite Hyp_min; Symmetry; Apply Rle_antisym. -Induction lf. -Simpl; Right; Symmetry; Assumption. -Assert H10 : (In (pos_Rl (cons_ORlist (cons r lf) lg) (0)) (cons_ORlist (cons r lf) lg)). -Elim (RList_P3 (cons_ORlist (cons r lf) lg) (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros _ H10; Apply H10; Exists O; Split; [Reflexivity | Rewrite RList_P11; Simpl; Apply lt_O_Sn]. -Elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros H12 _; Assert H13 := (H12 H10); Elim H13; Intro. -Elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros H11 _; Assert H14 := (H11 H8); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- H6; Elim (RList_P6 (cons r lf)); Intros; Apply H17; [Assumption | Apply le_O_n | Assumption]. -Elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros H11 _; Assert H14 := (H11 H8); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- H1; Elim (RList_P6 lg); Intros; Apply H17; [Assumption | Apply le_O_n | Assumption]. -Induction lf. -Simpl; Right; Assumption. -Assert H8 : (In a (cons_ORlist (cons r lf) lg)). -Elim (RList_P9 (cons r lf) lg a); Intros; Apply H10; Left; Elim (RList_P3 (cons r lf) a); Intros; Apply H12; Exists O; Split; [Symmetry; Assumption | Simpl; Apply lt_O_Sn]. -Apply RList_P5; [Apply RList_P2; Assumption | Assumption]. -Rewrite Hyp_max; Apply Rle_antisym. -Induction lf. -Simpl; Right; Assumption. -Assert H8 : (In (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg)))) (cons_ORlist (cons r lf) lg)). -Elim (RList_P3 (cons_ORlist (cons r lf) lg) (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros _ H10; Apply H10; Exists (pred (Rlength (cons_ORlist (cons r lf) lg))); Split; [Reflexivity | Rewrite RList_P11; Simpl; Apply lt_n_Sn]. -Elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros H10 _. -Assert H11 := (H10 H8); Elim H11; Intro. -Elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros H13 _; Assert H14 := (H13 H12); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- H5; Elim (RList_P6 (cons r lf)); Intros; Apply H17; [Assumption | Simpl; Simpl in H14; Apply lt_n_Sm_le; Assumption | Simpl; Apply lt_n_Sn]. -Elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros H13 _; Assert H14 := (H13 H12); Elim H14; Intros; Elim H15; Clear H15; Intros. -Rewrite H15; Assert H17 : (Rlength lg)=(S (pred (Rlength lg))). -Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H17 in H16; Elim (lt_n_O ? H16). -Rewrite <- H0; Elim (RList_P6 lg); Intros; Apply H18; [Assumption | Rewrite H17 in H16; Apply lt_n_Sm_le; Assumption | Apply lt_pred_n_n; Rewrite H17; Apply lt_O_Sn]. -Induction lf. -Simpl; Right; Symmetry; Assumption. -Assert H8 : (In b (cons_ORlist (cons r lf) lg)). -Elim (RList_P9 (cons r lf) lg b); Intros; Apply H10; Left; Elim (RList_P3 (cons r lf) b); Intros; Apply H12; Exists (pred (Rlength (cons r lf))); Split; [Symmetry; Assumption | Simpl; Apply lt_n_Sn]. -Apply RList_P7; [Apply RList_P2; Assumption | Assumption]. -Apply StepFun_P20; Rewrite RList_P11; Rewrite H2; Rewrite H7; Simpl; Apply lt_O_Sn. -Intros; Unfold constant_D_eq open_interval; Intros; Cut (EXT l:R | (constant_D_eq f (open_interval (pos_Rl (cons_ORlist lf lg) i) (pos_Rl (cons_ORlist lf lg) (S i))) l)). -Intros; Elim H11; Clear H11; Intros; Assert H12 := H11; Assert Hyp_cons : (EXT r:R | (EXT r0:Rlist | (cons_ORlist lf lg)==(cons r r0))). -Apply RList_P19; Red; Intro; Rewrite H13 in H8; Elim (lt_n_O ? H8). -Elim Hyp_cons; Clear Hyp_cons; Intros r [r0 Hyp_cons]; Rewrite Hyp_cons; Unfold FF; Rewrite RList_P12. -Change (f x)==(f (pos_Rl (mid_Rlist (cons r r0) r) (S i))); Rewrite <- Hyp_cons; Rewrite RList_P13. -Assert H13 := (RList_P2 ? ? H ? H8); Elim H13; Intro. -Unfold constant_D_eq open_interval in H11 H12; Rewrite (H11 x H10); Assert H15 : ``(pos_Rl (cons_ORlist lf lg) i)<((pos_Rl (cons_ORlist lf lg) i)+(pos_Rl (cons_ORlist lf lg) (S i)))/2<(pos_Rl (cons_ORlist lf lg) (S i))``. -Split. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Rewrite (Rplus_sym (pos_Rl (cons_ORlist lf lg) i)); Apply Rlt_compatibility; Assumption | DiscrR]]. -Rewrite (H11 ? H15); Reflexivity. -Elim H10; Intros; Rewrite H14 in H15; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H16 H15)). -Apply H8. -Rewrite RList_P14; Rewrite Hyp_cons in H8; Simpl in H8; Apply H8. -Assert H11 : ``a<b``. -Apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i). -Rewrite <- H6; Rewrite <- (RList_P15 lf lg). -Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H11. -Apply RList_P2; Assumption. -Apply le_O_n. -Apply lt_trans with (pred (Rlength (cons_ORlist lf lg))); [Assumption | Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H8; Elim (lt_n_O ? H8)]. -Assumption. -Assumption. -Rewrite H1; Assumption. -Apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)). -Elim H10; Intros; Apply Rlt_trans with x; Assumption. -Rewrite <- H5; Rewrite <- (RList_P16 lf lg); Try Assumption. -Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H11. -Apply RList_P2; Assumption. -Apply lt_n_Sm_le; Apply lt_n_S; Assumption. -Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H8; Elim (lt_n_O ? H8). -Rewrite H0; Assumption. -Pose I := [j:nat]``(pos_Rl lf j)<=(pos_Rl (cons_ORlist lf lg) i)``/\(lt j (Rlength lf)); Assert H12 : (Nbound I). -Unfold Nbound; Exists (Rlength lf); Intros; Unfold I in H12; Elim H12; Intros; Apply lt_le_weak; Assumption. -Assert H13 : (EX n:nat | (I n)). -Exists O; Unfold I; Split. -Apply Rle_trans with (pos_Rl (cons_ORlist lf lg) O). -Right; Symmetry. -Apply RList_P15; Try Assumption; Rewrite H1; Assumption. -Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H13. -Apply RList_P2; Assumption. -Apply le_O_n. -Apply lt_trans with (pred (Rlength (cons_ORlist lf lg))). -Assumption. -Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H15 in H8; Elim (lt_n_O ? H8). -Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H5; Rewrite <- H6 in H11; Rewrite <- H5 in H11; Elim (Rlt_antirefl ? H11). -Assert H14 := (Nzorn H13 H12); Elim H14; Clear H14; Intros x0 H14; Exists (pos_Rl lf0 x0); Unfold constant_D_eq open_interval; Intros; Assert H16 := (H9 x0); Assert H17 : (lt x0 (pred (Rlength lf))). -Elim H14; Clear H14; Intros; Unfold I in H14; Elim H14; Clear H14; Intros; Apply lt_S_n; Replace (S (pred (Rlength lf))) with (Rlength lf). -Inversion H18. -2:Apply lt_n_S; Assumption. -Cut x0=(pred (Rlength lf)). -Intro; Rewrite H19 in H14; Rewrite H5 in H14; Cut ``(pos_Rl (cons_ORlist lf lg) i)<b``. -Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H14 H21)). -Apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)). -Elim H10; Intros; Apply Rlt_trans with x; Assumption. -Rewrite <- H5; Apply Rle_trans with (pos_Rl (cons_ORlist lf lg) (pred (Rlength (cons_ORlist lf lg)))). -Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H21. -Apply RList_P2; Assumption. -Apply lt_n_Sm_le; Apply lt_n_S; Assumption. -Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H23 in H8; Elim (lt_n_O ? H8). -Right; Apply RList_P16; Try Assumption; Rewrite H0; Assumption. -Rewrite <- H20; Reflexivity. -Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H19 in H18; Elim (lt_n_O ? H18). -Assert H18 := (H16 H17); Unfold constant_D_eq open_interval in H18; Rewrite (H18 x1). -Reflexivity. -Elim H15; Clear H15; Intros; Elim H14; Clear H14; Intros; Unfold I in H14; Elim H14; Clear H14; Intros; Split. -Apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i); Assumption. -Apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)); Try Assumption. -Assert H22 : (lt (S x0) (Rlength lf)). -Replace (Rlength lf) with (S (pred (Rlength lf))); [Apply lt_n_S; Assumption | Symmetry; Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H22 in H21; Elim (lt_n_O ? H21)]. -Elim (total_order_Rle (pos_Rl lf (S x0)) (pos_Rl (cons_ORlist lf lg) i)); Intro. -Assert H23 : (le (S x0) x0). -Apply H20; Unfold I; Split; Assumption. -Elim (le_Sn_n ? H23). -Assert H23 : ``(pos_Rl (cons_ORlist lf lg) i)<(pos_Rl lf (S x0))``. -Auto with real. -Clear b0; Apply RList_P17; Try Assumption. -Apply RList_P2; Assumption. -Elim (RList_P9 lf lg (pos_Rl lf (S x0))); Intros; Apply H25; Left; Elim (RList_P3 lf (pos_Rl lf (S x0))); Intros; Apply H27; Exists (S x0); Split; [Reflexivity | Apply H22]. +Lemma StepFun_P22 : + forall (a b:R) (f g:R -> R) (lf lg:Rlist), + a <= b -> + is_subdivision f a b lf -> + is_subdivision g a b lg -> is_subdivision f a b (cons_ORlist lf lg). +unfold is_subdivision in |- *; intros a b f g lf lg Hyp X X0; elim X; elim X0; + clear X X0; intros lg0 p lf0 p0; assert (Hyp_min : Rmin a b = a). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +assert (Hyp_max : Rmax a b = b). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +apply existT with (FF (cons_ORlist lf lg) f); unfold adapted_couple in p, p0; + decompose [and] p; decompose [and] p0; clear p p0; + rewrite Hyp_min in H6; rewrite Hyp_min in H1; rewrite Hyp_max in H0; + rewrite Hyp_max in H5; unfold adapted_couple in |- *; + repeat split. +apply RList_P2; assumption. +rewrite Hyp_min; symmetry in |- *; apply Rle_antisym. +induction lf as [| r lf Hreclf]. +simpl in |- *; right; symmetry in |- *; assumption. +assert + (H10 : + In (pos_Rl (cons_ORlist (cons r lf) lg) 0) (cons_ORlist (cons r lf) lg)). +elim + (RList_P3 (cons_ORlist (cons r lf) lg) + (pos_Rl (cons_ORlist (cons r lf) lg) 0)); intros _ H10; + apply H10; exists 0%nat; split; + [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_O_Sn ]. +elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) 0)); + intros H12 _; assert (H13 := H12 H10); elim H13; intro. +elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) 0)); + intros H11 _; assert (H14 := H11 H8); elim H14; intros; + elim H15; clear H15; intros; rewrite H15; rewrite <- H6; + elim (RList_P6 (cons r lf)); intros; apply H17; + [ assumption | apply le_O_n | assumption ]. +elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) 0)); intros H11 _; + assert (H14 := H11 H8); elim H14; intros; elim H15; + clear H15; intros; rewrite H15; rewrite <- H1; elim (RList_P6 lg); + intros; apply H17; [ assumption | apply le_O_n | assumption ]. +induction lf as [| r lf Hreclf]. +simpl in |- *; right; assumption. +assert (H8 : In a (cons_ORlist (cons r lf) lg)). +elim (RList_P9 (cons r lf) lg a); intros; apply H10; left; + elim (RList_P3 (cons r lf) a); intros; apply H12; + exists 0%nat; split; + [ symmetry in |- *; assumption | simpl in |- *; apply lt_O_Sn ]. +apply RList_P5; [ apply RList_P2; assumption | assumption ]. +rewrite Hyp_max; apply Rle_antisym. +induction lf as [| r lf Hreclf]. +simpl in |- *; right; assumption. +assert + (H8 : + In + (pos_Rl (cons_ORlist (cons r lf) lg) + (pred (Rlength (cons_ORlist (cons r lf) lg)))) + (cons_ORlist (cons r lf) lg)). +elim + (RList_P3 (cons_ORlist (cons r lf) lg) + (pos_Rl (cons_ORlist (cons r lf) lg) + (pred (Rlength (cons_ORlist (cons r lf) lg))))); + intros _ H10; apply H10; + exists (pred (Rlength (cons_ORlist (cons r lf) lg))); + split; [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_n_Sn ]. +elim + (RList_P9 (cons r lf) lg + (pos_Rl (cons_ORlist (cons r lf) lg) + (pred (Rlength (cons_ORlist (cons r lf) lg))))); + intros H10 _. +assert (H11 := H10 H8); elim H11; intro. +elim + (RList_P3 (cons r lf) + (pos_Rl (cons_ORlist (cons r lf) lg) + (pred (Rlength (cons_ORlist (cons r lf) lg))))); + intros H13 _; assert (H14 := H13 H12); elim H14; intros; + elim H15; clear H15; intros; rewrite H15; rewrite <- H5; + elim (RList_P6 (cons r lf)); intros; apply H17; + [ assumption + | simpl in |- *; simpl in H14; apply lt_n_Sm_le; assumption + | simpl in |- *; apply lt_n_Sn ]. +elim + (RList_P3 lg + (pos_Rl (cons_ORlist (cons r lf) lg) + (pred (Rlength (cons_ORlist (cons r lf) lg))))); + intros H13 _; assert (H14 := H13 H12); elim H14; intros; + elim H15; clear H15; intros. +rewrite H15; assert (H17 : Rlength lg = S (pred (Rlength lg))). +apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro; + rewrite <- H17 in H16; elim (lt_n_O _ H16). +rewrite <- H0; elim (RList_P6 lg); intros; apply H18; + [ assumption + | rewrite H17 in H16; apply lt_n_Sm_le; assumption + | apply lt_pred_n_n; rewrite H17; apply lt_O_Sn ]. +induction lf as [| r lf Hreclf]. +simpl in |- *; right; symmetry in |- *; assumption. +assert (H8 : In b (cons_ORlist (cons r lf) lg)). +elim (RList_P9 (cons r lf) lg b); intros; apply H10; left; + elim (RList_P3 (cons r lf) b); intros; apply H12; + exists (pred (Rlength (cons r lf))); split; + [ symmetry in |- *; assumption | simpl in |- *; apply lt_n_Sn ]. +apply RList_P7; [ apply RList_P2; assumption | assumption ]. +apply StepFun_P20; rewrite RList_P11; rewrite H2; rewrite H7; simpl in |- *; + apply lt_O_Sn. +intros; unfold constant_D_eq, open_interval in |- *; intros; + cut + ( exists l : R + | constant_D_eq f + (open_interval (pos_Rl (cons_ORlist lf lg) i) + (pos_Rl (cons_ORlist lf lg) (S i))) l). +intros; elim H11; clear H11; intros; assert (H12 := H11); + assert + (Hyp_cons : + exists r : R | ( exists r0 : Rlist | cons_ORlist lf lg = cons r r0)). +apply RList_P19; red in |- *; intro; rewrite H13 in H8; elim (lt_n_O _ H8). +elim Hyp_cons; clear Hyp_cons; intros r [r0 Hyp_cons]; rewrite Hyp_cons; + unfold FF in |- *; rewrite RList_P12. +change (f x = f (pos_Rl (mid_Rlist (cons r r0) r) (S i))) in |- *; + rewrite <- Hyp_cons; rewrite RList_P13. +assert (H13 := RList_P2 _ _ H _ H8); elim H13; intro. +unfold constant_D_eq, open_interval in H11, H12; rewrite (H11 x H10); + assert + (H15 : + pos_Rl (cons_ORlist lf lg) i < + (pos_Rl (cons_ORlist lf lg) i + pos_Rl (cons_ORlist lf lg) (S i)) / 2 < + pos_Rl (cons_ORlist lf lg) (S i)). +split. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; + rewrite (Rplus_comm (pos_Rl (cons_ORlist lf lg) i)); + apply Rplus_lt_compat_l; assumption + | discrR ] ]. +rewrite (H11 _ H15); reflexivity. +elim H10; intros; rewrite H14 in H15; + elim (Rlt_irrefl _ (Rlt_trans _ _ _ H16 H15)). +apply H8. +rewrite RList_P14; rewrite Hyp_cons in H8; simpl in H8; apply H8. +assert (H11 : a < b). +apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i). +rewrite <- H6; rewrite <- (RList_P15 lf lg). +elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11. +apply RList_P2; assumption. +apply le_O_n. +apply lt_trans with (pred (Rlength (cons_ORlist lf lg))); + [ assumption + | apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; + rewrite <- H13 in H8; elim (lt_n_O _ H8) ]. +assumption. +assumption. +rewrite H1; assumption. +apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)). +elim H10; intros; apply Rlt_trans with x; assumption. +rewrite <- H5; rewrite <- (RList_P16 lf lg); try assumption. +elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11. +apply RList_P2; assumption. +apply lt_n_Sm_le; apply lt_n_S; assumption. +apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H8; + elim (lt_n_O _ H8). +rewrite H0; assumption. +pose + (I := + fun j:nat => + pos_Rl lf j <= pos_Rl (cons_ORlist lf lg) i /\ (j < Rlength lf)%nat); + assert (H12 : Nbound I). +unfold Nbound in |- *; exists (Rlength lf); intros; unfold I in H12; elim H12; + intros; apply lt_le_weak; assumption. +assert (H13 : exists n : nat | I n). +exists 0%nat; unfold I in |- *; split. +apply Rle_trans with (pos_Rl (cons_ORlist lf lg) 0). +right; symmetry in |- *. +apply RList_P15; try assumption; rewrite H1; assumption. +elim (RList_P6 (cons_ORlist lf lg)); intros; apply H13. +apply RList_P2; assumption. +apply le_O_n. +apply lt_trans with (pred (Rlength (cons_ORlist lf lg))). +assumption. +apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H15 in H8; + elim (lt_n_O _ H8). +apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H5; + rewrite <- H6 in H11; rewrite <- H5 in H11; elim (Rlt_irrefl _ H11). +assert (H14 := Nzorn H13 H12); elim H14; clear H14; intros x0 H14; + exists (pos_Rl lf0 x0); unfold constant_D_eq, open_interval in |- *; + intros; assert (H16 := H9 x0); assert (H17 : (x0 < pred (Rlength lf))%nat). +elim H14; clear H14; intros; unfold I in H14; elim H14; clear H14; intros; + apply lt_S_n; replace (S (pred (Rlength lf))) with (Rlength lf). +inversion H18. +2: apply lt_n_S; assumption. +cut (x0 = pred (Rlength lf)). +intro; rewrite H19 in H14; rewrite H5 in H14; + cut (pos_Rl (cons_ORlist lf lg) i < b). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H14 H21)). +apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)). +elim H10; intros; apply Rlt_trans with x; assumption. +rewrite <- H5; + apply Rle_trans with + (pos_Rl (cons_ORlist lf lg) (pred (Rlength (cons_ORlist lf lg)))). +elim (RList_P6 (cons_ORlist lf lg)); intros; apply H21. +apply RList_P2; assumption. +apply lt_n_Sm_le; apply lt_n_S; assumption. +apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H23 in H8; + elim (lt_n_O _ H8). +right; apply RList_P16; try assumption; rewrite H0; assumption. +rewrite <- H20; reflexivity. +apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro; + rewrite <- H19 in H18; elim (lt_n_O _ H18). +assert (H18 := H16 H17); unfold constant_D_eq, open_interval in H18; + rewrite (H18 x1). +reflexivity. +elim H15; clear H15; intros; elim H14; clear H14; intros; unfold I in H14; + elim H14; clear H14; intros; split. +apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i); assumption. +apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)); try assumption. +assert (H22 : (S x0 < Rlength lf)%nat). +replace (Rlength lf) with (S (pred (Rlength lf))); + [ apply lt_n_S; assumption + | symmetry in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *; + intro; rewrite <- H22 in H21; elim (lt_n_O _ H21) ]. +elim (Rle_dec (pos_Rl lf (S x0)) (pos_Rl (cons_ORlist lf lg) i)); intro. +assert (H23 : (S x0 <= x0)%nat). +apply H20; unfold I in |- *; split; assumption. +elim (le_Sn_n _ H23). +assert (H23 : pos_Rl (cons_ORlist lf lg) i < pos_Rl lf (S x0)). +auto with real. +clear b0; apply RList_P17; try assumption. +apply RList_P2; assumption. +elim (RList_P9 lf lg (pos_Rl lf (S x0))); intros; apply H25; left; + elim (RList_P3 lf (pos_Rl lf (S x0))); intros; apply H27; + exists (S x0); split; [ reflexivity | apply H22 ]. Qed. -Lemma StepFun_P23 : (a,b:R;f,g:R->R;lf,lg:Rlist) (is_subdivision f a b lf) -> (is_subdivision g a b lg) -> (is_subdivision f a b (cons_ORlist lf lg)). -Intros; Case (total_order_Rle a b); Intro; [Apply StepFun_P22 with g; Assumption | Apply StepFun_P5; Apply StepFun_P22 with g; [Auto with real | Apply StepFun_P5; Assumption | Apply StepFun_P5; Assumption]]. +Lemma StepFun_P23 : + forall (a b:R) (f g:R -> R) (lf lg:Rlist), + is_subdivision f a b lf -> + is_subdivision g a b lg -> is_subdivision f a b (cons_ORlist lf lg). +intros; case (Rle_dec a b); intro; + [ apply StepFun_P22 with g; assumption + | apply StepFun_P5; apply StepFun_P22 with g; + [ auto with real + | apply StepFun_P5; assumption + | apply StepFun_P5; assumption ] ]. Qed. -Lemma StepFun_P24 : (a,b:R;f,g:R->R;lf,lg:Rlist) ``a<=b`` -> (is_subdivision f a b lf) -> (is_subdivision g a b lg) -> (is_subdivision g a b (cons_ORlist lf lg)). -Unfold is_subdivision; Intros a b f g lf lg Hyp X X0; Elim X; Elim X0; Clear X X0; Intros lg0 p lf0 p0; Assert Hyp_min : (Rmin a b)==a. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Assert Hyp_max : (Rmax a b)==b. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Apply existTT with (FF (cons_ORlist lf lg) g); Unfold adapted_couple in p p0; Decompose [and] p; Decompose [and] p0; Clear p p0; Rewrite Hyp_min in H1; Rewrite Hyp_min in H6; Rewrite Hyp_max in H0; Rewrite Hyp_max in H5; Unfold adapted_couple; Repeat Split. -Apply RList_P2; Assumption. -Rewrite Hyp_min; Symmetry; Apply Rle_antisym. -Induction lf. -Simpl; Right; Symmetry; Assumption. -Assert H10 : (In (pos_Rl (cons_ORlist (cons r lf) lg) (0)) (cons_ORlist (cons r lf) lg)). -Elim (RList_P3 (cons_ORlist (cons r lf) lg) (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros _ H10; Apply H10; Exists O; Split; [Reflexivity | Rewrite RList_P11; Simpl; Apply lt_O_Sn]. -Elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros H12 _; Assert H13 := (H12 H10); Elim H13; Intro. -Elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros H11 _; Assert H14 := (H11 H8); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- H6; Elim (RList_P6 (cons r lf)); Intros; Apply H17; [Assumption | Apply le_O_n | Assumption]. -Elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) (0))); Intros H11 _; Assert H14 := (H11 H8); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- H1; Elim (RList_P6 lg); Intros; Apply H17; [Assumption | Apply le_O_n | Assumption]. -Induction lf. -Simpl; Right; Assumption. -Assert H8 : (In a (cons_ORlist (cons r lf) lg)). -Elim (RList_P9 (cons r lf) lg a); Intros; Apply H10; Left; Elim (RList_P3 (cons r lf) a); Intros; Apply H12; Exists O; Split; [Symmetry; Assumption | Simpl; Apply lt_O_Sn]. -Apply RList_P5; [Apply RList_P2; Assumption | Assumption]. -Rewrite Hyp_max; Apply Rle_antisym. -Induction lf. -Simpl; Right; Assumption. -Assert H8 : (In (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg)))) (cons_ORlist (cons r lf) lg)). -Elim (RList_P3 (cons_ORlist (cons r lf) lg) (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros _ H10; Apply H10; Exists (pred (Rlength (cons_ORlist (cons r lf) lg))); Split; [Reflexivity | Rewrite RList_P11; Simpl; Apply lt_n_Sn]. -Elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros H10 _; Assert H11 := (H10 H8); Elim H11; Intro. -Elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros H13 _; Assert H14 := (H13 H12); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Rewrite <- H5; Elim (RList_P6 (cons r lf)); Intros; Apply H17; [Assumption | Simpl; Simpl in H14; Apply lt_n_Sm_le; Assumption | Simpl; Apply lt_n_Sn]. -Elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) (pred (Rlength (cons_ORlist (cons r lf) lg))))); Intros H13 _; Assert H14 := (H13 H12); Elim H14; Intros; Elim H15; Clear H15; Intros; Rewrite H15; Assert H17 : (Rlength lg)=(S (pred (Rlength lg))). -Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H17 in H16; Elim (lt_n_O ? H16). -Rewrite <- H0; Elim (RList_P6 lg); Intros; Apply H18; [Assumption | Rewrite H17 in H16; Apply lt_n_Sm_le; Assumption | Apply lt_pred_n_n; Rewrite H17; Apply lt_O_Sn]. -Induction lf. -Simpl; Right; Symmetry; Assumption. -Assert H8 : (In b (cons_ORlist (cons r lf) lg)). -Elim (RList_P9 (cons r lf) lg b); Intros; Apply H10; Left; Elim (RList_P3 (cons r lf) b); Intros; Apply H12; Exists (pred (Rlength (cons r lf))); Split; [Symmetry; Assumption | Simpl; Apply lt_n_Sn]. -Apply RList_P7; [Apply RList_P2; Assumption | Assumption]. -Apply StepFun_P20; Rewrite RList_P11; Rewrite H7; Rewrite H2; Simpl; Apply lt_O_Sn. -Unfold constant_D_eq open_interval; Intros; Cut (EXT l:R | (constant_D_eq g (open_interval (pos_Rl (cons_ORlist lf lg) i) (pos_Rl (cons_ORlist lf lg) (S i))) l)). -Intros; Elim H11; Clear H11; Intros; Assert H12 := H11; Assert Hyp_cons : (EXT r:R | (EXT r0:Rlist | (cons_ORlist lf lg)==(cons r r0))). -Apply RList_P19; Red; Intro; Rewrite H13 in H8; Elim (lt_n_O ? H8). -Elim Hyp_cons; Clear Hyp_cons; Intros r [r0 Hyp_cons]; Rewrite Hyp_cons; Unfold FF; Rewrite RList_P12. -Change (g x)==(g (pos_Rl (mid_Rlist (cons r r0) r) (S i))); Rewrite <- Hyp_cons; Rewrite RList_P13. -Assert H13 := (RList_P2 ? ? H ? H8); Elim H13; Intro. -Unfold constant_D_eq open_interval in H11 H12; Rewrite (H11 x H10); Assert H15 : ``(pos_Rl (cons_ORlist lf lg) i)<((pos_Rl (cons_ORlist lf lg) i)+(pos_Rl (cons_ORlist lf lg) (S i)))/2<(pos_Rl (cons_ORlist lf lg) (S i))``. -Split. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Rewrite (Rplus_sym (pos_Rl (cons_ORlist lf lg) i)); Apply Rlt_compatibility; Assumption | DiscrR]]. -Rewrite (H11 ? H15); Reflexivity. -Elim H10; Intros; Rewrite H14 in H15; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H16 H15)). -Apply H8. -Rewrite RList_P14; Rewrite Hyp_cons in H8; Simpl in H8; Apply H8. -Assert H11 : ``a<b``. -Apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i). -Rewrite <- H6; Rewrite <- (RList_P15 lf lg); Try Assumption. -Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H11. -Apply RList_P2; Assumption. -Apply le_O_n. -Apply lt_trans with (pred (Rlength (cons_ORlist lf lg))); [Assumption | Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H8; Elim (lt_n_O ? H8)]. -Rewrite H1; Assumption. -Apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)). -Elim H10; Intros; Apply Rlt_trans with x; Assumption. -Rewrite <- H5; Rewrite <- (RList_P16 lf lg); Try Assumption. -Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H11. -Apply RList_P2; Assumption. -Apply lt_n_Sm_le; Apply lt_n_S; Assumption. -Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H8; Elim (lt_n_O ? H8). -Rewrite H0; Assumption. -Pose I := [j:nat]``(pos_Rl lg j)<=(pos_Rl (cons_ORlist lf lg) i)``/\(lt j (Rlength lg)); Assert H12 : (Nbound I). -Unfold Nbound; Exists (Rlength lg); Intros; Unfold I in H12; Elim H12; Intros; Apply lt_le_weak; Assumption. -Assert H13 : (EX n:nat | (I n)). -Exists O; Unfold I; Split. -Apply Rle_trans with (pos_Rl (cons_ORlist lf lg) O). -Right; Symmetry; Rewrite H1; Rewrite <- H6; Apply RList_P15; Try Assumption; Rewrite H1; Assumption. -Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H13; [Apply RList_P2; Assumption | Apply le_O_n | Apply lt_trans with (pred (Rlength (cons_ORlist lf lg))); [Assumption | Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H15 in H8; Elim (lt_n_O ? H8)]]. -Apply neq_O_lt; Red; Intro; Rewrite <- H13 in H0; Rewrite <- H1 in H11; Rewrite <- H0 in H11; Elim (Rlt_antirefl ? H11). -Assert H14 := (Nzorn H13 H12); Elim H14; Clear H14; Intros x0 H14; Exists (pos_Rl lg0 x0); Unfold constant_D_eq open_interval; Intros; Assert H16 := (H4 x0); Assert H17 : (lt x0 (pred (Rlength lg))). -Elim H14; Clear H14; Intros; Unfold I in H14; Elim H14; Clear H14; Intros; Apply lt_S_n; Replace (S (pred (Rlength lg))) with (Rlength lg). -Inversion H18. -2:Apply lt_n_S; Assumption. -Cut x0=(pred (Rlength lg)). -Intro; Rewrite H19 in H14; Rewrite H0 in H14; Cut ``(pos_Rl (cons_ORlist lf lg) i)<b``. -Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H14 H21)). -Apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)). -Elim H10; Intros; Apply Rlt_trans with x; Assumption. -Rewrite <- H0; Apply Rle_trans with (pos_Rl (cons_ORlist lf lg) (pred (Rlength (cons_ORlist lf lg)))). -Elim (RList_P6 (cons_ORlist lf lg)); Intros; Apply H21. -Apply RList_P2; Assumption. -Apply lt_n_Sm_le; Apply lt_n_S; Assumption. -Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H23 in H8; Elim (lt_n_O ? H8). -Right; Rewrite H0; Rewrite <- H5; Apply RList_P16; Try Assumption. -Rewrite H0; Assumption. -Rewrite <- H20; Reflexivity. -Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H19 in H18; Elim (lt_n_O ? H18). -Assert H18 := (H16 H17); Unfold constant_D_eq open_interval in H18; Rewrite (H18 x1). -Reflexivity. -Elim H15; Clear H15; Intros; Elim H14; Clear H14; Intros; Unfold I in H14; Elim H14; Clear H14; Intros; Split. -Apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i); Assumption. -Apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)); Try Assumption. -Assert H22 : (lt (S x0) (Rlength lg)). -Replace (Rlength lg) with (S (pred (Rlength lg))). -Apply lt_n_S; Assumption. -Symmetry; Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H22 in H21; Elim (lt_n_O ? H21). -Elim (total_order_Rle (pos_Rl lg (S x0)) (pos_Rl (cons_ORlist lf lg) i)); Intro. -Assert H23 : (le (S x0) x0); [Apply H20; Unfold I; Split; Assumption | Elim (le_Sn_n ? H23)]. -Assert H23 : ``(pos_Rl (cons_ORlist lf lg) i)<(pos_Rl lg (S x0))``. -Auto with real. -Clear b0; Apply RList_P17; Try Assumption; [Apply RList_P2; Assumption | Elim (RList_P9 lf lg (pos_Rl lg (S x0))); Intros; Apply H25; Right; Elim (RList_P3 lg (pos_Rl lg (S x0))); Intros; Apply H27; Exists (S x0); Split; [Reflexivity | Apply H22]]. +Lemma StepFun_P24 : + forall (a b:R) (f g:R -> R) (lf lg:Rlist), + a <= b -> + is_subdivision f a b lf -> + is_subdivision g a b lg -> is_subdivision g a b (cons_ORlist lf lg). +unfold is_subdivision in |- *; intros a b f g lf lg Hyp X X0; elim X; elim X0; + clear X X0; intros lg0 p lf0 p0; assert (Hyp_min : Rmin a b = a). +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +assert (Hyp_max : Rmax a b = b). +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +apply existT with (FF (cons_ORlist lf lg) g); unfold adapted_couple in p, p0; + decompose [and] p; decompose [and] p0; clear p p0; + rewrite Hyp_min in H1; rewrite Hyp_min in H6; rewrite Hyp_max in H0; + rewrite Hyp_max in H5; unfold adapted_couple in |- *; + repeat split. +apply RList_P2; assumption. +rewrite Hyp_min; symmetry in |- *; apply Rle_antisym. +induction lf as [| r lf Hreclf]. +simpl in |- *; right; symmetry in |- *; assumption. +assert + (H10 : + In (pos_Rl (cons_ORlist (cons r lf) lg) 0) (cons_ORlist (cons r lf) lg)). +elim + (RList_P3 (cons_ORlist (cons r lf) lg) + (pos_Rl (cons_ORlist (cons r lf) lg) 0)); intros _ H10; + apply H10; exists 0%nat; split; + [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_O_Sn ]. +elim (RList_P9 (cons r lf) lg (pos_Rl (cons_ORlist (cons r lf) lg) 0)); + intros H12 _; assert (H13 := H12 H10); elim H13; intro. +elim (RList_P3 (cons r lf) (pos_Rl (cons_ORlist (cons r lf) lg) 0)); + intros H11 _; assert (H14 := H11 H8); elim H14; intros; + elim H15; clear H15; intros; rewrite H15; rewrite <- H6; + elim (RList_P6 (cons r lf)); intros; apply H17; + [ assumption | apply le_O_n | assumption ]. +elim (RList_P3 lg (pos_Rl (cons_ORlist (cons r lf) lg) 0)); intros H11 _; + assert (H14 := H11 H8); elim H14; intros; elim H15; + clear H15; intros; rewrite H15; rewrite <- H1; elim (RList_P6 lg); + intros; apply H17; [ assumption | apply le_O_n | assumption ]. +induction lf as [| r lf Hreclf]. +simpl in |- *; right; assumption. +assert (H8 : In a (cons_ORlist (cons r lf) lg)). +elim (RList_P9 (cons r lf) lg a); intros; apply H10; left; + elim (RList_P3 (cons r lf) a); intros; apply H12; + exists 0%nat; split; + [ symmetry in |- *; assumption | simpl in |- *; apply lt_O_Sn ]. +apply RList_P5; [ apply RList_P2; assumption | assumption ]. +rewrite Hyp_max; apply Rle_antisym. +induction lf as [| r lf Hreclf]. +simpl in |- *; right; assumption. +assert + (H8 : + In + (pos_Rl (cons_ORlist (cons r lf) lg) + (pred (Rlength (cons_ORlist (cons r lf) lg)))) + (cons_ORlist (cons r lf) lg)). +elim + (RList_P3 (cons_ORlist (cons r lf) lg) + (pos_Rl (cons_ORlist (cons r lf) lg) + (pred (Rlength (cons_ORlist (cons r lf) lg))))); + intros _ H10; apply H10; + exists (pred (Rlength (cons_ORlist (cons r lf) lg))); + split; [ reflexivity | rewrite RList_P11; simpl in |- *; apply lt_n_Sn ]. +elim + (RList_P9 (cons r lf) lg + (pos_Rl (cons_ORlist (cons r lf) lg) + (pred (Rlength (cons_ORlist (cons r lf) lg))))); + intros H10 _; assert (H11 := H10 H8); elim H11; intro. +elim + (RList_P3 (cons r lf) + (pos_Rl (cons_ORlist (cons r lf) lg) + (pred (Rlength (cons_ORlist (cons r lf) lg))))); + intros H13 _; assert (H14 := H13 H12); elim H14; intros; + elim H15; clear H15; intros; rewrite H15; rewrite <- H5; + elim (RList_P6 (cons r lf)); intros; apply H17; + [ assumption + | simpl in |- *; simpl in H14; apply lt_n_Sm_le; assumption + | simpl in |- *; apply lt_n_Sn ]. +elim + (RList_P3 lg + (pos_Rl (cons_ORlist (cons r lf) lg) + (pred (Rlength (cons_ORlist (cons r lf) lg))))); + intros H13 _; assert (H14 := H13 H12); elim H14; intros; + elim H15; clear H15; intros; rewrite H15; + assert (H17 : Rlength lg = S (pred (Rlength lg))). +apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro; + rewrite <- H17 in H16; elim (lt_n_O _ H16). +rewrite <- H0; elim (RList_P6 lg); intros; apply H18; + [ assumption + | rewrite H17 in H16; apply lt_n_Sm_le; assumption + | apply lt_pred_n_n; rewrite H17; apply lt_O_Sn ]. +induction lf as [| r lf Hreclf]. +simpl in |- *; right; symmetry in |- *; assumption. +assert (H8 : In b (cons_ORlist (cons r lf) lg)). +elim (RList_P9 (cons r lf) lg b); intros; apply H10; left; + elim (RList_P3 (cons r lf) b); intros; apply H12; + exists (pred (Rlength (cons r lf))); split; + [ symmetry in |- *; assumption | simpl in |- *; apply lt_n_Sn ]. +apply RList_P7; [ apply RList_P2; assumption | assumption ]. +apply StepFun_P20; rewrite RList_P11; rewrite H7; rewrite H2; simpl in |- *; + apply lt_O_Sn. +unfold constant_D_eq, open_interval in |- *; intros; + cut + ( exists l : R + | constant_D_eq g + (open_interval (pos_Rl (cons_ORlist lf lg) i) + (pos_Rl (cons_ORlist lf lg) (S i))) l). +intros; elim H11; clear H11; intros; assert (H12 := H11); + assert + (Hyp_cons : + exists r : R | ( exists r0 : Rlist | cons_ORlist lf lg = cons r r0)). +apply RList_P19; red in |- *; intro; rewrite H13 in H8; elim (lt_n_O _ H8). +elim Hyp_cons; clear Hyp_cons; intros r [r0 Hyp_cons]; rewrite Hyp_cons; + unfold FF in |- *; rewrite RList_P12. +change (g x = g (pos_Rl (mid_Rlist (cons r r0) r) (S i))) in |- *; + rewrite <- Hyp_cons; rewrite RList_P13. +assert (H13 := RList_P2 _ _ H _ H8); elim H13; intro. +unfold constant_D_eq, open_interval in H11, H12; rewrite (H11 x H10); + assert + (H15 : + pos_Rl (cons_ORlist lf lg) i < + (pos_Rl (cons_ORlist lf lg) i + pos_Rl (cons_ORlist lf lg) (S i)) / 2 < + pos_Rl (cons_ORlist lf lg) (S i)). +split. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; + rewrite (Rplus_comm (pos_Rl (cons_ORlist lf lg) i)); + apply Rplus_lt_compat_l; assumption + | discrR ] ]. +rewrite (H11 _ H15); reflexivity. +elim H10; intros; rewrite H14 in H15; + elim (Rlt_irrefl _ (Rlt_trans _ _ _ H16 H15)). +apply H8. +rewrite RList_P14; rewrite Hyp_cons in H8; simpl in H8; apply H8. +assert (H11 : a < b). +apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i). +rewrite <- H6; rewrite <- (RList_P15 lf lg); try assumption. +elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11. +apply RList_P2; assumption. +apply le_O_n. +apply lt_trans with (pred (Rlength (cons_ORlist lf lg))); + [ assumption + | apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; + rewrite <- H13 in H8; elim (lt_n_O _ H8) ]. +rewrite H1; assumption. +apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)). +elim H10; intros; apply Rlt_trans with x; assumption. +rewrite <- H5; rewrite <- (RList_P16 lf lg); try assumption. +elim (RList_P6 (cons_ORlist lf lg)); intros; apply H11. +apply RList_P2; assumption. +apply lt_n_Sm_le; apply lt_n_S; assumption. +apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H8; + elim (lt_n_O _ H8). +rewrite H0; assumption. +pose + (I := + fun j:nat => + pos_Rl lg j <= pos_Rl (cons_ORlist lf lg) i /\ (j < Rlength lg)%nat); + assert (H12 : Nbound I). +unfold Nbound in |- *; exists (Rlength lg); intros; unfold I in H12; elim H12; + intros; apply lt_le_weak; assumption. +assert (H13 : exists n : nat | I n). +exists 0%nat; unfold I in |- *; split. +apply Rle_trans with (pos_Rl (cons_ORlist lf lg) 0). +right; symmetry in |- *; rewrite H1; rewrite <- H6; apply RList_P15; + try assumption; rewrite H1; assumption. +elim (RList_P6 (cons_ORlist lf lg)); intros; apply H13; + [ apply RList_P2; assumption + | apply le_O_n + | apply lt_trans with (pred (Rlength (cons_ORlist lf lg))); + [ assumption + | apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; + rewrite <- H15 in H8; elim (lt_n_O _ H8) ] ]. +apply neq_O_lt; red in |- *; intro; rewrite <- H13 in H0; + rewrite <- H1 in H11; rewrite <- H0 in H11; elim (Rlt_irrefl _ H11). +assert (H14 := Nzorn H13 H12); elim H14; clear H14; intros x0 H14; + exists (pos_Rl lg0 x0); unfold constant_D_eq, open_interval in |- *; + intros; assert (H16 := H4 x0); assert (H17 : (x0 < pred (Rlength lg))%nat). +elim H14; clear H14; intros; unfold I in H14; elim H14; clear H14; intros; + apply lt_S_n; replace (S (pred (Rlength lg))) with (Rlength lg). +inversion H18. +2: apply lt_n_S; assumption. +cut (x0 = pred (Rlength lg)). +intro; rewrite H19 in H14; rewrite H0 in H14; + cut (pos_Rl (cons_ORlist lf lg) i < b). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H14 H21)). +apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)). +elim H10; intros; apply Rlt_trans with x; assumption. +rewrite <- H0; + apply Rle_trans with + (pos_Rl (cons_ORlist lf lg) (pred (Rlength (cons_ORlist lf lg)))). +elim (RList_P6 (cons_ORlist lf lg)); intros; apply H21. +apply RList_P2; assumption. +apply lt_n_Sm_le; apply lt_n_S; assumption. +apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H23 in H8; + elim (lt_n_O _ H8). +right; rewrite H0; rewrite <- H5; apply RList_P16; try assumption. +rewrite H0; assumption. +rewrite <- H20; reflexivity. +apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro; + rewrite <- H19 in H18; elim (lt_n_O _ H18). +assert (H18 := H16 H17); unfold constant_D_eq, open_interval in H18; + rewrite (H18 x1). +reflexivity. +elim H15; clear H15; intros; elim H14; clear H14; intros; unfold I in H14; + elim H14; clear H14; intros; split. +apply Rle_lt_trans with (pos_Rl (cons_ORlist lf lg) i); assumption. +apply Rlt_le_trans with (pos_Rl (cons_ORlist lf lg) (S i)); try assumption. +assert (H22 : (S x0 < Rlength lg)%nat). +replace (Rlength lg) with (S (pred (Rlength lg))). +apply lt_n_S; assumption. +symmetry in |- *; apply S_pred with 0%nat; apply neq_O_lt; red in |- *; + intro; rewrite <- H22 in H21; elim (lt_n_O _ H21). +elim (Rle_dec (pos_Rl lg (S x0)) (pos_Rl (cons_ORlist lf lg) i)); intro. +assert (H23 : (S x0 <= x0)%nat); + [ apply H20; unfold I in |- *; split; assumption | elim (le_Sn_n _ H23) ]. +assert (H23 : pos_Rl (cons_ORlist lf lg) i < pos_Rl lg (S x0)). +auto with real. +clear b0; apply RList_P17; try assumption; + [ apply RList_P2; assumption + | elim (RList_P9 lf lg (pos_Rl lg (S x0))); intros; apply H25; right; + elim (RList_P3 lg (pos_Rl lg (S x0))); intros; + apply H27; exists (S x0); split; [ reflexivity | apply H22 ] ]. Qed. -Lemma StepFun_P25 : (a,b:R;f,g:R->R;lf,lg:Rlist) (is_subdivision f a b lf) -> (is_subdivision g a b lg) -> (is_subdivision g a b (cons_ORlist lf lg)). -Intros a b f g lf lg H H0; Case (total_order_Rle a b); Intro; [Apply StepFun_P24 with f; Assumption | Apply StepFun_P5; Apply StepFun_P24 with f; [Auto with real | Apply StepFun_P5; Assumption | Apply StepFun_P5; Assumption]]. +Lemma StepFun_P25 : + forall (a b:R) (f g:R -> R) (lf lg:Rlist), + is_subdivision f a b lf -> + is_subdivision g a b lg -> is_subdivision g a b (cons_ORlist lf lg). +intros a b f g lf lg H H0; case (Rle_dec a b); intro; + [ apply StepFun_P24 with f; assumption + | apply StepFun_P5; apply StepFun_P24 with f; + [ auto with real + | apply StepFun_P5; assumption + | apply StepFun_P5; assumption ] ]. Qed. -Lemma StepFun_P26 : (a,b,l:R;f,g:R->R;l1:Rlist) (is_subdivision f a b l1) -> (is_subdivision g a b l1) -> (is_subdivision [x:R]``(f x)+l*(g x)`` a b l1). -Intros a b l f g l1; Unfold is_subdivision; Intros; Elim X; Elim X0; Intros; Clear X X0; Unfold adapted_couple in p p0; Decompose [and] p; Decompose [and] p0; Clear p p0; Apply existTT with (FF l1 [x:R]``(f x)+l*(g x)``); Unfold adapted_couple; Repeat Split; Try Assumption. -Apply StepFun_P20; Apply neq_O_lt; Red; Intro; Rewrite <- H8 in H7; Discriminate. -Intros; Unfold constant_D_eq open_interval; Unfold constant_D_eq open_interval in H9 H4; Intros; Rewrite (H9 ? H8 ? H10); Rewrite (H4 ? H8 ? H10); Assert H11 : ~l1==nil. -Red; Intro; Rewrite H11 in H8; Elim (lt_n_O ? H8). -Assert H12 := (RList_P19 ? H11); Elim H12; Clear H12; Intros r [r0 H12]; Rewrite H12; Unfold FF; Change ``(pos_Rl x0 i)+l*(pos_Rl x i)`` == (pos_Rl (app_Rlist (mid_Rlist (cons r r0) r) [x2:R]``(f x2)+l*(g x2)``) (S i)); Rewrite RList_P12. -Rewrite RList_P13. -Rewrite <- H12; Rewrite (H9 ? H8); Try Rewrite (H4 ? H8); Reflexivity Orelse (Elim H10; Clear H10; Intros; Split; [Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Apply Rlt_trans with x1; Assumption | DiscrR]] | Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Rewrite (Rplus_sym (pos_Rl l1 i)); Apply Rlt_compatibility; Apply Rlt_trans with x1; Assumption | DiscrR]]]). -Rewrite <- H12; Assumption. -Rewrite RList_P14; Simpl; Rewrite H12 in H8; Simpl in H8; Apply lt_n_S; Apply H8. +Lemma StepFun_P26 : + forall (a b l:R) (f g:R -> R) (l1:Rlist), + is_subdivision f a b l1 -> + is_subdivision g a b l1 -> + is_subdivision (fun x:R => f x + l * g x) a b l1. +intros a b l f g l1; unfold is_subdivision in |- *; intros; elim X; elim X0; + intros; clear X X0; unfold adapted_couple in p, p0; + decompose [and] p; decompose [and] p0; clear p p0; + apply existT with (FF l1 (fun x:R => f x + l * g x)); + unfold adapted_couple in |- *; repeat split; try assumption. +apply StepFun_P20; apply neq_O_lt; red in |- *; intro; rewrite <- H8 in H7; + discriminate. +intros; unfold constant_D_eq, open_interval in |- *; + unfold constant_D_eq, open_interval in H9, H4; intros; + rewrite (H9 _ H8 _ H10); rewrite (H4 _ H8 _ H10); + assert (H11 : l1 <> nil). +red in |- *; intro; rewrite H11 in H8; elim (lt_n_O _ H8). +assert (H12 := RList_P19 _ H11); elim H12; clear H12; intros r [r0 H12]; + rewrite H12; unfold FF in |- *; + change + (pos_Rl x0 i + l * pos_Rl x i = + pos_Rl + (app_Rlist (mid_Rlist (cons r r0) r) (fun x2:R => f x2 + l * g x2)) + (S i)) in |- *; rewrite RList_P12. +rewrite RList_P13. +rewrite <- H12; rewrite (H9 _ H8); try rewrite (H4 _ H8); + reflexivity || + (elim H10; clear H10; intros; split; + [ apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; + apply Rlt_trans with x1; assumption + | discrR ] ] + | apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); + rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; + rewrite (Rplus_comm (pos_Rl l1 i)); apply Rplus_lt_compat_l; + apply Rlt_trans with x1; assumption + | discrR ] ] ]). +rewrite <- H12; assumption. +rewrite RList_P14; simpl in |- *; rewrite H12 in H8; simpl in H8; + apply lt_n_S; apply H8. Qed. -Lemma StepFun_P27 : (a,b,l:R;f,g:R->R;lf,lg:Rlist) (is_subdivision f a b lf) -> (is_subdivision g a b lg) -> (is_subdivision [x:R]``(f x)+l*(g x)`` a b (cons_ORlist lf lg)). -Intros a b l f g lf lg H H0; Apply StepFun_P26; [Apply StepFun_P23 with g; Assumption | Apply StepFun_P25 with f; Assumption]. +Lemma StepFun_P27 : + forall (a b l:R) (f g:R -> R) (lf lg:Rlist), + is_subdivision f a b lf -> + is_subdivision g a b lg -> + is_subdivision (fun x:R => f x + l * g x) a b (cons_ORlist lf lg). +intros a b l f g lf lg H H0; apply StepFun_P26; + [ apply StepFun_P23 with g; assumption + | apply StepFun_P25 with f; assumption ]. Qed. (* The set of step functions on [a,b] is a vectorial space *) -Lemma StepFun_P28 : (a,b,l:R;f,g:(StepFun a b)) (IsStepFun [x:R]``(f x)+l*(g x)`` a b). -Intros a b l f g; Unfold IsStepFun; Assert H := (pre f); Assert H0 := (pre g); Unfold IsStepFun in H H0; Elim H; Elim H0; Intros; Apply Specif.existT with (cons_ORlist x0 x); Apply StepFun_P27; Assumption. +Lemma StepFun_P28 : + forall (a b l:R) (f g:StepFun a b), IsStepFun (fun x:R => f x + l * g x) a b. +intros a b l f g; unfold IsStepFun in |- *; assert (H := pre f); + assert (H0 := pre g); unfold IsStepFun in H, H0; elim H; + elim H0; intros; apply existT with (cons_ORlist x0 x); + apply StepFun_P27; assumption. Qed. -Lemma StepFun_P29 : (a,b:R;f:(StepFun a b)) (is_subdivision f a b (subdivision f)). -Intros a b f; Unfold is_subdivision; Apply existTT with (subdivision_val f); Apply StepFun_P1. +Lemma StepFun_P29 : + forall (a b:R) (f:StepFun a b), is_subdivision f a b (subdivision f). +intros a b f; unfold is_subdivision in |- *; + apply existT with (subdivision_val f); apply StepFun_P1. Qed. -Lemma StepFun_P30 : (a,b,l:R;f,g:(StepFun a b)) ``(RiemannInt_SF (mkStepFun (StepFun_P28 l f g)))==(RiemannInt_SF f)+l*(RiemannInt_SF g)``. -Intros a b l f g; Unfold RiemannInt_SF; Case (total_order_Rle a b); (Intro; Replace ``(Int_SF (subdivision_val (mkStepFun (StepFun_P28 l f g))) (subdivision (mkStepFun (StepFun_P28 l f g))))`` with (Int_SF (FF (cons_ORlist (subdivision f) (subdivision g)) [x:R]``(f x)+l*(g x)``) (cons_ORlist (subdivision f) (subdivision g))); [Rewrite StepFun_P19; Replace (Int_SF (FF (cons_ORlist (subdivision f) (subdivision g)) f) (cons_ORlist (subdivision f) (subdivision g))) with (Int_SF (subdivision_val f) (subdivision f)); [Replace (Int_SF (FF (cons_ORlist (subdivision f) (subdivision g)) g) (cons_ORlist (subdivision f) (subdivision g))) with (Int_SF (subdivision_val g) (subdivision g)); [Ring | Apply StepFun_P17 with (fe g) a b; [Apply StepFun_P1 | Apply StepFun_P21; Apply StepFun_P25 with (fe f); Apply StepFun_P29]] | Apply StepFun_P17 with (fe f) a b; [Apply StepFun_P1 | Apply StepFun_P21; Apply StepFun_P23 with (fe g); Apply StepFun_P29]] | Apply StepFun_P17 with [x:R]``(f x)+l*(g x)`` a b; [Apply StepFun_P21; Apply StepFun_P27; Apply StepFun_P29 | Apply (StepFun_P1 (mkStepFun (StepFun_P28 l f g)))]]). +Lemma StepFun_P30 : + forall (a b l:R) (f g:StepFun a b), + RiemannInt_SF (mkStepFun (StepFun_P28 l f g)) = + RiemannInt_SF f + l * RiemannInt_SF g. +intros a b l f g; unfold RiemannInt_SF in |- *; case (Rle_dec a b); + (intro; + replace + (Int_SF (subdivision_val (mkStepFun (StepFun_P28 l f g))) + (subdivision (mkStepFun (StepFun_P28 l f g)))) with + (Int_SF + (FF (cons_ORlist (subdivision f) (subdivision g)) + (fun x:R => f x + l * g x)) + (cons_ORlist (subdivision f) (subdivision g))); + [ rewrite StepFun_P19; + replace + (Int_SF (FF (cons_ORlist (subdivision f) (subdivision g)) f) + (cons_ORlist (subdivision f) (subdivision g))) with + (Int_SF (subdivision_val f) (subdivision f)); + [ replace + (Int_SF (FF (cons_ORlist (subdivision f) (subdivision g)) g) + (cons_ORlist (subdivision f) (subdivision g))) with + (Int_SF (subdivision_val g) (subdivision g)); + [ ring + | apply StepFun_P17 with (fe g) a b; + [ apply StepFun_P1 + | apply StepFun_P21; apply StepFun_P25 with (fe f); + apply StepFun_P29 ] ] + | apply StepFun_P17 with (fe f) a b; + [ apply StepFun_P1 + | apply StepFun_P21; apply StepFun_P23 with (fe g); + apply StepFun_P29 ] ] + | apply StepFun_P17 with (fun x:R => f x + l * g x) a b; + [ apply StepFun_P21; apply StepFun_P27; apply StepFun_P29 + | apply (StepFun_P1 (mkStepFun (StepFun_P28 l f g))) ] ]). Qed. -Lemma StepFun_P31 : (a,b:R;f:R->R;l,lf:Rlist) (adapted_couple f a b l lf) -> (adapted_couple [x:R](Rabsolu (f x)) a b l (app_Rlist lf Rabsolu)). -Unfold adapted_couple; Intros; Decompose [and] H; Clear H; Repeat Split; Try Assumption. -Symmetry; Rewrite H3; Rewrite RList_P18; Reflexivity. -Intros; Unfold constant_D_eq open_interval; Unfold constant_D_eq open_interval in H5; Intros; Rewrite (H5 ? H ? H4); Rewrite RList_P12; [Reflexivity | Rewrite H3 in H; Simpl in H; Apply H]. +Lemma StepFun_P31 : + forall (a b:R) (f:R -> R) (l lf:Rlist), + adapted_couple f a b l lf -> + adapted_couple (fun x:R => Rabs (f x)) a b l (app_Rlist lf Rabs). +unfold adapted_couple in |- *; intros; decompose [and] H; clear H; + repeat split; try assumption. +symmetry in |- *; rewrite H3; rewrite RList_P18; reflexivity. +intros; unfold constant_D_eq, open_interval in |- *; + unfold constant_D_eq, open_interval in H5; intros; + rewrite (H5 _ H _ H4); rewrite RList_P12; + [ reflexivity | rewrite H3 in H; simpl in H; apply H ]. Qed. -Lemma StepFun_P32 : (a,b:R;f:(StepFun a b)) (IsStepFun [x:R](Rabsolu (f x)) a b). -Intros a b f; Unfold IsStepFun; Apply Specif.existT with (subdivision f); Unfold is_subdivision; Apply existTT with (app_Rlist (subdivision_val f) Rabsolu); Apply StepFun_P31; Apply StepFun_P1. +Lemma StepFun_P32 : + forall (a b:R) (f:StepFun a b), IsStepFun (fun x:R => Rabs (f x)) a b. +intros a b f; unfold IsStepFun in |- *; apply existT with (subdivision f); + unfold is_subdivision in |- *; + apply existT with (app_Rlist (subdivision_val f) Rabs); + apply StepFun_P31; apply StepFun_P1. Qed. -Lemma StepFun_P33 : (l2,l1:Rlist) (ordered_Rlist l1) -> ``(Rabsolu (Int_SF l2 l1))<=(Int_SF (app_Rlist l2 Rabsolu) l1)``. -Induction l2; Intros. -Simpl; Rewrite Rabsolu_R0; Right; Reflexivity. -Simpl; Induction l1. -Rewrite Rabsolu_R0; Right; Reflexivity. -Induction l1. -Rewrite Rabsolu_R0; Right; Reflexivity. -Apply Rle_trans with ``(Rabsolu (r*(r2-r1)))+(Rabsolu (Int_SF r0 (cons r2 l1)))``. -Apply Rabsolu_triang. -Rewrite Rabsolu_mult; Rewrite (Rabsolu_right ``r2-r1``); [Apply Rle_compatibility; Apply H; Apply RList_P4 with r1; Assumption | Apply Rge_minus; Apply Rle_sym1; Apply (H0 O); Simpl; Apply lt_O_Sn]. +Lemma StepFun_P33 : + forall l2 l1:Rlist, + ordered_Rlist l1 -> Rabs (Int_SF l2 l1) <= Int_SF (app_Rlist l2 Rabs) l1. +simple induction l2; intros. +simpl in |- *; rewrite Rabs_R0; right; reflexivity. +simpl in |- *; induction l1 as [| r1 l1 Hrecl1]. +rewrite Rabs_R0; right; reflexivity. +induction l1 as [| r2 l1 Hrecl0]. +rewrite Rabs_R0; right; reflexivity. +apply Rle_trans with (Rabs (r * (r2 - r1)) + Rabs (Int_SF r0 (cons r2 l1))). +apply Rabs_triang. +rewrite Rabs_mult; rewrite (Rabs_right (r2 - r1)); + [ apply Rplus_le_compat_l; apply H; apply RList_P4 with r1; assumption + | apply Rge_minus; apply Rle_ge; apply (H0 0%nat); simpl in |- *; + apply lt_O_Sn ]. Qed. -Lemma StepFun_P34 : (a,b:R;f:(StepFun a b)) ``a<=b`` -> ``(Rabsolu (RiemannInt_SF f))<=(RiemannInt_SF (mkStepFun (StepFun_P32 f)))``. -Intros; Unfold RiemannInt_SF; Case (total_order_Rle a b); Intro. -Replace (Int_SF (subdivision_val (mkStepFun (StepFun_P32 f))) (subdivision (mkStepFun (StepFun_P32 f)))) with (Int_SF (app_Rlist (subdivision_val f) Rabsolu) (subdivision f)). -Apply StepFun_P33; Assert H0 := (StepFun_P29 f); Unfold is_subdivision in H0; Elim H0; Intros; Unfold adapted_couple in p; Decompose [and] p; Assumption. -Apply StepFun_P17 with [x:R](Rabsolu (f x)) a b; [Apply StepFun_P31; Apply StepFun_P1 | Apply (StepFun_P1 (mkStepFun (StepFun_P32 f)))]. -Elim n; Assumption. +Lemma StepFun_P34 : + forall (a b:R) (f:StepFun a b), + a <= b -> + Rabs (RiemannInt_SF f) <= RiemannInt_SF (mkStepFun (StepFun_P32 f)). +intros; unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro. +replace + (Int_SF (subdivision_val (mkStepFun (StepFun_P32 f))) + (subdivision (mkStepFun (StepFun_P32 f)))) with + (Int_SF (app_Rlist (subdivision_val f) Rabs) (subdivision f)). +apply StepFun_P33; assert (H0 := StepFun_P29 f); unfold is_subdivision in H0; + elim H0; intros; unfold adapted_couple in p; decompose [and] p; + assumption. +apply StepFun_P17 with (fun x:R => Rabs (f x)) a b; + [ apply StepFun_P31; apply StepFun_P1 + | apply (StepFun_P1 (mkStepFun (StepFun_P32 f))) ]. +elim n; assumption. Qed. -Lemma StepFun_P35 : (l:Rlist;a,b:R;f,g:R->R) (ordered_Rlist l) -> (pos_Rl l O)==a -> (pos_Rl l (pred (Rlength l)))==b -> ((x:R)``a<x<b``->``(f x)<=(g x)``) -> ``(Int_SF (FF l f) l)<=(Int_SF (FF l g) l)``. -Induction l; Intros. -Right; Reflexivity. -Simpl; Induction r0. -Right; Reflexivity. -Simpl; Apply Rplus_le. -Case (Req_EM r r0); Intro. -Rewrite H4; Right; Ring. -Do 2 Rewrite <- (Rmult_sym ``r0-r``); Apply Rle_monotony. -Apply Rle_sym2; Apply Rge_minus; Apply Rle_sym1; Apply (H0 O); Simpl; Apply lt_O_Sn. -Apply H3; Split. -Apply Rlt_monotony_contra with ``2``. -Sup0. -Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Assert H5 : r==a. -Apply H1. -Rewrite H5; Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility. -Assert H6 := (H0 O (lt_O_Sn ?)). -Simpl in H6. -Elim H6; Intro. -Rewrite H5 in H7; Apply H7. -Elim H4; Assumption. -DiscrR. -Apply Rlt_monotony_contra with ``2``. -Sup0. -Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite double; Assert H5 : ``r0<=b``. -Replace b with (pos_Rl (cons r (cons r0 r1)) (pred (Rlength (cons r (cons r0 r1))))). -Replace r0 with (pos_Rl (cons r (cons r0 r1)) (S O)). -Elim (RList_P6 (cons r (cons r0 r1))); Intros; Apply H5. -Assumption. -Simpl; Apply le_n_S. -Apply le_O_n. -Simpl; Apply lt_n_Sn. -Reflexivity. -Apply Rle_lt_trans with ``r+b``. -Apply Rle_compatibility; Assumption. -Rewrite (Rplus_sym r); Apply Rlt_compatibility. -Apply Rlt_le_trans with r0. -Assert H6 := (H0 O (lt_O_Sn ?)). -Simpl in H6. -Elim H6; Intro. -Apply H7. -Elim H4; Assumption. -Assumption. -DiscrR. -Simpl in H; Apply H with r0 b. -Apply RList_P4 with r; Assumption. -Reflexivity. -Rewrite <- H2; Reflexivity. -Intros; Apply H3; Elim H4; Intros; Split; Try Assumption. -Apply Rle_lt_trans with r0; Try Assumption. -Rewrite <- H1. -Simpl; Apply (H0 O); Simpl; Apply lt_O_Sn. +Lemma StepFun_P35 : + forall (l:Rlist) (a b:R) (f g:R -> R), + ordered_Rlist l -> + pos_Rl l 0 = a -> + pos_Rl l (pred (Rlength l)) = b -> + (forall x:R, a < x < b -> f x <= g x) -> + Int_SF (FF l f) l <= Int_SF (FF l g) l. +simple induction l; intros. +right; reflexivity. +simpl in |- *; induction r0 as [| r0 r1 Hrecr0]. +right; reflexivity. +simpl in |- *; apply Rplus_le_compat. +case (Req_dec r r0); intro. +rewrite H4; right; ring. +do 2 rewrite <- (Rmult_comm (r0 - r)); apply Rmult_le_compat_l. +apply Rge_le; apply Rge_minus; apply Rle_ge; apply (H0 0%nat); simpl in |- *; + apply lt_O_Sn. +apply H3; split. +apply Rmult_lt_reg_l with 2. +prove_sup0. +unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +assert (H5 : r = a). +apply H1. +rewrite H5; rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l. +assert (H6 := H0 0%nat (lt_O_Sn _)). +simpl in H6. +elim H6; intro. +rewrite H5 in H7; apply H7. +elim H4; assumption. +discrR. +apply Rmult_lt_reg_l with 2. +prove_sup0. +unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite double; assert (H5 : r0 <= b). +replace b with + (pos_Rl (cons r (cons r0 r1)) (pred (Rlength (cons r (cons r0 r1))))). +replace r0 with (pos_Rl (cons r (cons r0 r1)) 1). +elim (RList_P6 (cons r (cons r0 r1))); intros; apply H5. +assumption. +simpl in |- *; apply le_n_S. +apply le_O_n. +simpl in |- *; apply lt_n_Sn. +reflexivity. +apply Rle_lt_trans with (r + b). +apply Rplus_le_compat_l; assumption. +rewrite (Rplus_comm r); apply Rplus_lt_compat_l. +apply Rlt_le_trans with r0. +assert (H6 := H0 0%nat (lt_O_Sn _)). +simpl in H6. +elim H6; intro. +apply H7. +elim H4; assumption. +assumption. +discrR. +simpl in H; apply H with r0 b. +apply RList_P4 with r; assumption. +reflexivity. +rewrite <- H2; reflexivity. +intros; apply H3; elim H4; intros; split; try assumption. +apply Rle_lt_trans with r0; try assumption. +rewrite <- H1. +simpl in |- *; apply (H0 0%nat); simpl in |- *; apply lt_O_Sn. Qed. -Lemma StepFun_P36 : (a,b:R;f,g:(StepFun a b);l:Rlist) ``a<=b`` -> (is_subdivision f a b l) -> (is_subdivision g a b l) -> ((x:R)``a<x<b``->``(f x)<=(g x)``) -> ``(RiemannInt_SF f) <= (RiemannInt_SF g)``. -Intros; Unfold RiemannInt_SF; Case (total_order_Rle a b); Intro. -Replace (Int_SF (subdivision_val f) (subdivision f)) with (Int_SF (FF l f) l). -Replace (Int_SF (subdivision_val g) (subdivision g)) with (Int_SF (FF l g) l). -Unfold is_subdivision in X; Elim X; Clear X; Intros; Unfold adapted_couple in p; Decompose [and] p; Clear p; Assert H5 : (Rmin a b)==a; [Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption] | Assert H7 : (Rmax a b)==b; [Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption] | Rewrite H5 in H3; Rewrite H7 in H2; EApply StepFun_P35 with a b; Assumption]]. -Apply StepFun_P17 with (fe g) a b; [Apply StepFun_P21; Assumption | Apply StepFun_P1]. -Apply StepFun_P17 with (fe f) a b; [Apply StepFun_P21; Assumption | Apply StepFun_P1]. -Elim n; Assumption. +Lemma StepFun_P36 : + forall (a b:R) (f g:StepFun a b) (l:Rlist), + a <= b -> + is_subdivision f a b l -> + is_subdivision g a b l -> + (forall x:R, a < x < b -> f x <= g x) -> + RiemannInt_SF f <= RiemannInt_SF g. +intros; unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro. +replace (Int_SF (subdivision_val f) (subdivision f)) with (Int_SF (FF l f) l). +replace (Int_SF (subdivision_val g) (subdivision g)) with (Int_SF (FF l g) l). +unfold is_subdivision in X; elim X; clear X; intros; + unfold adapted_couple in p; decompose [and] p; clear p; + assert (H5 : Rmin a b = a); + [ unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ] + | assert (H7 : Rmax a b = b); + [ unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ] + | rewrite H5 in H3; rewrite H7 in H2; eapply StepFun_P35 with a b; + assumption ] ]. +apply StepFun_P17 with (fe g) a b; + [ apply StepFun_P21; assumption | apply StepFun_P1 ]. +apply StepFun_P17 with (fe f) a b; + [ apply StepFun_P21; assumption | apply StepFun_P1 ]. +elim n; assumption. Qed. -Lemma StepFun_P37 : (a,b:R;f,g:(StepFun a b)) ``a<=b`` -> ((x:R)``a<x<b``->``(f x)<=(g x)``) -> ``(RiemannInt_SF f) <= (RiemannInt_SF g)``. -Intros; EApply StepFun_P36; Try Assumption. -EApply StepFun_P25; Apply StepFun_P29. -EApply StepFun_P23; Apply StepFun_P29. +Lemma StepFun_P37 : + forall (a b:R) (f g:StepFun a b), + a <= b -> + (forall x:R, a < x < b -> f x <= g x) -> + RiemannInt_SF f <= RiemannInt_SF g. +intros; eapply StepFun_P36; try assumption. +eapply StepFun_P25; apply StepFun_P29. +eapply StepFun_P23; apply StepFun_P29. Qed. -Lemma StepFun_P38 : (l:Rlist;a,b:R;f:R->R) (ordered_Rlist l) -> (pos_Rl l O)==a -> (pos_Rl l (pred (Rlength l)))==b -> (sigTT ? [g:(StepFun a b)](g b)==(f b)/\(i:nat)(lt i (pred (Rlength l)))->(constant_D_eq g (co_interval (pos_Rl l i) (pos_Rl l (S i))) (f (pos_Rl l i)))). -Intros l a b f; Generalize a; Clear a; NewInduction l. -Intros a H H0 H1; Simpl in H0; Simpl in H1; Exists (mkStepFun (StepFun_P4 a b (f b))); Split. -Reflexivity. -Intros; Elim (lt_n_O ? H2). -Intros; NewDestruct l as [|r1 l]. -Simpl in H1; Simpl in H0; Exists (mkStepFun (StepFun_P4 a b (f b))); Split. -Reflexivity. -Intros i H2; Elim (lt_n_O ? H2). -Intros; Assert H2 : (ordered_Rlist (cons r1 l)). -Apply RList_P4 with r; Assumption. -Assert H3 : (pos_Rl (cons r1 l) O)==r1. -Reflexivity. -Assert H4 : (pos_Rl (cons r1 l) (pred (Rlength (cons r1 l))))==b. -Rewrite <- H1; Reflexivity. -Elim (IHl r1 H2 H3 H4); Intros g [H5 H6]. -Pose g' := [x:R]Cases (total_order_Rle r1 x) of - | (leftT _) => (g x) - | (rightT _) => (f a) end. -Assert H7 : ``r1<=b``. -Rewrite <- H4; Apply RList_P7; [Assumption | Left; Reflexivity]. -Assert H8 : (IsStepFun g' a b). -Unfold IsStepFun; Assert H8 := (pre g); Unfold IsStepFun in H8; Elim H8; Intros lg H9; Unfold is_subdivision in H9; Elim H9; Clear H9; Intros lg2 H9; Split with (cons a lg); Unfold is_subdivision; Split with (cons (f a) lg2); Unfold adapted_couple in H9; Decompose [and] H9; Clear H9; Unfold adapted_couple; Repeat Split. -Unfold ordered_Rlist; Intros; Simpl in H9; Induction i. -Simpl; Rewrite H12; Replace (Rmin r1 b) with r1. -Simpl in H0; Rewrite <- H0; Apply (H O); Simpl; Apply lt_O_Sn. -Unfold Rmin; Case (total_order_Rle r1 b); Intro; [Reflexivity | Elim n; Assumption]. -Apply (H10 i); Apply lt_S_n. -Replace (S (pred (Rlength lg))) with (Rlength lg). -Apply H9. -Apply S_pred with O; Apply neq_O_lt; Intro; Rewrite <- H14 in H9; Elim (lt_n_O ? H9). -Simpl; Assert H14 : ``a<=b``. -Rewrite <- H1; Simpl in H0; Rewrite <- H0; Apply RList_P7; [Assumption | Left; Reflexivity]. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Assert H14 : ``a<=b``. -Rewrite <- H1; Simpl in H0; Rewrite <- H0; Apply RList_P7; [Assumption | Left; Reflexivity]. -Replace (Rmax a b) with (Rmax r1 b). -Rewrite <- H11; Induction lg. -Simpl in H13; Discriminate. -Reflexivity. -Unfold Rmax; Case (total_order_Rle a b); Case (total_order_Rle r1 b); Intros; Reflexivity Orelse Elim n; Assumption. -Simpl; Rewrite H13; Reflexivity. -Intros; Simpl in H9; Induction i. -Unfold constant_D_eq open_interval; Simpl; Intros; Assert H16 : (Rmin r1 b)==r1. -Unfold Rmin; Case (total_order_Rle r1 b); Intro; [Reflexivity | Elim n; Assumption]. -Rewrite H16 in H12; Rewrite H12 in H14; Elim H14; Clear H14; Intros _ H14; Unfold g'; Case (total_order_Rle r1 x); Intro r3. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r3 H14)). -Reflexivity. -Change (constant_D_eq g' (open_interval (pos_Rl lg i) (pos_Rl lg (S i))) (pos_Rl lg2 i)); Clear Hreci; Assert H16 := (H15 i); Assert H17 : (lt i (pred (Rlength lg))). -Apply lt_S_n. -Replace (S (pred (Rlength lg))) with (Rlength lg). -Assumption. -Apply S_pred with O; Apply neq_O_lt; Red; Intro; Rewrite <- H14 in H9; Elim (lt_n_O ? H9). -Assert H18 := (H16 H17); Unfold constant_D_eq open_interval in H18; Unfold constant_D_eq open_interval; Intros; Assert H19 := (H18 ? H14); Rewrite <- H19; Unfold g'; Case (total_order_Rle r1 x); Intro. -Reflexivity. -Elim n; Replace r1 with (Rmin r1 b). -Rewrite <- H12; Elim H14; Clear H14; Intros H14 _; Left; Apply Rle_lt_trans with (pos_Rl lg i); Try Assumption. -Apply RList_P5. -Assumption. -Elim (RList_P3 lg (pos_Rl lg i)); Intros; Apply H21; Exists i; Split. -Reflexivity. -Apply lt_trans with (pred (Rlength lg)); Try Assumption. -Apply lt_pred_n_n; Apply neq_O_lt; Red; Intro; Rewrite <- H22 in H17; Elim (lt_n_O ? H17). -Unfold Rmin; Case (total_order_Rle r1 b); Intro; [Reflexivity | Elim n0; Assumption]. -Exists (mkStepFun H8); Split. -Simpl; Unfold g'; Case (total_order_Rle r1 b); Intro. -Assumption. -Elim n; Assumption. -Intros; Simpl in H9; Induction i. -Unfold constant_D_eq co_interval; Simpl; Intros; Simpl in H0; Rewrite H0; Elim H10; Clear H10; Intros; Unfold g'; Case (total_order_Rle r1 x); Intro r3. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r3 H11)). -Reflexivity. -Clear Hreci; Change (constant_D_eq (mkStepFun H8) (co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i))) (f (pos_Rl (cons r1 l) i))); Assert H10 := (H6 i); Assert H11 : (lt i (pred (Rlength (cons r1 l)))). -Simpl; Apply lt_S_n; Assumption. -Assert H12 := (H10 H11); Unfold constant_D_eq co_interval in H12; Unfold constant_D_eq co_interval; Intros; Rewrite <- (H12 ? H13); Simpl; Unfold g'; Case (total_order_Rle r1 x); Intro. -Reflexivity. -Elim n; Elim H13; Clear H13; Intros; Apply Rle_trans with (pos_Rl (cons r1 l) i); Try Assumption; Change ``(pos_Rl (cons r1 l) O)<=(pos_Rl (cons r1 l) i)``; Elim (RList_P6 (cons r1 l)); Intros; Apply H15; [Assumption | Apply le_O_n | Simpl; Apply lt_trans with (Rlength l); [Apply lt_S_n; Assumption | Apply lt_n_Sn]]. +Lemma StepFun_P38 : + forall (l:Rlist) (a b:R) (f:R -> R), + ordered_Rlist l -> + pos_Rl l 0 = a -> + pos_Rl l (pred (Rlength l)) = b -> + sigT + (fun g:StepFun a b => + g b = f b /\ + (forall i:nat, + (i < pred (Rlength l))%nat -> + constant_D_eq g (co_interval (pos_Rl l i) (pos_Rl l (S i))) + (f (pos_Rl l i)))). +intros l a b f; generalize a; clear a; induction l. +intros a H H0 H1; simpl in H0; simpl in H1; + exists (mkStepFun (StepFun_P4 a b (f b))); split. +reflexivity. +intros; elim (lt_n_O _ H2). +intros; destruct l as [| r1 l]. +simpl in H1; simpl in H0; exists (mkStepFun (StepFun_P4 a b (f b))); split. +reflexivity. +intros i H2; elim (lt_n_O _ H2). +intros; assert (H2 : ordered_Rlist (cons r1 l)). +apply RList_P4 with r; assumption. +assert (H3 : pos_Rl (cons r1 l) 0 = r1). +reflexivity. +assert (H4 : pos_Rl (cons r1 l) (pred (Rlength (cons r1 l))) = b). +rewrite <- H1; reflexivity. +elim (IHl r1 H2 H3 H4); intros g [H5 H6]. +pose + (g' := + fun x:R => match Rle_dec r1 x with + | left _ => g x + | right _ => f a + end). +assert (H7 : r1 <= b). +rewrite <- H4; apply RList_P7; [ assumption | left; reflexivity ]. +assert (H8 : IsStepFun g' a b). +unfold IsStepFun in |- *; assert (H8 := pre g); unfold IsStepFun in H8; + elim H8; intros lg H9; unfold is_subdivision in H9; + elim H9; clear H9; intros lg2 H9; split with (cons a lg); + unfold is_subdivision in |- *; split with (cons (f a) lg2); + unfold adapted_couple in H9; decompose [and] H9; clear H9; + unfold adapted_couple in |- *; repeat split. +unfold ordered_Rlist in |- *; intros; simpl in H9; + induction i as [| i Hreci]. +simpl in |- *; rewrite H12; replace (Rmin r1 b) with r1. +simpl in H0; rewrite <- H0; apply (H 0%nat); simpl in |- *; apply lt_O_Sn. +unfold Rmin in |- *; case (Rle_dec r1 b); intro; + [ reflexivity | elim n; assumption ]. +apply (H10 i); apply lt_S_n. +replace (S (pred (Rlength lg))) with (Rlength lg). +apply H9. +apply S_pred with 0%nat; apply neq_O_lt; intro; rewrite <- H14 in H9; + elim (lt_n_O _ H9). +simpl in |- *; assert (H14 : a <= b). +rewrite <- H1; simpl in H0; rewrite <- H0; apply RList_P7; + [ assumption | left; reflexivity ]. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +assert (H14 : a <= b). +rewrite <- H1; simpl in H0; rewrite <- H0; apply RList_P7; + [ assumption | left; reflexivity ]. +replace (Rmax a b) with (Rmax r1 b). +rewrite <- H11; induction lg as [| r0 lg Hreclg]. +simpl in H13; discriminate. +reflexivity. +unfold Rmax in |- *; case (Rle_dec a b); case (Rle_dec r1 b); intros; + reflexivity || elim n; assumption. +simpl in |- *; rewrite H13; reflexivity. +intros; simpl in H9; induction i as [| i Hreci]. +unfold constant_D_eq, open_interval in |- *; simpl in |- *; intros; + assert (H16 : Rmin r1 b = r1). +unfold Rmin in |- *; case (Rle_dec r1 b); intro; + [ reflexivity | elim n; assumption ]. +rewrite H16 in H12; rewrite H12 in H14; elim H14; clear H14; intros _ H14; + unfold g' in |- *; case (Rle_dec r1 x); intro r3. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H14)). +reflexivity. +change + (constant_D_eq g' (open_interval (pos_Rl lg i) (pos_Rl lg (S i))) + (pos_Rl lg2 i)) in |- *; clear Hreci; assert (H16 := H15 i); + assert (H17 : (i < pred (Rlength lg))%nat). +apply lt_S_n. +replace (S (pred (Rlength lg))) with (Rlength lg). +assumption. +apply S_pred with 0%nat; apply neq_O_lt; red in |- *; intro; + rewrite <- H14 in H9; elim (lt_n_O _ H9). +assert (H18 := H16 H17); unfold constant_D_eq, open_interval in H18; + unfold constant_D_eq, open_interval in |- *; intros; + assert (H19 := H18 _ H14); rewrite <- H19; unfold g' in |- *; + case (Rle_dec r1 x); intro. +reflexivity. +elim n; replace r1 with (Rmin r1 b). +rewrite <- H12; elim H14; clear H14; intros H14 _; left; + apply Rle_lt_trans with (pos_Rl lg i); try assumption. +apply RList_P5. +assumption. +elim (RList_P3 lg (pos_Rl lg i)); intros; apply H21; exists i; split. +reflexivity. +apply lt_trans with (pred (Rlength lg)); try assumption. +apply lt_pred_n_n; apply neq_O_lt; red in |- *; intro; rewrite <- H22 in H17; + elim (lt_n_O _ H17). +unfold Rmin in |- *; case (Rle_dec r1 b); intro; + [ reflexivity | elim n0; assumption ]. +exists (mkStepFun H8); split. +simpl in |- *; unfold g' in |- *; case (Rle_dec r1 b); intro. +assumption. +elim n; assumption. +intros; simpl in H9; induction i as [| i Hreci]. +unfold constant_D_eq, co_interval in |- *; simpl in |- *; intros; simpl in H0; + rewrite H0; elim H10; clear H10; intros; unfold g' in |- *; + case (Rle_dec r1 x); intro r3. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r3 H11)). +reflexivity. +clear Hreci; + change + (constant_D_eq (mkStepFun H8) + (co_interval (pos_Rl (cons r1 l) i) (pos_Rl (cons r1 l) (S i))) + (f (pos_Rl (cons r1 l) i))) in |- *; assert (H10 := H6 i); + assert (H11 : (i < pred (Rlength (cons r1 l)))%nat). +simpl in |- *; apply lt_S_n; assumption. +assert (H12 := H10 H11); unfold constant_D_eq, co_interval in H12; + unfold constant_D_eq, co_interval in |- *; intros; + rewrite <- (H12 _ H13); simpl in |- *; unfold g' in |- *; + case (Rle_dec r1 x); intro. +reflexivity. +elim n; elim H13; clear H13; intros; + apply Rle_trans with (pos_Rl (cons r1 l) i); try assumption; + change (pos_Rl (cons r1 l) 0 <= pos_Rl (cons r1 l) i) in |- *; + elim (RList_P6 (cons r1 l)); intros; apply H15; + [ assumption + | apply le_O_n + | simpl in |- *; apply lt_trans with (Rlength l); + [ apply lt_S_n; assumption | apply lt_n_Sn ] ]. Qed. -Lemma StepFun_P39 : (a,b:R;f:(StepFun a b)) (RiemannInt_SF f)==(Ropp (RiemannInt_SF (mkStepFun (StepFun_P6 (pre f))))). -Intros; Unfold RiemannInt_SF; Case (total_order_Rle a b); Case (total_order_Rle b a); Intros. -Assert H : (adapted_couple f a b (subdivision f) (subdivision_val f)); [Apply StepFun_P1 | Assert H0 : (adapted_couple (mkStepFun (StepFun_P6 (pre f))) b a (subdivision (mkStepFun (StepFun_P6 (pre f)))) (subdivision_val (mkStepFun (StepFun_P6 (pre f))))); [Apply StepFun_P1 | Assert H1 : a==b; [Apply Rle_antisym; Assumption | Rewrite (StepFun_P8 H H1); Assert H2 : b==a; [Symmetry; Apply H1 | Rewrite (StepFun_P8 H0 H2); Ring]]]]. -Rewrite Ropp_Ropp; EApply StepFun_P17; [Apply StepFun_P1 | Apply StepFun_P2; Assert H := (StepFun_P6 (pre f)); Unfold IsStepFun in H; Elim H; Intros; Unfold is_subdivision; Elim p; Intros; Apply p0]. -Apply eq_Ropp; EApply StepFun_P17; [Apply StepFun_P1 | Apply StepFun_P2; Assert H := (StepFun_P6 (pre f)); Unfold IsStepFun in H; Elim H; Intros; Unfold is_subdivision; Elim p; Intros; Apply p0]. -Assert H : ``a<b``; [Auto with real | Assert H0 : ``b<a``; [Auto with real | Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H H0))]]. +Lemma StepFun_P39 : + forall (a b:R) (f:StepFun a b), + RiemannInt_SF f = - RiemannInt_SF (mkStepFun (StepFun_P6 (pre f))). +intros; unfold RiemannInt_SF in |- *; case (Rle_dec a b); case (Rle_dec b a); + intros. +assert (H : adapted_couple f a b (subdivision f) (subdivision_val f)); + [ apply StepFun_P1 + | assert + (H0 : + adapted_couple (mkStepFun (StepFun_P6 (pre f))) b a + (subdivision (mkStepFun (StepFun_P6 (pre f)))) + (subdivision_val (mkStepFun (StepFun_P6 (pre f))))); + [ apply StepFun_P1 + | assert (H1 : a = b); + [ apply Rle_antisym; assumption + | rewrite (StepFun_P8 H H1); assert (H2 : b = a); + [ symmetry in |- *; apply H1 | rewrite (StepFun_P8 H0 H2); ring ] ] ] ]. +rewrite Ropp_involutive; eapply StepFun_P17; + [ apply StepFun_P1 + | apply StepFun_P2; assert (H := StepFun_P6 (pre f)); unfold IsStepFun in H; + elim H; intros; unfold is_subdivision in |- *; + elim p; intros; apply p0 ]. +apply Ropp_eq_compat; eapply StepFun_P17; + [ apply StepFun_P1 + | apply StepFun_P2; assert (H := StepFun_P6 (pre f)); unfold IsStepFun in H; + elim H; intros; unfold is_subdivision in |- *; + elim p; intros; apply p0 ]. +assert (H : a < b); + [ auto with real + | assert (H0 : b < a); + [ auto with real | elim (Rlt_irrefl _ (Rlt_trans _ _ _ H H0)) ] ]. Qed. -Lemma StepFun_P40 : (f:R->R;a,b,c:R;l1,l2,lf1,lf2:Rlist) ``a<b`` -> ``b<c`` -> (adapted_couple f a b l1 lf1) -> (adapted_couple f b c l2 lf2) -> (adapted_couple f a c (cons_Rlist l1 l2) (FF (cons_Rlist l1 l2) f)). -Intros f a b c l1 l2 lf1 lf2 H H0 H1 H2; Unfold adapted_couple in H1 H2; Unfold adapted_couple; Decompose [and] H1; Decompose [and] H2; Clear H1 H2; Repeat Split. -Apply RList_P25; Try Assumption. -Rewrite H10; Rewrite H4; Unfold Rmin Rmax; Case (total_order_Rle a b); Case (total_order_Rle b c); Intros; (Right; Reflexivity) Orelse (Elim n; Left; Assumption). -Rewrite RList_P22. -Rewrite H5; Unfold Rmin Rmax; Case (total_order_Rle a b); Case (total_order_Rle a c); Intros; [Reflexivity | Elim n; Apply Rle_trans with b; Left; Assumption | Elim n; Left; Assumption | Elim n0; Left; Assumption]. -Red; Intro; Rewrite H1 in H6; Discriminate. -Rewrite RList_P24. -Rewrite H9; Unfold Rmin Rmax; Case (total_order_Rle b c); Case (total_order_Rle a c); Intros; [Reflexivity | Elim n; Apply Rle_trans with b; Left; Assumption | Elim n; Left; Assumption | Elim n0; Left; Assumption]. -Red; Intro; Rewrite H1 in H11; Discriminate. -Apply StepFun_P20. -Rewrite RList_P23; Apply neq_O_lt; Red; Intro. -Assert H2 : (plus (Rlength l1) (Rlength l2))=O. -Symmetry; Apply H1. -Elim (plus_is_O ? ? H2); Intros; Rewrite H12 in H6; Discriminate. -Unfold constant_D_eq open_interval; Intros; Elim (le_or_lt (S (S i)) (Rlength l1)); Intro. -Assert H14 : (pos_Rl (cons_Rlist l1 l2) i) == (pos_Rl l1 i). -Apply RList_P26; Apply lt_S_n; Apply le_lt_n_Sm; Apply le_S_n; Apply le_trans with (Rlength l1); [Assumption | Apply le_n_Sn]. -Assert H15 : (pos_Rl (cons_Rlist l1 l2) (S i))==(pos_Rl l1 (S i)). -Apply RList_P26; Apply lt_S_n; Apply le_lt_n_Sm; Assumption. -Rewrite H14 in H2; Rewrite H15 in H2; Assert H16 : (le (2) (Rlength l1)). -Apply le_trans with (S (S i)); [Repeat Apply le_n_S; Apply le_O_n | Assumption]. -Elim (RList_P20 ? H16); Intros r1 [r2 [r3 H17]]; Rewrite H17; Change (f x)==(pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i); Rewrite RList_P12. -Induction i. -Simpl; Assert H18 := (H8 O); Unfold constant_D_eq open_interval in H18; Assert H19 : (lt O (pred (Rlength l1))). -Rewrite H17; Simpl; Apply lt_O_Sn. -Assert H20 := (H18 H19); Repeat Rewrite H20. -Reflexivity. -Assert H21 : ``r1<=r2``. -Rewrite H17 in H3; Apply (H3 O). -Simpl; Apply lt_O_Sn. -Elim H21; Intro. -Split. -Rewrite H17; Simpl; Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Rewrite H17; Simpl; Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite (Rplus_sym r1); Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Elim H2; Intros; Rewrite H17 in H23; Rewrite H17 in H24; Simpl in H24; Simpl in H23; Rewrite H22 in H23; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H23 H24)). -Assumption. -Clear Hreci; Rewrite RList_P13. -Rewrite H17 in H14; Rewrite H17 in H15; Change (pos_Rl (cons_Rlist (cons r2 r3) l2) i)== (pos_Rl (cons r1 (cons r2 r3)) (S i)) in H14; Rewrite H14; Change (pos_Rl (cons_Rlist (cons r2 r3) l2) (S i))==(pos_Rl (cons r1 (cons r2 r3)) (S (S i))) in H15; Rewrite H15; Assert H18 := (H8 (S i)); Unfold constant_D_eq open_interval in H18; Assert H19 : (lt (S i) (pred (Rlength l1))). -Apply lt_pred; Apply lt_S_n; Apply le_lt_n_Sm; Assumption. -Assert H20 := (H18 H19); Repeat Rewrite H20. -Reflexivity. -Rewrite <- H17; Assert H21 : ``(pos_Rl l1 (S i))<=(pos_Rl l1 (S (S i)))``. -Apply (H3 (S i)); Apply lt_pred; Apply lt_S_n; Apply le_lt_n_Sm; Assumption. -Elim H21; Intro. -Split. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite (Rplus_sym (pos_Rl l1 (S i))); Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Elim H2; Intros; Rewrite H22 in H23; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H23 H24)). -Assumption. -Simpl; Rewrite H17 in H1; Simpl in H1; Apply lt_S_n; Assumption. -Rewrite RList_P14; Rewrite H17 in H1; Simpl in H1; Apply H1. -Inversion H12. -Assert H16 : (pos_Rl (cons_Rlist l1 l2) (S i))==b. -Rewrite RList_P29. -Rewrite H15; Rewrite <- minus_n_n; Rewrite H10; Unfold Rmin; Case (total_order_Rle b c); Intro; [Reflexivity | Elim n; Left; Assumption]. -Rewrite H15; Apply le_n. -Induction l1. -Simpl in H15; Discriminate. -Clear Hrecl1; Simpl in H1; Simpl; Apply lt_n_S; Assumption. -Assert H17 : (pos_Rl (cons_Rlist l1 l2) i)==b. -Rewrite RList_P26. -Replace i with (pred (Rlength l1)); [Rewrite H4; Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Left; Assumption] | Rewrite H15; Reflexivity]. -Rewrite H15; Apply lt_n_Sn. -Rewrite H16 in H2; Rewrite H17 in H2; Elim H2; Intros; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H14 H18)). -Assert H16 : (pos_Rl (cons_Rlist l1 l2) i) == (pos_Rl l2 (minus i (Rlength l1))). -Apply RList_P29. -Apply le_S_n; Assumption. -Apply lt_le_trans with (pred (Rlength (cons_Rlist l1 l2))); [Assumption | Apply le_pred_n]. -Assert H17 : (pos_Rl (cons_Rlist l1 l2) (S i))==(pos_Rl l2 (S (minus i (Rlength l1)))). -Replace (S (minus i (Rlength l1))) with (minus (S i) (Rlength l1)). -Apply RList_P29. -Apply le_S_n; Apply le_trans with (S i); [Assumption | Apply le_n_Sn]. -Induction l1. -Simpl in H6; Discriminate. -Clear Hrecl1; Simpl in H1; Simpl; Apply lt_n_S; Assumption. -Symmetry; Apply minus_Sn_m; Apply le_S_n; Assumption. -Assert H18 : (le (2) (Rlength l1)). -Clear f c l2 lf2 H0 H3 H8 H7 H10 H9 H11 H13 i H1 x H2 H12 m H14 H15 H16 H17; Induction l1. -Discriminate. -Clear Hrecl1; Induction l1. -Simpl in H5; Simpl in H4; Assert H0 : ``(Rmin a b)<(Rmax a b)``. -Unfold Rmin Rmax; Case (total_order_Rle a b); Intro; [Assumption | Elim n; Left; Assumption]. -Rewrite <- H5 in H0; Rewrite <- H4 in H0; Elim (Rlt_antirefl ? H0). -Clear Hrecl1; Simpl; Repeat Apply le_n_S; Apply le_O_n. -Elim (RList_P20 ? H18); Intros r1 [r2 [r3 H19]]; Rewrite H19; Change (f x)==(pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i); Rewrite RList_P12. -Induction i. -Assert H20 := (le_S_n ? ? H15); Assert H21 := (le_trans ? ? ? H18 H20); Elim (le_Sn_O ? H21). -Clear Hreci; Rewrite RList_P13. -Rewrite H19 in H16; Rewrite H19 in H17; Change (pos_Rl (cons_Rlist (cons r2 r3) l2) i)== (pos_Rl l2 (minus (S i) (Rlength (cons r1 (cons r2 r3))))) in H16; Rewrite H16; Change (pos_Rl (cons_Rlist (cons r2 r3) l2) (S i))== (pos_Rl l2 (S (minus (S i) (Rlength (cons r1 (cons r2 r3)))))) in H17; Rewrite H17; Assert H20 := (H13 (minus (S i) (Rlength l1))); Unfold constant_D_eq open_interval in H20; Assert H21 : (lt (minus (S i) (Rlength l1)) (pred (Rlength l2))). -Apply lt_pred; Rewrite minus_Sn_m. -Apply simpl_lt_plus_l with (Rlength l1); Rewrite <- le_plus_minus. -Rewrite H19 in H1; Simpl in H1; Rewrite H19; Simpl; Rewrite RList_P23 in H1; Apply lt_n_S; Assumption. -Apply le_trans with (S i); [Apply le_S_n; Assumption | Apply le_n_Sn]. -Apply le_S_n; Assumption. -Assert H22 := (H20 H21); Repeat Rewrite H22. -Reflexivity. -Rewrite <- H19; Assert H23 : ``(pos_Rl l2 (minus (S i) (Rlength l1)))<=(pos_Rl l2 (S (minus (S i) (Rlength l1))))``. -Apply H7; Apply lt_pred. -Rewrite minus_Sn_m. -Apply simpl_lt_plus_l with (Rlength l1); Rewrite <- le_plus_minus. -Rewrite H19 in H1; Simpl in H1; Rewrite H19; Simpl; Rewrite RList_P23 in H1; Apply lt_n_S; Assumption. -Apply le_trans with (S i); [Apply le_S_n; Assumption | Apply le_n_Sn]. -Apply le_S_n; Assumption. -Elim H23; Intro. -Split. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Apply Rlt_monotony_contra with ``2``; [Sup0 | Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1l; Rewrite (Rplus_sym (pos_Rl l2 (minus (S i) (Rlength l1)))); Rewrite double; Apply Rlt_compatibility; Assumption | DiscrR]]. -Rewrite <- H19 in H16; Rewrite <- H19 in H17; Elim H2; Intros; Rewrite H19 in H25; Rewrite H19 in H26; Simpl in H25; Simpl in H16; Rewrite H16 in H25; Simpl in H26; Simpl in H17; Rewrite H17 in H26; Simpl in H24; Rewrite H24 in H25; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H25 H26)). -Assert H23 : (pos_Rl (cons_Rlist l1 l2) (S i))==(pos_Rl l2 (minus (S i) (Rlength l1))). -Rewrite H19; Simpl; Simpl in H16; Apply H16. -Assert H24 : (pos_Rl (cons_Rlist l1 l2) (S (S i)))==(pos_Rl l2 (S (minus (S i) (Rlength l1)))). -Rewrite H19; Simpl; Simpl in H17; Apply H17. -Rewrite <- H23; Rewrite <- H24; Assumption. -Simpl; Rewrite H19 in H1; Simpl in H1; Apply lt_S_n; Assumption. -Rewrite RList_P14; Rewrite H19 in H1; Simpl in H1; Simpl; Apply H1. +Lemma StepFun_P40 : + forall (f:R -> R) (a b c:R) (l1 l2 lf1 lf2:Rlist), + a < b -> + b < c -> + adapted_couple f a b l1 lf1 -> + adapted_couple f b c l2 lf2 -> + adapted_couple f a c (cons_Rlist l1 l2) (FF (cons_Rlist l1 l2) f). +intros f a b c l1 l2 lf1 lf2 H H0 H1 H2; unfold adapted_couple in H1, H2; + unfold adapted_couple in |- *; decompose [and] H1; + decompose [and] H2; clear H1 H2; repeat split. +apply RList_P25; try assumption. +rewrite H10; rewrite H4; unfold Rmin, Rmax in |- *; case (Rle_dec a b); + case (Rle_dec b c); intros; + (right; reflexivity) || (elim n; left; assumption). +rewrite RList_P22. +rewrite H5; unfold Rmin, Rmax in |- *; case (Rle_dec a b); case (Rle_dec a c); + intros; + [ reflexivity + | elim n; apply Rle_trans with b; left; assumption + | elim n; left; assumption + | elim n0; left; assumption ]. +red in |- *; intro; rewrite H1 in H6; discriminate. +rewrite RList_P24. +rewrite H9; unfold Rmin, Rmax in |- *; case (Rle_dec b c); case (Rle_dec a c); + intros; + [ reflexivity + | elim n; apply Rle_trans with b; left; assumption + | elim n; left; assumption + | elim n0; left; assumption ]. +red in |- *; intro; rewrite H1 in H11; discriminate. +apply StepFun_P20. +rewrite RList_P23; apply neq_O_lt; red in |- *; intro. +assert (H2 : (Rlength l1 + Rlength l2)%nat = 0%nat). +symmetry in |- *; apply H1. +elim (plus_is_O _ _ H2); intros; rewrite H12 in H6; discriminate. +unfold constant_D_eq, open_interval in |- *; intros; + elim (le_or_lt (S (S i)) (Rlength l1)); intro. +assert (H14 : pos_Rl (cons_Rlist l1 l2) i = pos_Rl l1 i). +apply RList_P26; apply lt_S_n; apply le_lt_n_Sm; apply le_S_n; + apply le_trans with (Rlength l1); [ assumption | apply le_n_Sn ]. +assert (H15 : pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l1 (S i)). +apply RList_P26; apply lt_S_n; apply le_lt_n_Sm; assumption. +rewrite H14 in H2; rewrite H15 in H2; assert (H16 : (2 <= Rlength l1)%nat). +apply le_trans with (S (S i)); + [ repeat apply le_n_S; apply le_O_n | assumption ]. +elim (RList_P20 _ H16); intros r1 [r2 [r3 H17]]; rewrite H17; + change + (f x = pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i) + in |- *; rewrite RList_P12. +induction i as [| i Hreci]. +simpl in |- *; assert (H18 := H8 0%nat); + unfold constant_D_eq, open_interval in H18; + assert (H19 : (0 < pred (Rlength l1))%nat). +rewrite H17; simpl in |- *; apply lt_O_Sn. +assert (H20 := H18 H19); repeat rewrite H20. +reflexivity. +assert (H21 : r1 <= r2). +rewrite H17 in H3; apply (H3 0%nat). +simpl in |- *; apply lt_O_Sn. +elim H21; intro. +split. +rewrite H17; simpl in |- *; apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +rewrite H17; simpl in |- *; apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite (Rplus_comm r1); rewrite double; + apply Rplus_lt_compat_l; assumption + | discrR ] ]. +elim H2; intros; rewrite H17 in H23; rewrite H17 in H24; simpl in H24; + simpl in H23; rewrite H22 in H23; + elim (Rlt_irrefl _ (Rlt_trans _ _ _ H23 H24)). +assumption. +clear Hreci; rewrite RList_P13. +rewrite H17 in H14; rewrite H17 in H15; + change + (pos_Rl (cons_Rlist (cons r2 r3) l2) i = + pos_Rl (cons r1 (cons r2 r3)) (S i)) in H14; rewrite H14; + change + (pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) = + pos_Rl (cons r1 (cons r2 r3)) (S (S i))) in H15; + rewrite H15; assert (H18 := H8 (S i)); + unfold constant_D_eq, open_interval in H18; + assert (H19 : (S i < pred (Rlength l1))%nat). +apply lt_pred; apply lt_S_n; apply le_lt_n_Sm; assumption. +assert (H20 := H18 H19); repeat rewrite H20. +reflexivity. +rewrite <- H17; assert (H21 : pos_Rl l1 (S i) <= pos_Rl l1 (S (S i))). +apply (H3 (S i)); apply lt_pred; apply lt_S_n; apply le_lt_n_Sm; assumption. +elim H21; intro. +split. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite (Rplus_comm (pos_Rl l1 (S i))); + rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +elim H2; intros; rewrite H22 in H23; + elim (Rlt_irrefl _ (Rlt_trans _ _ _ H23 H24)). +assumption. +simpl in |- *; rewrite H17 in H1; simpl in H1; apply lt_S_n; assumption. +rewrite RList_P14; rewrite H17 in H1; simpl in H1; apply H1. +inversion H12. +assert (H16 : pos_Rl (cons_Rlist l1 l2) (S i) = b). +rewrite RList_P29. +rewrite H15; rewrite <- minus_n_n; rewrite H10; unfold Rmin in |- *; + case (Rle_dec b c); intro; [ reflexivity | elim n; left; assumption ]. +rewrite H15; apply le_n. +induction l1 as [| r l1 Hrecl1]. +simpl in H15; discriminate. +clear Hrecl1; simpl in H1; simpl in |- *; apply lt_n_S; assumption. +assert (H17 : pos_Rl (cons_Rlist l1 l2) i = b). +rewrite RList_P26. +replace i with (pred (Rlength l1)); + [ rewrite H4; unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; left; assumption ] + | rewrite H15; reflexivity ]. +rewrite H15; apply lt_n_Sn. +rewrite H16 in H2; rewrite H17 in H2; elim H2; intros; + elim (Rlt_irrefl _ (Rlt_trans _ _ _ H14 H18)). +assert (H16 : pos_Rl (cons_Rlist l1 l2) i = pos_Rl l2 (i - Rlength l1)). +apply RList_P29. +apply le_S_n; assumption. +apply lt_le_trans with (pred (Rlength (cons_Rlist l1 l2))); + [ assumption | apply le_pred_n ]. +assert + (H17 : pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l2 (S (i - Rlength l1))). +replace (S (i - Rlength l1)) with (S i - Rlength l1)%nat. +apply RList_P29. +apply le_S_n; apply le_trans with (S i); [ assumption | apply le_n_Sn ]. +induction l1 as [| r l1 Hrecl1]. +simpl in H6; discriminate. +clear Hrecl1; simpl in H1; simpl in |- *; apply lt_n_S; assumption. +symmetry in |- *; apply minus_Sn_m; apply le_S_n; assumption. +assert (H18 : (2 <= Rlength l1)%nat). +clear f c l2 lf2 H0 H3 H8 H7 H10 H9 H11 H13 i H1 x H2 H12 m H14 H15 H16 H17; + induction l1 as [| r l1 Hrecl1]. +discriminate. +clear Hrecl1; induction l1 as [| r0 l1 Hrecl1]. +simpl in H5; simpl in H4; assert (H0 : Rmin a b < Rmax a b). +unfold Rmin, Rmax in |- *; case (Rle_dec a b); intro; + [ assumption | elim n; left; assumption ]. +rewrite <- H5 in H0; rewrite <- H4 in H0; elim (Rlt_irrefl _ H0). +clear Hrecl1; simpl in |- *; repeat apply le_n_S; apply le_O_n. +elim (RList_P20 _ H18); intros r1 [r2 [r3 H19]]; rewrite H19; + change + (f x = pos_Rl (app_Rlist (mid_Rlist (cons_Rlist (cons r2 r3) l2) r1) f) i) + in |- *; rewrite RList_P12. +induction i as [| i Hreci]. +assert (H20 := le_S_n _ _ H15); assert (H21 := le_trans _ _ _ H18 H20); + elim (le_Sn_O _ H21). +clear Hreci; rewrite RList_P13. +rewrite H19 in H16; rewrite H19 in H17; + change + (pos_Rl (cons_Rlist (cons r2 r3) l2) i = + pos_Rl l2 (S i - Rlength (cons r1 (cons r2 r3)))) + in H16; rewrite H16; + change + (pos_Rl (cons_Rlist (cons r2 r3) l2) (S i) = + pos_Rl l2 (S (S i - Rlength (cons r1 (cons r2 r3))))) + in H17; rewrite H17; assert (H20 := H13 (S i - Rlength l1)%nat); + unfold constant_D_eq, open_interval in H20; + assert (H21 : (S i - Rlength l1 < pred (Rlength l2))%nat). +apply lt_pred; rewrite minus_Sn_m. +apply plus_lt_reg_l with (Rlength l1); rewrite <- le_plus_minus. +rewrite H19 in H1; simpl in H1; rewrite H19; simpl in |- *; + rewrite RList_P23 in H1; apply lt_n_S; assumption. +apply le_trans with (S i); [ apply le_S_n; assumption | apply le_n_Sn ]. +apply le_S_n; assumption. +assert (H22 := H20 H21); repeat rewrite H22. +reflexivity. +rewrite <- H19; + assert + (H23 : pos_Rl l2 (S i - Rlength l1) <= pos_Rl l2 (S (S i - Rlength l1))). +apply H7; apply lt_pred. +rewrite minus_Sn_m. +apply plus_lt_reg_l with (Rlength l1); rewrite <- le_plus_minus. +rewrite H19 in H1; simpl in H1; rewrite H19; simpl in |- *; + rewrite RList_P23 in H1; apply lt_n_S; assumption. +apply le_trans with (S i); [ apply le_S_n; assumption | apply le_n_Sn ]. +apply le_S_n; assumption. +elim H23; intro. +split. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +apply Rmult_lt_reg_l with 2; + [ prove_sup0 + | unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym; + [ rewrite Rmult_1_l; rewrite (Rplus_comm (pos_Rl l2 (S i - Rlength l1))); + rewrite double; apply Rplus_lt_compat_l; assumption + | discrR ] ]. +rewrite <- H19 in H16; rewrite <- H19 in H17; elim H2; intros; + rewrite H19 in H25; rewrite H19 in H26; simpl in H25; + simpl in H16; rewrite H16 in H25; simpl in H26; simpl in H17; + rewrite H17 in H26; simpl in H24; rewrite H24 in H25; + elim (Rlt_irrefl _ (Rlt_trans _ _ _ H25 H26)). +assert (H23 : pos_Rl (cons_Rlist l1 l2) (S i) = pos_Rl l2 (S i - Rlength l1)). +rewrite H19; simpl in |- *; simpl in H16; apply H16. +assert + (H24 : + pos_Rl (cons_Rlist l1 l2) (S (S i)) = pos_Rl l2 (S (S i - Rlength l1))). +rewrite H19; simpl in |- *; simpl in H17; apply H17. +rewrite <- H23; rewrite <- H24; assumption. +simpl in |- *; rewrite H19 in H1; simpl in H1; apply lt_S_n; assumption. +rewrite RList_P14; rewrite H19 in H1; simpl in H1; simpl in |- *; apply H1. Qed. -Lemma StepFun_P41 : (f:R->R;a,b,c:R) ``a<=b``->``b<=c``->(IsStepFun f a b) -> (IsStepFun f b c) -> (IsStepFun f a c). -Unfold IsStepFun; Unfold is_subdivision; Intros; Elim X; Clear X; Intros l1 [lf1 H1]; Elim X0; Clear X0; Intros l2 [lf2 H2]; Case (total_order_T a b); Intro. -Elim s; Intro. -Case (total_order_T b c); Intro. -Elim s0; Intro. -Split with (cons_Rlist l1 l2); Split with (FF (cons_Rlist l1 l2) f); Apply StepFun_P40 with b lf1 lf2; Assumption. -Split with l1; Split with lf1; Rewrite b0 in H1; Assumption. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H0 r)). -Split with l2; Split with lf2; Rewrite <- b0 in H2; Assumption. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Lemma StepFun_P41 : + forall (f:R -> R) (a b c:R), + a <= b -> b <= c -> IsStepFun f a b -> IsStepFun f b c -> IsStepFun f a c. +unfold IsStepFun in |- *; unfold is_subdivision in |- *; intros; elim X; + clear X; intros l1 [lf1 H1]; elim X0; clear X0; intros l2 [lf2 H2]; + case (total_order_T a b); intro. +elim s; intro. +case (total_order_T b c); intro. +elim s0; intro. +split with (cons_Rlist l1 l2); split with (FF (cons_Rlist l1 l2) f); + apply StepFun_P40 with b lf1 lf2; assumption. +split with l1; split with lf1; rewrite b0 in H1; assumption. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 r)). +split with l2; split with lf2; rewrite <- b0 in H2; assumption. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). Qed. -Lemma StepFun_P42 : (l1,l2:Rlist;f:R->R) (pos_Rl l1 (pred (Rlength l1)))==(pos_Rl l2 O) -> ``(Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2)) == (Int_SF (FF l1 f) l1) + (Int_SF (FF l2 f) l2)``. -Intros l1 l2 f; NewInduction l1 as [|r l1 IHl1]; Intros H; [ Simpl; Ring | NewDestruct l1; [Simpl in H; Simpl; NewDestruct l2; [Simpl; Ring | Simpl; Simpl in H; Rewrite H; Ring] | Simpl; Rewrite Rplus_assoc; Apply Rplus_plus_r; Apply IHl1; Rewrite <- H; Reflexivity]]. +Lemma StepFun_P42 : + forall (l1 l2:Rlist) (f:R -> R), + pos_Rl l1 (pred (Rlength l1)) = pos_Rl l2 0 -> + Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2) = + Int_SF (FF l1 f) l1 + Int_SF (FF l2 f) l2. +intros l1 l2 f; induction l1 as [| r l1 IHl1]; intros H; + [ simpl in |- *; ring + | destruct l1 as [| r0 r1]; + [ simpl in H; simpl in |- *; destruct l2 as [| r0 r1]; + [ simpl in |- *; ring | simpl in |- *; simpl in H; rewrite H; ring ] + | simpl in |- *; rewrite Rplus_assoc; apply Rplus_eq_compat_l; apply IHl1; + rewrite <- H; reflexivity ] ]. Qed. -Lemma StepFun_P43 : (f:R->R;a,b,c:R;pr1:(IsStepFun f a b);pr2:(IsStepFun f b c);pr3:(IsStepFun f a c)) ``(RiemannInt_SF (mkStepFun pr1))+(RiemannInt_SF (mkStepFun pr2))==(RiemannInt_SF (mkStepFun pr3))``. -Intros f; Intros; Assert H1 : (SigT ? [l:Rlist](sigTT ? [l0:Rlist](adapted_couple f a b l l0))). -Apply pr1. -Assert H2 : (SigT ? [l:Rlist](sigTT ? [l0:Rlist](adapted_couple f b c l l0))). -Apply pr2. -Assert H3 : (SigT ? [l:Rlist](sigTT ? [l0:Rlist](adapted_couple f a c l l0))). -Apply pr3. -Elim H1; Clear H1; Intros l1 [lf1 H1]; Elim H2; Clear H2; Intros l2 [lf2 H2]; Elim H3; Clear H3; Intros l3 [lf3 H3]. -Replace (RiemannInt_SF (mkStepFun pr1)) with (Cases (total_order_Rle a b) of (leftT _) => (Int_SF lf1 l1) | (rightT _) => ``-(Int_SF lf1 l1)`` end). -Replace (RiemannInt_SF (mkStepFun pr2)) with (Cases (total_order_Rle b c) of (leftT _) => (Int_SF lf2 l2) | (rightT _) => ``-(Int_SF lf2 l2)`` end). -Replace (RiemannInt_SF (mkStepFun pr3)) with (Cases (total_order_Rle a c) of (leftT _) => (Int_SF lf3 l3) | (rightT _) => ``-(Int_SF lf3 l3)`` end). -Case (total_order_Rle a b); Case (total_order_Rle b c); Case (total_order_Rle a c); Intros. -Elim r1; Intro. -Elim r0; Intro. -Replace (Int_SF lf3 l3) with (Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2)). -Replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). -Replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). -Symmetry; Apply StepFun_P42. -Unfold adapted_couple in H1 H2; Decompose [and] H1; Decompose [and] H2; Clear H1 H2; Rewrite H11; Rewrite H5; Unfold Rmax Rmin; Case (total_order_Rle a b); Case (total_order_Rle b c); Intros; Reflexivity Orelse Elim n; Assumption. -EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf2; Apply H2; Assumption | Assumption]. -EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf1; Apply H1 | Assumption]. -EApply StepFun_P17; [Apply (StepFun_P40 H H0 H1 H2) | Apply H3]. -Replace (Int_SF lf2 l2) with R0. -Rewrite Rplus_Or; EApply StepFun_P17; [Apply H1 | Rewrite <- H0 in H3; Apply H3]. -Symmetry; EApply StepFun_P8; [Apply H2 | Assumption]. -Replace (Int_SF lf1 l1) with R0. -Rewrite Rplus_Ol; EApply StepFun_P17; [Apply H2 | Rewrite H in H3; Apply H3]. -Symmetry; EApply StepFun_P8; [Apply H1 | Assumption]. -Elim n; Apply Rle_trans with b; Assumption. -Apply r_Rplus_plus with (Int_SF lf2 l2); Replace ``(Int_SF lf2 l2)+((Int_SF lf1 l1)+ -(Int_SF lf2 l2))`` with (Int_SF lf1 l1); [Idtac | Ring]. -Assert H : ``c<b``. -Auto with real. -Elim r; Intro. -Rewrite Rplus_sym; Replace (Int_SF lf1 l1) with (Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2)). -Replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). -Replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). -Apply StepFun_P42. -Unfold adapted_couple in H2 H3; Decompose [and] H2; Decompose [and] H3; Clear H3 H2; Rewrite H10; Rewrite H6; Unfold Rmax Rmin; Case (total_order_Rle a c); Case (total_order_Rle b c); Intros; [Elim n; Assumption | Reflexivity | Elim n0; Assumption | Elim n1; Assumption]. -EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf2; Apply H2 | Assumption]. -EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf3; Apply H3 | Assumption]. -EApply StepFun_P17; [Apply (StepFun_P40 H0 H H3 (StepFun_P2 H2)) | Apply H1]. -Replace (Int_SF lf3 l3) with R0. -Rewrite Rplus_Or; EApply StepFun_P17; [Apply H1 | Apply StepFun_P2; Rewrite <- H0 in H2; Apply H2]. -Symmetry; EApply StepFun_P8; [Apply H3 | Assumption]. -Replace (Int_SF lf2 l2) with ``(Int_SF lf3 l3)+(Int_SF lf1 l1)``. -Ring. -Elim r; Intro. -Replace (Int_SF lf2 l2) with (Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1)). -Replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). -Replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). -Symmetry; Apply StepFun_P42. -Unfold adapted_couple in H1 H3; Decompose [and] H1; Decompose [and] H3; Clear H3 H1; Rewrite H9; Rewrite H5; Unfold Rmax Rmin; Case (total_order_Rle a c); Case (total_order_Rle a b); Intros; [Elim n; Assumption | Elim n1; Assumption | Reflexivity | Elim n1; Assumption]. -EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf1; Apply H1 | Assumption]. -EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf3; Apply H3 | Assumption]. -EApply StepFun_P17. -Assert H0 : ``c<a``. -Auto with real. -Apply (StepFun_P40 H0 H (StepFun_P2 H3) H1). -Apply StepFun_P2; Apply H2. -Replace (Int_SF lf1 l1) with R0. -Rewrite Rplus_Or; EApply StepFun_P17; [Apply H3 | Rewrite <- H in H2; Apply H2]. -Symmetry; EApply StepFun_P8; [Apply H1 | Assumption]. -Assert H : ``b<a``. -Auto with real. -Replace (Int_SF lf2 l2) with ``(Int_SF lf3 l3)+(Int_SF lf1 l1)``. -Ring. -Rewrite Rplus_sym; Elim r; Intro. -Replace (Int_SF lf2 l2) with (Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3)). -Replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). -Replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). -Symmetry; Apply StepFun_P42. -Unfold adapted_couple in H1 H3; Decompose [and] H1; Decompose [and] H3; Clear H3 H1; Rewrite H11; Rewrite H5; Unfold Rmax Rmin; Case (total_order_Rle a c); Case (total_order_Rle a b); Intros; [Elim n; Assumption | Reflexivity | Elim n0; Assumption | Elim n1; Assumption]. -EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf1; Apply H1 | Assumption]. -EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf3; Apply H3 | Assumption]. -EApply StepFun_P17. -Apply (StepFun_P40 H H0 (StepFun_P2 H1) H3). -Apply H2. -Replace (Int_SF lf3 l3) with R0. -Rewrite Rplus_Or; EApply StepFun_P17; [Apply H1 | Rewrite <- H0 in H2; Apply StepFun_P2; Apply H2]. -Symmetry; EApply StepFun_P8; [Apply H3 | Assumption]. -Assert H : ``c<a``. -Auto with real. -Replace (Int_SF lf1 l1) with ``(Int_SF lf2 l2)+(Int_SF lf3 l3)``. -Ring. -Elim r; Intro. -Replace (Int_SF lf1 l1) with (Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3)). -Replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). -Replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). -Symmetry; Apply StepFun_P42. -Unfold adapted_couple in H2 H3; Decompose [and] H2; Decompose [and] H3; Clear H3 H2; Rewrite H11; Rewrite H5; Unfold Rmax Rmin; Case (total_order_Rle a c); Case (total_order_Rle b c); Intros; [Elim n; Assumption | Elim n1; Assumption | Reflexivity | Elim n1; Assumption]. -EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf2; Apply H2 | Assumption]. -EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf3; Apply H3 | Assumption]. -EApply StepFun_P17. -Apply (StepFun_P40 H0 H H2 (StepFun_P2 H3)). -Apply StepFun_P2; Apply H1. -Replace (Int_SF lf2 l2) with R0. -Rewrite Rplus_Ol; EApply StepFun_P17; [Apply H3 | Rewrite H0 in H1; Apply H1]. -Symmetry; EApply StepFun_P8; [Apply H2 | Assumption]. -Elim n; Apply Rle_trans with a; Try Assumption. -Auto with real. -Assert H : ``c<b``. -Auto with real. -Assert H0 : ``b<a``. -Auto with real. -Replace (Int_SF lf3 l3) with ``(Int_SF lf2 l2)+(Int_SF lf1 l1)``. -Ring. -Replace (Int_SF lf3 l3) with (Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1)). -Replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). -Replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). -Symmetry; Apply StepFun_P42. -Unfold adapted_couple in H2 H1; Decompose [and] H2; Decompose [and] H1; Clear H1 H2; Rewrite H11; Rewrite H5; Unfold Rmax Rmin; Case (total_order_Rle a b); Case (total_order_Rle b c); Intros; [Elim n1; Assumption | Elim n1; Assumption | Elim n0; Assumption | Reflexivity]. -EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf2; Apply H2 | Assumption]. -EApply StepFun_P17; [Apply StepFun_P21; Unfold is_subdivision; Split with lf1; Apply H1 | Assumption]. -EApply StepFun_P17. -Apply (StepFun_P40 H H0 (StepFun_P2 H2) (StepFun_P2 H1)). -Apply StepFun_P2; Apply H3. -Unfold RiemannInt_SF; Case (total_order_Rle a c); Intro. -EApply StepFun_P17. -Apply H3. -Change (adapted_couple (mkStepFun pr3) a c (subdivision (mkStepFun 1!a 2!c 3!f pr3)) (subdivision_val (mkStepFun 1!a 2!c 3!f pr3))); Apply StepFun_P1. -Apply eq_Ropp; EApply StepFun_P17. -Apply H3. -Change (adapted_couple (mkStepFun pr3) a c (subdivision (mkStepFun 1!a 2!c 3!f pr3)) (subdivision_val (mkStepFun 1!a 2!c 3!f pr3))); Apply StepFun_P1. -Unfold RiemannInt_SF; Case (total_order_Rle b c); Intro. -EApply StepFun_P17. -Apply H2. -Change (adapted_couple (mkStepFun pr2) b c (subdivision (mkStepFun 1!b 2!c 3!f pr2)) (subdivision_val (mkStepFun 1!b 2!c 3!f pr2))); Apply StepFun_P1. -Apply eq_Ropp; EApply StepFun_P17. -Apply H2. -Change (adapted_couple (mkStepFun pr2) b c (subdivision (mkStepFun 1!b 2!c 3!f pr2)) (subdivision_val (mkStepFun 1!b 2!c 3!f pr2))); Apply StepFun_P1. -Unfold RiemannInt_SF; Case (total_order_Rle a b); Intro. -EApply StepFun_P17. -Apply H1. -Change (adapted_couple (mkStepFun pr1) a b (subdivision (mkStepFun 1!a 2!b 3!f pr1)) (subdivision_val (mkStepFun 1!a 2!b 3!f pr1))); Apply StepFun_P1. -Apply eq_Ropp; EApply StepFun_P17. -Apply H1. -Change (adapted_couple (mkStepFun pr1) a b (subdivision (mkStepFun 1!a 2!b 3!f pr1)) (subdivision_val (mkStepFun 1!a 2!b 3!f pr1))); Apply StepFun_P1. +Lemma StepFun_P43 : + forall (f:R -> R) (a b c:R) (pr1:IsStepFun f a b) + (pr2:IsStepFun f b c) (pr3:IsStepFun f a c), + RiemannInt_SF (mkStepFun pr1) + RiemannInt_SF (mkStepFun pr2) = + RiemannInt_SF (mkStepFun pr3). +intros f; intros; + assert + (H1 : + sigT (fun l:Rlist => sigT (fun l0:Rlist => adapted_couple f a b l l0))). +apply pr1. +assert + (H2 : + sigT (fun l:Rlist => sigT (fun l0:Rlist => adapted_couple f b c l l0))). +apply pr2. +assert + (H3 : + sigT (fun l:Rlist => sigT (fun l0:Rlist => adapted_couple f a c l l0))). +apply pr3. +elim H1; clear H1; intros l1 [lf1 H1]; elim H2; clear H2; intros l2 [lf2 H2]; + elim H3; clear H3; intros l3 [lf3 H3]. +replace (RiemannInt_SF (mkStepFun pr1)) with + match Rle_dec a b with + | left _ => Int_SF lf1 l1 + | right _ => - Int_SF lf1 l1 + end. +replace (RiemannInt_SF (mkStepFun pr2)) with + match Rle_dec b c with + | left _ => Int_SF lf2 l2 + | right _ => - Int_SF lf2 l2 + end. +replace (RiemannInt_SF (mkStepFun pr3)) with + match Rle_dec a c with + | left _ => Int_SF lf3 l3 + | right _ => - Int_SF lf3 l3 + end. +case (Rle_dec a b); case (Rle_dec b c); case (Rle_dec a c); intros. +elim r1; intro. +elim r0; intro. +replace (Int_SF lf3 l3) with + (Int_SF (FF (cons_Rlist l1 l2) f) (cons_Rlist l1 l2)). +replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). +replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). +symmetry in |- *; apply StepFun_P42. +unfold adapted_couple in H1, H2; decompose [and] H1; decompose [and] H2; + clear H1 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin in |- *; + case (Rle_dec a b); case (Rle_dec b c); intros; reflexivity || elim n; + assumption. +eapply StepFun_P17; + [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf2; apply H2; + assumption + | assumption ]. +eapply StepFun_P17; + [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf1; apply H1 + | assumption ]. +eapply StepFun_P17; [ apply (StepFun_P40 H H0 H1 H2) | apply H3 ]. +replace (Int_SF lf2 l2) with 0. +rewrite Rplus_0_r; eapply StepFun_P17; + [ apply H1 | rewrite <- H0 in H3; apply H3 ]. +symmetry in |- *; eapply StepFun_P8; [ apply H2 | assumption ]. +replace (Int_SF lf1 l1) with 0. +rewrite Rplus_0_l; eapply StepFun_P17; + [ apply H2 | rewrite H in H3; apply H3 ]. +symmetry in |- *; eapply StepFun_P8; [ apply H1 | assumption ]. +elim n; apply Rle_trans with b; assumption. +apply Rplus_eq_reg_l with (Int_SF lf2 l2); + replace (Int_SF lf2 l2 + (Int_SF lf1 l1 + - Int_SF lf2 l2)) with + (Int_SF lf1 l1); [ idtac | ring ]. +assert (H : c < b). +auto with real. +elim r; intro. +rewrite Rplus_comm; + replace (Int_SF lf1 l1) with + (Int_SF (FF (cons_Rlist l3 l2) f) (cons_Rlist l3 l2)). +replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). +replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). +apply StepFun_P42. +unfold adapted_couple in H2, H3; decompose [and] H2; decompose [and] H3; + clear H3 H2; rewrite H10; rewrite H6; unfold Rmax, Rmin in |- *; + case (Rle_dec a c); case (Rle_dec b c); intros; + [ elim n; assumption + | reflexivity + | elim n0; assumption + | elim n1; assumption ]. +eapply StepFun_P17; + [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf2; apply H2 + | assumption ]. +eapply StepFun_P17; + [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf3; apply H3 + | assumption ]. +eapply StepFun_P17; + [ apply (StepFun_P40 H0 H H3 (StepFun_P2 H2)) | apply H1 ]. +replace (Int_SF lf3 l3) with 0. +rewrite Rplus_0_r; eapply StepFun_P17; + [ apply H1 | apply StepFun_P2; rewrite <- H0 in H2; apply H2 ]. +symmetry in |- *; eapply StepFun_P8; [ apply H3 | assumption ]. +replace (Int_SF lf2 l2) with (Int_SF lf3 l3 + Int_SF lf1 l1). +ring. +elim r; intro. +replace (Int_SF lf2 l2) with + (Int_SF (FF (cons_Rlist l3 l1) f) (cons_Rlist l3 l1)). +replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). +replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). +symmetry in |- *; apply StepFun_P42. +unfold adapted_couple in H1, H3; decompose [and] H1; decompose [and] H3; + clear H3 H1; rewrite H9; rewrite H5; unfold Rmax, Rmin in |- *; + case (Rle_dec a c); case (Rle_dec a b); intros; + [ elim n; assumption + | elim n1; assumption + | reflexivity + | elim n1; assumption ]. +eapply StepFun_P17; + [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf1; apply H1 + | assumption ]. +eapply StepFun_P17; + [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf3; apply H3 + | assumption ]. +eapply StepFun_P17. +assert (H0 : c < a). +auto with real. +apply (StepFun_P40 H0 H (StepFun_P2 H3) H1). +apply StepFun_P2; apply H2. +replace (Int_SF lf1 l1) with 0. +rewrite Rplus_0_r; eapply StepFun_P17; + [ apply H3 | rewrite <- H in H2; apply H2 ]. +symmetry in |- *; eapply StepFun_P8; [ apply H1 | assumption ]. +assert (H : b < a). +auto with real. +replace (Int_SF lf2 l2) with (Int_SF lf3 l3 + Int_SF lf1 l1). +ring. +rewrite Rplus_comm; elim r; intro. +replace (Int_SF lf2 l2) with + (Int_SF (FF (cons_Rlist l1 l3) f) (cons_Rlist l1 l3)). +replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). +replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). +symmetry in |- *; apply StepFun_P42. +unfold adapted_couple in H1, H3; decompose [and] H1; decompose [and] H3; + clear H3 H1; rewrite H11; rewrite H5; unfold Rmax, Rmin in |- *; + case (Rle_dec a c); case (Rle_dec a b); intros; + [ elim n; assumption + | reflexivity + | elim n0; assumption + | elim n1; assumption ]. +eapply StepFun_P17; + [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf1; apply H1 + | assumption ]. +eapply StepFun_P17; + [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf3; apply H3 + | assumption ]. +eapply StepFun_P17. +apply (StepFun_P40 H H0 (StepFun_P2 H1) H3). +apply H2. +replace (Int_SF lf3 l3) with 0. +rewrite Rplus_0_r; eapply StepFun_P17; + [ apply H1 | rewrite <- H0 in H2; apply StepFun_P2; apply H2 ]. +symmetry in |- *; eapply StepFun_P8; [ apply H3 | assumption ]. +assert (H : c < a). +auto with real. +replace (Int_SF lf1 l1) with (Int_SF lf2 l2 + Int_SF lf3 l3). +ring. +elim r; intro. +replace (Int_SF lf1 l1) with + (Int_SF (FF (cons_Rlist l2 l3) f) (cons_Rlist l2 l3)). +replace (Int_SF lf3 l3) with (Int_SF (FF l3 f) l3). +replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). +symmetry in |- *; apply StepFun_P42. +unfold adapted_couple in H2, H3; decompose [and] H2; decompose [and] H3; + clear H3 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin in |- *; + case (Rle_dec a c); case (Rle_dec b c); intros; + [ elim n; assumption + | elim n1; assumption + | reflexivity + | elim n1; assumption ]. +eapply StepFun_P17; + [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf2; apply H2 + | assumption ]. +eapply StepFun_P17; + [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf3; apply H3 + | assumption ]. +eapply StepFun_P17. +apply (StepFun_P40 H0 H H2 (StepFun_P2 H3)). +apply StepFun_P2; apply H1. +replace (Int_SF lf2 l2) with 0. +rewrite Rplus_0_l; eapply StepFun_P17; + [ apply H3 | rewrite H0 in H1; apply H1 ]. +symmetry in |- *; eapply StepFun_P8; [ apply H2 | assumption ]. +elim n; apply Rle_trans with a; try assumption. +auto with real. +assert (H : c < b). +auto with real. +assert (H0 : b < a). +auto with real. +replace (Int_SF lf3 l3) with (Int_SF lf2 l2 + Int_SF lf1 l1). +ring. +replace (Int_SF lf3 l3) with + (Int_SF (FF (cons_Rlist l2 l1) f) (cons_Rlist l2 l1)). +replace (Int_SF lf1 l1) with (Int_SF (FF l1 f) l1). +replace (Int_SF lf2 l2) with (Int_SF (FF l2 f) l2). +symmetry in |- *; apply StepFun_P42. +unfold adapted_couple in H2, H1; decompose [and] H2; decompose [and] H1; + clear H1 H2; rewrite H11; rewrite H5; unfold Rmax, Rmin in |- *; + case (Rle_dec a b); case (Rle_dec b c); intros; + [ elim n1; assumption + | elim n1; assumption + | elim n0; assumption + | reflexivity ]. +eapply StepFun_P17; + [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf2; apply H2 + | assumption ]. +eapply StepFun_P17; + [ apply StepFun_P21; unfold is_subdivision in |- *; split with lf1; apply H1 + | assumption ]. +eapply StepFun_P17. +apply (StepFun_P40 H H0 (StepFun_P2 H2) (StepFun_P2 H1)). +apply StepFun_P2; apply H3. +unfold RiemannInt_SF in |- *; case (Rle_dec a c); intro. +eapply StepFun_P17. +apply H3. +change + (adapted_couple (mkStepFun pr3) a c (subdivision (mkStepFun pr3)) + (subdivision_val (mkStepFun pr3))) in |- *; apply StepFun_P1. +apply Ropp_eq_compat; eapply StepFun_P17. +apply H3. +change + (adapted_couple (mkStepFun pr3) a c (subdivision (mkStepFun pr3)) + (subdivision_val (mkStepFun pr3))) in |- *; apply StepFun_P1. +unfold RiemannInt_SF in |- *; case (Rle_dec b c); intro. +eapply StepFun_P17. +apply H2. +change + (adapted_couple (mkStepFun pr2) b c (subdivision (mkStepFun pr2)) + (subdivision_val (mkStepFun pr2))) in |- *; apply StepFun_P1. +apply Ropp_eq_compat; eapply StepFun_P17. +apply H2. +change + (adapted_couple (mkStepFun pr2) b c (subdivision (mkStepFun pr2)) + (subdivision_val (mkStepFun pr2))) in |- *; apply StepFun_P1. +unfold RiemannInt_SF in |- *; case (Rle_dec a b); intro. +eapply StepFun_P17. +apply H1. +change + (adapted_couple (mkStepFun pr1) a b (subdivision (mkStepFun pr1)) + (subdivision_val (mkStepFun pr1))) in |- *; apply StepFun_P1. +apply Ropp_eq_compat; eapply StepFun_P17. +apply H1. +change + (adapted_couple (mkStepFun pr1) a b (subdivision (mkStepFun pr1)) + (subdivision_val (mkStepFun pr1))) in |- *; apply StepFun_P1. Qed. -Lemma StepFun_P44 : (f:R->R;a,b,c:R) (IsStepFun f a b) -> ``a<=c<=b`` -> (IsStepFun f a c). -Intros f; Intros; Assert H0 : ``a<=b``. -Elim H; Intros; Apply Rle_trans with c; Assumption. -Elim H; Clear H; Intros; Unfold IsStepFun in X; Unfold is_subdivision in X; Elim X; Clear X; Intros l1 [lf1 H2]; Cut (l1,lf1:Rlist;a,b,c:R;f:R->R) (adapted_couple f a b l1 lf1) -> ``a<=c<=b`` -> (SigT ? [l:Rlist](sigTT ? [l0:Rlist](adapted_couple f a c l l0))). -Intros; Unfold IsStepFun; Unfold is_subdivision; EApply X. -Apply H2. -Split; Assumption. -Clear f a b c H0 H H1 H2 l1 lf1; Induction l1. -Intros; Unfold adapted_couple in H; Decompose [and] H; Clear H; Simpl in H4; Discriminate. -Induction r0. -Intros; Assert H1 : ``a==b``. -Unfold adapted_couple in H; Decompose [and] H; Clear H; Simpl in H3; Simpl in H2; Assert H7 : ``a<=b``. -Elim H0; Intros; Apply Rle_trans with c; Assumption. -Replace a with (Rmin a b). -Pattern 2 b; Replace b with (Rmax a b). -Rewrite <- H2; Rewrite H3; Reflexivity. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Split with (cons r nil); Split with lf1; Assert H2 : ``c==b``. -Rewrite H1 in H0; Elim H0; Intros; Apply Rle_antisym; Assumption. -Rewrite H2; Assumption. -Intros; Clear X; Induction lf1. -Unfold adapted_couple in H; Decompose [and] H; Clear H; Simpl in H4; Discriminate. -Clear Hreclf1; Assert H1 : (sumboolT ``c<=r1`` ``r1<c``). -Case (total_order_Rle c r1); Intro; [Left; Assumption | Right; Auto with real]. -Elim H1; Intro. -Split with (cons r (cons c nil)); Split with (cons r3 nil); Unfold adapted_couple in H; Decompose [and] H; Clear H; Assert H6 : ``r==a``. -Simpl in H4; Rewrite H4; Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Elim H0; Intros; Apply Rle_trans with c; Assumption]. -Elim H0; Clear H0; Intros; Unfold adapted_couple; Repeat Split. -Rewrite H6; Unfold ordered_Rlist; Intros; Simpl in H8; Inversion H8; [Simpl; Assumption | Elim (le_Sn_O ? H10)]. -Simpl; Unfold Rmin; Case (total_order_Rle a c); Intro; [Assumption | Elim n; Assumption]. -Simpl; Unfold Rmax; Case (total_order_Rle a c); Intro; [Reflexivity | Elim n; Assumption]. -Unfold constant_D_eq open_interval; Intros; Simpl in H8; Inversion H8. -Simpl; Assert H10 := (H7 O); Assert H12 : (lt (0) (pred (Rlength (cons r (cons r1 r2))))). -Simpl; Apply lt_O_Sn. -Apply (H10 H12); Unfold open_interval; Simpl; Rewrite H11 in H9; Simpl in H9; Elim H9; Clear H9; Intros; Split; Try Assumption. -Apply Rlt_le_trans with c; Assumption. -Elim (le_Sn_O ? H11). -Cut (adapted_couple f r1 b (cons r1 r2) lf1). -Cut ``r1<=c<=b``. -Intros. -Elim (X0 ? ? ? ? ? H3 H2); Intros l1' [lf1' H4]; Split with (cons r l1'); Split with (cons r3 lf1'); Unfold adapted_couple in H H4; Decompose [and] H; Decompose [and] H4; Clear H H4 X0; Assert H14 : ``a<=b``. -Elim H0; Intros; Apply Rle_trans with c; Assumption. -Assert H16 : ``r==a``. -Simpl in H7; Rewrite H7; Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Induction l1'. -Simpl in H13; Discriminate. -Clear Hrecl1'; Unfold adapted_couple; Repeat Split. -Unfold ordered_Rlist; Intros; Simpl in H; Induction i. -Simpl; Replace r4 with r1. -Apply (H5 O). -Simpl; Apply lt_O_Sn. -Simpl in H12; Rewrite H12; Unfold Rmin; Case (total_order_Rle r1 c); Intro; [Reflexivity | Elim n; Left; Assumption]. -Apply (H9 i); Simpl; Apply lt_S_n; Assumption. -Simpl; Unfold Rmin; Case (total_order_Rle a c); Intro; [Assumption | Elim n; Elim H0; Intros; Assumption]. -Replace (Rmax a c) with (Rmax r1 c). -Rewrite <- H11; Reflexivity. -Unfold Rmax; Case (total_order_Rle r1 c); Case (total_order_Rle a c); Intros; [Reflexivity | Elim n; Elim H0; Intros; Assumption | Elim n; Left; Assumption | Elim n0; Left; Assumption]. -Simpl; Simpl in H13; Rewrite H13; Reflexivity. -Intros; Simpl in H; Unfold constant_D_eq open_interval; Intros; Induction i. -Simpl; Assert H17 := (H10 O); Assert H18 : (lt (0) (pred (Rlength (cons r (cons r1 r2))))). -Simpl; Apply lt_O_Sn. -Apply (H17 H18); Unfold open_interval; Simpl; Simpl in H4; Elim H4; Clear H4; Intros; Split; Try Assumption; Replace r1 with r4. -Assumption. -Simpl in H12; Rewrite H12; Unfold Rmin; Case (total_order_Rle r1 c); Intro; [Reflexivity | Elim n; Left; Assumption]. -Clear Hreci; Simpl; Apply H15. -Simpl; Apply lt_S_n; Assumption. -Unfold open_interval; Apply H4. -Split. -Left; Assumption. -Elim H0; Intros; Assumption. -EApply StepFun_P7; [Elim H0; Intros; Apply Rle_trans with c; [Apply H2 | Apply H3] | Apply H]. +Lemma StepFun_P44 : + forall (f:R -> R) (a b c:R), + IsStepFun f a b -> a <= c <= b -> IsStepFun f a c. +intros f; intros; assert (H0 : a <= b). +elim H; intros; apply Rle_trans with c; assumption. +elim H; clear H; intros; unfold IsStepFun in X; unfold is_subdivision in X; + elim X; clear X; intros l1 [lf1 H2]; + cut + (forall (l1 lf1:Rlist) (a b c:R) (f:R -> R), + adapted_couple f a b l1 lf1 -> + a <= c <= b -> + sigT (fun l:Rlist => sigT (fun l0:Rlist => adapted_couple f a c l l0))). +intros; unfold IsStepFun in |- *; unfold is_subdivision in |- *; eapply X. +apply H2. +split; assumption. +clear f a b c H0 H H1 H2 l1 lf1; simple induction l1. +intros; unfold adapted_couple in H; decompose [and] H; clear H; simpl in H4; + discriminate. +simple induction r0. +intros; assert (H1 : a = b). +unfold adapted_couple in H; decompose [and] H; clear H; simpl in H3; + simpl in H2; assert (H7 : a <= b). +elim H0; intros; apply Rle_trans with c; assumption. +replace a with (Rmin a b). +pattern b at 2 in |- *; replace b with (Rmax a b). +rewrite <- H2; rewrite H3; reflexivity. +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +split with (cons r nil); split with lf1; assert (H2 : c = b). +rewrite H1 in H0; elim H0; intros; apply Rle_antisym; assumption. +rewrite H2; assumption. +intros; clear X; induction lf1 as [| r3 lf1 Hreclf1]. +unfold adapted_couple in H; decompose [and] H; clear H; simpl in H4; + discriminate. +clear Hreclf1; assert (H1 : {c <= r1} + {r1 < c}). +case (Rle_dec c r1); intro; [ left; assumption | right; auto with real ]. +elim H1; intro. +split with (cons r (cons c nil)); split with (cons r3 nil); + unfold adapted_couple in H; decompose [and] H; clear H; + assert (H6 : r = a). +simpl in H4; rewrite H4; unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity + | elim n; elim H0; intros; apply Rle_trans with c; assumption ]. +elim H0; clear H0; intros; unfold adapted_couple in |- *; repeat split. +rewrite H6; unfold ordered_Rlist in |- *; intros; simpl in H8; inversion H8; + [ simpl in |- *; assumption | elim (le_Sn_O _ H10) ]. +simpl in |- *; unfold Rmin in |- *; case (Rle_dec a c); intro; + [ assumption | elim n; assumption ]. +simpl in |- *; unfold Rmax in |- *; case (Rle_dec a c); intro; + [ reflexivity | elim n; assumption ]. +unfold constant_D_eq, open_interval in |- *; intros; simpl in H8; + inversion H8. +simpl in |- *; assert (H10 := H7 0%nat); + assert (H12 : (0 < pred (Rlength (cons r (cons r1 r2))))%nat). +simpl in |- *; apply lt_O_Sn. +apply (H10 H12); unfold open_interval in |- *; simpl in |- *; + rewrite H11 in H9; simpl in H9; elim H9; clear H9; + intros; split; try assumption. +apply Rlt_le_trans with c; assumption. +elim (le_Sn_O _ H11). +cut (adapted_couple f r1 b (cons r1 r2) lf1). +cut (r1 <= c <= b). +intros. +elim (X0 _ _ _ _ _ H3 H2); intros l1' [lf1' H4]; split with (cons r l1'); + split with (cons r3 lf1'); unfold adapted_couple in H, H4; + decompose [and] H; decompose [and] H4; clear H H4 X0; + assert (H14 : a <= b). +elim H0; intros; apply Rle_trans with c; assumption. +assert (H16 : r = a). +simpl in H7; rewrite H7; unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +induction l1' as [| r4 l1' Hrecl1']. +simpl in H13; discriminate. +clear Hrecl1'; unfold adapted_couple in |- *; repeat split. +unfold ordered_Rlist in |- *; intros; simpl in H; induction i as [| i Hreci]. +simpl in |- *; replace r4 with r1. +apply (H5 0%nat). +simpl in |- *; apply lt_O_Sn. +simpl in H12; rewrite H12; unfold Rmin in |- *; case (Rle_dec r1 c); intro; + [ reflexivity | elim n; left; assumption ]. +apply (H9 i); simpl in |- *; apply lt_S_n; assumption. +simpl in |- *; unfold Rmin in |- *; case (Rle_dec a c); intro; + [ assumption | elim n; elim H0; intros; assumption ]. +replace (Rmax a c) with (Rmax r1 c). +rewrite <- H11; reflexivity. +unfold Rmax in |- *; case (Rle_dec r1 c); case (Rle_dec a c); intros; + [ reflexivity + | elim n; elim H0; intros; assumption + | elim n; left; assumption + | elim n0; left; assumption ]. +simpl in |- *; simpl in H13; rewrite H13; reflexivity. +intros; simpl in H; unfold constant_D_eq, open_interval in |- *; intros; + induction i as [| i Hreci]. +simpl in |- *; assert (H17 := H10 0%nat); + assert (H18 : (0 < pred (Rlength (cons r (cons r1 r2))))%nat). +simpl in |- *; apply lt_O_Sn. +apply (H17 H18); unfold open_interval in |- *; simpl in |- *; simpl in H4; + elim H4; clear H4; intros; split; try assumption; + replace r1 with r4. +assumption. +simpl in H12; rewrite H12; unfold Rmin in |- *; case (Rle_dec r1 c); intro; + [ reflexivity | elim n; left; assumption ]. +clear Hreci; simpl in |- *; apply H15. +simpl in |- *; apply lt_S_n; assumption. +unfold open_interval in |- *; apply H4. +split. +left; assumption. +elim H0; intros; assumption. +eapply StepFun_P7; + [ elim H0; intros; apply Rle_trans with c; [ apply H2 | apply H3 ] + | apply H ]. Qed. -Lemma StepFun_P45 : (f:R->R;a,b,c:R) (IsStepFun f a b) -> ``a<=c<=b`` -> (IsStepFun f c b). -Intros f; Intros; Assert H0 : ``a<=b``. -Elim H; Intros; Apply Rle_trans with c; Assumption. -Elim H; Clear H; Intros; Unfold IsStepFun in X; Unfold is_subdivision in X; Elim X; Clear X; Intros l1 [lf1 H2]; Cut (l1,lf1:Rlist;a,b,c:R;f:R->R) (adapted_couple f a b l1 lf1) -> ``a<=c<=b`` -> (SigT ? [l:Rlist](sigTT ? [l0:Rlist](adapted_couple f c b l l0))). -Intros; Unfold IsStepFun; Unfold is_subdivision; EApply X; [Apply H2 | Split; Assumption]. -Clear f a b c H0 H H1 H2 l1 lf1; Induction l1. -Intros; Unfold adapted_couple in H; Decompose [and] H; Clear H; Simpl in H4; Discriminate. -Induction r0. -Intros; Assert H1 : ``a==b``. -Unfold adapted_couple in H; Decompose [and] H; Clear H; Simpl in H3; Simpl in H2; Assert H7 : ``a<=b``. -Elim H0; Intros; Apply Rle_trans with c; Assumption. -Replace a with (Rmin a b). -Pattern 2 b; Replace b with (Rmax a b). -Rewrite <- H2; Rewrite H3; Reflexivity. -Unfold Rmax; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Unfold Rmin; Case (total_order_Rle a b); Intro; [Reflexivity | Elim n; Assumption]. -Split with (cons r nil); Split with lf1; Assert H2 : ``c==b``. -Rewrite H1 in H0; Elim H0; Intros; Apply Rle_antisym; Assumption. -Rewrite <- H2 in H1; Rewrite <- H1; Assumption. -Intros; Clear X; Induction lf1. -Unfold adapted_couple in H; Decompose [and] H; Clear H; Simpl in H4; Discriminate. -Clear Hreclf1; Assert H1 : (sumboolT ``c<=r1`` ``r1<c``). -Case (total_order_Rle c r1); Intro; [Left; Assumption | Right; Auto with real]. -Elim H1; Intro. -Split with (cons c (cons r1 r2)); Split with (cons r3 lf1); Unfold adapted_couple in H; Decompose [and] H; Clear H; Unfold adapted_couple; Repeat Split. -Unfold ordered_Rlist; Intros; Simpl in H; Induction i. -Simpl; Assumption. -Clear Hreci; Apply (H2 (S i)); Simpl; Assumption. -Simpl; Unfold Rmin; Case (total_order_Rle c b); Intro; [Reflexivity | Elim n; Elim H0; Intros; Assumption]. -Replace (Rmax c b) with (Rmax a b). -Rewrite <- H3; Reflexivity. -Unfold Rmax; Case (total_order_Rle a b); Case (total_order_Rle c b); Intros; [Reflexivity | Elim n; Elim H0; Intros; Assumption | Elim n; Elim H0; Intros; Apply Rle_trans with c; Assumption | Elim n0; Elim H0; Intros; Apply Rle_trans with c; Assumption]. -Simpl; Simpl in H5; Apply H5. -Intros; Simpl in H; Induction i. -Unfold constant_D_eq open_interval; Intros; Simpl; Apply (H7 O). -Simpl; Apply lt_O_Sn. -Unfold open_interval; Simpl; Simpl in H6; Elim H6; Clear H6; Intros; Split; Try Assumption; Apply Rle_lt_trans with c; Try Assumption; Replace r with a. -Elim H0; Intros; Assumption. -Simpl in H4; Rewrite H4; Unfold Rmin; Case (total_order_Rle a b); Intros; [Reflexivity | Elim n; Elim H0; Intros; Apply Rle_trans with c; Assumption]. -Clear Hreci; Apply (H7 (S i)); Simpl; Assumption. -Cut (adapted_couple f r1 b (cons r1 r2) lf1). -Cut ``r1<=c<=b``. -Intros; Elim (X0 ? ? ? ? ? H3 H2); Intros l1' [lf1' H4]; Split with l1'; Split with lf1'; Assumption. -Split; [Left; Assumption | Elim H0; Intros; Assumption]. -EApply StepFun_P7; [Elim H0; Intros; Apply Rle_trans with c; [Apply H2 | Apply H3] | Apply H]. +Lemma StepFun_P45 : + forall (f:R -> R) (a b c:R), + IsStepFun f a b -> a <= c <= b -> IsStepFun f c b. +intros f; intros; assert (H0 : a <= b). +elim H; intros; apply Rle_trans with c; assumption. +elim H; clear H; intros; unfold IsStepFun in X; unfold is_subdivision in X; + elim X; clear X; intros l1 [lf1 H2]; + cut + (forall (l1 lf1:Rlist) (a b c:R) (f:R -> R), + adapted_couple f a b l1 lf1 -> + a <= c <= b -> + sigT (fun l:Rlist => sigT (fun l0:Rlist => adapted_couple f c b l l0))). +intros; unfold IsStepFun in |- *; unfold is_subdivision in |- *; eapply X; + [ apply H2 | split; assumption ]. +clear f a b c H0 H H1 H2 l1 lf1; simple induction l1. +intros; unfold adapted_couple in H; decompose [and] H; clear H; simpl in H4; + discriminate. +simple induction r0. +intros; assert (H1 : a = b). +unfold adapted_couple in H; decompose [and] H; clear H; simpl in H3; + simpl in H2; assert (H7 : a <= b). +elim H0; intros; apply Rle_trans with c; assumption. +replace a with (Rmin a b). +pattern b at 2 in |- *; replace b with (Rmax a b). +rewrite <- H2; rewrite H3; reflexivity. +unfold Rmax in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +unfold Rmin in |- *; case (Rle_dec a b); intro; + [ reflexivity | elim n; assumption ]. +split with (cons r nil); split with lf1; assert (H2 : c = b). +rewrite H1 in H0; elim H0; intros; apply Rle_antisym; assumption. +rewrite <- H2 in H1; rewrite <- H1; assumption. +intros; clear X; induction lf1 as [| r3 lf1 Hreclf1]. +unfold adapted_couple in H; decompose [and] H; clear H; simpl in H4; + discriminate. +clear Hreclf1; assert (H1 : {c <= r1} + {r1 < c}). +case (Rle_dec c r1); intro; [ left; assumption | right; auto with real ]. +elim H1; intro. +split with (cons c (cons r1 r2)); split with (cons r3 lf1); + unfold adapted_couple in H; decompose [and] H; clear H; + unfold adapted_couple in |- *; repeat split. +unfold ordered_Rlist in |- *; intros; simpl in H; induction i as [| i Hreci]. +simpl in |- *; assumption. +clear Hreci; apply (H2 (S i)); simpl in |- *; assumption. +simpl in |- *; unfold Rmin in |- *; case (Rle_dec c b); intro; + [ reflexivity | elim n; elim H0; intros; assumption ]. +replace (Rmax c b) with (Rmax a b). +rewrite <- H3; reflexivity. +unfold Rmax in |- *; case (Rle_dec a b); case (Rle_dec c b); intros; + [ reflexivity + | elim n; elim H0; intros; assumption + | elim n; elim H0; intros; apply Rle_trans with c; assumption + | elim n0; elim H0; intros; apply Rle_trans with c; assumption ]. +simpl in |- *; simpl in H5; apply H5. +intros; simpl in H; induction i as [| i Hreci]. +unfold constant_D_eq, open_interval in |- *; intros; simpl in |- *; + apply (H7 0%nat). +simpl in |- *; apply lt_O_Sn. +unfold open_interval in |- *; simpl in |- *; simpl in H6; elim H6; clear H6; + intros; split; try assumption; apply Rle_lt_trans with c; + try assumption; replace r with a. +elim H0; intros; assumption. +simpl in H4; rewrite H4; unfold Rmin in |- *; case (Rle_dec a b); intros; + [ reflexivity + | elim n; elim H0; intros; apply Rle_trans with c; assumption ]. +clear Hreci; apply (H7 (S i)); simpl in |- *; assumption. +cut (adapted_couple f r1 b (cons r1 r2) lf1). +cut (r1 <= c <= b). +intros; elim (X0 _ _ _ _ _ H3 H2); intros l1' [lf1' H4]; split with l1'; + split with lf1'; assumption. +split; [ left; assumption | elim H0; intros; assumption ]. +eapply StepFun_P7; + [ elim H0; intros; apply Rle_trans with c; [ apply H2 | apply H3 ] + | apply H ]. Qed. -Lemma StepFun_P46 : (f:R->R;a,b,c:R) (IsStepFun f a b) -> (IsStepFun f b c) -> (IsStepFun f a c). -Intros f; Intros; Case (total_order_Rle a b); Case (total_order_Rle b c); Intros. -Apply StepFun_P41 with b; Assumption. -Case (total_order_Rle a c); Intro. -Apply StepFun_P44 with b; Try Assumption. -Split; [Assumption | Auto with real]. -Apply StepFun_P6; Apply StepFun_P44 with b. -Apply StepFun_P6; Assumption. -Split; Auto with real. -Case (total_order_Rle a c); Intro. -Apply StepFun_P45 with b; Try Assumption. -Split; Auto with real. -Apply StepFun_P6; Apply StepFun_P45 with b. -Apply StepFun_P6; Assumption. -Split; [Assumption | Auto with real]. -Apply StepFun_P6; Apply StepFun_P41 with b; Auto with real Orelse Apply StepFun_P6; Assumption. -Qed. +Lemma StepFun_P46 : + forall (f:R -> R) (a b c:R), + IsStepFun f a b -> IsStepFun f b c -> IsStepFun f a c. +intros f; intros; case (Rle_dec a b); case (Rle_dec b c); intros. +apply StepFun_P41 with b; assumption. +case (Rle_dec a c); intro. +apply StepFun_P44 with b; try assumption. +split; [ assumption | auto with real ]. +apply StepFun_P6; apply StepFun_P44 with b. +apply StepFun_P6; assumption. +split; auto with real. +case (Rle_dec a c); intro. +apply StepFun_P45 with b; try assumption. +split; auto with real. +apply StepFun_P6; apply StepFun_P45 with b. +apply StepFun_P6; assumption. +split; [ assumption | auto with real ]. +apply StepFun_P6; apply StepFun_P41 with b; + auto with real || apply StepFun_P6; assumption. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Rlimit.v b/theories/Reals/Rlimit.v index 6ad02e50c..5fb50822b 100644 --- a/theories/Reals/Rlimit.v +++ b/theories/Reals/Rlimit.v @@ -13,136 +13,124 @@ (* *) (*********************************************************) -Require Rbase. -Require Rfunctions. -Require Classical_Prop. -Require Fourier. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rfunctions. +Require Import Classical_Prop. +Require Import Fourier. 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. +Lemma eps2_Rgt_R0 : forall eps:R, eps > 0 -> eps * / 2 > 0. +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. +Lemma eps2 : forall eps:R, eps * / 2 + eps * / 2 = eps. +intro esp. +assert (H := double_var esp). +unfold Rdiv in H. +symmetry in |- *; 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. +Lemma eps4 : forall eps:R, eps * / (2 + 2) + eps * / (2 + 2) = eps * / 2. +intro eps. +replace (2 + 2) with 4. +pattern eps at 3 in |- *; rewrite double_var. +rewrite (Rmult_plus_distr_r (eps / 2) (eps / 2) (/ 2)). +unfold Rdiv in |- *. +repeat rewrite Rmult_assoc. +rewrite <- Rinv_mult_distr. +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. +Lemma Rlt_eps2_eps : forall eps:R, eps > 0 -> eps * / 2 < eps. +intros. +pattern eps at 2 in |- *; rewrite <- Rmult_1_r. +repeat rewrite (Rmult_comm eps). +apply Rmult_lt_compat_r. +exact H. +apply Rmult_lt_reg_l with 2. +fourier. +rewrite Rmult_1_r; 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. +Lemma Rlt_eps4_eps : forall eps:R, eps > 0 -> eps * / (2 + 2) < eps. +intros. +replace (2 + 2) with 4. +pattern eps at 2 in |- *; rewrite <- Rmult_1_r. +repeat rewrite (Rmult_comm eps). +apply Rmult_lt_compat_r. +exact H. +apply Rmult_lt_reg_l with 4. +replace 4 with 4. +apply Rmult_lt_0_compat; fourier. +ring. +rewrite Rmult_1_r; 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. +Lemma prop_eps : forall r:R, (forall eps:R, eps > 0 -> r < eps) -> r <= 0. +intros; elim (Rtotal_order r 0); intro. +apply Rlt_le; assumption. +elim H0; intro. +apply Req_le; assumption. +clear H0; generalize (H r H1); intro; generalize (Rlt_irrefl r); intro; + elimtype False; auto. Qed. (*********) -Definition mul_factor := [l,l':R](Rinv (Rplus R1 (Rplus (Rabsolu l) - (Rabsolu l')))). +Definition mul_factor (l l':R) := / (1 + (Rabs l + Rabs 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. +Lemma mul_factor_wd : forall l l':R, 1 + (Rabs l + Rabs l') <> 0. +intros; rewrite (Rplus_comm 1 (Rabs l + Rabs l')); apply tech_Rplus. +cut (Rabs (l + l') <= Rabs l + Rabs l'). +cut (0 <= Rabs (l + l')). +exact (Rle_trans _ _ _). +exact (Rabs_pos (l + l')). +exact (Rabs_triang _ _). +exact Rlt_0_1. 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. +Lemma mul_factor_gt : forall eps l l':R, eps > 0 -> eps * mul_factor l l' > 0. +intros; unfold Rgt in |- *; rewrite <- (Rmult_0_r eps); + apply Rmult_lt_compat_l. +assumption. +unfold mul_factor in |- *; apply Rinv_0_lt_compat; + cut (1 <= 1 + (Rabs l + Rabs l')). +cut (0 < 1). +exact (Rlt_le_trans _ _ _). +exact Rlt_0_1. +replace (1 <= 1 + (Rabs l + Rabs l')) with (1 + 0 <= 1 + (Rabs l + Rabs l')). +apply Rplus_le_compat_l. +cut (Rabs (l + l') <= Rabs l + Rabs l'). +cut (0 <= Rabs (l + l')). +exact (Rle_trans _ _ _). +exact (Rabs_pos _). +exact (Rabs_triang _ _). +rewrite (proj1 (Rplus_ne 1)); 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). +Lemma mul_factor_gt_f : + forall eps l l':R, eps > 0 -> Rmin 1 (eps * mul_factor l l') > 0. +intros; apply Rmin_Rgt_r; split. +exact Rlt_0_1. +exact (mul_factor_gt eps l l' H). Qed. @@ -151,389 +139,419 @@ Qed. (*******************************) (*********) -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))) }. +Record Metric_Space : Type := + {Base : Type; + dist : Base -> Base -> R; + dist_pos : forall x y:Base, dist x y >= 0; + dist_sym : forall x y:Base, dist x y = dist y x; + dist_refl : forall x y:Base, dist x y = 0 <-> x = y; + dist_tri : forall x y z:Base, dist x y <= 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)). +Definition limit_in (X X':Metric_Space) (f:Base X -> Base X') + (D:Base X -> Prop) (x0:Base X) (l:Base X') := + forall eps:R, + eps > 0 -> + exists alp : R + | alp > 0 /\ + (forall x:Base X, D x /\ dist X x x0 < alp -> 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). +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 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). +Definition limit1_in (f:R -> R) (D:R -> Prop) (l x0:R) : Prop := + 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. +Lemma tech_limit : + forall (f:R -> R) (D:R -> Prop) (l x0:R), + D x0 -> limit1_in f D l x0 -> l = f x0. +intros f D l x0 H H0. +case (Rabs_pos (f x0 - l)); intros H1. +absurd (dist R_met (f x0) l < dist R_met (f x0) l). +apply Rlt_irrefl. +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_eq; 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. +Lemma tech_limit_contr : + forall (f:R -> R) (D:R -> Prop) (l 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. +Lemma lim_x : forall (D:R -> Prop) (x0:R), limit1_in (fun x:R => x) D x0 x0. +unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; 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). +Lemma limit_plus : + forall (f g:R -> R) (D:R -> Prop) (l l' x0:R), + limit1_in f D l x0 -> + limit1_in g D l' x0 -> limit1_in (fun x:R => f x + g x) D (l + l') x0. +intros; unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; + intros; elim (H (eps * / 2) (eps2_Rgt_R0 eps H1)); + elim (H0 (eps * / 2) (eps2_Rgt_R0 eps H1)); simpl in |- *; + clear H H0; intros; elim H; elim H0; clear H H0; intros; + split with (Rmin x1 x); split. +exact (Rmin_Rgt_r x1 x 0 (conj H H2)). +intros; elim H4; clear H4; intros; + cut (R_dist (f x2) l + R_dist (g x2) l' < eps). + cut (R_dist (f x2 + g x2) (l + l') <= 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 H4 H6)); generalize (H0 x2 (conj H4 H5)); intros; + replace eps with (eps * / 2 + eps * / 2). +exact (Rplus_lt_compat _ _ _ _ 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. +Lemma limit_Ropp : + forall (f:R -> R) (D:R -> Prop) (l x0:R), + limit1_in f D l x0 -> limit1_in (fun x:R => - f x) D (- l) x0. +unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; 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 in |- *; unfold Rminus in |- *; + rewrite (Ropp_involutive l); rewrite (Rplus_comm (- f x1) l); + fold (l - f x1) in |- *; fold (R_dist l (f x1)) in |- *; + 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). +Lemma limit_minus : + forall (f g:R -> R) (D:R -> Prop) (l l' x0:R), + limit1_in f D l x0 -> + limit1_in g D l' x0 -> limit1_in (fun x:R => f x - g x) D (l - l') x0. +intros; unfold Rminus in |- *; generalize (limit_Ropp g D l' x0 H0); intro; + exact (limit_plus f (fun x:R => - g x) D l (- 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. +Lemma limit_free : + forall (f:R -> R) (D:R -> Prop) (x x0:R), + limit1_in (fun h:R => f x) D (f x) x0. +unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; intros; + split with eps; split; auto; intros; elim (R_dist_refl (f x) (f x)); + intros a b; rewrite (b (refl_equal (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. +Lemma limit_mul : + forall (f g:R -> R) (D:R -> Prop) (l l' x0:R), + limit1_in f D l x0 -> + limit1_in g D l' x0 -> limit1_in (fun x:R => f x * g x) D (l * l') x0. +intros; unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; + intros; + elim (H (Rmin 1 (eps * mul_factor l l')) (mul_factor_gt_f eps l l' H1)); + elim (H0 (eps * mul_factor l l') (mul_factor_gt eps l l' H1)); + clear H H0; simpl in |- *; intros; elim H; elim H0; + clear H H0; intros; split with (Rmin x1 x); split. +exact (Rmin_Rgt_r x1 x 0 (conj H H2)). +intros; elim H4; clear H4; intros; unfold R_dist in |- *; + replace (f x2 * g x2 - l * l') with (f x2 * (g x2 - l') + l' * (f x2 - l)). +cut (Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l)) < eps). +cut + (Rabs (f x2 * (g x2 - l') + l' * (f x2 - l)) <= + Rabs (f x2 * (g x2 - l')) + Rabs (l' * (f x2 - l))). +exact (Rle_lt_trans _ _ _). +exact (Rabs_triang _ _). +rewrite (Rabs_mult (f x2) (g x2 - l')); rewrite (Rabs_mult l' (f x2 - l)); + cut + ((1 + Rabs l) * (eps * mul_factor l l') + Rabs l' * (eps * mul_factor l l') <= + eps). +cut + (Rabs (f x2) * Rabs (g x2 - l') + Rabs l' * Rabs (f x2 - l) < + (1 + Rabs l) * (eps * mul_factor l l') + Rabs l' * (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 H4 H5)); intro; generalize (Rmin_Rgt_l _ _ _ H7); + intro; elim H8; intros; clear H0 H8; apply Rplus_lt_le_compat. +apply Rmult_ge_0_gt_0_lt_compat. +apply Rle_ge. +exact (Rabs_pos (g x2 - l')). +rewrite (Rplus_comm 1 (Rabs l)); unfold Rgt in |- *; apply Rle_lt_0_plus_1; + exact (Rabs_pos l). +unfold R_dist in H9; + apply (Rplus_lt_reg_r (- Rabs l) (Rabs (f x2)) (1 + Rabs l)). +rewrite <- (Rplus_assoc (- Rabs l) 1 (Rabs l)); + rewrite (Rplus_comm (- Rabs l) 1); + rewrite (Rplus_assoc 1 (- Rabs l) (Rabs l)); rewrite (Rplus_opp_l (Rabs l)); + rewrite (proj1 (Rplus_ne 1)); rewrite (Rplus_comm (- Rabs l) (Rabs (f x2))); + generalize H9; cut (Rabs (f x2) - Rabs l <= Rabs (f x2 - l)). +exact (Rle_lt_trans _ _ _). +exact (Rabs_triang_inv _ _). +generalize (H3 x2 (conj H4 H6)); trivial. +apply Rmult_le_compat_l. +exact (Rabs_pos l'). +unfold Rle in |- *; left; assumption. +rewrite (Rmult_comm (1 + Rabs l) (eps * mul_factor l l')); + rewrite (Rmult_comm (Rabs l') (eps * mul_factor l l')); + rewrite <- + (Rmult_plus_distr_l (eps * mul_factor l l') (1 + Rabs l) (Rabs l')) + ; rewrite (Rmult_assoc eps (mul_factor l l') (1 + Rabs l + Rabs l')); + rewrite (Rplus_assoc 1 (Rabs l) (Rabs l')); unfold mul_factor in |- *; + rewrite (Rinv_l (1 + (Rabs l + Rabs l')) (mul_factor_wd l l')); + rewrite (proj1 (Rmult_ne eps)); apply Req_le; 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)). +Definition adhDa (D:R -> Prop) (a:R) : Prop := + forall alp:R, alp > 0 -> exists x : R | D x /\ 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. +Lemma single_limit : + forall (f:R -> R) (D:R -> Prop) (l l' x0:R), + adhDa D x0 -> limit1_in f D l x0 -> limit1_in f D l' x0 -> l = l'. +unfold limit1_in in |- *; unfold limit_in in |- *; intros. +cut (forall eps:R, eps > 0 -> dist R_met l l' < 2 * eps). +clear H0 H1; unfold dist in |- *; unfold R_met in |- *; unfold R_dist in |- *; + unfold Rabs in |- *; case (Rcase_abs (l - l')); intros. +cut (forall eps:R, eps > 0 -> - (l - l') < eps). +intro; generalize (prop_eps (- (l - l')) H1); intro; + generalize (Ropp_gt_lt_0_contravar (l - l') r); intro; + unfold Rgt in H3; generalize (Rgt_not_le (- (l - l')) 0 H3); + intro; elimtype False; auto. +intros; cut (eps * / 2 > 0). +intro; generalize (H0 (eps * / 2) H2); rewrite (Rmult_comm eps (/ 2)); + rewrite <- (Rmult_assoc 2 (/ 2) eps); rewrite (Rinv_r 2). +elim (Rmult_ne eps); intros a b; rewrite b; clear a b; trivial. +apply (Rlt_dichotomy_converse 2 0); right; generalize Rlt_0_1; intro; + unfold Rgt in |- *; generalize (Rplus_lt_compat_l 1 0 1 H3); + intro; elim (Rplus_ne 1); intros a b; rewrite a in H4; + clear a b; apply (Rlt_trans 0 1 2 H3 H4). +unfold Rgt in |- *; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2)); + rewrite <- (Rmult_0_r (/ 2)); apply (Rmult_lt_compat_l (/ 2) 0 eps); + auto. +apply (Rinv_0_lt_compat 2); cut (1 < 2). +intro; apply (Rlt_trans 0 1 2 Rlt_0_1 H2). +generalize (Rplus_lt_compat_l 1 0 1 Rlt_0_1); elim (Rplus_ne 1); 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. +cut (forall eps:R, eps > 0 -> l - l' < eps). +intro; generalize (prop_eps (l - l') H1); intro; elim (Rle_le_eq (l - l') 0); + intros a b; clear b; apply (Rminus_diag_uniq l l'); + apply a; split. +assumption. +apply (Rge_le (l - l') 0 r). +intros; cut (eps * / 2 > 0). +intro; generalize (H0 (eps * / 2) H2); rewrite (Rmult_comm eps (/ 2)); + rewrite <- (Rmult_assoc 2 (/ 2) eps); rewrite (Rinv_r 2). +elim (Rmult_ne eps); intros a b; rewrite b; clear a b; trivial. +apply (Rlt_dichotomy_converse 2 0); right; generalize Rlt_0_1; intro; + unfold Rgt in |- *; generalize (Rplus_lt_compat_l 1 0 1 H3); + intro; elim (Rplus_ne 1); intros a b; rewrite a in H4; + clear a b; apply (Rlt_trans 0 1 2 H3 H4). +unfold Rgt in |- *; unfold Rgt in H1; rewrite (Rmult_comm eps (/ 2)); + rewrite <- (Rmult_0_r (/ 2)); apply (Rmult_lt_compat_l (/ 2) 0 eps); + auto. +apply (Rinv_0_lt_compat 2); cut (1 < 2). +intro; apply (Rlt_trans 0 1 2 Rlt_0_1 H2). +generalize (Rplus_lt_compat_l 1 0 1 Rlt_0_1); elim (Rplus_ne 1); 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). +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 in |- *; simpl in H1, H4; generalize (Rmin_Rgt x x1 0); + intro; elim H5; intros; clear H5; elim (H (Rmin x x1) (H7 (conj 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 H8 H6)); + generalize (H4 x2 (conj H8 H)); clear H8 H H6 H1 H4 H0 H3; + intros; + generalize + (Rplus_lt_compat (R_dist (f x2) l) eps (R_dist (f x2) l') eps H H0); + unfold R_dist in |- *; intros; rewrite (Rabs_minus_sym (f x2) l) in H1; + rewrite (Rmult_comm 2 eps); rewrite (Rmult_plus_distr_l eps 1 1); + elim (Rmult_ne eps); intros a b; rewrite a; clear a b; + generalize (R_dist_tri l l' (f x2)); unfold R_dist in |- *; + intros; + apply + (Rle_lt_trans (Rabs (l - l')) (Rabs (l - f x2) + Rabs (f x2 - l')) + (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. +Lemma limit_comp : + forall (f g:R -> R) (Df Dg:R -> Prop) (l l' x0:R), + limit1_in f Df l x0 -> + limit1_in g Dg l' l -> limit1_in (fun x:R => g (f x)) (Dgf Df Dg f) l' x0. +unfold limit1_in, limit_in, Dgf in |- *; simpl in |- *. +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. +Lemma limit_inv : + forall (f:R -> R) (D:R -> Prop) (l x0:R), + limit1_in f D l x0 -> l <> 0 -> limit1_in (fun x:R => / f x) D (/ l) x0. +unfold limit1_in in |- *; unfold limit_in in |- *; simpl in |- *; + unfold R_dist in |- *; intros; elim (H (Rabs 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 in |- *; case (Rle_dec 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 /\ Rabs (x - x0) < delta1). +cut (D x /\ Rabs (x - x0) < delta2). +intros; generalize (H5 H11); clear H5; intro H5; generalize (H7 H12); + clear H7; intro H7; generalize (Rabs_triang_inv l (f x)); + intro; rewrite Rabs_minus_sym in H7; + generalize + (Rle_lt_trans (Rabs l - Rabs (f x)) (Rabs (l - f x)) (Rabs l / 2) H13 H7); + intro; + generalize + (Rplus_lt_compat_l (Rabs (f x) - Rabs l / 2) (Rabs l - Rabs (f x)) + (Rabs l / 2) H14); + replace (Rabs (f x) - Rabs l / 2 + (Rabs l - Rabs (f x))) with (Rabs l / 2). +unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_r; intro; cut (f x <> 0). +intro; replace (/ f x + - / l) with ((l - f x) * / (l * f x)). +rewrite Rabs_mult; rewrite Rabs_Rinv. +cut (/ Rabs (l * f x) < 2 / Rsqr l). +intro; rewrite Rabs_minus_sym in H5; cut (0 <= / Rabs (l * f x)). +intro; + generalize + (Rmult_le_0_lt_compat (Rabs (l - f x)) (eps * (Rsqr l / 2)) + (/ Rabs (l * f x)) (2 / Rsqr l) (Rabs_pos (l - f x)) H18 H5 H17); + replace (eps * (Rsqr l / 2) * (2 / Rsqr l)) with eps. +intro; assumption. +unfold Rdiv in |- *; unfold Rsqr in |- *; rewrite Rinv_mult_distr. +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm l). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite (Rmult_comm l). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; reflexivity. +discrR. +exact H0. +exact H0. +exact H0. +exact H0. +left; apply Rinv_0_lt_compat; apply Rabs_pos_lt; apply prod_neq_R0; + assumption. +rewrite Rmult_comm; rewrite Rabs_mult; rewrite Rinv_mult_distr. +rewrite (Rsqr_abs l); unfold Rsqr in |- *; unfold Rdiv in |- *; + rewrite Rinv_mult_distr. +repeat rewrite <- Rmult_assoc; apply Rmult_lt_compat_r. +apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. +apply Rmult_lt_reg_l with (Rabs (f x) * Rabs l * / 2). +repeat apply Rmult_lt_0_compat. +apply Rabs_pos_lt; assumption. +apply Rabs_pos_lt; assumption. +apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); + [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR in |- *; + intro H18; assumption + | discriminate ]. +replace (Rabs (f x) * Rabs l * / 2 * / Rabs (f x)) with (Rabs l / 2). +replace (Rabs (f x) * Rabs l * / 2 * (2 * / Rabs l)) with (Rabs (f x)). +assumption. +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (Rabs l)). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; reflexivity. +discrR. +apply Rabs_no_R0. +assumption. +unfold Rdiv in |- *. +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (Rabs (f x))). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +reflexivity. +apply Rabs_no_R0; assumption. +apply Rabs_no_R0; assumption. +apply Rabs_no_R0; assumption. +apply Rabs_no_R0; assumption. +apply Rabs_no_R0; assumption. +apply prod_neq_R0; assumption. +rewrite (Rinv_mult_distr _ _ H0 H16). +unfold Rminus in |- *; rewrite Rmult_plus_distr_r. +rewrite <- Rmult_assoc. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_l. +rewrite Ropp_mult_distr_l_reverse. +rewrite (Rmult_comm (f x)). +rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +reflexivity. +assumption. +assumption. +red in |- *; intro; rewrite H16 in H15; rewrite Rabs_R0 in H15; + cut (0 < Rabs l / 2). +intro; elim (Rlt_irrefl 0 (Rlt_trans 0 (Rabs l / 2) 0 H17 H15)). +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply Rabs_pos_lt; assumption. +apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); + [ intro H17; generalize (lt_INR_0 2 (neq_O_lt 2 H17)); unfold INR in |- *; + intro; assumption + | discriminate ]. +pattern (Rabs l) at 3 in |- *; 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)) in |- *; unfold Rdiv in |- *; + repeat rewrite Rmult_assoc; repeat apply Rmult_lt_0_compat. +assumption. +apply Rsqr_pos_lt; assumption. +apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); + [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR in |- *; + intro; assumption + | discriminate ]. +change (0 < Rabs l / 2) in |- *; unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply Rabs_pos_lt; assumption + | apply Rinv_0_lt_compat; cut (0%nat <> 2%nat); + [ intro H3; generalize (lt_INR_0 2 (neq_O_lt 2 H3)); unfold INR in |- *; + intro; assumption + | discriminate ] ]. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Rpower.v b/theories/Reals/Rpower.v index c4cb1a8eb..7c31bbe61 100644 --- a/theories/Reals/Rpower.v +++ b/theories/Reals/Rpower.v @@ -13,548 +13,649 @@ (* Definitions of log and Rpower : R->R->R; main properties *) (************************************************************) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo. -Require Ranalysis1. -Require Exp_prop. -Require Rsqrt_def. -Require R_sqrt. -Require MVT. -Require Ranalysis4. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import Ranalysis1. +Require Import Exp_prop. +Require Import Rsqrt_def. +Require Import R_sqrt. +Require Import MVT. +Require Import Ranalysis4. Open Local Scope R_scope. -Lemma P_Rmin: (P : R -> Prop) (x, y : R) (P x) -> (P y) -> (P (Rmin x y)). -Intros P x y H1 H2; Unfold Rmin; Case (total_order_Rle x y); Intro; Assumption. +Lemma P_Rmin : forall (P:R -> Prop) (x y:R), P x -> P y -> P (Rmin x y). +intros P x y H1 H2; unfold Rmin in |- *; case (Rle_dec x y); intro; + assumption. Qed. -Lemma exp_le_3 : ``(exp 1)<=3``. -Assert exp_1 : ``(exp 1)<>0``. -Assert H0 := (exp_pos R1); Red; Intro; Rewrite H in H0; Elim (Rlt_antirefl ? H0). -Apply Rle_monotony_contra with ``/(exp 1)``. -Apply Rlt_Rinv; Apply exp_pos. -Rewrite <- Rinv_l_sym. -Apply Rle_monotony_contra with ``/3``. -Apply Rlt_Rinv; Sup0. -Rewrite Rmult_1r; Rewrite <- (Rmult_sym ``3``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Replace ``/(exp 1)`` with ``(exp (-1))``. -Unfold exp; Case (exist_exp ``-1``); Intros; Simpl; Unfold exp_in in e; Assert H := (alternated_series_ineq [i:nat]``/(INR (fact i))`` x (S O)). -Cut ``(sum_f_R0 (tg_alt [([i:nat]``/(INR (fact i))``)]) (S (mult (S (S O)) (S O)))) <= x <= (sum_f_R0 (tg_alt [([i:nat]``/(INR (fact i))``)]) (mult (S (S O)) (S O)))``. -Intro; Elim H0; Clear H0; Intros H0 _; Simpl in H0; Unfold tg_alt in H0; Simpl in H0. -Replace ``/3`` with ``1*/1+ -1*1*/1+ -1*( -1*1)*/2+ -1*( -1*( -1*1))*/(2+1+1+1+1)``. -Apply H0. -Repeat Rewrite Rinv_R1; Repeat Rewrite Rmult_1r; Rewrite Ropp_mul1; Rewrite Rmult_1l; Rewrite Ropp_Ropp; Rewrite Rplus_Ropp_r; Rewrite Rmult_1r; Rewrite Rplus_Ol; Rewrite Rmult_1l; Apply r_Rmult_mult with ``6``. -Rewrite Rmult_Rplus_distr; Replace ``2+1+1+1+1`` with ``6``. -Rewrite <- (Rmult_sym ``/6``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Replace ``6`` with ``2*3``. -Do 2 Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Rewrite (Rmult_sym ``3``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Ring. -DiscrR. -DiscrR. -Ring. -DiscrR. -Ring. -DiscrR. -Apply H. -Unfold Un_decreasing; Intros; Apply Rle_monotony_contra with ``(INR (fact n))``. -Apply INR_fact_lt_0. -Apply Rle_monotony_contra with ``(INR (fact (S n)))``. -Apply INR_fact_lt_0. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Apply le_INR; Apply fact_growing; Apply le_n_Sn. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Assert H0 := (cv_speed_pow_fact R1); Unfold Un_cv; Unfold Un_cv in H0; Intros; Elim (H0 ? H1); Intros; Exists x0; Intros; Unfold R_dist in H2; Unfold R_dist; Replace ``/(INR (fact n))`` with ``(pow 1 n)/(INR (fact n))``. -Apply (H2 ? H3). -Unfold Rdiv; Rewrite pow1; Rewrite Rmult_1l; Reflexivity. -Unfold infinit_sum in e; Unfold Un_cv tg_alt; Intros; Elim (e ? H0); Intros; Exists x0; Intros; Replace (sum_f_R0 ([i:nat]``(pow ( -1) i)*/(INR (fact i))``) n) with (sum_f_R0 ([i:nat]``/(INR (fact i))*(pow ( -1) i)``) n). -Apply (H1 ? H2). -Apply sum_eq; Intros; Apply Rmult_sym. -Apply r_Rmult_mult with ``(exp 1)``. -Rewrite <- exp_plus; Rewrite Rplus_Ropp_r; Rewrite exp_0; Rewrite <- Rinv_r_sym. -Reflexivity. -Assumption. -Assumption. -DiscrR. -Assumption. +Lemma exp_le_3 : exp 1 <= 3. +assert (exp_1 : exp 1 <> 0). +assert (H0 := exp_pos 1); red in |- *; intro; rewrite H in H0; + elim (Rlt_irrefl _ H0). +apply Rmult_le_reg_l with (/ exp 1). +apply Rinv_0_lt_compat; apply exp_pos. +rewrite <- Rinv_l_sym. +apply Rmult_le_reg_l with (/ 3). +apply Rinv_0_lt_compat; prove_sup0. +rewrite Rmult_1_r; rewrite <- (Rmult_comm 3); rewrite <- Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; replace (/ exp 1) with (exp (-1)). +unfold exp in |- *; case (exist_exp (-1)); intros; simpl in |- *; + unfold exp_in in e; + assert (H := alternated_series_ineq (fun i:nat => / INR (fact i)) x 1). +cut + (sum_f_R0 (tg_alt (fun i:nat => / INR (fact i))) (S (2 * 1)) <= x <= + sum_f_R0 (tg_alt (fun i:nat => / INR (fact i))) (2 * 1)). +intro; elim H0; clear H0; intros H0 _; simpl in H0; unfold tg_alt in H0; + simpl in H0. +replace (/ 3) with + (1 * / 1 + -1 * 1 * / 1 + -1 * (-1 * 1) * / 2 + + -1 * (-1 * (-1 * 1)) * / (2 + 1 + 1 + 1 + 1)). +apply H0. +repeat rewrite Rinv_1; repeat rewrite Rmult_1_r; + rewrite Ropp_mult_distr_l_reverse; rewrite Rmult_1_l; + rewrite Ropp_involutive; rewrite Rplus_opp_r; rewrite Rmult_1_r; + rewrite Rplus_0_l; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 6. +rewrite Rmult_plus_distr_l; replace (2 + 1 + 1 + 1 + 1) with 6. +rewrite <- (Rmult_comm (/ 6)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; replace 6 with 6. +do 2 rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; rewrite (Rmult_comm 3); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +ring. +discrR. +discrR. +ring. +discrR. +ring. +discrR. +apply H. +unfold Un_decreasing in |- *; intros; + apply Rmult_le_reg_l with (INR (fact n)). +apply INR_fact_lt_0. +apply Rmult_le_reg_l with (INR (fact (S n))). +apply INR_fact_lt_0. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; apply le_INR; apply fact_le; apply le_n_Sn. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +assert (H0 := cv_speed_pow_fact 1); unfold Un_cv in |- *; unfold Un_cv in H0; + intros; elim (H0 _ H1); intros; exists x0; intros; + unfold R_dist in H2; unfold R_dist in |- *; + replace (/ INR (fact n)) with (1 ^ n / INR (fact n)). +apply (H2 _ H3). +unfold Rdiv in |- *; rewrite pow1; rewrite Rmult_1_l; reflexivity. +unfold infinit_sum in e; unfold Un_cv, tg_alt in |- *; intros; elim (e _ H0); + intros; exists x0; intros; + replace (sum_f_R0 (fun i:nat => (-1) ^ i * / INR (fact i)) n) with + (sum_f_R0 (fun i:nat => / INR (fact i) * (-1) ^ i) n). +apply (H1 _ H2). +apply sum_eq; intros; apply Rmult_comm. +apply Rmult_eq_reg_l with (exp 1). +rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0; + rewrite <- Rinv_r_sym. +reflexivity. +assumption. +assumption. +discrR. +assumption. Qed. (******************************************************************) (* Properties of Exp *) (******************************************************************) -Theorem exp_increasing: (x, y : R) ``x<y`` -> ``(exp x)<(exp y)``. -Intros x y H. -Assert H0 : (derivable exp). -Apply derivable_exp. -Assert H1 := (positive_derivative ? H0). -Unfold strict_increasing in H1. -Apply H1. -Intro. -Replace (derive_pt exp x0 (H0 x0)) with (exp x0). -Apply exp_pos. -Symmetry; Apply derive_pt_eq_0. -Apply (derivable_pt_lim_exp x0). -Apply H. +Theorem exp_increasing : forall x y:R, x < y -> exp x < exp y. +intros x y H. +assert (H0 : derivable exp). +apply derivable_exp. +assert (H1 := positive_derivative _ H0). +unfold strict_increasing in H1. +apply H1. +intro. +replace (derive_pt exp x0 (H0 x0)) with (exp x0). +apply exp_pos. +symmetry in |- *; apply derive_pt_eq_0. +apply (derivable_pt_lim_exp x0). +apply H. Qed. -Theorem exp_lt_inv: (x, y : R) ``(exp x)<(exp y)`` -> ``x<y``. -Intros x y H; Case (total_order x y); [Intros H1 | Intros [H1|H1]]. -Assumption. -Rewrite H1 in H; Elim (Rlt_antirefl ? H). -Assert H2 := (exp_increasing ? ? H1). -Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H H2)). +Theorem exp_lt_inv : forall x y:R, exp x < exp y -> x < y. +intros x y H; case (Rtotal_order x y); [ intros H1 | intros [H1| H1] ]. +assumption. +rewrite H1 in H; elim (Rlt_irrefl _ H). +assert (H2 := exp_increasing _ _ H1). +elim (Rlt_irrefl _ (Rlt_trans _ _ _ H H2)). Qed. -Lemma exp_ineq1 : (x:R) ``0<x`` -> ``1+x < (exp x)``. -Intros; Apply Rlt_anti_compatibility with ``-(exp 0)``; Rewrite <- (Rplus_sym (exp x)); Assert H0 := (MVT_cor1 exp R0 x derivable_exp H); Elim H0; Intros; Elim H1; Intros; Unfold Rminus in H2; Rewrite H2; Rewrite Ropp_O; Rewrite Rplus_Or; Replace (derive_pt exp x0 (derivable_exp x0)) with (exp x0). -Rewrite exp_0; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Pattern 1 x; Rewrite <- Rmult_1r; Rewrite (Rmult_sym (exp x0)); Apply Rlt_monotony. -Apply H. -Rewrite <- exp_0; Apply exp_increasing; Elim H3; Intros; Assumption. -Symmetry; Apply derive_pt_eq_0; Apply derivable_pt_lim_exp. +Lemma exp_ineq1 : forall x:R, 0 < x -> 1 + x < exp x. +intros; apply Rplus_lt_reg_r with (- exp 0); rewrite <- (Rplus_comm (exp x)); + assert (H0 := MVT_cor1 exp 0 x derivable_exp H); elim H0; + intros; elim H1; intros; unfold Rminus in H2; rewrite H2; + rewrite Ropp_0; rewrite Rplus_0_r; + replace (derive_pt exp x0 (derivable_exp x0)) with (exp x0). +rewrite exp_0; rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; + pattern x at 1 in |- *; rewrite <- Rmult_1_r; rewrite (Rmult_comm (exp x0)); + apply Rmult_lt_compat_l. +apply H. +rewrite <- exp_0; apply exp_increasing; elim H3; intros; assumption. +symmetry in |- *; apply derive_pt_eq_0; apply derivable_pt_lim_exp. Qed. -Lemma ln_exists1 : (y:R) ``0<y``->``1<=y``->(sigTT R [z:R]``y==(exp z)``). -Intros; Pose f := [x:R]``(exp x)-y``; Cut ``(f 0)<=0``. -Intro; Cut (continuity f). -Intro; Cut ``0<=(f y)``. -Intro; Cut ``(f 0)*(f y)<=0``. -Intro; Assert X := (IVT_cor f R0 y H2 (Rlt_le ? ? H) H4); Elim X; Intros t H5; Apply existTT with t; Elim H5; Intros; Unfold f in H7; Apply Rminus_eq_right; Exact H7. -Pattern 2 R0; Rewrite <- (Rmult_Or (f y)); Rewrite (Rmult_sym (f R0)); Apply Rle_monotony; Assumption. -Unfold f; Apply Rle_anti_compatibility with y; Left; Apply Rlt_trans with ``1+y``. -Rewrite <- (Rplus_sym y); Apply Rlt_compatibility; Apply Rlt_R0_R1. -Replace ``y+((exp y)-y)`` with (exp y); [Apply (exp_ineq1 y H) | Ring]. -Unfold f; Change (continuity (minus_fct exp (fct_cte y))); Apply continuity_minus; [Apply derivable_continuous; Apply derivable_exp | Apply derivable_continuous; Apply derivable_const]. -Unfold f; Rewrite exp_0; Apply Rle_anti_compatibility with y; Rewrite Rplus_Or; Replace ``y+(1-y)`` with R1; [Apply H0 | Ring]. +Lemma ln_exists1 : forall y:R, 0 < y -> 1 <= y -> sigT (fun z:R => y = exp z). +intros; pose (f := fun x:R => exp x - y); cut (f 0 <= 0). +intro; cut (continuity f). +intro; cut (0 <= f y). +intro; cut (f 0 * f y <= 0). +intro; assert (X := IVT_cor f 0 y H2 (Rlt_le _ _ H) H4); elim X; intros t H5; + apply existT with t; elim H5; intros; unfold f in H7; + apply Rminus_diag_uniq_sym; exact H7. +pattern 0 at 2 in |- *; rewrite <- (Rmult_0_r (f y)); + rewrite (Rmult_comm (f 0)); apply Rmult_le_compat_l; + assumption. +unfold f in |- *; apply Rplus_le_reg_l with y; left; + apply Rlt_trans with (1 + y). +rewrite <- (Rplus_comm y); apply Rplus_lt_compat_l; apply Rlt_0_1. +replace (y + (exp y - y)) with (exp y); [ apply (exp_ineq1 y H) | ring ]. +unfold f in |- *; change (continuity (exp - fct_cte y)) in |- *; + apply continuity_minus; + [ apply derivable_continuous; apply derivable_exp + | apply derivable_continuous; apply derivable_const ]. +unfold f in |- *; rewrite exp_0; apply Rplus_le_reg_l with y; + rewrite Rplus_0_r; replace (y + (1 - y)) with 1; [ apply H0 | ring ]. Qed. (**********) -Lemma ln_exists : (y:R) ``0<y`` -> (sigTT R [z:R]``y==(exp z)``). -Intros; Case (total_order_Rle R1 y); Intro. -Apply (ln_exists1 ? H r). -Assert H0 : ``1<=/y``. -Apply Rle_monotony_contra with y. -Apply H. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Left; Apply (not_Rle ? ? n). -Red; Intro; Rewrite H0 in H; Elim (Rlt_antirefl ? H). -Assert H1 : ``0</y``. -Apply Rlt_Rinv; Apply H. -Assert H2 := (ln_exists1 ? H1 H0); Elim H2; Intros; Apply existTT with ``-x``; Apply r_Rmult_mult with ``(exp x)/y``. -Unfold Rdiv; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite <- (Rmult_sym ``/y``); Rewrite Rmult_assoc; Rewrite <- exp_plus; Rewrite Rplus_Ropp_r; Rewrite exp_0; Rewrite Rmult_1r; Symmetry; Apply p. -Red; Intro; Rewrite H3 in H; Elim (Rlt_antirefl ? H). -Unfold Rdiv; Apply prod_neq_R0. -Assert H3 := (exp_pos x); Red; Intro; Rewrite H4 in H3; Elim (Rlt_antirefl ? H3). -Apply Rinv_neq_R0; Red; Intro; Rewrite H3 in H; Elim (Rlt_antirefl ? H). +Lemma ln_exists : forall y:R, 0 < y -> sigT (fun z:R => y = exp z). +intros; case (Rle_dec 1 y); intro. +apply (ln_exists1 _ H r). +assert (H0 : 1 <= / y). +apply Rmult_le_reg_l with y. +apply H. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; left; apply (Rnot_le_lt _ _ n). +red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). +assert (H1 : 0 < / y). +apply Rinv_0_lt_compat; apply H. +assert (H2 := ln_exists1 _ H1 H0); elim H2; intros; apply existT with (- x); + apply Rmult_eq_reg_l with (exp x / y). +unfold Rdiv in |- *; rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite <- (Rmult_comm (/ y)); rewrite Rmult_assoc; + rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0; + rewrite Rmult_1_r; symmetry in |- *; apply p. +red in |- *; intro; rewrite H3 in H; elim (Rlt_irrefl _ H). +unfold Rdiv in |- *; apply prod_neq_R0. +assert (H3 := exp_pos x); red in |- *; intro; rewrite H4 in H3; + elim (Rlt_irrefl _ H3). +apply Rinv_neq_0_compat; red in |- *; intro; rewrite H3 in H; + elim (Rlt_irrefl _ H). Qed. (* Definition of log R+* -> R *) -Definition Rln [y:posreal] : R := Cases (ln_exists (pos y) (RIneq.cond_pos y)) of (existTT a b) => a end. +Definition Rln (y:posreal) : R := + match ln_exists (pos y) (cond_pos y) with + | existT a b => a + end. (* Extension on R *) -Definition ln : R->R := [x:R](Cases (total_order_Rlt R0 x) of - (leftT a) => (Rln (mkposreal x a)) - | (rightT a) => R0 end). +Definition ln (x:R) : R := + match Rlt_dec 0 x with + | left a => Rln (mkposreal x a) + | right a => 0 + end. -Lemma exp_ln : (x : R) ``0<x`` -> (exp (ln x)) == x. -Intros; Unfold ln; Case (total_order_Rlt R0 x); Intro. -Unfold Rln; Case (ln_exists (mkposreal x r) (RIneq.cond_pos (mkposreal x r))); Intros. -Simpl in e; Symmetry; Apply e. -Elim n; Apply H. +Lemma exp_ln : forall x:R, 0 < x -> exp (ln x) = x. +intros; unfold ln in |- *; case (Rlt_dec 0 x); intro. +unfold Rln in |- *; + case (ln_exists (mkposreal x r) (cond_pos (mkposreal x r))); + intros. +simpl in e; symmetry in |- *; apply e. +elim n; apply H. Qed. -Theorem exp_inv: (x, y : R) (exp x) == (exp y) -> x == y. -Intros x y H; Case (total_order x y); [Intros H1 | Intros [H1|H1]]; Auto; Assert H2 := (exp_increasing ? ? H1); Rewrite H in H2; Elim (Rlt_antirefl ? H2). +Theorem exp_inv : forall x y:R, exp x = exp y -> x = y. +intros x y H; case (Rtotal_order x y); [ intros H1 | intros [H1| H1] ]; auto; + assert (H2 := exp_increasing _ _ H1); rewrite H in H2; + elim (Rlt_irrefl _ H2). Qed. -Theorem exp_Ropp: (x : R) ``(exp (-x)) == /(exp x)``. -Intros x; Assert H : ``(exp x)<>0``. -Assert H := (exp_pos x); Red; Intro; Rewrite H0 in H; Elim (Rlt_antirefl ? H). -Apply r_Rmult_mult with r := (exp x). -Rewrite <- exp_plus; Rewrite Rplus_Ropp_r; Rewrite exp_0. -Apply Rinv_r_sym. -Apply H. -Apply H. +Theorem exp_Ropp : forall x:R, exp (- x) = / exp x. +intros x; assert (H : exp x <> 0). +assert (H := exp_pos x); red in |- *; intro; rewrite H0 in H; + elim (Rlt_irrefl _ H). +apply Rmult_eq_reg_l with (r := exp x). +rewrite <- exp_plus; rewrite Rplus_opp_r; rewrite exp_0. +apply Rinv_r_sym. +apply H. +apply H. Qed. (******************************************************************) (* Properties of Ln *) (******************************************************************) -Theorem ln_increasing: - (x, y : R) ``0<x`` -> ``x<y`` -> ``(ln x) < (ln y)``. -Intros x y H H0; Apply exp_lt_inv. -Repeat Rewrite exp_ln. -Apply H0. -Apply Rlt_trans with x; Assumption. -Apply H. +Theorem ln_increasing : forall x y:R, 0 < x -> x < y -> ln x < ln y. +intros x y H H0; apply exp_lt_inv. +repeat rewrite exp_ln. +apply H0. +apply Rlt_trans with x; assumption. +apply H. Qed. -Theorem ln_exp: (x : R) (ln (exp x)) == x. -Intros x; Apply exp_inv. -Apply exp_ln. -Apply exp_pos. +Theorem ln_exp : forall x:R, ln (exp x) = x. +intros x; apply exp_inv. +apply exp_ln. +apply exp_pos. Qed. -Theorem ln_1: ``(ln 1) == 0``. -Rewrite <- exp_0; Rewrite ln_exp; Reflexivity. +Theorem ln_1 : ln 1 = 0. +rewrite <- exp_0; rewrite ln_exp; reflexivity. Qed. -Theorem ln_lt_inv: - (x, y : R) ``0<x`` -> ``0<y`` -> ``(ln x)<(ln y)`` -> ``x<y``. -Intros x y H H0 H1; Rewrite <- (exp_ln x); Try Rewrite <- (exp_ln y). -Apply exp_increasing; Apply H1. -Assumption. -Assumption. +Theorem ln_lt_inv : forall x y:R, 0 < x -> 0 < y -> ln x < ln y -> x < y. +intros x y H H0 H1; rewrite <- (exp_ln x); try rewrite <- (exp_ln y). +apply exp_increasing; apply H1. +assumption. +assumption. Qed. -Theorem ln_inv: (x, y : R) ``0<x`` -> ``0<y`` -> (ln x) == (ln y) -> x == y. -Intros x y H H0 H'0; Case (total_order x y); [Intros H1 | Intros [H1|H1]]; Auto. -Assert H2 := (ln_increasing ? ? H H1); Rewrite H'0 in H2; Elim (Rlt_antirefl ? H2). -Assert H2 := (ln_increasing ? ? H0 H1); Rewrite H'0 in H2; Elim (Rlt_antirefl ? H2). +Theorem ln_inv : forall x y:R, 0 < x -> 0 < y -> ln x = ln y -> x = y. +intros x y H H0 H'0; case (Rtotal_order x y); [ intros H1 | intros [H1| H1] ]; + auto. +assert (H2 := ln_increasing _ _ H H1); rewrite H'0 in H2; + elim (Rlt_irrefl _ H2). +assert (H2 := ln_increasing _ _ H0 H1); rewrite H'0 in H2; + elim (Rlt_irrefl _ H2). Qed. -Theorem ln_mult: (x, y : R) ``0<x`` -> ``0<y`` -> ``(ln (x*y)) == (ln x)+(ln y)``. -Intros x y H H0; Apply exp_inv. -Rewrite exp_plus. -Repeat Rewrite exp_ln. -Reflexivity. -Assumption. -Assumption. -Apply Rmult_lt_pos; Assumption. +Theorem ln_mult : forall x y:R, 0 < x -> 0 < y -> ln (x * y) = ln x + ln y. +intros x y H H0; apply exp_inv. +rewrite exp_plus. +repeat rewrite exp_ln. +reflexivity. +assumption. +assumption. +apply Rmult_lt_0_compat; assumption. Qed. -Theorem ln_Rinv: (x : R) ``0<x`` -> ``(ln (/x)) == -(ln x)``. -Intros x H; Apply exp_inv; Repeat (Rewrite exp_ln Orelse Rewrite exp_Ropp). -Reflexivity. -Assumption. -Apply Rlt_Rinv; Assumption. +Theorem ln_Rinv : forall x:R, 0 < x -> ln (/ x) = - ln x. +intros x H; apply exp_inv; repeat rewrite exp_ln || rewrite exp_Ropp. +reflexivity. +assumption. +apply Rinv_0_lt_compat; assumption. Qed. -Theorem ln_continue: - (y : R) ``0<y`` -> (continue_in ln [x : R] (Rlt R0 x) y). -Intros y H. -Unfold continue_in limit1_in limit_in; Intros eps Heps. -Cut (Rlt R1 (exp eps)); [Intros H1 | Idtac]. -Cut (Rlt (exp (Ropp eps)) R1); [Intros H2 | Idtac]. -Exists - (Rmin (Rmult y (Rminus (exp eps) R1)) (Rmult y (Rminus R1 (exp (Ropp eps))))); - Split. -Red; Apply P_Rmin. -Apply Rmult_lt_pos. -Assumption. -Apply Rlt_anti_compatibility with R1. -Rewrite Rplus_Or; Replace ``(1+((exp eps)-1))`` with (exp eps); [Apply H1 | Ring]. -Apply Rmult_lt_pos. -Assumption. -Apply Rlt_anti_compatibility with ``(exp (-eps))``. -Rewrite Rplus_Or; Replace ``(exp ( -eps))+(1-(exp ( -eps)))`` with R1; [Apply H2 | Ring]. -Unfold dist R_met R_dist; Simpl. -Intros x ((H3, H4), H5). -Cut (Rmult y (Rmult x (Rinv y))) == x. -Intro Hxyy. -Replace (Rminus (ln x) (ln y)) with (ln (Rmult x (Rinv y))). -Case (total_order x y); [Intros Hxy | Intros [Hxy|Hxy]]. -Rewrite Rabsolu_left. -Apply Ropp_Rlt; Rewrite Ropp_Ropp. -Apply exp_lt_inv. -Rewrite exp_ln. -Apply Rlt_monotony_contra with z := y. -Apply H. -Rewrite Hxyy. -Apply Ropp_Rlt. -Apply Rlt_anti_compatibility with r := y. -Replace (Rplus y (Ropp (Rmult y (exp (Ropp eps))))) - with (Rmult y (Rminus R1 (exp (Ropp eps)))); [Idtac | Ring]. -Replace (Rplus y (Ropp x)) with (Rabsolu (Rminus x y)); [Idtac | Ring]. -Apply Rlt_le_trans with 1 := H5; Apply Rmin_r. -Rewrite Rabsolu_left; [Ring | Idtac]. -Apply (Rlt_minus ? ? Hxy). -Apply Rmult_lt_pos; [Apply H3 | Apply (Rlt_Rinv ? H)]. -Rewrite <- ln_1. -Apply ln_increasing. -Apply Rmult_lt_pos; [Apply H3 | Apply (Rlt_Rinv ? H)]. -Apply Rlt_monotony_contra with z := y. -Apply H. -Rewrite Hxyy; Rewrite Rmult_1r; Apply Hxy. -Rewrite Hxy; Rewrite Rinv_r. -Rewrite ln_1; Rewrite Rabsolu_R0; Apply Heps. -Red; Intro; Rewrite H0 in H; Elim (Rlt_antirefl ? H). -Rewrite Rabsolu_right. -Apply exp_lt_inv. -Rewrite exp_ln. -Apply Rlt_monotony_contra with z := y. -Apply H. -Rewrite Hxyy. -Apply Rlt_anti_compatibility with r := (Ropp y). -Replace (Rplus (Ropp y) (Rmult y (exp eps))) - with (Rmult y (Rminus (exp eps) R1)); [Idtac | Ring]. -Replace (Rplus (Ropp y) x) with (Rabsolu (Rminus x y)); [Idtac | Ring]. -Apply Rlt_le_trans with 1 := H5; Apply Rmin_l. -Rewrite Rabsolu_right; [Ring | Idtac]. -Left; Apply (Rgt_minus ? ? Hxy). -Apply Rmult_lt_pos; [Apply H3 | Apply (Rlt_Rinv ? H)]. -Rewrite <- ln_1. -Apply Rgt_ge; Red; Apply ln_increasing. -Apply Rlt_R0_R1. -Apply Rlt_monotony_contra with z := y. -Apply H. -Rewrite Hxyy; Rewrite Rmult_1r; Apply Hxy. -Rewrite ln_mult. -Rewrite ln_Rinv. -Ring. -Assumption. -Assumption. -Apply Rlt_Rinv; Assumption. -Rewrite (Rmult_sym x); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Ring. -Red; Intro; Rewrite H0 in H; Elim (Rlt_antirefl ? H). -Apply Rlt_monotony_contra with (exp eps). -Apply exp_pos. -Rewrite <- exp_plus; Rewrite Rmult_1r; Rewrite Rplus_Ropp_r; Rewrite exp_0; Apply H1. -Rewrite <- exp_0. -Apply exp_increasing; Apply Heps. +Theorem ln_continue : + forall y:R, 0 < y -> continue_in ln (fun x:R => 0 < x) y. +intros y H. +unfold continue_in, limit1_in, limit_in in |- *; intros eps Heps. +cut (1 < exp eps); [ intros H1 | idtac ]. +cut (exp (- eps) < 1); [ intros H2 | idtac ]. +exists (Rmin (y * (exp eps - 1)) (y * (1 - exp (- eps)))); split. +red in |- *; apply P_Rmin. +apply Rmult_lt_0_compat. +assumption. +apply Rplus_lt_reg_r with 1. +rewrite Rplus_0_r; replace (1 + (exp eps - 1)) with (exp eps); + [ apply H1 | ring ]. +apply Rmult_lt_0_compat. +assumption. +apply Rplus_lt_reg_r with (exp (- eps)). +rewrite Rplus_0_r; replace (exp (- eps) + (1 - exp (- eps))) with 1; + [ apply H2 | ring ]. +unfold dist, R_met, R_dist in |- *; simpl in |- *. +intros x [[H3 H4] H5]. +cut (y * (x * / y) = x). +intro Hxyy. +replace (ln x - ln y) with (ln (x * / y)). +case (Rtotal_order x y); [ intros Hxy | intros [Hxy| Hxy] ]. +rewrite Rabs_left. +apply Ropp_lt_cancel; rewrite Ropp_involutive. +apply exp_lt_inv. +rewrite exp_ln. +apply Rmult_lt_reg_l with (r := y). +apply H. +rewrite Hxyy. +apply Ropp_lt_cancel. +apply Rplus_lt_reg_r with (r := y). +replace (y + - (y * exp (- eps))) with (y * (1 - exp (- eps))); + [ idtac | ring ]. +replace (y + - x) with (Rabs (x - y)); [ idtac | ring ]. +apply Rlt_le_trans with (1 := H5); apply Rmin_r. +rewrite Rabs_left; [ ring | idtac ]. +apply (Rlt_minus _ _ Hxy). +apply Rmult_lt_0_compat; [ apply H3 | apply (Rinv_0_lt_compat _ H) ]. +rewrite <- ln_1. +apply ln_increasing. +apply Rmult_lt_0_compat; [ apply H3 | apply (Rinv_0_lt_compat _ H) ]. +apply Rmult_lt_reg_l with (r := y). +apply H. +rewrite Hxyy; rewrite Rmult_1_r; apply Hxy. +rewrite Hxy; rewrite Rinv_r. +rewrite ln_1; rewrite Rabs_R0; apply Heps. +red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). +rewrite Rabs_right. +apply exp_lt_inv. +rewrite exp_ln. +apply Rmult_lt_reg_l with (r := y). +apply H. +rewrite Hxyy. +apply Rplus_lt_reg_r with (r := - y). +replace (- y + y * exp eps) with (y * (exp eps - 1)); [ idtac | ring ]. +replace (- y + x) with (Rabs (x - y)); [ idtac | ring ]. +apply Rlt_le_trans with (1 := H5); apply Rmin_l. +rewrite Rabs_right; [ ring | idtac ]. +left; apply (Rgt_minus _ _ Hxy). +apply Rmult_lt_0_compat; [ apply H3 | apply (Rinv_0_lt_compat _ H) ]. +rewrite <- ln_1. +apply Rgt_ge; red in |- *; apply ln_increasing. +apply Rlt_0_1. +apply Rmult_lt_reg_l with (r := y). +apply H. +rewrite Hxyy; rewrite Rmult_1_r; apply Hxy. +rewrite ln_mult. +rewrite ln_Rinv. +ring. +assumption. +assumption. +apply Rinv_0_lt_compat; assumption. +rewrite (Rmult_comm x); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +ring. +red in |- *; intro; rewrite H0 in H; elim (Rlt_irrefl _ H). +apply Rmult_lt_reg_l with (exp eps). +apply exp_pos. +rewrite <- exp_plus; rewrite Rmult_1_r; rewrite Rplus_opp_r; rewrite exp_0; + apply H1. +rewrite <- exp_0. +apply exp_increasing; apply Heps. Qed. (******************************************************************) (* Definition of Rpower *) (******************************************************************) -Definition Rpower := [x : R] [y : R] ``(exp (y*(ln x)))``. +Definition Rpower (x y:R) := exp (y * ln x). -Infix Local "^R" Rpower (at level 2, left associativity) : R_scope. +Infix Local "^R" := Rpower (at level 30, left associativity) : R_scope. (******************************************************************) (* Properties of Rpower *) (******************************************************************) -Theorem Rpower_plus: - (x, y, z : R) ``(Rpower z (x+y)) == (Rpower z x)*(Rpower z y)``. -Intros x y z; Unfold Rpower. -Rewrite Rmult_Rplus_distrl; Rewrite exp_plus; Auto. +Theorem Rpower_plus : forall x y z:R, z ^R (x + y) = z ^R x * z ^R y. +intros x y z; unfold Rpower in |- *. +rewrite Rmult_plus_distr_r; rewrite exp_plus; auto. Qed. -Theorem Rpower_mult: - (x, y, z : R) ``(Rpower (Rpower x y) z) == (Rpower x (y*z))``. -Intros x y z; Unfold Rpower. -Rewrite ln_exp. -Replace (Rmult z (Rmult y (ln x))) with (Rmult (Rmult y z) (ln x)). -Reflexivity. -Ring. +Theorem Rpower_mult : forall x y z:R, x ^R y ^R z = x ^R (y * z). +intros x y z; unfold Rpower in |- *. +rewrite ln_exp. +replace (z * (y * ln x)) with (y * z * ln x). +reflexivity. +ring. Qed. -Theorem Rpower_O: (x : R) ``0<x`` -> ``(Rpower x 0) == 1``. -Intros x H; Unfold Rpower. -Rewrite Rmult_Ol; Apply exp_0. +Theorem Rpower_O : forall x:R, 0 < x -> x ^R 0 = 1. +intros x H; unfold Rpower in |- *. +rewrite Rmult_0_l; apply exp_0. Qed. -Theorem Rpower_1: (x : R) ``0<x`` -> ``(Rpower x 1) == x``. -Intros x H; Unfold Rpower. -Rewrite Rmult_1l; Apply exp_ln; Apply H. +Theorem Rpower_1 : forall x:R, 0 < x -> x ^R 1 = x. +intros x H; unfold Rpower in |- *. +rewrite Rmult_1_l; apply exp_ln; apply H. Qed. -Theorem Rpower_pow: - (n : nat) (x : R) ``0<x`` -> (Rpower x (INR n)) == (pow x n). -Intros n; Elim n; Simpl; Auto; Fold INR. -Intros x H; Apply Rpower_O; Auto. -Intros n1; Case n1. -Intros H x H0; Simpl; Rewrite Rmult_1r; Apply Rpower_1; Auto. -Intros n0 H x H0; Rewrite Rpower_plus; Rewrite H; Try Rewrite Rpower_1; Try Apply Rmult_sym Orelse Assumption. +Theorem Rpower_pow : forall (n:nat) (x:R), 0 < x -> x ^R INR n = x ^ n. +intros n; elim n; simpl in |- *; auto; fold INR in |- *. +intros x H; apply Rpower_O; auto. +intros n1; case n1. +intros H x H0; simpl in |- *; rewrite Rmult_1_r; apply Rpower_1; auto. +intros n0 H x H0; rewrite Rpower_plus; rewrite H; try rewrite Rpower_1; + try apply Rmult_comm || assumption. Qed. -Theorem Rpower_lt: (x, y, z : R) ``1<x`` -> ``0<=y`` -> ``y<z`` -> ``(Rpower x y) < (Rpower x z)``. -Intros x y z H H0 H1. -Unfold Rpower. -Apply exp_increasing. -Apply Rlt_monotony_r. -Rewrite <- ln_1; Apply ln_increasing. -Apply Rlt_R0_R1. -Apply H. -Apply H1. +Theorem Rpower_lt : + forall x y z:R, 1 < x -> 0 <= y -> y < z -> x ^R y < x ^R z. +intros x y z H H0 H1. +unfold Rpower in |- *. +apply exp_increasing. +apply Rmult_lt_compat_r. +rewrite <- ln_1; apply ln_increasing. +apply Rlt_0_1. +apply H. +apply H1. Qed. -Theorem Rpower_sqrt: (x : R) ``0<x`` -> ``(Rpower x (/2)) == (sqrt x)``. -Intros x H. -Apply ln_inv. -Unfold Rpower; Apply exp_pos. -Apply sqrt_lt_R0; Apply H. -Apply r_Rmult_mult with (INR (S (S O))). -Apply exp_inv. -Fold Rpower. -Cut (Rpower (Rpower x (Rinv (Rplus R1 R1))) (INR (S (S O)))) == (Rpower (sqrt x) (INR (S (S O)))). -Unfold Rpower; Auto. -Rewrite Rpower_mult. -Rewrite Rinv_l. -Replace R1 with (INR (S O)); Auto. -Repeat Rewrite Rpower_pow; Simpl. -Pattern 1 x; Rewrite <- (sqrt_sqrt x (Rlt_le ? ? H)). -Ring. -Apply sqrt_lt_R0; Apply H. -Apply H. -Apply not_O_INR; Discriminate. -Apply not_O_INR; Discriminate. +Theorem Rpower_sqrt : forall x:R, 0 < x -> x ^R (/ 2) = sqrt x. +intros x H. +apply ln_inv. +unfold Rpower in |- *; apply exp_pos. +apply sqrt_lt_R0; apply H. +apply Rmult_eq_reg_l with (INR 2). +apply exp_inv. +fold Rpower in |- *. +cut (x ^R (/ 2) ^R INR 2 = sqrt x ^R INR 2). +unfold Rpower in |- *; auto. +rewrite Rpower_mult. +rewrite Rinv_l. +replace 1 with (INR 1); auto. +repeat rewrite Rpower_pow; simpl in |- *. +pattern x at 1 in |- *; rewrite <- (sqrt_sqrt x (Rlt_le _ _ H)). +ring. +apply sqrt_lt_R0; apply H. +apply H. +apply not_O_INR; discriminate. +apply not_O_INR; discriminate. Qed. -Theorem Rpower_Ropp: (x, y : R) ``(Rpower x (-y)) == /(Rpower x y)``. -Unfold Rpower. -Intros x y; Rewrite Ropp_mul1. -Apply exp_Ropp. +Theorem Rpower_Ropp : forall x y:R, x ^R (- y) = / x ^R y. +unfold Rpower in |- *. +intros x y; rewrite Ropp_mult_distr_l_reverse. +apply exp_Ropp. Qed. -Theorem Rle_Rpower: (e,n,m : R) ``1<e`` -> ``0<=n`` -> ``n<=m`` -> ``(Rpower e n)<=(Rpower e m)``. -Intros e n m H H0 H1; Case H1. -Intros H2; Left; Apply Rpower_lt; Assumption. -Intros H2; Rewrite H2; Right; Reflexivity. +Theorem Rle_Rpower : + forall e n m:R, 1 < e -> 0 <= n -> n <= m -> e ^R n <= e ^R m. +intros e n m H H0 H1; case H1. +intros H2; left; apply Rpower_lt; assumption. +intros H2; rewrite H2; right; reflexivity. Qed. -Theorem ln_lt_2: ``/2<(ln 2)``. -Apply Rlt_monotony_contra with z := (Rplus R1 R1). -Sup0. -Rewrite Rinv_r. -Apply exp_lt_inv. -Apply Rle_lt_trans with 1 := exp_le_3. -Change (Rlt (Rplus R1 (Rplus R1 R1)) (Rpower (Rplus R1 R1) (Rplus R1 R1))). -Repeat Rewrite Rpower_plus; Repeat Rewrite Rpower_1. -Repeat Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_Rplus_distr; - Repeat Rewrite Rmult_1l. -Pattern 1 ``3``; Rewrite <- Rplus_Or; Replace ``2+2`` with ``3+1``; [Apply Rlt_compatibility; Apply Rlt_R0_R1 | Ring]. -Sup0. -DiscrR. +Theorem ln_lt_2 : / 2 < ln 2. +apply Rmult_lt_reg_l with (r := 2). +prove_sup0. +rewrite Rinv_r. +apply exp_lt_inv. +apply Rle_lt_trans with (1 := exp_le_3). +change (3 < 2 ^R 2) in |- *. +repeat rewrite Rpower_plus; repeat rewrite Rpower_1. +repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l; + repeat rewrite Rmult_1_l. +pattern 3 at 1 in |- *; rewrite <- Rplus_0_r; replace (2 + 2) with (3 + 1); + [ apply Rplus_lt_compat_l; apply Rlt_0_1 | ring ]. +prove_sup0. +discrR. Qed. (**************************************) (* Differentiability of Ln and Rpower *) (**************************************) -Theorem limit1_ext: (f, g : R -> R)(D : R -> Prop)(l, x : R) ((x : R) (D x) -> (f x) == (g x)) -> (limit1_in f D l x) -> (limit1_in g D l x). -Intros f g D l x H; Unfold limit1_in limit_in. -Intros H0 eps H1; Case (H0 eps); Auto. -Intros x0 (H2, H3); Exists x0; Split; Auto. -Intros x1 (H4, H5); Rewrite <- H; Auto. +Theorem limit1_ext : + forall (f g:R -> R) (D:R -> Prop) (l x:R), + (forall x:R, D x -> f x = g x) -> limit1_in f D l x -> limit1_in g D l x. +intros f g D l x H; unfold limit1_in, limit_in in |- *. +intros H0 eps H1; case (H0 eps); auto. +intros x0 [H2 H3]; exists x0; split; auto. +intros x1 [H4 H5]; rewrite <- H; auto. Qed. -Theorem limit1_imp: (f : R -> R)(D, D1 : R -> Prop)(l, x : R) ((x : R) (D1 x) -> (D x)) -> (limit1_in f D l x) -> (limit1_in f D1 l x). -Intros f D D1 l x H; Unfold limit1_in limit_in. -Intros H0 eps H1; Case (H0 eps H1); Auto. -Intros alpha (H2, H3); Exists alpha; Split; Auto. -Intros d (H4, H5); Apply H3; Split; Auto. +Theorem limit1_imp : + forall (f:R -> R) (D D1:R -> Prop) (l x:R), + (forall x:R, D1 x -> D x) -> limit1_in f D l x -> limit1_in f D1 l x. +intros f D D1 l x H; unfold limit1_in, limit_in in |- *. +intros H0 eps H1; case (H0 eps H1); auto. +intros alpha [H2 H3]; exists alpha; split; auto. +intros d [H4 H5]; apply H3; split; auto. Qed. -Theorem Rinv_Rdiv: (x, y : R) ``x<>0`` -> ``y<>0`` -> ``/(x/y) == y/x``. -Intros x y H1 H2; Unfold Rdiv; Rewrite Rinv_Rmult. -Rewrite Rinv_Rinv. -Apply Rmult_sym. -Assumption. -Assumption. -Apply Rinv_neq_R0; Assumption. +Theorem Rinv_Rdiv : forall x y:R, x <> 0 -> y <> 0 -> / (x / y) = y / x. +intros x y H1 H2; unfold Rdiv in |- *; rewrite Rinv_mult_distr. +rewrite Rinv_involutive. +apply Rmult_comm. +assumption. +assumption. +apply Rinv_neq_0_compat; assumption. Qed. -Theorem Dln: (y : R) ``0<y`` -> (D_in ln Rinv [x:R]``0<x`` y). -Intros y Hy; Unfold D_in. -Apply limit1_ext with f := [x : R](Rinv (Rdiv (Rminus (exp (ln x)) (exp (ln y))) (Rminus (ln x) (ln y)))). -Intros x (HD1, HD2); Repeat Rewrite exp_ln. -Unfold Rdiv; Rewrite Rinv_Rmult. -Rewrite Rinv_Rinv. -Apply Rmult_sym. -Apply Rminus_eq_contra. -Red; Intros H2; Case HD2. -Symmetry; Apply (ln_inv ? ? HD1 Hy H2). -Apply Rminus_eq_contra; Apply (not_sym ? ? HD2). -Apply Rinv_neq_R0; Apply Rminus_eq_contra; Red; Intros H2; Case HD2; Apply ln_inv; Auto. -Assumption. -Assumption. -Apply limit_inv with f := [x : R] (Rdiv (Rminus (exp (ln x)) (exp (ln y))) (Rminus (ln x) (ln y))). -Apply limit1_imp with f := [x : R] ([x : R] (Rdiv (Rminus (exp x) (exp (ln y))) (Rminus x (ln y))) (ln x)) D := (Dgf (D_x [x : R] (Rlt R0 x) y) (D_x [x : R] True (ln y)) ln). -Intros x (H1, H2); Split. -Split; Auto. -Split; Auto. -Red; Intros H3; Case H2; Apply ln_inv; Auto. -Apply limit_comp with l := (ln y) g := [x : R] (Rdiv (Rminus (exp x) (exp (ln y))) (Rminus x (ln y))) f := ln. -Apply ln_continue; Auto. -Assert H0 := (derivable_pt_lim_exp (ln y)); Unfold derivable_pt_lim in H0; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Elim (H0 ? H); Intros; Exists (pos x); Split. -Apply (RIneq.cond_pos x). -Intros; Pattern 3 y; Rewrite <- exp_ln. -Pattern 1 x0; Replace x0 with ``(ln y)+(x0-(ln y))``; [Idtac | Ring]. -Apply H1. -Elim H2; Intros H3 _; Unfold D_x in H3; Elim H3; Clear H3; Intros _ H3; Apply Rminus_eq_contra; Apply not_sym; Apply H3. -Elim H2; Clear H2; Intros _ H2; Apply H2. -Assumption. -Red; Intro; Rewrite H in Hy; Elim (Rlt_antirefl ? Hy). +Theorem Dln : forall y:R, 0 < y -> D_in ln Rinv (fun x:R => 0 < x) y. +intros y Hy; unfold D_in in |- *. +apply limit1_ext with + (f := fun x:R => / ((exp (ln x) - exp (ln y)) / (ln x - ln y))). +intros x [HD1 HD2]; repeat rewrite exp_ln. +unfold Rdiv in |- *; rewrite Rinv_mult_distr. +rewrite Rinv_involutive. +apply Rmult_comm. +apply Rminus_eq_contra. +red in |- *; intros H2; case HD2. +symmetry in |- *; apply (ln_inv _ _ HD1 Hy H2). +apply Rminus_eq_contra; apply (sym_not_eq HD2). +apply Rinv_neq_0_compat; apply Rminus_eq_contra; red in |- *; intros H2; + case HD2; apply ln_inv; auto. +assumption. +assumption. +apply limit_inv with + (f := fun x:R => (exp (ln x) - exp (ln y)) / (ln x - ln y)). +apply limit1_imp with + (f := fun x:R => (fun x:R => (exp x - exp (ln y)) / (x - ln y)) (ln x)) + (D := Dgf (D_x (fun x:R => 0 < x) y) (D_x (fun x:R => True) (ln y)) ln). +intros x [H1 H2]; split. +split; auto. +split; auto. +red in |- *; intros H3; case H2; apply ln_inv; auto. +apply limit_comp with + (l := ln y) (g := fun x:R => (exp x - exp (ln y)) / (x - ln y)) (f := ln). +apply ln_continue; auto. +assert (H0 := derivable_pt_lim_exp (ln y)); unfold derivable_pt_lim in H0; + unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros; elim (H0 _ H); + intros; exists (pos x); split. +apply (cond_pos x). +intros; pattern y at 3 in |- *; rewrite <- exp_ln. +pattern x0 at 1 in |- *; replace x0 with (ln y + (x0 - ln y)); + [ idtac | ring ]. +apply H1. +elim H2; intros H3 _; unfold D_x in H3; elim H3; clear H3; intros _ H3; + apply Rminus_eq_contra; apply (sym_not_eq (A:=R)); + apply H3. +elim H2; clear H2; intros _ H2; apply H2. +assumption. +red in |- *; intro; rewrite H in Hy; elim (Rlt_irrefl _ Hy). Qed. -Lemma derivable_pt_lim_ln : (x:R) ``0<x`` -> (derivable_pt_lim ln x ``/x``). -Intros; Assert H0 := (Dln x H); Unfold D_in in H0; Unfold limit1_in in H0; Unfold limit_in in H0; Simpl in H0; Unfold R_dist in H0; Unfold derivable_pt_lim; Intros; Elim (H0 ? H1); Intros; Elim H2; Clear H2; Intros; Pose alp := (Rmin x0 ``x/2``); Assert H4 : ``0<alp``. -Unfold alp; Unfold Rmin; Case (total_order_Rle x0 ``x/2``); Intro. -Apply H2. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Exists (mkposreal ? H4); Intros; Pattern 2 h; Replace h with ``(x+h)-x``; [Idtac | Ring]. -Apply H3; Split. -Unfold D_x; Split. -Case (case_Rabsolu h); Intro. -Assert H7 : ``(Rabsolu h)<x/2``. -Apply Rlt_le_trans with alp. -Apply H6. -Unfold alp; Apply Rmin_r. -Apply Rlt_trans with ``x/2``. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. -Rewrite Rabsolu_left in H7. -Apply Rlt_anti_compatibility with ``-h-x/2``. -Replace ``-h-x/2+x/2`` with ``-h``; [Idtac | Ring]. -Pattern 2 x; Rewrite double_var. -Replace ``-h-x/2+(x/2+x/2+h)`` with ``x/2``; [Apply H7 | Ring]. -Apply r. -Apply gt0_plus_ge0_is_gt0; [Assumption | Apply Rle_sym2; Apply r]. -Apply not_sym; Apply Rminus_not_eq; Replace ``x+h-x`` with h; [Apply H5 | Ring]. -Replace ``x+h-x`` with h; [Apply Rlt_le_trans with alp; [Apply H6 | Unfold alp; Apply Rmin_l] | Ring]. +Lemma derivable_pt_lim_ln : forall x:R, 0 < x -> derivable_pt_lim ln x (/ x). +intros; assert (H0 := Dln x H); unfold D_in in H0; unfold limit1_in in H0; + unfold limit_in in H0; simpl in H0; unfold R_dist in H0; + unfold derivable_pt_lim in |- *; intros; elim (H0 _ H1); + intros; elim H2; clear H2; intros; pose (alp := Rmin x0 (x / 2)); + assert (H4 : 0 < alp). +unfold alp in |- *; unfold Rmin in |- *; case (Rle_dec x0 (x / 2)); intro. +apply H2. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +exists (mkposreal _ H4); intros; pattern h at 2 in |- *; + replace h with (x + h - x); [ idtac | ring ]. +apply H3; split. +unfold D_x in |- *; split. +case (Rcase_abs h); intro. +assert (H7 : Rabs h < x / 2). +apply Rlt_le_trans with alp. +apply H6. +unfold alp in |- *; apply Rmin_r. +apply Rlt_trans with (x / 2). +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. +rewrite Rabs_left in H7. +apply Rplus_lt_reg_r with (- h - x / 2). +replace (- h - x / 2 + x / 2) with (- h); [ idtac | ring ]. +pattern x at 2 in |- *; rewrite double_var. +replace (- h - x / 2 + (x / 2 + x / 2 + h)) with (x / 2); [ apply H7 | ring ]. +apply r. +apply Rplus_lt_le_0_compat; [ assumption | apply Rge_le; apply r ]. +apply (sym_not_eq (A:=R)); apply Rminus_not_eq; replace (x + h - x) with h; + [ apply H5 | ring ]. +replace (x + h - x) with h; + [ apply Rlt_le_trans with alp; + [ apply H6 | unfold alp in |- *; apply Rmin_l ] + | ring ]. Qed. -Theorem D_in_imp: (f, g : R -> R)(D, D1 : R -> Prop)(x : R) ((x : R) (D1 x) -> (D x)) -> (D_in f g D x) -> (D_in f g D1 x). -Intros f g D D1 x H; Unfold D_in. -Intros H0; Apply limit1_imp with D := (D_x D x); Auto. -Intros x1 (H1, H2); Split; Auto. +Theorem D_in_imp : + forall (f g:R -> R) (D D1:R -> Prop) (x:R), + (forall x:R, D1 x -> D x) -> D_in f g D x -> D_in f g D1 x. +intros f g D D1 x H; unfold D_in in |- *. +intros H0; apply limit1_imp with (D := D_x D x); auto. +intros x1 [H1 H2]; split; auto. Qed. -Theorem D_in_ext: (f, g, h : R -> R)(D : R -> Prop) (x : R) (f x) == (g x) -> (D_in h f D x) -> (D_in h g D x). -Intros f g h D x H; Unfold D_in. -Rewrite H; Auto. +Theorem D_in_ext : + forall (f g h:R -> R) (D:R -> Prop) (x:R), + f x = g x -> D_in h f D x -> D_in h g D x. +intros f g h D x H; unfold D_in in |- *. +rewrite H; auto. Qed. -Theorem Dpower: (y, z : R) ``0<y`` -> (D_in [x:R](Rpower x z) [x:R](Rmult z (Rpower x (Rminus z R1))) [x:R]``0<x`` y). -Intros y z H; Apply D_in_imp with D := (Dgf [x : R] (Rlt R0 x) [x : R] True ln). -Intros x H0; Repeat Split. -Assumption. -Apply D_in_ext with f := [x : R] (Rmult (Rinv x) (Rmult z (exp (Rmult z (ln x))))). -Unfold Rminus; Rewrite Rpower_plus; Rewrite Rpower_Ropp; Rewrite (Rpower_1 ? H); Ring. -Apply Dcomp with f := ln g := [x : R] (exp (Rmult z x)) df := Rinv dg := [x : R] (Rmult z (exp (Rmult z x))). -Apply (Dln ? H). -Apply D_in_imp with D := (Dgf [x : R] True [x : R] True [x : R] (Rmult z x)). -Intros x H1; Repeat Split; Auto. -Apply (Dcomp [_ : R] True [_ : R] True [x : ?] z exp [x : R] (Rmult z x) exp); Simpl. -Apply D_in_ext with f := [x : R] (Rmult z R1). -Apply Rmult_1r. -Apply (Dmult_const [x : ?] True [x : ?] x [x : ?] R1); Apply Dx. -Assert H0 := (derivable_pt_lim_D_in exp exp ``z*(ln y)``); Elim H0; Clear H0; Intros _ H0; Apply H0; Apply derivable_pt_lim_exp. +Theorem Dpower : + forall y z:R, + 0 < y -> + D_in (fun x:R => x ^R z) (fun x:R => z * x ^R (z - 1)) ( + fun x:R => 0 < x) y. +intros y z H; + apply D_in_imp with (D := Dgf (fun x:R => 0 < x) (fun x:R => True) ln). +intros x H0; repeat split. +assumption. +apply D_in_ext with (f := fun x:R => / x * (z * exp (z * ln x))). +unfold Rminus in |- *; rewrite Rpower_plus; rewrite Rpower_Ropp; + rewrite (Rpower_1 _ H); ring. +apply Dcomp with + (f := ln) + (g := fun x:R => exp (z * x)) + (df := Rinv) + (dg := fun x:R => z * exp (z * x)). +apply (Dln _ H). +apply D_in_imp with + (D := Dgf (fun x:R => True) (fun x:R => True) (fun x:R => z * x)). +intros x H1; repeat split; auto. +apply + (Dcomp (fun _:R => True) (fun _:R => True) (fun x => z) exp + (fun x:R => z * x) exp); simpl in |- *. +apply D_in_ext with (f := fun x:R => z * 1). +apply Rmult_1_r. +apply (Dmult_const (fun x => True) (fun x => x) (fun x => 1)); apply Dx. +assert (H0 := derivable_pt_lim_D_in exp exp (z * ln y)); elim H0; clear H0; + intros _ H0; apply H0; apply derivable_pt_lim_exp. Qed. -Theorem derivable_pt_lim_power: (x, y : R) (Rlt R0 x) -> (derivable_pt_lim [x : ?] (Rpower x y) x (Rmult y (Rpower x (Rminus y R1)))). -Intros x y H. -Unfold Rminus; Rewrite Rpower_plus. -Rewrite Rpower_Ropp. -Rewrite Rpower_1; Auto. -Rewrite <- Rmult_assoc. -Unfold Rpower. -Apply derivable_pt_lim_comp with f1 := ln f2 := [x : ?] (exp (Rmult y x)). -Apply derivable_pt_lim_ln; Assumption. -Rewrite (Rmult_sym y). -Apply derivable_pt_lim_comp with f1 := [x : ?] (Rmult y x) f2 := exp. -Pattern 2 y; Replace y with (Rplus (Rmult R0 (ln x)) (Rmult y R1)). -Apply derivable_pt_lim_mult with f1 := [x : R] y f2 := [x : R] x. -Apply derivable_pt_lim_const with a := y. -Apply derivable_pt_lim_id. -Ring. -Apply derivable_pt_lim_exp. -Qed. +Theorem derivable_pt_lim_power : + forall x y:R, + 0 < x -> derivable_pt_lim (fun x => x ^R y) x (y * x ^R (y - 1)). +intros x y H. +unfold Rminus in |- *; rewrite Rpower_plus. +rewrite Rpower_Ropp. +rewrite Rpower_1; auto. +rewrite <- Rmult_assoc. +unfold Rpower in |- *. +apply derivable_pt_lim_comp with (f1 := ln) (f2 := fun x => exp (y * x)). +apply derivable_pt_lim_ln; assumption. +rewrite (Rmult_comm y). +apply derivable_pt_lim_comp with (f1 := fun x => y * x) (f2 := exp). +pattern y at 2 in |- *; replace y with (0 * ln x + y * 1). +apply derivable_pt_lim_mult with (f1 := fun x:R => y) (f2 := fun x:R => x). +apply derivable_pt_lim_const with (a := y). +apply derivable_pt_lim_id. +ring. +apply derivable_pt_lim_exp. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Rprod.v b/theories/Reals/Rprod.v index c613c7647..9d962e125 100644 --- a/theories/Reals/Rprod.v +++ b/theories/Reals/Rprod.v @@ -8,157 +8,184 @@ (*i $Id$ i*) -Require Compare. -Require Rbase. -Require Rfunctions. -Require Rseries. -Require PartSum. -Require Binomial. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Compare. +Require Import Rbase. +Require Import Rfunctions. +Require Import Rseries. +Require Import PartSum. +Require Import Binomial. Open Local Scope R_scope. (* TT Ak; 1<=k<=N *) -Fixpoint prod_f_SO [An:nat->R;N:nat] : R := Cases N of - O => R1 -| (S p) => ``(prod_f_SO An p)*(An (S p))`` -end. +Fixpoint prod_f_SO (An:nat -> R) (N:nat) {struct N} : R := + match N with + | O => 1 + | S p => prod_f_SO An p * An (S p) + end. (**********) -Lemma prod_SO_split : (An:nat->R;n,k:nat) (le k n) -> (prod_f_SO An n)==(Rmult (prod_f_SO An k) (prod_f_SO [l:nat](An (plus k l)) (minus n k))). -Intros; Induction n. -Cut k=O; [Intro; Rewrite H0; Simpl; Ring | Inversion H; Reflexivity]. -Cut k=(S n)\/(le k n). -Intro; Elim H0; Intro. -Rewrite H1; Simpl; Rewrite <- minus_n_n; Simpl; Ring. -Replace (minus (S n) k) with (S (minus n k)). -Simpl; Replace (plus k (S (minus n k))) with (S n). -Rewrite Hrecn; [Ring | Assumption]. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite S_INR; Rewrite minus_INR; [Ring | Assumption]. -Apply INR_eq; Rewrite S_INR; Repeat Rewrite minus_INR. -Rewrite S_INR; Ring. -Apply le_trans with n; [Assumption | Apply le_n_Sn]. -Assumption. -Inversion H; [Left; Reflexivity | Right; Assumption]. +Lemma prod_SO_split : + forall (An:nat -> R) (n k:nat), + (k <= n)%nat -> + prod_f_SO An n = + prod_f_SO An k * prod_f_SO (fun l:nat => An (k + l)%nat) (n - k). +intros; induction n as [| n Hrecn]. +cut (k = 0%nat); + [ intro; rewrite H0; simpl in |- *; ring | inversion H; reflexivity ]. +cut (k = S n \/ (k <= n)%nat). +intro; elim H0; intro. +rewrite H1; simpl in |- *; rewrite <- minus_n_n; simpl in |- *; ring. +replace (S n - k)%nat with (S (n - k)). +simpl in |- *; replace (k + S (n - k))%nat with (S n). +rewrite Hrecn; [ ring | assumption ]. +apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite S_INR; + rewrite minus_INR; [ ring | assumption ]. +apply INR_eq; rewrite S_INR; repeat rewrite minus_INR. +rewrite S_INR; ring. +apply le_trans with n; [ assumption | apply le_n_Sn ]. +assumption. +inversion H; [ left; reflexivity | right; assumption ]. Qed. (**********) -Lemma prod_SO_pos : (An:nat->R;N:nat) ((n:nat)(le n N)->``0<=(An n)``) -> ``0<=(prod_f_SO An N)``. -Intros; Induction N. -Simpl; Left; Apply Rlt_R0_R1. -Simpl; Apply Rmult_le_pos. -Apply HrecN; Intros; Apply H; Apply le_trans with N; [Assumption | Apply le_n_Sn]. -Apply H; Apply le_n. +Lemma prod_SO_pos : + forall (An:nat -> R) (N:nat), + (forall n:nat, (n <= N)%nat -> 0 <= An n) -> 0 <= prod_f_SO An N. +intros; induction N as [| N HrecN]. +simpl in |- *; left; apply Rlt_0_1. +simpl in |- *; apply Rmult_le_pos. +apply HrecN; intros; apply H; apply le_trans with N; + [ assumption | apply le_n_Sn ]. +apply H; apply le_n. Qed. (**********) -Lemma prod_SO_Rle : (An,Bn:nat->R;N:nat) ((n:nat)(le n N)->``0<=(An n)<=(Bn n)``) -> ``(prod_f_SO An N)<=(prod_f_SO Bn N)``. -Intros; Induction N. -Right; Reflexivity. -Simpl; Apply Rle_trans with ``(prod_f_SO An N)*(Bn (S N))``. -Apply Rle_monotony. -Apply prod_SO_pos; Intros; Elim (H n (le_trans ? ? ? H0 (le_n_Sn N))); Intros; Assumption. -Elim (H (S N) (le_n (S N))); Intros; Assumption. -Do 2 Rewrite <- (Rmult_sym (Bn (S N))); Apply Rle_monotony. -Elim (H (S N) (le_n (S N))); Intros. -Apply Rle_trans with (An (S N)); Assumption. -Apply HrecN; Intros; Elim (H n (le_trans ? ? ? H0 (le_n_Sn N))); Intros; Split; Assumption. +Lemma prod_SO_Rle : + forall (An Bn:nat -> R) (N:nat), + (forall n:nat, (n <= N)%nat -> 0 <= An n <= Bn n) -> + prod_f_SO An N <= prod_f_SO Bn N. +intros; induction N as [| N HrecN]. +right; reflexivity. +simpl in |- *; apply Rle_trans with (prod_f_SO An N * Bn (S N)). +apply Rmult_le_compat_l. +apply prod_SO_pos; intros; elim (H n (le_trans _ _ _ H0 (le_n_Sn N))); intros; + assumption. +elim (H (S N) (le_n (S N))); intros; assumption. +do 2 rewrite <- (Rmult_comm (Bn (S N))); apply Rmult_le_compat_l. +elim (H (S N) (le_n (S N))); intros. +apply Rle_trans with (An (S N)); assumption. +apply HrecN; intros; elim (H n (le_trans _ _ _ H0 (le_n_Sn N))); intros; + split; assumption. Qed. (* Application to factorial *) -Lemma fact_prodSO : (n:nat) (INR (fact n))==(prod_f_SO [k:nat](INR k) n). -Intro; Induction n. -Reflexivity. -Change (INR (mult (S n) (fact n)))==(prod_f_SO ([k:nat](INR k)) (S n)). -Rewrite mult_INR; Rewrite Rmult_sym; Rewrite Hrecn; Reflexivity. +Lemma fact_prodSO : + forall n:nat, INR (fact n) = prod_f_SO (fun k:nat => INR k) n. +intro; induction n as [| n Hrecn]. +reflexivity. +change (INR (S n * fact n) = prod_f_SO (fun k:nat => INR k) (S n)) in |- *. +rewrite mult_INR; rewrite Rmult_comm; rewrite Hrecn; reflexivity. Qed. -Lemma le_n_2n : (n:nat) (le n (mult (2) n)). -Induction n. -Replace (mult (2) (O)) with O; [Apply le_n | Ring]. -Intros; Replace (mult (2) (S n0)) with (S (S (mult (2) n0))). -Apply le_n_S; Apply le_S; Assumption. -Replace (S (S (mult (2) n0))) with (plus (mult (2) n0) (2)); [Idtac | Ring]. -Replace (S n0) with (plus n0 (1)); [Idtac | Ring]. -Ring. +Lemma le_n_2n : forall n:nat, (n <= 2 * n)%nat. +simple induction n. +replace (2 * 0)%nat with 0%nat; [ apply le_n | ring ]. +intros; replace (2 * S n0)%nat with (S (S (2 * n0))). +apply le_n_S; apply le_S; assumption. +replace (S (S (2 * n0))) with (2 * n0 + 2)%nat; [ idtac | ring ]. +replace (S n0) with (n0 + 1)%nat; [ idtac | ring ]. +ring. Qed. (* We prove that (N!)²<=(2N-k)!*k! forall k in [|O;2N|] *) -Lemma RfactN_fact2N_factk : (N,k:nat) (le k (mult (2) N)) -> ``(Rsqr (INR (fact N)))<=(INR (fact (minus (mult (S (S O)) N) k)))*(INR (fact k))``. -Intros; Unfold Rsqr; Repeat Rewrite fact_prodSO. -Cut (le k N)\/(le N k). -Intro; Elim H0; Intro. -Rewrite (prod_SO_split [l:nat](INR l) (minus (mult (2) N) k) N). -Rewrite Rmult_assoc; Apply Rle_monotony. -Apply prod_SO_pos; Intros; Apply pos_INR. -Replace (minus (minus (mult (2) N) k) N) with (minus N k). -Rewrite Rmult_sym; Rewrite (prod_SO_split [l:nat](INR l) N k). -Apply Rle_monotony. -Apply prod_SO_pos; Intros; Apply pos_INR. -Apply prod_SO_Rle; Intros; Split. -Apply pos_INR. -Apply le_INR; Apply le_reg_r; Assumption. -Assumption. -Apply INR_eq; Repeat Rewrite minus_INR. -Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Apply le_trans with N; [Assumption | Apply le_n_2n]. -Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. -Replace (mult (2) N) with (plus N N); [Idtac | Ring]. -Apply le_reg_r; Assumption. -Assumption. -Assumption. -Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. -Replace (mult (2) N) with (plus N N); [Idtac | Ring]. -Apply le_reg_r; Assumption. -Assumption. -Rewrite <- (Rmult_sym (prod_f_SO [l:nat](INR l) k)); Rewrite (prod_SO_split [l:nat](INR l) k N). -Rewrite Rmult_assoc; Apply Rle_monotony. -Apply prod_SO_pos; Intros; Apply pos_INR. -Rewrite Rmult_sym; Rewrite (prod_SO_split [l:nat](INR l) N (minus (mult (2) N) k)). -Apply Rle_monotony. -Apply prod_SO_pos; Intros; Apply pos_INR. -Replace (minus N (minus (mult (2) N) k)) with (minus k N). -Apply prod_SO_Rle; Intros; Split. -Apply pos_INR. -Apply le_INR; Apply le_reg_r. -Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. -Replace (mult (2) N) with (plus N N); [Idtac | Ring]; Apply le_reg_r; Assumption. -Assumption. -Apply INR_eq; Repeat Rewrite minus_INR. -Rewrite mult_INR; Do 2 Rewrite S_INR; Ring. -Assumption. -Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. -Replace (mult (2) N) with (plus N N); [Idtac | Ring]; Apply le_reg_r; Assumption. -Assumption. -Assumption. -Apply simpl_le_plus_l with k; Rewrite <- le_plus_minus. -Replace (mult (2) N) with (plus N N); [Idtac | Ring]; Apply le_reg_r; Assumption. -Assumption. -Assumption. -Elim (le_dec k N); Intro; [Left; Assumption | Right; Assumption]. +Lemma RfactN_fact2N_factk : + forall N k:nat, + (k <= 2 * N)%nat -> + Rsqr (INR (fact N)) <= INR (fact (2 * N - k)) * INR (fact k). +intros; unfold Rsqr in |- *; repeat rewrite fact_prodSO. +cut ((k <= N)%nat \/ (N <= k)%nat). +intro; elim H0; intro. +rewrite (prod_SO_split (fun l:nat => INR l) (2 * N - k) N). +rewrite Rmult_assoc; apply Rmult_le_compat_l. +apply prod_SO_pos; intros; apply pos_INR. +replace (2 * N - k - N)%nat with (N - k)%nat. +rewrite Rmult_comm; rewrite (prod_SO_split (fun l:nat => INR l) N k). +apply Rmult_le_compat_l. +apply prod_SO_pos; intros; apply pos_INR. +apply prod_SO_Rle; intros; split. +apply pos_INR. +apply le_INR; apply plus_le_compat_r; assumption. +assumption. +apply INR_eq; repeat rewrite minus_INR. +rewrite mult_INR; repeat rewrite S_INR; ring. +apply le_trans with N; [ assumption | apply le_n_2n ]. +apply (fun p n m:nat => plus_le_reg_l n m p) with k; rewrite <- le_plus_minus. +replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ]. +apply plus_le_compat_r; assumption. +assumption. +assumption. +apply (fun p n m:nat => plus_le_reg_l n m p) with k; rewrite <- le_plus_minus. +replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ]. +apply plus_le_compat_r; assumption. +assumption. +rewrite <- (Rmult_comm (prod_f_SO (fun l:nat => INR l) k)); + rewrite (prod_SO_split (fun l:nat => INR l) k N). +rewrite Rmult_assoc; apply Rmult_le_compat_l. +apply prod_SO_pos; intros; apply pos_INR. +rewrite Rmult_comm; + rewrite (prod_SO_split (fun l:nat => INR l) N (2 * N - k)). +apply Rmult_le_compat_l. +apply prod_SO_pos; intros; apply pos_INR. +replace (N - (2 * N - k))%nat with (k - N)%nat. +apply prod_SO_Rle; intros; split. +apply pos_INR. +apply le_INR; apply plus_le_compat_r. +apply (fun p n m:nat => plus_le_reg_l n m p) with k; rewrite <- le_plus_minus. +replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ]; + apply plus_le_compat_r; assumption. +assumption. +apply INR_eq; repeat rewrite minus_INR. +rewrite mult_INR; do 2 rewrite S_INR; ring. +assumption. +apply (fun p n m:nat => plus_le_reg_l n m p) with k; rewrite <- le_plus_minus. +replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ]; + apply plus_le_compat_r; assumption. +assumption. +assumption. +apply (fun p n m:nat => plus_le_reg_l n m p) with k; rewrite <- le_plus_minus. +replace (2 * N)%nat with (N + N)%nat; [ idtac | ring ]; + apply plus_le_compat_r; assumption. +assumption. +assumption. +elim (le_dec k N); intro; [ left; assumption | right; assumption ]. Qed. (**********) -Lemma INR_fact_lt_0 : (n:nat) ``0<(INR (fact n))``. -Intro; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Elim (fact_neq_0 n); Symmetry; Assumption. +Lemma INR_fact_lt_0 : forall n:nat, 0 < INR (fact n). +intro; apply lt_INR_0; apply neq_O_lt; red in |- *; intro; + elim (fact_neq_0 n); symmetry in |- *; assumption. Qed. (* We have the following inequality : (C 2N k) <= (C 2N N) forall k in [|O;2N|] *) -Lemma C_maj : (N,k:nat) (le k (mult (2) N)) -> ``(C (mult (S (S O)) N) k)<=(C (mult (S (S O)) N) N)``. -Intros; Unfold C; Unfold Rdiv; Apply Rle_monotony. -Apply pos_INR. -Replace (minus (mult (2) N) N) with N. -Apply Rle_monotony_contra with ``((INR (fact N))*(INR (fact N)))``. -Apply Rmult_lt_pos; Apply INR_fact_lt_0. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_sym; Apply Rle_monotony_contra with ``((INR (fact k))* - (INR (fact (minus (mult (S (S O)) N) k))))``. -Apply Rmult_lt_pos; Apply INR_fact_lt_0. -Rewrite Rmult_1r; Rewrite <- mult_INR; Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite mult_INR; Rewrite (Rmult_sym (INR (fact k))); Replace ``(INR (fact N))*(INR (fact N))`` with (Rsqr (INR (fact N))). -Apply RfactN_fact2N_factk. -Assumption. -Reflexivity. -Rewrite mult_INR; Apply prod_neq_R0; Apply INR_fact_neq_0. -Apply prod_neq_R0; Apply INR_fact_neq_0. -Apply INR_eq; Rewrite minus_INR; [Rewrite mult_INR; Do 2 Rewrite S_INR; Ring | Apply le_n_2n]. -Qed. +Lemma C_maj : forall N k:nat, (k <= 2 * N)%nat -> C (2 * N) k <= C (2 * N) N. +intros; unfold C in |- *; unfold Rdiv in |- *; apply Rmult_le_compat_l. +apply pos_INR. +replace (2 * N - N)%nat with N. +apply Rmult_le_reg_l with (INR (fact N) * INR (fact N)). +apply Rmult_lt_0_compat; apply INR_fact_lt_0. +rewrite <- Rinv_r_sym. +rewrite Rmult_comm; + apply Rmult_le_reg_l with (INR (fact k) * INR (fact (2 * N - k))). +apply Rmult_lt_0_compat; apply INR_fact_lt_0. +rewrite Rmult_1_r; rewrite <- mult_INR; rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite mult_INR; rewrite (Rmult_comm (INR (fact k))); + replace (INR (fact N) * INR (fact N)) with (Rsqr (INR (fact N))). +apply RfactN_fact2N_factk. +assumption. +reflexivity. +rewrite mult_INR; apply prod_neq_R0; apply INR_fact_neq_0. +apply prod_neq_R0; apply INR_fact_neq_0. +apply INR_eq; rewrite minus_INR; + [ rewrite mult_INR; do 2 rewrite S_INR; ring | apply le_n_2n ]. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Rseries.v b/theories/Reals/Rseries.v index 032524771..03544af4b 100644 --- a/theories/Reals/Rseries.v +++ b/theories/Reals/Rseries.v @@ -8,14 +8,13 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Classical. -Require Compare. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import Classical. +Require Import Compare. Open Local Scope R_scope. -Implicit Variable Type r:R. +Implicit Type r : R. (* classical is needed for [Un_cv_crit] *) (*********************************************************) @@ -26,144 +25,153 @@ Implicit Variable Type r:R. Section sequence. (*********) -Variable Un:nat->R. +Variable Un : nat -> R. (*********) -Fixpoint Rmax_N [N:nat]:R:= - Cases N of - O => (Un O) - |(S n) => (Rmax (Un (S n)) (Rmax_N n)) - end. +Fixpoint Rmax_N (N:nat) : R := + match N with + | O => Un 0 + | S n => Rmax (Un (S n)) (Rmax_N n) + end. (*********) -Definition EUn:R->Prop:=[r:R](Ex [i:nat] (r==(Un i))). +Definition EUn r : Prop := exists i : nat | r = Un i. (*********) -Definition Un_cv:R->Prop:=[l:R] - (eps:R)(Rgt eps R0)->(Ex[N:nat](n:nat)(ge n N)-> - (Rlt (R_dist (Un n) l) eps)). +Definition Un_cv (l:R) : Prop := + forall eps:R, + eps > 0 -> + exists N : nat | (forall n:nat, (n >= N)%nat -> R_dist (Un n) l < eps). (*********) -Definition Cauchy_crit:Prop:=(eps:R)(Rgt eps R0)-> - (Ex[N:nat] (n,m:nat)(ge n N)->(ge m N)-> - (Rlt (R_dist (Un n) (Un m)) eps)). +Definition Cauchy_crit : Prop := + forall eps:R, + eps > 0 -> + exists N : nat + | (forall n m:nat, + (n >= N)%nat -> (m >= N)%nat -> R_dist (Un n) (Un m) < eps). (*********) -Definition Un_growing:Prop:=(n:nat)(Rle (Un n) (Un (S n))). +Definition Un_growing : Prop := forall n:nat, Un n <= Un (S n). (*********) -Lemma EUn_noempty:(ExT [r:R] (EUn r)). -Unfold EUn;Split with (Un O);Split with O;Trivial. +Lemma EUn_noempty : exists r : R | EUn r. +unfold EUn in |- *; split with (Un 0); split with 0%nat; trivial. Qed. (*********) -Lemma Un_in_EUn:(n:nat)(EUn (Un n)). -Intro;Unfold EUn;Split with n;Trivial. +Lemma Un_in_EUn : forall n:nat, EUn (Un n). +intro; unfold EUn in |- *; split with n; trivial. Qed. (*********) -Lemma Un_bound_imp:(x:R)((n:nat)(Rle (Un n) x))->(is_upper_bound EUn x). -Intros;Unfold is_upper_bound;Intros;Unfold EUn in H0;Elim H0;Clear H0; - Intros;Generalize (H x1);Intro;Rewrite <- H0 in H1;Trivial. +Lemma Un_bound_imp : + forall x:R, (forall n:nat, Un n <= x) -> is_upper_bound EUn x. +intros; unfold is_upper_bound in |- *; intros; unfold EUn in H0; elim H0; + clear H0; intros; generalize (H x1); intro; rewrite <- H0 in H1; + trivial. Qed. (*********) -Lemma growing_prop:(n,m:nat)Un_growing->(ge n m)->(Rge (Un n) (Un m)). -Double Induction n m;Intros. -Unfold Rge;Right;Trivial. -ElimType False;Unfold ge in H1;Generalize (le_Sn_O n0);Intro;Auto. -Cut (ge n0 (0)). -Generalize H0;Intros;Unfold Un_growing in H0; - Apply (Rge_trans (Un (S n0)) (Un n0) (Un (0)) - (Rle_sym1 (Un n0) (Un (S n0)) (H0 n0)) (H O H2 H3)). -Elim n0;Auto. -Elim (lt_eq_lt_dec n1 n0);Intro y. -Elim y;Clear y;Intro y. -Unfold ge in H2;Generalize (le_not_lt n0 n1 (le_S_n n0 n1 H2));Intro; - ElimType False;Auto. -Rewrite y;Unfold Rge;Right;Trivial. -Unfold ge in H0;Generalize (H0 (S n0) H1 (lt_le_S n0 n1 y));Intro; - Unfold Un_growing in H1; - Apply (Rge_trans (Un (S n1)) (Un n1) (Un (S n0)) - (Rle_sym1 (Un n1) (Un (S n1)) (H1 n1)) H3). +Lemma growing_prop : + forall n m:nat, Un_growing -> (n >= m)%nat -> Un n >= Un m. +double induction n m; intros. +unfold Rge in |- *; right; trivial. +elimtype False; unfold ge in H1; generalize (le_Sn_O n0); intro; auto. +cut (n0 >= 0)%nat. +generalize H0; intros; unfold Un_growing in H0; + apply + (Rge_trans (Un (S n0)) (Un n0) (Un 0) (Rle_ge (Un n0) (Un (S n0)) (H0 n0)) + (H 0%nat H2 H3)). +elim n0; auto. +elim (lt_eq_lt_dec n1 n0); intro y. +elim y; clear y; intro y. +unfold ge in H2; generalize (le_not_lt n0 n1 (le_S_n n0 n1 H2)); intro; + elimtype False; auto. +rewrite y; unfold Rge in |- *; right; trivial. +unfold ge in H0; generalize (H0 (S n0) H1 (lt_le_S n0 n1 y)); intro; + unfold Un_growing in H1; + apply + (Rge_trans (Un (S n1)) (Un n1) (Un (S n0)) + (Rle_ge (Un n1) (Un (S n1)) (H1 n1)) H3). Qed. (* classical is needed: [not_all_not_ex] *) (*********) -Lemma Un_cv_crit:Un_growing->(bound EUn)->(ExT [l:R] (Un_cv l)). -Unfold Un_growing Un_cv;Intros; - Generalize (complet_weak EUn H0 EUn_noempty);Intro; - Elim H1;Clear H1;Intros;Split with x;Intros; - Unfold is_lub in H1;Unfold bound in H0;Unfold is_upper_bound in H0 H1; - Elim H0;Clear H0;Intros;Elim H1;Clear H1;Intros; - Generalize (H3 x0 H0);Intro;Cut (n:nat)(Rle (Un n) x);Intro. -Cut (Ex [N:nat] (Rlt (Rminus x eps) (Un N))). -Intro;Elim H6;Clear H6;Intros;Split with x1. -Intros;Unfold R_dist;Apply (Rabsolu_def1 (Rminus (Un n) x) eps). -Unfold Rgt in H2; - Apply (Rle_lt_trans (Rminus (Un n) x) R0 eps - (Rle_minus (Un n) x (H5 n)) H2). -Fold Un_growing in H;Generalize (growing_prop n x1 H H7);Intro; - Generalize (Rlt_le_trans (Rminus x eps) (Un x1) (Un n) H6 - (Rle_sym2 (Un x1) (Un n) H8));Intro; - Generalize (Rlt_compatibility (Ropp x) (Rminus x eps) (Un n) H9); - Unfold Rminus;Rewrite <-(Rplus_assoc (Ropp x) x (Ropp eps)); - Rewrite (Rplus_sym (Ropp x) (Un n));Fold (Rminus (Un n) x); - Rewrite Rplus_Ropp_l;Rewrite (let (H1,H2)=(Rplus_ne (Ropp eps)) in H2); - Trivial. -Cut ~((N:nat)(Rge (Rminus x eps) (Un N))). -Intro;Apply (not_all_not_ex nat ([N:nat](Rlt (Rminus x eps) (Un N)))); - Red;Intro;Red in H6;Elim H6;Clear H6;Intro; - Apply (Rlt_not_ge (Rminus x eps) (Un N) (H7 N)). -Red;Intro;Cut (N:nat)(Rle (Un N) (Rminus x eps)). -Intro;Generalize (Un_bound_imp (Rminus x eps) H7);Intro; - Unfold is_upper_bound in H8;Generalize (H3 (Rminus x eps) H8);Intro; - Generalize (Rle_minus x (Rminus x eps) H9);Unfold Rminus; - Rewrite Ropp_distr1;Rewrite <- Rplus_assoc;Rewrite Rplus_Ropp_r; - Rewrite (let (H1,H2)=(Rplus_ne (Ropp (Ropp eps))) in H2); - Rewrite Ropp_Ropp;Intro;Unfold Rgt in H2; - Generalize (Rle_not eps R0 H2);Intro;Auto. -Intro;Elim (H6 N);Intro;Unfold Rle. -Left;Unfold Rgt in H7;Assumption. -Right;Auto. -Apply (H1 (Un n) (Un_in_EUn n)). +Lemma Un_cv_crit : Un_growing -> bound EUn -> exists l : R | Un_cv l. +unfold Un_growing, Un_cv in |- *; intros; + generalize (completeness_weak EUn H0 EUn_noempty); + intro; elim H1; clear H1; intros; split with x; intros; + unfold is_lub in H1; unfold bound in H0; unfold is_upper_bound in H0, H1; + elim H0; clear H0; intros; elim H1; clear H1; intros; + generalize (H3 x0 H0); intro; cut (forall n:nat, Un n <= x); + intro. +cut ( exists N : nat | x - eps < Un N). +intro; elim H6; clear H6; intros; split with x1. +intros; unfold R_dist in |- *; apply (Rabs_def1 (Un n - x) eps). +unfold Rgt in H2; + apply (Rle_lt_trans (Un n - x) 0 eps (Rle_minus (Un n) x (H5 n)) H2). +fold Un_growing in H; generalize (growing_prop n x1 H H7); intro; + generalize + (Rlt_le_trans (x - eps) (Un x1) (Un n) H6 (Rge_le (Un n) (Un x1) H8)); + intro; generalize (Rplus_lt_compat_l (- x) (x - eps) (Un n) H9); + unfold Rminus in |- *; rewrite <- (Rplus_assoc (- x) x (- eps)); + rewrite (Rplus_comm (- x) (Un n)); fold (Un n - x) in |- *; + rewrite Rplus_opp_l; rewrite (let (H1, H2) := Rplus_ne (- eps) in H2); + trivial. +cut (~ (forall N:nat, x - eps >= Un N)). +intro; apply (not_all_not_ex nat (fun N:nat => x - eps < Un N)); red in |- *; + intro; red in H6; elim H6; clear H6; intro; + apply (Rnot_lt_ge (x - eps) (Un N) (H7 N)). +red in |- *; intro; cut (forall N:nat, Un N <= x - eps). +intro; generalize (Un_bound_imp (x - eps) H7); intro; + unfold is_upper_bound in H8; generalize (H3 (x - eps) H8); + intro; generalize (Rle_minus x (x - eps) H9); unfold Rminus in |- *; + rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r; + rewrite (let (H1, H2) := Rplus_ne (- - eps) in H2); + rewrite Ropp_involutive; intro; unfold Rgt in H2; + generalize (Rgt_not_le eps 0 H2); intro; auto. +intro; elim (H6 N); intro; unfold Rle in |- *. +left; unfold Rgt in H7; assumption. +right; auto. +apply (H1 (Un n) (Un_in_EUn n)). Qed. (*********) -Lemma finite_greater:(N:nat)(ExT [M:R] (n:nat)(le n N)->(Rle (Un n) M)). -Intro;Induction N. -Split with (Un O);Intros;Rewrite (le_n_O_eq n H); - Apply (eq_Rle (Un (n)) (Un (n)) (refl_eqT R (Un (n)))). -Elim HrecN;Clear HrecN;Intros;Split with (Rmax (Un (S N)) x);Intros; - Elim (Rmax_Rle (Un (S N)) x (Un n));Intros;Clear H1;Inversion H0. -Rewrite <-H1;Rewrite <-H1 in H2; - Apply (H2 (or_introl (Rle (Un n) (Un n)) (Rle (Un n) x) - (eq_Rle (Un n) (Un n) (refl_eqT R (Un n))))). -Apply (H2 (or_intror (Rle (Un n) (Un (S N))) (Rle (Un n) x) - (H n H3))). +Lemma finite_greater : + forall N:nat, exists M : R | (forall n:nat, (n <= N)%nat -> Un n <= M). +intro; induction N as [| N HrecN]. +split with (Un 0); intros; rewrite (le_n_O_eq n H); + apply (Req_le (Un n) (Un n) (refl_equal (Un n))). +elim HrecN; clear HrecN; intros; split with (Rmax (Un (S N)) x); intros; + elim (Rmax_Rle (Un (S N)) x (Un n)); intros; clear H1; + inversion H0. +rewrite <- H1; rewrite <- H1 in H2; + apply + (H2 (or_introl (Un n <= x) (Req_le (Un n) (Un n) (refl_equal (Un n))))). +apply (H2 (or_intror (Un n <= Un (S N)) (H n H3))). Qed. (*********) -Lemma cauchy_bound:Cauchy_crit->(bound EUn). -Unfold Cauchy_crit bound;Intros;Unfold is_upper_bound; - Unfold Rgt in H;Elim (H R1 Rlt_R0_R1);Clear H;Intros; - Generalize (H x);Intro;Generalize (le_dec x);Intro; - Elim (finite_greater x);Intros;Split with (Rmax x0 (Rplus (Un x) R1)); - Clear H;Intros;Unfold EUn in H;Elim H;Clear H;Intros;Elim (H1 x2); - Clear H1;Intro y. -Unfold ge in H0;Generalize (H0 x2 (le_n x) y);Clear H0;Intro; - Rewrite <- H in H0;Unfold R_dist in H0; - Elim (Rabsolu_def2 (Rminus (Un x) x1) R1 H0);Clear H0;Intros; - Elim (Rmax_Rle x0 (Rplus (Un x) R1) x1);Intros;Apply H4;Clear H3 H4; - Right;Clear H H0 y;Apply (Rlt_le x1 (Rplus (Un x) R1)); - Generalize (Rlt_minus (Ropp R1) (Rminus (Un x) x1) H1);Clear H1; - Intro;Apply (Rminus_lt x1 (Rplus (Un x) R1)); - Cut (Rminus (Ropp R1) (Rminus (Un x) x1))== - (Rminus x1 (Rplus (Un x) R1));[Intro;Rewrite H0 in H;Assumption|Ring]. -Generalize (H2 x2 y);Clear H2 H0;Intro;Rewrite<-H in H0; - Elim (Rmax_Rle x0 (Rplus (Un x) R1) x1);Intros;Clear H1;Apply H2; - Left;Assumption. +Lemma cauchy_bound : Cauchy_crit -> bound EUn. +unfold Cauchy_crit, bound in |- *; intros; unfold is_upper_bound in |- *; + unfold Rgt in H; elim (H 1 Rlt_0_1); clear H; intros; + generalize (H x); intro; generalize (le_dec x); intro; + elim (finite_greater x); intros; split with (Rmax x0 (Un x + 1)); + clear H; intros; unfold EUn in H; elim H; clear H; + intros; elim (H1 x2); clear H1; intro y. +unfold ge in H0; generalize (H0 x2 (le_n x) y); clear H0; intro; + rewrite <- H in H0; unfold R_dist in H0; elim (Rabs_def2 (Un x - x1) 1 H0); + clear H0; intros; elim (Rmax_Rle x0 (Un x + 1) x1); + intros; apply H4; clear H3 H4; right; clear H H0 y; + apply (Rlt_le x1 (Un x + 1)); generalize (Rlt_minus (-1) (Un x - x1) H1); + clear H1; intro; apply (Rminus_lt x1 (Un x + 1)); + cut (-1 - (Un x - x1) = x1 - (Un x + 1)); + [ intro; rewrite H0 in H; assumption | ring ]. +generalize (H2 x2 y); clear H2 H0; intro; rewrite <- H in H0; + elim (Rmax_Rle x0 (Un x + 1) x1); intros; clear H1; + apply H2; left; assumption. Qed. End sequence. @@ -176,104 +184,92 @@ End sequence. Section Isequence. (*********) -Variable An:nat->R. +Variable An : nat -> R. (*********) -Definition Pser:R->R->Prop:=[x,l:R] - (infinit_sum [n:nat](Rmult (An n) (pow x n)) l). +Definition Pser (x l:R) : Prop := infinit_sum (fun n:nat => An n * x ^ n) l. End Isequence. -Lemma GP_infinite: - (x:R) (Rlt (Rabsolu x) R1) - -> (Pser ([n:nat] R1) x (Rinv(Rminus R1 x))). -Intros;Unfold Pser; Unfold infinit_sum;Intros;Elim (Req_EM x R0). -Intros;Exists O; Intros;Rewrite H1;Rewrite minus_R0;Rewrite Rinv_R1; - Cut (sum_f_R0 [n0:nat](Rmult R1 (pow R0 n0)) n)==R1. -Intros; Rewrite H3;Rewrite R_dist_eq;Auto. -Elim n; Simpl. -Ring. -Intros;Rewrite H3;Ring. -Intro;Cut (Rlt R0 - (Rmult eps (Rmult (Rabsolu (Rminus R1 x)) - (Rabsolu (Rinv x))))). -Intro;Elim (pow_lt_1_zero x H - (Rmult eps (Rmult (Rabsolu (Rminus R1 x)) - (Rabsolu (Rinv x)))) - H2);Intro N; Intros;Exists N; Intros; - Cut (sum_f_R0 [n0:nat](Rmult R1 (pow x n0)) n)== - (sum_f_R0 [n0:nat](pow x n0) n). -Intros; Rewrite H5;Apply (Rlt_monotony_rev - (Rabsolu (Rminus R1 x)) - (R_dist (sum_f_R0 [n0:nat](pow x n0) n) - (Rinv (Rminus R1 x))) - eps). -Apply Rabsolu_pos_lt. -Apply Rminus_eq_contra. -Apply imp_not_Req. -Right; Unfold Rgt. -Apply (Rle_lt_trans x (Rabsolu x) R1). -Apply Rle_Rabsolu. -Assumption. -Unfold R_dist; Rewrite <- Rabsolu_mult. -Rewrite Rminus_distr. -Cut (Rmult (Rminus R1 x) (sum_f_R0 [n0:nat](pow x n0) n))== - (Ropp (Rmult(sum_f_R0 [n0:nat](pow x n0) n) - (Rminus x R1))). -Intro; Rewrite H6. -Rewrite GP_finite. -Rewrite Rinv_r. -Cut (Rminus (Ropp (Rminus (pow x (plus n (1))) R1)) R1)== - (Ropp (pow x (plus n (1)))). -Intro; Rewrite H7. -Rewrite Rabsolu_Ropp;Cut (plus n (S O))=(S n);Auto. -Intro H8;Rewrite H8;Simpl;Rewrite Rabsolu_mult; - Apply (Rlt_le_trans (Rmult (Rabsolu x) (Rabsolu (pow x n))) - (Rmult (Rabsolu x) - (Rmult eps - (Rmult (Rabsolu (Rminus R1 x)) - (Rabsolu (Rinv x))))) - (Rmult (Rabsolu (Rminus R1 x)) eps)). -Apply Rlt_monotony. -Apply Rabsolu_pos_lt. -Assumption. -Auto. -Cut (Rmult (Rabsolu x) - (Rmult eps (Rmult (Rabsolu (Rminus R1 x)) - (Rabsolu (Rinv x)))))== - (Rmult (Rmult (Rabsolu x) (Rabsolu (Rinv x))) - (Rmult eps (Rabsolu (Rminus R1 x)))). -Clear H8;Intros; Rewrite H8;Rewrite <- Rabsolu_mult;Rewrite Rinv_r. -Rewrite Rabsolu_R1;Cut (Rmult R1 (Rmult eps (Rabsolu (Rminus R1 x))))== - (Rmult (Rabsolu (Rminus R1 x)) eps). -Intros; Rewrite H9;Unfold Rle; Right; Reflexivity. -Ring. -Assumption. -Ring. -Ring. -Ring. -Apply Rminus_eq_contra. -Apply imp_not_Req. -Right; Unfold Rgt. -Apply (Rle_lt_trans x (Rabsolu x) R1). -Apply Rle_Rabsolu. -Assumption. -Ring; Ring. -Elim n; Simpl. -Ring. -Intros; Rewrite H5. -Ring. -Apply Rmult_lt_pos. -Auto. -Apply Rmult_lt_pos. -Apply Rabsolu_pos_lt. -Apply Rminus_eq_contra. -Apply imp_not_Req. -Right; Unfold Rgt. -Apply (Rle_lt_trans x (Rabsolu x) R1). -Apply Rle_Rabsolu. -Assumption. -Apply Rabsolu_pos_lt. -Apply Rinv_neq_R0. -Assumption. -Qed. +Lemma GP_infinite : + forall x:R, Rabs x < 1 -> Pser (fun n:nat => 1) x (/ (1 - x)). +intros; unfold Pser in |- *; unfold infinit_sum in |- *; intros; + elim (Req_dec x 0). +intros; exists 0%nat; intros; rewrite H1; rewrite Rminus_0_r; rewrite Rinv_1; + cut (sum_f_R0 (fun n0:nat => 1 * 0 ^ n0) n = 1). +intros; rewrite H3; rewrite R_dist_eq; auto. +elim n; simpl in |- *. +ring. +intros; rewrite H3; ring. +intro; cut (0 < eps * (Rabs (1 - x) * Rabs (/ x))). +intro; elim (pow_lt_1_zero x H (eps * (Rabs (1 - x) * Rabs (/ x))) H2); + intro N; intros; exists N; intros; + cut + (sum_f_R0 (fun n0:nat => 1 * x ^ n0) n = sum_f_R0 (fun n0:nat => x ^ n0) n). +intros; rewrite H5; + apply + (Rmult_lt_reg_l (Rabs (1 - x)) + (R_dist (sum_f_R0 (fun n0:nat => x ^ n0) n) (/ (1 - x))) eps). +apply Rabs_pos_lt. +apply Rminus_eq_contra. +apply Rlt_dichotomy_converse. +right; unfold Rgt in |- *. +apply (Rle_lt_trans x (Rabs x) 1). +apply RRle_abs. +assumption. +unfold R_dist in |- *; rewrite <- Rabs_mult. +rewrite Rmult_minus_distr_l. +cut + ((1 - x) * sum_f_R0 (fun n0:nat => x ^ n0) n = + - (sum_f_R0 (fun n0:nat => x ^ n0) n * (x - 1))). +intro; rewrite H6. +rewrite GP_finite. +rewrite Rinv_r. +cut (- (x ^ (n + 1) - 1) - 1 = - x ^ (n + 1)). +intro; rewrite H7. +rewrite Rabs_Ropp; cut ((n + 1)%nat = S n); auto. +intro H8; rewrite H8; simpl in |- *; rewrite Rabs_mult; + apply + (Rlt_le_trans (Rabs x * Rabs (x ^ n)) + (Rabs x * (eps * (Rabs (1 - x) * Rabs (/ x)))) ( + Rabs (1 - x) * eps)). +apply Rmult_lt_compat_l. +apply Rabs_pos_lt. +assumption. +auto. +cut + (Rabs x * (eps * (Rabs (1 - x) * Rabs (/ x))) = + Rabs x * Rabs (/ x) * (eps * Rabs (1 - x))). +clear H8; intros; rewrite H8; rewrite <- Rabs_mult; rewrite Rinv_r. +rewrite Rabs_R1; cut (1 * (eps * Rabs (1 - x)) = Rabs (1 - x) * eps). +intros; rewrite H9; unfold Rle in |- *; right; reflexivity. +ring. +assumption. +ring. +ring. +ring. +apply Rminus_eq_contra. +apply Rlt_dichotomy_converse. +right; unfold Rgt in |- *. +apply (Rle_lt_trans x (Rabs x) 1). +apply RRle_abs. +assumption. +ring; ring. +elim n; simpl in |- *. +ring. +intros; rewrite H5. +ring. +apply Rmult_lt_0_compat. +auto. +apply Rmult_lt_0_compat. +apply Rabs_pos_lt. +apply Rminus_eq_contra. +apply Rlt_dichotomy_converse. +right; unfold Rgt in |- *. +apply (Rle_lt_trans x (Rabs x) 1). +apply RRle_abs. +assumption. +apply Rabs_pos_lt. +apply Rinv_neq_0_compat. +assumption. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Rsigma.v b/theories/Reals/Rsigma.v index fd14d2c8c..592ddf68f 100644 --- a/theories/Reals/Rsigma.v +++ b/theories/Reals/Rsigma.v @@ -8,110 +8,133 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Rseries. -Require PartSum. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import Rseries. +Require Import PartSum. Open Local Scope R_scope. Set Implicit Arguments. Section Sigma. -Variable f : nat->R. +Variable f : nat -> R. -Definition sigma [low,high:nat] : R := (sum_f_R0 [k:nat](f (plus low k)) (minus high low)). +Definition sigma (low high:nat) : R := + sum_f_R0 (fun k:nat => f (low + k)) (high - low). -Theorem sigma_split : (low,high,k:nat) (le low k)->(lt k high)->``(sigma low high)==(sigma low k)+(sigma (S k) high)``. -Intros; Induction k. -Cut low = O. -Intro; Rewrite H1; Unfold sigma; Rewrite <- minus_n_n; Rewrite <- minus_n_O; Simpl; Replace (minus high (S O)) with (pred high). -Apply (decomp_sum [k:nat](f k)). -Assumption. -Apply pred_of_minus. -Inversion H; Reflexivity. -Cut (le low k)\/low=(S k). -Intro; Elim H1; Intro. -Replace (sigma low (S k)) with ``(sigma low k)+(f (S k))``. -Rewrite Rplus_assoc; Replace ``(f (S k))+(sigma (S (S k)) high)`` with (sigma (S k) high). -Apply Hreck. -Assumption. -Apply lt_trans with (S k); [Apply lt_n_Sn | Assumption]. -Unfold sigma; Replace (minus high (S (S k))) with (pred (minus high (S k))). -Pattern 3 (S k); Replace (S k) with (plus (S k) O); [Idtac | Ring]. -Replace (sum_f_R0 [k0:nat](f (plus (S (S k)) k0)) (pred (minus high (S k)))) with (sum_f_R0 [k0:nat](f (plus (S k) (S k0))) (pred (minus high (S k)))). -Apply (decomp_sum [i:nat](f (plus (S k) i))). -Apply lt_minus_O_lt; Assumption. -Apply sum_eq; Intros; Replace (plus (S k) (S i)) with (plus (S (S k)) i). -Reflexivity. -Apply INR_eq; Do 2 Rewrite plus_INR; Do 3 Rewrite S_INR; Ring. -Replace (minus high (S (S k))) with (minus (minus high (S k)) (S O)). -Apply pred_of_minus. -Apply INR_eq; Repeat Rewrite minus_INR. -Do 4 Rewrite S_INR; Ring. -Apply lt_le_S; Assumption. -Apply lt_le_weak; Assumption. -Apply lt_le_S; Apply lt_minus_O_lt; Assumption. -Unfold sigma; Replace (minus (S k) low) with (S (minus k low)). -Pattern 1 (S k); Replace (S k) with (plus low (S (minus k low))). -Symmetry; Apply (tech5 [i:nat](f (plus low i))). -Apply INR_eq; Rewrite plus_INR; Do 2 Rewrite S_INR; Rewrite minus_INR. -Ring. -Assumption. -Apply minus_Sn_m; Assumption. -Rewrite <- H2; Unfold sigma; Rewrite <- minus_n_n; Simpl; Replace (minus high (S low)) with (pred (minus high low)). -Replace (sum_f_R0 [k0:nat](f (S (plus low k0))) (pred (minus high low))) with (sum_f_R0 [k0:nat](f (plus low (S k0))) (pred (minus high low))). -Apply (decomp_sum [k0:nat](f (plus low k0))). -Apply lt_minus_O_lt. -Apply le_lt_trans with (S k); [Rewrite H2; Apply le_n | Assumption]. -Apply sum_eq; Intros; Replace (S (plus low i)) with (plus low (S i)). -Reflexivity. -Apply INR_eq; Rewrite plus_INR; Do 2 Rewrite S_INR; Rewrite plus_INR; Ring. -Replace (minus high (S low)) with (minus (minus high low) (S O)). -Apply pred_of_minus. -Apply INR_eq; Repeat Rewrite minus_INR. -Do 2 Rewrite S_INR; Ring. -Apply lt_le_S; Rewrite H2; Assumption. -Rewrite H2; Apply lt_le_weak; Assumption. -Apply lt_le_S; Apply lt_minus_O_lt; Rewrite H2; Assumption. -Inversion H; [ - Right; Reflexivity -| Left; Assumption]. +Theorem sigma_split : + forall low high k:nat, + (low <= k)%nat -> + (k < high)%nat -> sigma low high = sigma low k + sigma (S k) high. +intros; induction k as [| k Hreck]. +cut (low = 0%nat). +intro; rewrite H1; unfold sigma in |- *; rewrite <- minus_n_n; + rewrite <- minus_n_O; simpl in |- *; replace (high - 1)%nat with (pred high). +apply (decomp_sum (fun k:nat => f k)). +assumption. +apply pred_of_minus. +inversion H; reflexivity. +cut ((low <= k)%nat \/ low = S k). +intro; elim H1; intro. +replace (sigma low (S k)) with (sigma low k + f (S k)). +rewrite Rplus_assoc; + replace (f (S k) + sigma (S (S k)) high) with (sigma (S k) high). +apply Hreck. +assumption. +apply lt_trans with (S k); [ apply lt_n_Sn | assumption ]. +unfold sigma in |- *; replace (high - S (S k))%nat with (pred (high - S k)). +pattern (S k) at 3 in |- *; replace (S k) with (S k + 0)%nat; + [ idtac | ring ]. +replace (sum_f_R0 (fun k0:nat => f (S (S k) + k0)) (pred (high - S k))) with + (sum_f_R0 (fun k0:nat => f (S k + S k0)) (pred (high - S k))). +apply (decomp_sum (fun i:nat => f (S k + i))). +apply lt_minus_O_lt; assumption. +apply sum_eq; intros; replace (S k + S i)%nat with (S (S k) + i)%nat. +reflexivity. +apply INR_eq; do 2 rewrite plus_INR; do 3 rewrite S_INR; ring. +replace (high - S (S k))%nat with (high - S k - 1)%nat. +apply pred_of_minus. +apply INR_eq; repeat rewrite minus_INR. +do 4 rewrite S_INR; ring. +apply lt_le_S; assumption. +apply lt_le_weak; assumption. +apply lt_le_S; apply lt_minus_O_lt; assumption. +unfold sigma in |- *; replace (S k - low)%nat with (S (k - low)). +pattern (S k) at 1 in |- *; replace (S k) with (low + S (k - low))%nat. +symmetry in |- *; apply (tech5 (fun i:nat => f (low + i))). +apply INR_eq; rewrite plus_INR; do 2 rewrite S_INR; rewrite minus_INR. +ring. +assumption. +apply minus_Sn_m; assumption. +rewrite <- H2; unfold sigma in |- *; rewrite <- minus_n_n; simpl in |- *; + replace (high - S low)%nat with (pred (high - low)). +replace (sum_f_R0 (fun k0:nat => f (S (low + k0))) (pred (high - low))) with + (sum_f_R0 (fun k0:nat => f (low + S k0)) (pred (high - low))). +apply (decomp_sum (fun k0:nat => f (low + k0))). +apply lt_minus_O_lt. +apply le_lt_trans with (S k); [ rewrite H2; apply le_n | assumption ]. +apply sum_eq; intros; replace (S (low + i)) with (low + S i)%nat. +reflexivity. +apply INR_eq; rewrite plus_INR; do 2 rewrite S_INR; rewrite plus_INR; ring. +replace (high - S low)%nat with (high - low - 1)%nat. +apply pred_of_minus. +apply INR_eq; repeat rewrite minus_INR. +do 2 rewrite S_INR; ring. +apply lt_le_S; rewrite H2; assumption. +rewrite H2; apply lt_le_weak; assumption. +apply lt_le_S; apply lt_minus_O_lt; rewrite H2; assumption. +inversion H; [ right; reflexivity | left; assumption ]. Qed. -Theorem sigma_diff : (low,high,k:nat) (le low k) -> (lt k high )->``(sigma low high)-(sigma low k)==(sigma (S k) high)``. -Intros low high k H1 H2; Symmetry; Rewrite -> (sigma_split H1 H2); Ring. +Theorem sigma_diff : + forall low high k:nat, + (low <= k)%nat -> + (k < high)%nat -> sigma low high - sigma low k = sigma (S k) high. +intros low high k H1 H2; symmetry in |- *; rewrite (sigma_split H1 H2); ring. Qed. -Theorem sigma_diff_neg : (low,high,k:nat) (le low k) -> (lt k high)-> ``(sigma low k)-(sigma low high)==-(sigma (S k) high)``. -Intros low high k H1 H2; Rewrite -> (sigma_split H1 H2); Ring. +Theorem sigma_diff_neg : + forall low high k:nat, + (low <= k)%nat -> + (k < high)%nat -> sigma low k - sigma low high = - sigma (S k) high. +intros low high k H1 H2; rewrite (sigma_split H1 H2); ring. Qed. -Theorem sigma_first : (low,high:nat) (lt low high) -> ``(sigma low high)==(f low)+(sigma (S low) high)``. -Intros low high H1; Generalize (lt_le_S low high H1); Intro H2; Generalize (lt_le_weak low high H1); Intro H3; Replace ``(f low)`` with ``(sigma low low)``. -Apply sigma_split. -Apply le_n. -Assumption. -Unfold sigma; Rewrite <- minus_n_n. -Simpl. -Replace (plus low O) with low; [Reflexivity | Ring]. +Theorem sigma_first : + forall low high:nat, + (low < high)%nat -> sigma low high = f low + sigma (S low) high. +intros low high H1; generalize (lt_le_S low high H1); intro H2; + generalize (lt_le_weak low high H1); intro H3; + replace (f low) with (sigma low low). +apply sigma_split. +apply le_n. +assumption. +unfold sigma in |- *; rewrite <- minus_n_n. +simpl in |- *. +replace (low + 0)%nat with low; [ reflexivity | ring ]. Qed. -Theorem sigma_last : (low,high:nat) (lt low high) -> ``(sigma low high)==(f high)+(sigma low (pred high))``. -Intros low high H1; Generalize (lt_le_S low high H1); Intro H2; Generalize (lt_le_weak low high H1); Intro H3; Replace ``(f high)`` with ``(sigma high high)``. -Rewrite Rplus_sym; Cut high = (S (pred high)). -Intro; Pattern 3 high; Rewrite H. -Apply sigma_split. -Apply le_S_n; Rewrite <- H; Apply lt_le_S; Assumption. -Apply lt_pred_n_n; Apply le_lt_trans with low; [Apply le_O_n | Assumption]. -Apply S_pred with O; Apply le_lt_trans with low; [Apply le_O_n | Assumption]. -Unfold sigma; Rewrite <- minus_n_n; Simpl; Replace (plus high O) with high; [Reflexivity | Ring]. +Theorem sigma_last : + forall low high:nat, + (low < high)%nat -> sigma low high = f high + sigma low (pred high). +intros low high H1; generalize (lt_le_S low high H1); intro H2; + generalize (lt_le_weak low high H1); intro H3; + replace (f high) with (sigma high high). +rewrite Rplus_comm; cut (high = S (pred high)). +intro; pattern high at 3 in |- *; rewrite H. +apply sigma_split. +apply le_S_n; rewrite <- H; apply lt_le_S; assumption. +apply lt_pred_n_n; apply le_lt_trans with low; [ apply le_O_n | assumption ]. +apply S_pred with 0%nat; apply le_lt_trans with low; + [ apply le_O_n | assumption ]. +unfold sigma in |- *; rewrite <- minus_n_n; simpl in |- *; + replace (high + 0)%nat with high; [ reflexivity | ring ]. Qed. -Theorem sigma_eq_arg : (low:nat) (sigma low low)==(f low). -Intro; Unfold sigma; Rewrite <- minus_n_n. -Simpl; Replace (plus low O) with low; [Reflexivity | Ring]. +Theorem sigma_eq_arg : forall low:nat, sigma low low = f low. +intro; unfold sigma in |- *; rewrite <- minus_n_n. +simpl in |- *; replace (low + 0)%nat with low; [ reflexivity | ring ]. Qed. -End Sigma. +End Sigma.
\ No newline at end of file diff --git a/theories/Reals/Rsqrt_def.v b/theories/Reals/Rsqrt_def.v index ebdece374..b123f1bb7 100644 --- a/theories/Reals/Rsqrt_def.v +++ b/theories/Reals/Rsqrt_def.v @@ -8,681 +8,755 @@ (*i $Id$ i*) -Require Sumbool. -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Ranalysis1. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Sumbool. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Ranalysis1. Open Local Scope R_scope. -Fixpoint Dichotomy_lb [x,y:R;P:R->bool;N:nat] : R := -Cases N of - O => x -| (S n) => let down = (Dichotomy_lb x y P n) in let up = (Dichotomy_ub x y P n) in let z = ``(down+up)/2`` in if (P z) then down else z -end -with Dichotomy_ub [x,y:R;P:R->bool;N:nat] : R := -Cases N of - O => y -| (S n) => let down = (Dichotomy_lb x y P n) in let up = (Dichotomy_ub x y P n) in let z = ``(down+up)/2`` in if (P z) then z else up -end. +Fixpoint Dichotomy_lb (x y:R) (P:R -> bool) (N:nat) {struct N} : R := + match N with + | O => x + | S n => + let down := Dichotomy_lb x y P n in + let up := Dichotomy_ub x y P n in + let z := (down + up) / 2 in if P z then down else z + end + + with Dichotomy_ub (x y:R) (P:R -> bool) (N:nat) {struct N} : R := + match N with + | O => y + | S n => + let down := Dichotomy_lb x y P n in + let up := Dichotomy_ub x y P n in + let z := (down + up) / 2 in if P z then z else up + end. -Definition dicho_lb [x,y:R;P:R->bool] : nat->R := [N:nat](Dichotomy_lb x y P N). -Definition dicho_up [x,y:R;P:R->bool] : nat->R := [N:nat](Dichotomy_ub x y P N). +Definition dicho_lb (x y:R) (P:R -> bool) (N:nat) : R := Dichotomy_lb x y P N. +Definition dicho_up (x y:R) (P:R -> bool) (N:nat) : R := Dichotomy_ub x y P N. (**********) -Lemma dicho_comp : (x,y:R;P:R->bool;n:nat) ``x<=y`` -> ``(dicho_lb x y P n)<=(dicho_up x y P n)``. -Intros. -Induction n. -Simpl; Assumption. -Simpl. -Case (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``). -Unfold Rdiv; Apply Rle_monotony_contra with ``2``. -Sup0. -Pattern 1 ``2``; Rewrite Rmult_sym. -Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]. -Rewrite Rmult_1r. -Rewrite double. -Apply Rle_compatibility. -Assumption. -Unfold Rdiv; Apply Rle_monotony_contra with ``2``. -Sup0. -Pattern 3 ``2``; Rewrite Rmult_sym. -Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]. -Rewrite Rmult_1r. -Rewrite double. -Rewrite <- (Rplus_sym (Dichotomy_ub x y P n)). -Apply Rle_compatibility. -Assumption. +Lemma dicho_comp : + forall (x y:R) (P:R -> bool) (n:nat), + x <= y -> dicho_lb x y P n <= dicho_up x y P n. +intros. +induction n as [| n Hrecn]. +simpl in |- *; assumption. +simpl in |- *. +case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)). +unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. +prove_sup0. +pattern 2 at 1 in |- *; rewrite Rmult_comm. +rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]. +rewrite Rmult_1_r. +rewrite double. +apply Rplus_le_compat_l. +assumption. +unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. +prove_sup0. +pattern 2 at 3 in |- *; rewrite Rmult_comm. +rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]. +rewrite Rmult_1_r. +rewrite double. +rewrite <- (Rplus_comm (Dichotomy_ub x y P n)). +apply Rplus_le_compat_l. +assumption. Qed. -Lemma dicho_lb_growing : (x,y:R;P:R->bool) ``x<=y`` -> (Un_growing (dicho_lb x y P)). -Intros. -Unfold Un_growing. -Intro. -Simpl. -Case (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``). -Right; Reflexivity. -Unfold Rdiv; Apply Rle_monotony_contra with ``2``. -Sup0. -Pattern 1 ``2``; Rewrite Rmult_sym. -Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]. -Rewrite Rmult_1r. -Rewrite double. -Apply Rle_compatibility. -Replace (Dichotomy_ub x y P n) with (dicho_up x y P n); [Apply dicho_comp; Assumption | Reflexivity]. +Lemma dicho_lb_growing : + forall (x y:R) (P:R -> bool), x <= y -> Un_growing (dicho_lb x y P). +intros. +unfold Un_growing in |- *. +intro. +simpl in |- *. +case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)). +right; reflexivity. +unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. +prove_sup0. +pattern 2 at 1 in |- *; rewrite Rmult_comm. +rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]. +rewrite Rmult_1_r. +rewrite double. +apply Rplus_le_compat_l. +replace (Dichotomy_ub x y P n) with (dicho_up x y P n); + [ apply dicho_comp; assumption | reflexivity ]. Qed. -Lemma dicho_up_decreasing : (x,y:R;P:R->bool) ``x<=y`` -> (Un_decreasing (dicho_up x y P)). -Intros. -Unfold Un_decreasing. -Intro. -Simpl. -Case (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``). -Unfold Rdiv; Apply Rle_monotony_contra with ``2``. -Sup0. -Pattern 3 ``2``; Rewrite Rmult_sym. -Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]. -Rewrite Rmult_1r. -Rewrite double. -Replace (Dichotomy_ub x y P n) with (dicho_up x y P n); [Idtac | Reflexivity]. -Replace (Dichotomy_lb x y P n) with (dicho_lb x y P n); [Idtac | Reflexivity]. -Rewrite <- (Rplus_sym ``(dicho_up x y P n)``). -Apply Rle_compatibility. -Apply dicho_comp; Assumption. -Right; Reflexivity. +Lemma dicho_up_decreasing : + forall (x y:R) (P:R -> bool), x <= y -> Un_decreasing (dicho_up x y P). +intros. +unfold Un_decreasing in |- *. +intro. +simpl in |- *. +case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)). +unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. +prove_sup0. +pattern 2 at 3 in |- *; rewrite Rmult_comm. +rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ idtac | discrR ]. +rewrite Rmult_1_r. +rewrite double. +replace (Dichotomy_ub x y P n) with (dicho_up x y P n); + [ idtac | reflexivity ]. +replace (Dichotomy_lb x y P n) with (dicho_lb x y P n); + [ idtac | reflexivity ]. +rewrite <- (Rplus_comm (dicho_up x y P n)). +apply Rplus_le_compat_l. +apply dicho_comp; assumption. +right; reflexivity. Qed. -Lemma dicho_lb_maj_y : (x,y:R;P:R->bool) ``x<=y`` -> (n:nat)``(dicho_lb x y P n)<=y``. -Intros. -Induction n. -Simpl; Assumption. -Simpl. -Case (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``). -Assumption. -Unfold Rdiv; Apply Rle_monotony_contra with ``2``. -Sup0. -Pattern 3 ``2``; Rewrite Rmult_sym. -Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Rewrite Rmult_1r | DiscrR]. -Rewrite double; Apply Rplus_le. -Assumption. -Pattern 2 y; Replace y with (Dichotomy_ub x y P O); [Idtac | Reflexivity]. -Apply decreasing_prop. -Assert H0 := (dicho_up_decreasing x y P H). -Assumption. -Apply le_O_n. +Lemma dicho_lb_maj_y : + forall (x y:R) (P:R -> bool), x <= y -> forall n:nat, dicho_lb x y P n <= y. +intros. +induction n as [| n Hrecn]. +simpl in |- *; assumption. +simpl in |- *. +case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)). +assumption. +unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. +prove_sup0. +pattern 2 at 3 in |- *; rewrite Rmult_comm. +rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ rewrite Rmult_1_r | discrR ]. +rewrite double; apply Rplus_le_compat. +assumption. +pattern y at 2 in |- *; replace y with (Dichotomy_ub x y P 0); + [ idtac | reflexivity ]. +apply decreasing_prop. +assert (H0 := dicho_up_decreasing x y P H). +assumption. +apply le_O_n. Qed. -Lemma dicho_lb_maj : (x,y:R;P:R->bool) ``x<=y`` -> (has_ub (dicho_lb x y P)). -Intros. -Cut (n:nat)``(dicho_lb x y P n)<=y``. -Intro. -Unfold has_ub. -Unfold bound. -Exists y. -Unfold is_upper_bound. -Intros. -Elim H1; Intros. -Rewrite H2; Apply H0. -Apply dicho_lb_maj_y; Assumption. +Lemma dicho_lb_maj : + forall (x y:R) (P:R -> bool), x <= y -> has_ub (dicho_lb x y P). +intros. +cut (forall n:nat, dicho_lb x y P n <= y). +intro. +unfold has_ub in |- *. +unfold bound in |- *. +exists y. +unfold is_upper_bound in |- *. +intros. +elim H1; intros. +rewrite H2; apply H0. +apply dicho_lb_maj_y; assumption. Qed. -Lemma dicho_up_min_x : (x,y:R;P:R->bool) ``x<=y`` -> (n:nat)``x<=(dicho_up x y P n)``. -Intros. -Induction n. -Simpl; Assumption. -Simpl. -Case (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``). -Unfold Rdiv; Apply Rle_monotony_contra with ``2``. -Sup0. -Pattern 1 ``2``; Rewrite Rmult_sym. -Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Rewrite Rmult_1r | DiscrR]. -Rewrite double; Apply Rplus_le. -Pattern 1 x; Replace x with (Dichotomy_lb x y P O); [Idtac | Reflexivity]. -Apply tech9. -Assert H0 := (dicho_lb_growing x y P H). -Assumption. -Apply le_O_n. -Assumption. -Assumption. +Lemma dicho_up_min_x : + forall (x y:R) (P:R -> bool), x <= y -> forall n:nat, x <= dicho_up x y P n. +intros. +induction n as [| n Hrecn]. +simpl in |- *; assumption. +simpl in |- *. +case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)). +unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. +prove_sup0. +pattern 2 at 1 in |- *; rewrite Rmult_comm. +rewrite Rmult_assoc; rewrite <- Rinv_l_sym; [ rewrite Rmult_1_r | discrR ]. +rewrite double; apply Rplus_le_compat. +pattern x at 1 in |- *; replace x with (Dichotomy_lb x y P 0); + [ idtac | reflexivity ]. +apply tech9. +assert (H0 := dicho_lb_growing x y P H). +assumption. +apply le_O_n. +assumption. +assumption. Qed. -Lemma dicho_up_min : (x,y:R;P:R->bool) ``x<=y`` -> (has_lb (dicho_up x y P)). -Intros. -Cut (n:nat)``x<=(dicho_up x y P n)``. -Intro. -Unfold has_lb. -Unfold bound. -Exists ``-x``. -Unfold is_upper_bound. -Intros. -Elim H1; Intros. -Rewrite H2. -Unfold opp_seq. -Apply Rle_Ropp1. -Apply H0. -Apply dicho_up_min_x; Assumption. +Lemma dicho_up_min : + forall (x y:R) (P:R -> bool), x <= y -> has_lb (dicho_up x y P). +intros. +cut (forall n:nat, x <= dicho_up x y P n). +intro. +unfold has_lb in |- *. +unfold bound in |- *. +exists (- x). +unfold is_upper_bound in |- *. +intros. +elim H1; intros. +rewrite H2. +unfold opp_seq in |- *. +apply Ropp_le_contravar. +apply H0. +apply dicho_up_min_x; assumption. Qed. -Lemma dicho_lb_cv : (x,y:R;P:R->bool) ``x<=y`` -> (sigTT R [l:R](Un_cv (dicho_lb x y P) l)). -Intros. -Apply growing_cv. -Apply dicho_lb_growing; Assumption. -Apply dicho_lb_maj; Assumption. +Lemma dicho_lb_cv : + forall (x y:R) (P:R -> bool), + x <= y -> sigT (fun l:R => Un_cv (dicho_lb x y P) l). +intros. +apply growing_cv. +apply dicho_lb_growing; assumption. +apply dicho_lb_maj; assumption. Qed. -Lemma dicho_up_cv : (x,y:R;P:R->bool) ``x<=y`` -> (sigTT R [l:R](Un_cv (dicho_up x y P) l)). -Intros. -Apply decreasing_cv. -Apply dicho_up_decreasing; Assumption. -Apply dicho_up_min; Assumption. +Lemma dicho_up_cv : + forall (x y:R) (P:R -> bool), + x <= y -> sigT (fun l:R => Un_cv (dicho_up x y P) l). +intros. +apply decreasing_cv. +apply dicho_up_decreasing; assumption. +apply dicho_up_min; assumption. Qed. -Lemma dicho_lb_dicho_up : (x,y:R;P:R->bool;n:nat) ``x<=y`` -> ``(dicho_up x y P n)-(dicho_lb x y P n)==(y-x)/(pow 2 n)``. -Intros. -Induction n. -Simpl. -Unfold Rdiv; Rewrite Rinv_R1; Ring. -Simpl. -Case (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``). -Unfold Rdiv. -Replace ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))*/2- - (Dichotomy_lb x y P n)`` with ``((dicho_up x y P n)-(dicho_lb x y P n))/2``. -Unfold Rdiv; Rewrite Hrecn. -Unfold Rdiv. -Rewrite Rinv_Rmult. -Ring. -DiscrR. -Apply pow_nonzero; DiscrR. -Pattern 2 (Dichotomy_lb x y P n); Rewrite (double_var (Dichotomy_lb x y P n)); Unfold dicho_up dicho_lb Rminus Rdiv; Ring. -Replace ``(Dichotomy_ub x y P n)-((Dichotomy_lb x y P n)+ - (Dichotomy_ub x y P n))/2`` with ``((dicho_up x y P n)-(dicho_lb x y P n))/2``. -Unfold Rdiv; Rewrite Hrecn. -Unfold Rdiv. -Rewrite Rinv_Rmult. -Ring. -DiscrR. -Apply pow_nonzero; DiscrR. -Pattern 1 (Dichotomy_ub x y P n); Rewrite (double_var (Dichotomy_ub x y P n)); Unfold dicho_up dicho_lb Rminus Rdiv; Ring. +Lemma dicho_lb_dicho_up : + forall (x y:R) (P:R -> bool) (n:nat), + x <= y -> dicho_up x y P n - dicho_lb x y P n = (y - x) / 2 ^ n. +intros. +induction n as [| n Hrecn]. +simpl in |- *. +unfold Rdiv in |- *; rewrite Rinv_1; ring. +simpl in |- *. +case (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2)). +unfold Rdiv in |- *. +replace + ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) * / 2 - Dichotomy_lb x y P n) + with ((dicho_up x y P n - dicho_lb x y P n) / 2). +unfold Rdiv in |- *; rewrite Hrecn. +unfold Rdiv in |- *. +rewrite Rinv_mult_distr. +ring. +discrR. +apply pow_nonzero; discrR. +pattern (Dichotomy_lb x y P n) at 2 in |- *; + rewrite (double_var (Dichotomy_lb x y P n)); + unfold dicho_up, dicho_lb, Rminus, Rdiv in |- *; ring. +replace + (Dichotomy_ub x y P n - (Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2) + with ((dicho_up x y P n - dicho_lb x y P n) / 2). +unfold Rdiv in |- *; rewrite Hrecn. +unfold Rdiv in |- *. +rewrite Rinv_mult_distr. +ring. +discrR. +apply pow_nonzero; discrR. +pattern (Dichotomy_ub x y P n) at 1 in |- *; + rewrite (double_var (Dichotomy_ub x y P n)); + unfold dicho_up, dicho_lb, Rminus, Rdiv in |- *; ring. Qed. -Definition pow_2_n := [n:nat](pow ``2`` n). +Definition pow_2_n (n:nat) := 2 ^ n. -Lemma pow_2_n_neq_R0 : (n:nat) ``(pow_2_n n)<>0``. -Intro. -Unfold pow_2_n. -Apply pow_nonzero. -DiscrR. +Lemma pow_2_n_neq_R0 : forall n:nat, pow_2_n n <> 0. +intro. +unfold pow_2_n in |- *. +apply pow_nonzero. +discrR. Qed. -Lemma pow_2_n_growing : (Un_growing pow_2_n). -Unfold Un_growing. -Intro. -Replace (S n) with (plus n (1)); [Unfold pow_2_n; Rewrite pow_add | Ring]. -Pattern 1 (pow ``2`` n); Rewrite <- Rmult_1r. -Apply Rle_monotony. -Left; Apply pow_lt; Sup0. -Simpl. -Rewrite Rmult_1r. -Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Apply Rlt_R0_R1. +Lemma pow_2_n_growing : Un_growing pow_2_n. +unfold Un_growing in |- *. +intro. +replace (S n) with (n + 1)%nat; + [ unfold pow_2_n in |- *; rewrite pow_add | ring ]. +pattern (2 ^ n) at 1 in |- *; rewrite <- Rmult_1_r. +apply Rmult_le_compat_l. +left; apply pow_lt; prove_sup0. +simpl in |- *. +rewrite Rmult_1_r. +pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + apply Rlt_0_1. Qed. -Lemma pow_2_n_infty : (cv_infty pow_2_n). -Cut (N:nat)``(INR N)<=(pow 2 N)``. -Intros. -Unfold cv_infty. -Intro. -Case (total_order_T R0 M); Intro. -Elim s; Intro. -Pose N := (up M). -Cut `0<=N`. -Intro. -Elim (IZN N H0); Intros N0 H1. -Exists N0. -Intros. -Apply Rlt_le_trans with (INR N0). -Rewrite INR_IZR_INZ. -Rewrite <- H1. -Unfold N. -Assert H3 := (archimed M). -Elim H3; Intros; Assumption. -Apply Rle_trans with (pow_2_n N0). -Unfold pow_2_n; Apply H. -Apply Rle_sym2. -Apply growing_prop. -Apply pow_2_n_growing. -Assumption. -Apply le_IZR. -Unfold N. -Simpl. -Assert H0 := (archimed M); Elim H0; Intros. -Left; Apply Rlt_trans with M; Assumption. -Exists O; Intros. -Rewrite <- b. -Unfold pow_2_n; Apply pow_lt; Sup0. -Exists O; Intros. -Apply Rlt_trans with R0. -Assumption. -Unfold pow_2_n; Apply pow_lt; Sup0. -Induction N. -Simpl. -Left; Apply Rlt_R0_R1. -Intros. -Pattern 2 (S n); Replace (S n) with (plus n (1)); [Idtac | Ring]. -Rewrite S_INR; Rewrite pow_add. -Simpl. -Rewrite Rmult_1r. -Apply Rle_trans with ``(pow 2 n)``. -Rewrite <- (Rplus_sym R1). -Rewrite <- (Rmult_1r (INR n)). -Apply (poly n R1). -Apply Rlt_R0_R1. -Pattern 1 (pow ``2`` n); Rewrite <- Rplus_Or. -Rewrite <- (Rmult_sym ``2``). -Rewrite double. -Apply Rle_compatibility. -Left; Apply pow_lt; Sup0. +Lemma pow_2_n_infty : cv_infty pow_2_n. +cut (forall N:nat, INR N <= 2 ^ N). +intros. +unfold cv_infty in |- *. +intro. +case (total_order_T 0 M); intro. +elim s; intro. +pose (N := up M). +cut (0 <= N)%Z. +intro. +elim (IZN N H0); intros N0 H1. +exists N0. +intros. +apply Rlt_le_trans with (INR N0). +rewrite INR_IZR_INZ. +rewrite <- H1. +unfold N in |- *. +assert (H3 := archimed M). +elim H3; intros; assumption. +apply Rle_trans with (pow_2_n N0). +unfold pow_2_n in |- *; apply H. +apply Rge_le. +apply growing_prop. +apply pow_2_n_growing. +assumption. +apply le_IZR. +unfold N in |- *. +simpl in |- *. +assert (H0 := archimed M); elim H0; intros. +left; apply Rlt_trans with M; assumption. +exists 0%nat; intros. +rewrite <- b. +unfold pow_2_n in |- *; apply pow_lt; prove_sup0. +exists 0%nat; intros. +apply Rlt_trans with 0. +assumption. +unfold pow_2_n in |- *; apply pow_lt; prove_sup0. +simple induction N. +simpl in |- *. +left; apply Rlt_0_1. +intros. +pattern (S n) at 2 in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. +rewrite S_INR; rewrite pow_add. +simpl in |- *. +rewrite Rmult_1_r. +apply Rle_trans with (2 ^ n). +rewrite <- (Rplus_comm 1). +rewrite <- (Rmult_1_r (INR n)). +apply (poly n 1). +apply Rlt_0_1. +pattern (2 ^ n) at 1 in |- *; rewrite <- Rplus_0_r. +rewrite <- (Rmult_comm 2). +rewrite double. +apply Rplus_le_compat_l. +left; apply pow_lt; prove_sup0. Qed. -Lemma cv_dicho : (x,y,l1,l2:R;P:R->bool) ``x<=y`` -> (Un_cv (dicho_lb x y P) l1) -> (Un_cv (dicho_up x y P) l2) -> l1==l2. -Intros. -Assert H2 := (CV_minus ? ? ? ? H0 H1). -Cut (Un_cv [i:nat]``(dicho_lb x y P i)-(dicho_up x y P i)`` R0). -Intro. -Assert H4 := (UL_sequence ? ? ? H2 H3). -Symmetry; Apply Rminus_eq_right; Assumption. -Unfold Un_cv; Unfold R_dist. -Intros. -Assert H4 := (cv_infty_cv_R0 pow_2_n pow_2_n_neq_R0 pow_2_n_infty). -Case (total_order_T x y); Intro. -Elim s; Intro. -Unfold Un_cv in H4; Unfold R_dist in H4. -Cut ``0<y-x``. -Intro Hyp. -Cut ``0<eps/(y-x)``. -Intro. -Elim (H4 ``eps/(y-x)`` H5); Intros N H6. -Exists N; Intros. -Replace ``(dicho_lb x y P n)-(dicho_up x y P n)-0`` with ``(dicho_lb x y P n)-(dicho_up x y P n)``; [Idtac | Ring]. -Rewrite <- Rabsolu_Ropp. -Rewrite Ropp_distr3. -Rewrite dicho_lb_dicho_up. -Unfold Rdiv; Rewrite Rabsolu_mult. -Rewrite (Rabsolu_right ``y-x``). -Apply Rlt_monotony_contra with ``/(y-x)``. -Apply Rlt_Rinv; Assumption. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Replace ``/(pow 2 n)`` with ``/(pow 2 n)-0``; [Unfold pow_2_n Rdiv in H6; Rewrite <- (Rmult_sym eps); Apply H6; Assumption | Ring]. -Red; Intro; Rewrite H8 in Hyp; Elim (Rlt_antirefl ? Hyp). -Apply Rle_sym1. -Apply Rle_anti_compatibility with x; Rewrite Rplus_Or. -Replace ``x+(y-x)`` with y; [Assumption | Ring]. -Assumption. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Assumption]. -Apply Rlt_anti_compatibility with x; Rewrite Rplus_Or. -Replace ``x+(y-x)`` with y; [Assumption | Ring]. -Exists O; Intros. -Replace ``(dicho_lb x y P n)-(dicho_up x y P n)-0`` with ``(dicho_lb x y P n)-(dicho_up x y P n)``; [Idtac | Ring]. -Rewrite <- Rabsolu_Ropp. -Rewrite Ropp_distr3. -Rewrite dicho_lb_dicho_up. -Rewrite b. -Unfold Rminus Rdiv; Rewrite Rplus_Ropp_r; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Assumption. -Assumption. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H r)). +Lemma cv_dicho : + forall (x y l1 l2:R) (P:R -> bool), + x <= y -> + Un_cv (dicho_lb x y P) l1 -> Un_cv (dicho_up x y P) l2 -> l1 = l2. +intros. +assert (H2 := CV_minus _ _ _ _ H0 H1). +cut (Un_cv (fun i:nat => dicho_lb x y P i - dicho_up x y P i) 0). +intro. +assert (H4 := UL_sequence _ _ _ H2 H3). +symmetry in |- *; apply Rminus_diag_uniq_sym; assumption. +unfold Un_cv in |- *; unfold R_dist in |- *. +intros. +assert (H4 := cv_infty_cv_R0 pow_2_n pow_2_n_neq_R0 pow_2_n_infty). +case (total_order_T x y); intro. +elim s; intro. +unfold Un_cv in H4; unfold R_dist in H4. +cut (0 < y - x). +intro Hyp. +cut (0 < eps / (y - x)). +intro. +elim (H4 (eps / (y - x)) H5); intros N H6. +exists N; intros. +replace (dicho_lb x y P n - dicho_up x y P n - 0) with + (dicho_lb x y P n - dicho_up x y P n); [ idtac | ring ]. +rewrite <- Rabs_Ropp. +rewrite Ropp_minus_distr'. +rewrite dicho_lb_dicho_up. +unfold Rdiv in |- *; rewrite Rabs_mult. +rewrite (Rabs_right (y - x)). +apply Rmult_lt_reg_l with (/ (y - x)). +apply Rinv_0_lt_compat; assumption. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +replace (/ 2 ^ n) with (/ 2 ^ n - 0); + [ unfold pow_2_n, Rdiv in H6; rewrite <- (Rmult_comm eps); apply H6; + assumption + | ring ]. +red in |- *; intro; rewrite H8 in Hyp; elim (Rlt_irrefl _ Hyp). +apply Rle_ge. +apply Rplus_le_reg_l with x; rewrite Rplus_0_r. +replace (x + (y - x)) with y; [ assumption | ring ]. +assumption. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; assumption ]. +apply Rplus_lt_reg_r with x; rewrite Rplus_0_r. +replace (x + (y - x)) with y; [ assumption | ring ]. +exists 0%nat; intros. +replace (dicho_lb x y P n - dicho_up x y P n - 0) with + (dicho_lb x y P n - dicho_up x y P n); [ idtac | ring ]. +rewrite <- Rabs_Ropp. +rewrite Ropp_minus_distr'. +rewrite dicho_lb_dicho_up. +rewrite b. +unfold Rminus, Rdiv in |- *; rewrite Rplus_opp_r; rewrite Rmult_0_l; + rewrite Rabs_R0; assumption. +assumption. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). Qed. -Definition cond_positivity [x:R] : bool := Cases (total_order_Rle R0 x) of - (leftT _) => true -| (rightT _) => false end. +Definition cond_positivity (x:R) : bool := + match Rle_dec 0 x with + | left _ => true + | right _ => false + end. (* Sequential caracterisation of continuity *) -Lemma continuity_seq : (f:R->R;Un:nat->R;l:R) (continuity_pt f l) -> (Un_cv Un l) -> (Un_cv [i:nat](f (Un i)) (f l)). -Unfold continuity_pt Un_cv; Unfold continue_in. -Unfold limit1_in. -Unfold limit_in. -Unfold dist. -Simpl. -Unfold R_dist. -Intros. -Elim (H eps H1); Intros alp H2. -Elim H2; Intros. -Elim (H0 alp H3); Intros N H5. -Exists N; Intros. -Case (Req_EM (Un n) l); Intro. -Rewrite H7; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Apply H4. -Split. -Unfold D_x no_cond. -Split. -Trivial. -Apply not_sym; Assumption. -Apply H5; Assumption. +Lemma continuity_seq : + forall (f:R -> R) (Un:nat -> R) (l:R), + continuity_pt f l -> Un_cv Un l -> Un_cv (fun i:nat => f (Un i)) (f l). +unfold continuity_pt, Un_cv in |- *; unfold continue_in in |- *. +unfold limit1_in in |- *. +unfold limit_in in |- *. +unfold dist in |- *. +simpl in |- *. +unfold R_dist in |- *. +intros. +elim (H eps H1); intros alp H2. +elim H2; intros. +elim (H0 alp H3); intros N H5. +exists N; intros. +case (Req_dec (Un n) l); intro. +rewrite H7; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + assumption. +apply H4. +split. +unfold D_x, no_cond in |- *. +split. +trivial. +apply (sym_not_eq (A:=R)); assumption. +apply H5; assumption. Qed. -Lemma dicho_lb_car : (x,y:R;P:R->bool;n:nat) (P x)=false -> (P (dicho_lb x y P n))=false. -Intros. -Induction n. -Simpl. -Assumption. -Simpl. -Assert X := (sumbool_of_bool (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``)). -Elim X; Intro. -Rewrite a. -Unfold dicho_lb in Hrecn; Assumption. -Rewrite b. -Assumption. +Lemma dicho_lb_car : + forall (x y:R) (P:R -> bool) (n:nat), + P x = false -> P (dicho_lb x y P n) = false. +intros. +induction n as [| n Hrecn]. +simpl in |- *. +assumption. +simpl in |- *. +assert + (X := + sumbool_of_bool (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2))). +elim X; intro. +rewrite a. +unfold dicho_lb in Hrecn; assumption. +rewrite b. +assumption. Qed. -Lemma dicho_up_car : (x,y:R;P:R->bool;n:nat) (P y)=true -> (P (dicho_up x y P n))=true. -Intros. -Induction n. -Simpl. -Assumption. -Simpl. -Assert X := (sumbool_of_bool (P ``((Dichotomy_lb x y P n)+(Dichotomy_ub x y P n))/2``)). -Elim X; Intro. -Rewrite a. -Unfold dicho_lb in Hrecn; Assumption. -Rewrite b. -Assumption. +Lemma dicho_up_car : + forall (x y:R) (P:R -> bool) (n:nat), + P y = true -> P (dicho_up x y P n) = true. +intros. +induction n as [| n Hrecn]. +simpl in |- *. +assumption. +simpl in |- *. +assert + (X := + sumbool_of_bool (P ((Dichotomy_lb x y P n + Dichotomy_ub x y P n) / 2))). +elim X; intro. +rewrite a. +unfold dicho_lb in Hrecn; assumption. +rewrite b. +assumption. Qed. (* Intermediate Value Theorem *) -Lemma IVT : (f:R->R;x,y:R) (continuity f) -> ``x<y`` -> ``(f x)<0`` -> ``0<(f y)`` -> (sigTT R [z:R]``x<=z<=y``/\``(f z)==0``). -Intros. -Cut ``x<=y``. -Intro. -Generalize (dicho_lb_cv x y [z:R](cond_positivity (f z)) H3). -Generalize (dicho_up_cv x y [z:R](cond_positivity (f z)) H3). -Intros. -Elim X; Intros. -Elim X0; Intros. -Assert H4 := (cv_dicho ? ? ? ? ? H3 p0 p). -Rewrite H4 in p0. -Apply existTT with x0. -Split. -Split. -Apply Rle_trans with (dicho_lb x y [z:R](cond_positivity (f z)) O). -Simpl. -Right; Reflexivity. -Apply growing_ineq. -Apply dicho_lb_growing; Assumption. -Assumption. -Apply Rle_trans with (dicho_up x y [z:R](cond_positivity (f z)) O). -Apply decreasing_ineq. -Apply dicho_up_decreasing; Assumption. -Assumption. -Right; Reflexivity. -2:Left; Assumption. -Pose Vn := [n:nat](dicho_lb x y [z:R](cond_positivity (f z)) n). -Pose Wn := [n:nat](dicho_up x y [z:R](cond_positivity (f z)) n). -Cut ((n:nat)``(f (Vn n))<=0``)->``(f x0)<=0``. -Cut ((n:nat)``0<=(f (Wn n))``)->``0<=(f x0)``. -Intros. -Cut (n:nat)``(f (Vn n))<=0``. -Cut (n:nat)``0<=(f (Wn n))``. -Intros. -Assert H9 := (H6 H8). -Assert H10 := (H5 H7). -Apply Rle_antisym; Assumption. -Intro. -Unfold Wn. -Cut (z:R) (cond_positivity z)=true <-> ``0<=z``. -Intro. -Assert H8 := (dicho_up_car x y [z:R](cond_positivity (f z)) n). -Elim (H7 (f (dicho_up x y [z:R](cond_positivity (f z)) n))); Intros. -Apply H9. -Apply H8. -Elim (H7 (f y)); Intros. -Apply H12. -Left; Assumption. -Intro. -Unfold cond_positivity. -Case (total_order_Rle R0 z); Intro. -Split. -Intro; Assumption. -Intro; Reflexivity. -Split. -Intro; Elim diff_false_true; Assumption. -Intro. -Elim n0; Assumption. -Unfold Vn. -Cut (z:R) (cond_positivity z)=false <-> ``z<0``. -Intros. -Assert H8 := (dicho_lb_car x y [z:R](cond_positivity (f z)) n). -Left. -Elim (H7 (f (dicho_lb x y [z:R](cond_positivity (f z)) n))); Intros. -Apply H9. -Apply H8. -Elim (H7 (f x)); Intros. -Apply H12. -Assumption. -Intro. -Unfold cond_positivity. -Case (total_order_Rle R0 z); Intro. -Split. -Intro; Elim diff_true_false; Assumption. -Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H7)). -Split. -Intro; Auto with real. -Intro; Reflexivity. -Cut (Un_cv Wn x0). -Intros. -Assert H7 := (continuity_seq f Wn x0 (H x0) H5). -Case (total_order_T R0 (f x0)); Intro. -Elim s; Intro. -Left; Assumption. -Rewrite <- b; Right; Reflexivity. -Unfold Un_cv in H7; Unfold R_dist in H7. -Cut ``0< -(f x0)``. -Intro. -Elim (H7 ``-(f x0)`` H8); Intros. -Cut (ge x2 x2); [Intro | Unfold ge; Apply le_n]. -Assert H11 := (H9 x2 H10). -Rewrite Rabsolu_right in H11. -Pattern 1 ``-(f x0)`` in H11; Rewrite <- Rplus_Or in H11. -Unfold Rminus in H11; Rewrite (Rplus_sym (f (Wn x2))) in H11. -Assert H12 := (Rlt_anti_compatibility ? ? ? H11). -Assert H13 := (H6 x2). -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H13 H12)). -Apply Rle_sym1; Left; Unfold Rminus; Apply ge0_plus_gt0_is_gt0. -Apply H6. -Exact H8. -Apply Rgt_RO_Ropp; Assumption. -Unfold Wn; Assumption. -Cut (Un_cv Vn x0). -Intros. -Assert H7 := (continuity_seq f Vn x0 (H x0) H5). -Case (total_order_T R0 (f x0)); Intro. -Elim s; Intro. -Unfold Un_cv in H7; Unfold R_dist in H7. -Elim (H7 ``(f x0)`` a); Intros. -Cut (ge x2 x2); [Intro | Unfold ge; Apply le_n]. -Assert H10 := (H8 x2 H9). -Rewrite Rabsolu_left in H10. -Pattern 2 ``(f x0)`` in H10; Rewrite <- Rplus_Or in H10. -Rewrite Ropp_distr3 in H10. -Unfold Rminus in H10. -Assert H11 := (Rlt_anti_compatibility ? ? ? H10). -Assert H12 := (H6 x2). -Cut ``0<(f (Vn x2))``. -Intro. -Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H13 H12)). -Rewrite <- (Ropp_Ropp (f (Vn x2))). -Apply Rgt_RO_Ropp; Assumption. -Apply Rlt_anti_compatibility with ``(f x0)-(f (Vn x2))``. -Rewrite Rplus_Or; Replace ``(f x0)-(f (Vn x2))+((f (Vn x2))-(f x0))`` with R0; [Unfold Rminus; Apply gt0_plus_ge0_is_gt0 | Ring]. -Assumption. -Apply Rge_RO_Ropp; Apply Rle_sym1; Apply H6. -Right; Rewrite <- b; Reflexivity. -Left; Assumption. -Unfold Vn; Assumption. +Lemma IVT : + forall (f:R -> R) (x y:R), + continuity f -> + x < y -> f x < 0 -> 0 < f y -> sigT (fun z:R => x <= z <= y /\ f z = 0). +intros. +cut (x <= y). +intro. +generalize (dicho_lb_cv x y (fun z:R => cond_positivity (f z)) H3). +generalize (dicho_up_cv x y (fun z:R => cond_positivity (f z)) H3). +intros. +elim X; intros. +elim X0; intros. +assert (H4 := cv_dicho _ _ _ _ _ H3 p0 p). +rewrite H4 in p0. +apply existT with x0. +split. +split. +apply Rle_trans with (dicho_lb x y (fun z:R => cond_positivity (f z)) 0). +simpl in |- *. +right; reflexivity. +apply growing_ineq. +apply dicho_lb_growing; assumption. +assumption. +apply Rle_trans with (dicho_up x y (fun z:R => cond_positivity (f z)) 0). +apply decreasing_ineq. +apply dicho_up_decreasing; assumption. +assumption. +right; reflexivity. +2: left; assumption. +pose (Vn := fun n:nat => dicho_lb x y (fun z:R => cond_positivity (f z)) n). +pose (Wn := fun n:nat => dicho_up x y (fun z:R => cond_positivity (f z)) n). +cut ((forall n:nat, f (Vn n) <= 0) -> f x0 <= 0). +cut ((forall n:nat, 0 <= f (Wn n)) -> 0 <= f x0). +intros. +cut (forall n:nat, f (Vn n) <= 0). +cut (forall n:nat, 0 <= f (Wn n)). +intros. +assert (H9 := H6 H8). +assert (H10 := H5 H7). +apply Rle_antisym; assumption. +intro. +unfold Wn in |- *. +cut (forall z:R, cond_positivity z = true <-> 0 <= z). +intro. +assert (H8 := dicho_up_car x y (fun z:R => cond_positivity (f z)) n). +elim (H7 (f (dicho_up x y (fun z:R => cond_positivity (f z)) n))); intros. +apply H9. +apply H8. +elim (H7 (f y)); intros. +apply H12. +left; assumption. +intro. +unfold cond_positivity in |- *. +case (Rle_dec 0 z); intro. +split. +intro; assumption. +intro; reflexivity. +split. +intro; elim diff_false_true; assumption. +intro. +elim n0; assumption. +unfold Vn in |- *. +cut (forall z:R, cond_positivity z = false <-> z < 0). +intros. +assert (H8 := dicho_lb_car x y (fun z:R => cond_positivity (f z)) n). +left. +elim (H7 (f (dicho_lb x y (fun z:R => cond_positivity (f z)) n))); intros. +apply H9. +apply H8. +elim (H7 (f x)); intros. +apply H12. +assumption. +intro. +unfold cond_positivity in |- *. +case (Rle_dec 0 z); intro. +split. +intro; elim diff_true_false; assumption. +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H7)). +split. +intro; auto with real. +intro; reflexivity. +cut (Un_cv Wn x0). +intros. +assert (H7 := continuity_seq f Wn x0 (H x0) H5). +case (total_order_T 0 (f x0)); intro. +elim s; intro. +left; assumption. +rewrite <- b; right; reflexivity. +unfold Un_cv in H7; unfold R_dist in H7. +cut (0 < - f x0). +intro. +elim (H7 (- f x0) H8); intros. +cut (x2 >= x2)%nat; [ intro | unfold ge in |- *; apply le_n ]. +assert (H11 := H9 x2 H10). +rewrite Rabs_right in H11. +pattern (- f x0) at 1 in H11; rewrite <- Rplus_0_r in H11. +unfold Rminus in H11; rewrite (Rplus_comm (f (Wn x2))) in H11. +assert (H12 := Rplus_lt_reg_r _ _ _ H11). +assert (H13 := H6 x2). +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H13 H12)). +apply Rle_ge; left; unfold Rminus in |- *; apply Rplus_le_lt_0_compat. +apply H6. +exact H8. +apply Ropp_0_gt_lt_contravar; assumption. +unfold Wn in |- *; assumption. +cut (Un_cv Vn x0). +intros. +assert (H7 := continuity_seq f Vn x0 (H x0) H5). +case (total_order_T 0 (f x0)); intro. +elim s; intro. +unfold Un_cv in H7; unfold R_dist in H7. +elim (H7 (f x0) a); intros. +cut (x2 >= x2)%nat; [ intro | unfold ge in |- *; apply le_n ]. +assert (H10 := H8 x2 H9). +rewrite Rabs_left in H10. +pattern (f x0) at 2 in H10; rewrite <- Rplus_0_r in H10. +rewrite Ropp_minus_distr' in H10. +unfold Rminus in H10. +assert (H11 := Rplus_lt_reg_r _ _ _ H10). +assert (H12 := H6 x2). +cut (0 < f (Vn x2)). +intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H13 H12)). +rewrite <- (Ropp_involutive (f (Vn x2))). +apply Ropp_0_gt_lt_contravar; assumption. +apply Rplus_lt_reg_r with (f x0 - f (Vn x2)). +rewrite Rplus_0_r; replace (f x0 - f (Vn x2) + (f (Vn x2) - f x0)) with 0; + [ unfold Rminus in |- *; apply Rplus_lt_le_0_compat | ring ]. +assumption. +apply Ropp_0_ge_le_contravar; apply Rle_ge; apply H6. +right; rewrite <- b; reflexivity. +left; assumption. +unfold Vn in |- *; assumption. Qed. -Lemma IVT_cor : (f:R->R;x,y:R) (continuity f) -> ``x<=y`` -> ``(f x)*(f y)<=0`` -> (sigTT R [z:R]``x<=z<=y``/\``(f z)==0``). -Intros. -Case (total_order_T R0 (f x)); Intro. -Case (total_order_T R0 (f y)); Intro. -Elim s; Intro. -Elim s0; Intro. -Cut ``0<(f x)*(f y)``; [Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 H2)) | Apply Rmult_lt_pos; Assumption]. -Exists y. -Split. -Split; [Assumption | Right; Reflexivity]. -Symmetry; Exact b. -Exists x. -Split. -Split; [Right; Reflexivity | Assumption]. -Symmetry; Exact b. -Elim s; Intro. -Cut ``x<y``. -Intro. -Assert H3 := (IVT (opp_fct f) x y (continuity_opp f H) H2). -Cut ``(opp_fct f x)<0``. -Cut ``0<(opp_fct f y)``. -Intros. -Elim (H3 H5 H4); Intros. -Apply existTT with x0. -Elim p; Intros. -Split. -Assumption. -Unfold opp_fct in H7. -Rewrite <- (Ropp_Ropp (f x0)). -Apply eq_RoppO; Assumption. -Unfold opp_fct; Apply Rgt_RO_Ropp; Assumption. -Unfold opp_fct. -Apply Rlt_anti_compatibility with (f x); Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Assumption. -Inversion H0. -Assumption. -Rewrite H2 in a. -Elim (Rlt_antirefl ? (Rlt_trans ? ? ? r a)). -Apply existTT with x. -Split. -Split; [Right; Reflexivity | Assumption]. -Symmetry; Assumption. -Case (total_order_T R0 (f y)); Intro. -Elim s; Intro. -Cut ``x<y``. -Intro. -Apply IVT; Assumption. -Inversion H0. -Assumption. -Rewrite H2 in r. -Elim (Rlt_antirefl ? (Rlt_trans ? ? ? r a)). -Apply existTT with y. -Split. -Split; [Assumption | Right; Reflexivity]. -Symmetry; Assumption. -Cut ``0<(f x)*(f y)``. -Intro. -Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H2 H1)). -Rewrite <- Ropp_mul2; Apply Rmult_lt_pos; Apply Rgt_RO_Ropp; Assumption. +Lemma IVT_cor : + forall (f:R -> R) (x y:R), + continuity f -> + x <= y -> f x * f y <= 0 -> sigT (fun z:R => x <= z <= y /\ f z = 0). +intros. +case (total_order_T 0 (f x)); intro. +case (total_order_T 0 (f y)); intro. +elim s; intro. +elim s0; intro. +cut (0 < f x * f y); + [ intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 H2)) + | apply Rmult_lt_0_compat; assumption ]. +exists y. +split. +split; [ assumption | right; reflexivity ]. +symmetry in |- *; exact b. +exists x. +split. +split; [ right; reflexivity | assumption ]. +symmetry in |- *; exact b. +elim s; intro. +cut (x < y). +intro. +assert (H3 := IVT (- f)%F x y (continuity_opp f H) H2). +cut ((- f)%F x < 0). +cut (0 < (- f)%F y). +intros. +elim (H3 H5 H4); intros. +apply existT with x0. +elim p; intros. +split. +assumption. +unfold opp_fct in H7. +rewrite <- (Ropp_involutive (f x0)). +apply Ropp_eq_0_compat; assumption. +unfold opp_fct in |- *; apply Ropp_0_gt_lt_contravar; assumption. +unfold opp_fct in |- *. +apply Rplus_lt_reg_r with (f x); rewrite Rplus_opp_r; rewrite Rplus_0_r; + assumption. +inversion H0. +assumption. +rewrite H2 in a. +elim (Rlt_irrefl _ (Rlt_trans _ _ _ r a)). +apply existT with x. +split. +split; [ right; reflexivity | assumption ]. +symmetry in |- *; assumption. +case (total_order_T 0 (f y)); intro. +elim s; intro. +cut (x < y). +intro. +apply IVT; assumption. +inversion H0. +assumption. +rewrite H2 in r. +elim (Rlt_irrefl _ (Rlt_trans _ _ _ r a)). +apply existT with y. +split. +split; [ assumption | right; reflexivity ]. +symmetry in |- *; assumption. +cut (0 < f x * f y). +intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H2 H1)). +rewrite <- Rmult_opp_opp; apply Rmult_lt_0_compat; + apply Ropp_0_gt_lt_contravar; assumption. Qed. (* We can now define the square root function as the reciprocal transformation of the square root function *) -Lemma Rsqrt_exists : (y:R) ``0<=y`` -> (sigTT R [z:R]``0<=z``/\``y==(Rsqr z)``). -Intros. -Pose f := [x:R]``(Rsqr x)-y``. -Cut ``(f 0)<=0``. -Intro. -Cut (continuity f). -Intro. -Case (total_order_T y R1); Intro. -Elim s; Intro. -Cut ``0<=(f 1)``. -Intro. -Cut ``(f 0)*(f 1)<=0``. -Intro. -Assert X := (IVT_cor f R0 R1 H1 (Rlt_le ? ? Rlt_R0_R1) H3). -Elim X; Intros t H4. -Apply existTT with t. -Elim H4; Intros. -Split. -Elim H5; Intros; Assumption. -Unfold f in H6. -Apply Rminus_eq_right; Exact H6. -Rewrite Rmult_sym; Pattern 2 R0; Rewrite <- (Rmult_Or (f R1)). -Apply Rle_monotony; Assumption. -Unfold f. -Rewrite Rsqr_1. -Apply Rle_anti_compatibility with y. -Rewrite Rplus_Or; Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Left; Assumption. -Apply existTT with R1. -Split. -Left; Apply Rlt_R0_R1. -Rewrite b; Symmetry; Apply Rsqr_1. -Cut ``0<=(f y)``. -Intro. -Cut ``(f 0)*(f y)<=0``. -Intro. -Assert X := (IVT_cor f R0 y H1 H H3). -Elim X; Intros t H4. -Apply existTT with t. -Elim H4; Intros. -Split. -Elim H5; Intros; Assumption. -Unfold f in H6. -Apply Rminus_eq_right; Exact H6. -Rewrite Rmult_sym; Pattern 2 R0; Rewrite <- (Rmult_Or (f y)). -Apply Rle_monotony; Assumption. -Unfold f. -Apply Rle_anti_compatibility with y. -Rewrite Rplus_Or; Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or. -Pattern 1 y; Rewrite <- Rmult_1r. -Unfold Rsqr; Apply Rle_monotony. -Assumption. -Left; Exact r. -Replace f with (minus_fct Rsqr (fct_cte y)). -Apply continuity_minus. -Apply derivable_continuous; Apply derivable_Rsqr. -Apply derivable_continuous; Apply derivable_const. -Reflexivity. -Unfold f; Rewrite Rsqr_O. -Unfold Rminus; Rewrite Rplus_Ol. -Apply Rle_sym2. -Apply Rle_RO_Ropp; Assumption. +Lemma Rsqrt_exists : + forall y:R, 0 <= y -> sigT (fun z:R => 0 <= z /\ y = Rsqr z). +intros. +pose (f := fun x:R => Rsqr x - y). +cut (f 0 <= 0). +intro. +cut (continuity f). +intro. +case (total_order_T y 1); intro. +elim s; intro. +cut (0 <= f 1). +intro. +cut (f 0 * f 1 <= 0). +intro. +assert (X := IVT_cor f 0 1 H1 (Rlt_le _ _ Rlt_0_1) H3). +elim X; intros t H4. +apply existT with t. +elim H4; intros. +split. +elim H5; intros; assumption. +unfold f in H6. +apply Rminus_diag_uniq_sym; exact H6. +rewrite Rmult_comm; pattern 0 at 2 in |- *; rewrite <- (Rmult_0_r (f 1)). +apply Rmult_le_compat_l; assumption. +unfold f in |- *. +rewrite Rsqr_1. +apply Rplus_le_reg_l with y. +rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *; + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; + left; assumption. +apply existT with 1. +split. +left; apply Rlt_0_1. +rewrite b; symmetry in |- *; apply Rsqr_1. +cut (0 <= f y). +intro. +cut (f 0 * f y <= 0). +intro. +assert (X := IVT_cor f 0 y H1 H H3). +elim X; intros t H4. +apply existT with t. +elim H4; intros. +split. +elim H5; intros; assumption. +unfold f in H6. +apply Rminus_diag_uniq_sym; exact H6. +rewrite Rmult_comm; pattern 0 at 2 in |- *; rewrite <- (Rmult_0_r (f y)). +apply Rmult_le_compat_l; assumption. +unfold f in |- *. +apply Rplus_le_reg_l with y. +rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *; + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r. +pattern y at 1 in |- *; rewrite <- Rmult_1_r. +unfold Rsqr in |- *; apply Rmult_le_compat_l. +assumption. +left; exact r. +replace f with (Rsqr - fct_cte y)%F. +apply continuity_minus. +apply derivable_continuous; apply derivable_Rsqr. +apply derivable_continuous; apply derivable_const. +reflexivity. +unfold f in |- *; rewrite Rsqr_0. +unfold Rminus in |- *; rewrite Rplus_0_l. +apply Rge_le. +apply Ropp_0_le_ge_contravar; assumption. Qed. (* Definition of the square root: R+->R *) -Definition Rsqrt [y:nonnegreal] : R := Cases (Rsqrt_exists (nonneg y) (cond_nonneg y)) of (existTT a b) => a end. +Definition Rsqrt (y:nonnegreal) : R := + match Rsqrt_exists (nonneg y) (cond_nonneg y) with + | existT a b => a + end. (**********) -Lemma Rsqrt_positivity : (x:nonnegreal) ``0<=(Rsqrt x)``. -Intro. -Assert X := (Rsqrt_exists (nonneg x) (cond_nonneg x)). -Elim X; Intros. -Cut x0==(Rsqrt x). -Intros. -Elim p; Intros. -Rewrite H in H0; Assumption. -Unfold Rsqrt. -Case (Rsqrt_exists x (cond_nonneg x)). -Intros. -Elim p; Elim a; Intros. -Apply Rsqr_inj. -Assumption. -Assumption. -Rewrite <- H0; Rewrite <- H2; Reflexivity. +Lemma Rsqrt_positivity : forall x:nonnegreal, 0 <= Rsqrt x. +intro. +assert (X := Rsqrt_exists (nonneg x) (cond_nonneg x)). +elim X; intros. +cut (x0 = Rsqrt x). +intros. +elim p; intros. +rewrite H in H0; assumption. +unfold Rsqrt in |- *. +case (Rsqrt_exists x (cond_nonneg x)). +intros. +elim p; elim a; intros. +apply Rsqr_inj. +assumption. +assumption. +rewrite <- H0; rewrite <- H2; reflexivity. Qed. (**********) -Lemma Rsqrt_Rsqrt : (x:nonnegreal) ``(Rsqrt x)*(Rsqrt x)==x``. -Intros. -Assert X := (Rsqrt_exists (nonneg x) (cond_nonneg x)). -Elim X; Intros. -Cut x0==(Rsqrt x). -Intros. -Rewrite <- H. -Elim p; Intros. -Rewrite H1; Reflexivity. -Unfold Rsqrt. -Case (Rsqrt_exists x (cond_nonneg x)). -Intros. -Elim p; Elim a; Intros. -Apply Rsqr_inj. -Assumption. -Assumption. -Rewrite <- H0; Rewrite <- H2; Reflexivity. -Qed. +Lemma Rsqrt_Rsqrt : forall x:nonnegreal, Rsqrt x * Rsqrt x = x. +intros. +assert (X := Rsqrt_exists (nonneg x) (cond_nonneg x)). +elim X; intros. +cut (x0 = Rsqrt x). +intros. +rewrite <- H. +elim p; intros. +rewrite H1; reflexivity. +unfold Rsqrt in |- *. +case (Rsqrt_exists x (cond_nonneg x)). +intros. +elim p; elim a; intros. +apply Rsqr_inj. +assumption. +assumption. +rewrite <- H0; rewrite <- H2; reflexivity. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Rsyntax.v b/theories/Reals/Rsyntax.v index 53f8aec07..b453ef9db 100644 --- a/theories/Reals/Rsyntax.v +++ b/theories/Reals/Rsyntax.v @@ -9,228 +9,5 @@ Require Export Rdefinitions. -Axiom NRplus : R->R. -Axiom NRmult : R->R. - -V7only[ -Grammar rnatural ident := - nat_id [ prim:var($id) ] -> [$id] - -with rnegnumber : constr := - neg_expr [ "-" rnumber ($c) ] -> [ (Ropp $c) ] - -with rnumber := - -with rformula : constr := - form_expr [ rexpr($p) ] -> [ $p ] -(* | form_eq [ rexpr($p) "==" rexpr($c) ] -> [ (eqT R $p $c) ] *) -| form_eq [ rexpr($p) "==" rexpr($c) ] -> [ (eqT ? $p $c) ] -| form_eq2 [ rexpr($p) "=" rexpr($c) ] -> [ (eqT ? $p $c) ] -| form_le [ rexpr($p) "<=" rexpr($c) ] -> [ (Rle $p $c) ] -| form_lt [ rexpr($p) "<" rexpr($c) ] -> [ (Rlt $p $c) ] -| form_ge [ rexpr($p) ">=" rexpr($c) ] -> [ (Rge $p $c) ] -| form_gt [ rexpr($p) ">" rexpr($c) ] -> [ (Rgt $p $c) ] -(* -| form_eq_eq [ rexpr($p) "==" rexpr($c) "==" rexpr($c1) ] - -> [ (eqT R $p $c)/\(eqT R $c $c1) ] -*) -| form_eq_eq [ rexpr($p) "==" rexpr($c) "==" rexpr($c1) ] - -> [ (eqT ? $p $c)/\(eqT ? $c $c1) ] -| form_le_le [ rexpr($p) "<=" rexpr($c) "<=" rexpr($c1) ] - -> [ (Rle $p $c)/\(Rle $c $c1) ] -| form_le_lt [ rexpr($p) "<=" rexpr($c) "<" rexpr($c1) ] - -> [ (Rle $p $c)/\(Rlt $c $c1) ] -| form_lt_le [ rexpr($p) "<" rexpr($c) "<=" rexpr($c1) ] - -> [ (Rlt $p $c)/\(Rle $c $c1) ] -| form_lt_lt [ rexpr($p) "<" rexpr($c) "<" rexpr($c1) ] - -> [ (Rlt $p $c)/\(Rlt $c $c1) ] -| form_neq [ rexpr($p) "<>" rexpr($c) ] -> [ ~(eqT ? $p $c) ] - -with rexpr : constr := - expr_plus [ rexpr($p) "+" rexpr($c) ] -> [ (Rplus $p $c) ] -| expr_minus [ rexpr($p) "-" rexpr($c) ] -> [ (Rminus $p $c) ] -| rexpr2 [ rexpr2($e) ] -> [ $e ] - -with rexpr2 : constr := - expr_mult [ rexpr2($p) "*" rexpr2($c) ] -> [ (Rmult $p $c) ] -| rexpr0 [ rexpr0($e) ] -> [ $e ] - - -with rexpr0 : constr := - expr_id [ constr:global($c) ] -> [ $c ] -| expr_com [ "[" constr:constr($c) "]" ] -> [ $c ] -| expr_appl [ "(" rapplication($a) ")" ] -> [ $a ] -| expr_num [ rnumber($s) ] -> [ $s ] -| expr_negnum [ "-" rnegnumber($n) ] -> [ $n ] -| expr_div [ rexpr0($p) "/" rexpr0($c) ] -> [ (Rdiv $p $c) ] -| expr_opp [ "-" rexpr0($c) ] -> [ (Ropp $c) ] -| expr_inv [ "/" rexpr0($c) ] -> [ (Rinv $c) ] -| expr_meta [ meta($m) ] -> [ $m ] - -with meta := -| rimpl [ "?" ] -> [ ? ] -| rmeta0 [ "?" "0" ] -> [ ?0 ] -| rmeta1 [ "?" "1" ] -> [ ?1 ] -| rmeta2 [ "?" "2" ] -> [ ?2 ] -| rmeta3 [ "?" "3" ] -> [ ?3 ] -| rmeta4 [ "?" "4" ] -> [ ?4 ] -| rmeta5 [ "?" "5" ] -> [ ?5 ] - -with rapplication : constr := - apply [ rapplication($p) rexpr($c1) ] -> [ ($p $c1) ] -| pair [ rexpr($p) "," rexpr($c) ] -> [ ($p, $c) ] -| appl0 [ rexpr($a) ] -> [ $a ]. - -Grammar constr constr0 := - r_in_com [ "``" rnatural:rformula($c) "``" ] -> [ $c ]. - -Grammar constr atomic_pattern := - r_in_pattern [ "``" rnatural:rnumber($c) "``" ] -> [ $c ]. - -(*i* pp **) - -Syntax constr - level 0: - Rle [ (Rle $n1 $n2) ] -> - [[<hov 0> "``" (REXPR $n1) [1 0] "<= " (REXPR $n2) "``"]] - | Rlt [ (Rlt $n1 $n2) ] -> - [[<hov 0> "``" (REXPR $n1) [1 0] "< "(REXPR $n2) "``" ]] - | Rge [ (Rge $n1 $n2) ] -> - [[<hov 0> "``" (REXPR $n1) [1 0] ">= "(REXPR $n2) "``" ]] - | Rgt [ (Rgt $n1 $n2) ] -> - [[<hov 0> "``" (REXPR $n1) [1 0] "> "(REXPR $n2) "``" ]] - | Req [ (eqT R $n1 $n2) ] -> - [[<hov 0> "``" (REXPR $n1) [1 0] "= "(REXPR $n2)"``"]] - | Rneq [ ~(eqT R $n1 $n2) ] -> - [[<hov 0> "``" (REXPR $n1) [1 0] "<> "(REXPR $n2) "``"]] - | Rle_Rle [ (Rle $n1 $n2)/\(Rle $n2 $n3) ] -> - [[<hov 0> "``" (REXPR $n1) [1 0] "<= " (REXPR $n2) - [1 0] "<= " (REXPR $n3) "``"]] - | Rle_Rlt [ (Rle $n1 $n2)/\(Rlt $n2 $n3) ] -> - [[<hov 0> "``" (REXPR $n1) [1 0] "<= "(REXPR $n2) - [1 0] "< " (REXPR $n3) "``"]] - | Rlt_Rle [ (Rlt $n1 $n2)/\(Rle $n2 $n3) ] -> - [[<hov 0> "``" (REXPR $n1) [1 0] "< " (REXPR $n2) - [1 0] "<= " (REXPR $n3) "``"]] - | Rlt_Rlt [ (Rlt $n1 $n2)/\(Rlt $n2 $n3) ] -> - [[<hov 0> "``" (REXPR $n1) [1 0] "< " (REXPR $n2) - [1 0] "< " (REXPR $n3) "``"]] - | Rzero [ R0 ] -> [ "``0``" ] - | Rone [ R1 ] -> [ "``1``" ] - ; - - level 7: - Rplus [ (Rplus $n1 $n2) ] - -> [ [<hov 0> "``"(REXPR $n1):E "+" [0 0] (REXPR $n2):L "``"] ] - | Rodd_outside [(Rplus R1 $r)] -> [ $r:"r_printer_odd_outside"] - | Rminus [ (Rminus $n1 $n2) ] - -> [ [<hov 0> "``"(REXPR $n1):E "-" [0 0] (REXPR $n2):L "``"] ] - ; - - level 6: - Rmult [ (Rmult $n1 $n2) ] - -> [ [<hov 0> "``"(REXPR $n1):E "*" [0 0] (REXPR $n2):L "``"] ] - | Reven_outside [ (Rmult (Rplus R1 R1) $r) ] -> [ $r:"r_printer_even_outside"] - | Rdiv [ (Rdiv $n1 $n2) ] - -> [ [<hov 0> "``"(REXPR $n1):E "/" [0 0] (REXPR $n2):L "``"] ] - ; - - level 8: - Ropp [(Ropp $n1)] -> [ [<hov 0> "``" "-"(REXPR $n1):E "``"] ] - | Rinv [(Rinv $n1)] -> [ [<hov 0> "``" "/"(REXPR $n1):E "``"] ] - ; - - level 0: - rescape_inside [<< (REXPR $r) >>] -> [ "[" $r:E "]" ] - ; - - level 4: - Rappl_inside [<<(REXPR (APPLIST $h ($LIST $t)))>>] - -> [ [<hov 0> "("(REXPR $h):E [1 0] (RAPPLINSIDETAIL ($LIST $t)):E ")"] ] - | Rappl_inside_tail [<<(RAPPLINSIDETAIL $h ($LIST $t))>>] - -> [(REXPR $h):E [1 0] (RAPPLINSIDETAIL ($LIST $t)):E] - | Rappl_inside_one [<<(RAPPLINSIDETAIL $e)>>] ->[(REXPR $e):E] - | rpair_inside [<<(REXPR <<(pair $s1 $s2 $r1 $r2)>>)>>] - -> [ [<hov 0> "("(REXPR $r1):E "," [1 0] (REXPR $r2):E ")"] ] - ; - - level 3: - rvar_inside [<<(REXPR ($VAR $i))>>] -> [$i] - | rsecvar_inside [<<(REXPR (SECVAR $i))>>] -> [(SECVAR $i)] - | rconst_inside [<<(REXPR (CONST $c))>>] -> [(CONST $c)] - | rmutind_inside [<<(REXPR (MUTIND $i $n))>>] - -> [(MUTIND $i $n)] - | rmutconstruct_inside [<<(REXPR (MUTCONSTRUCT $c1 $c2 $c3))>>] - -> [ (MUTCONSTRUCT $c1 $c2 $c3) ] - | rimplicit_head_inside [<<(REXPR (XTRA "!" $c))>>] -> [ $c ] - | rimplicit_arg_inside [<<(REXPR (XTRA "!" $n $c))>>] -> [ ] - - ; - - - level 7: - Rplus_inside - [<<(REXPR <<(Rplus $n1 $n2)>>)>>] - -> [ (REXPR $n1):E "+" [0 0] (REXPR $n2):L ] - | Rminus_inside - [<<(REXPR <<(Rminus $n1 $n2)>>)>>] - -> [ (REXPR $n1):E "-" [0 0] (REXPR $n2):L ] - | NRplus_inside - [<<(REXPR <<(NRplus $r)>>)>>] -> [ "(" "1" "+" (REXPR $r):L ")"] - ; - - level 6: - Rmult_inside - [<<(REXPR <<(Rmult $n1 $n2)>>)>>] - -> [ (REXPR $n1):E "*" (REXPR $n2):L ] - | NRmult_inside - [<<(REXPR <<(NRmult $r)>>)>>] -> [ "(" "2" "*" (REXPR $r):L ")"] - ; - - level 5: - Ropp_inside [<<(REXPR <<(Ropp $n1)>>)>>] -> [ " -" (REXPR $n1):E ] - | Rinv_inside [<<(REXPR <<(Rinv $n1)>>)>>] -> [ "/" (REXPR $n1):E ] - | Rdiv_inside - [<<(REXPR <<(Rdiv $n1 $n2)>>)>>] - -> [ (REXPR $n1):E "/" [0 0] (REXPR $n2):L ] - ; - - level 0: - Rzero_inside [<<(REXPR <<R0>>)>>] -> ["0"] - | Rone_inside [<<(REXPR <<R1>>)>>] -> ["1"] - | Rodd_inside [<<(REXPR <<(Rplus R1 $r)>>)>>] -> [ $r:"r_printer_odd" ] - | Reven_inside [<<(REXPR <<(Rmult (Rplus R1 R1) $r)>>)>>] -> [ $r:"r_printer_even" ] -. - -(* For parsing/printing based on scopes *) -Module R_scope. - -Infix "<=" Rle (at level 5, no associativity) : R_scope V8only. -Infix "<" Rlt (at level 5, no associativity) : R_scope V8only. -Infix ">=" Rge (at level 5, no associativity) : R_scope V8only. -Infix ">" Rgt (at level 5, no associativity) : R_scope V8only. -Infix "+" Rplus (at level 4) : R_scope V8only. -Infix "-" Rminus (at level 4) : R_scope V8only. -Infix "*" Rmult (at level 3) : R_scope V8only. -Infix "/" Rdiv (at level 3) : R_scope V8only. -Notation "- x" := (Ropp x) (at level 0) : R_scope V8only. -Notation "x == y == z" := (eqT R x y)/\(eqT R y z) - (at level 5, y at level 4, no associtivity): R_scope. -Notation "x <= y <= z" := (Rle x y)/\(Rle y z) - (at level 5, y at level 4) : R_scope - V8only. -Notation "x <= y < z" := (Rle x y)/\(Rlt y z) - (at level 5, y at level 4) : R_scope - V8only. -Notation "x < y < z" := (Rlt x y)/\(Rlt y z) - (at level 5, y at level 4) : R_scope - V8only. -Notation "x < y <= z" := (Rlt x y)/\(Rle y z) - (at level 5, y at level 4) : R_scope - V8only. -Notation "/ x" := (Rinv x) (at level 0): R_scope - V8only. - -Open Local Scope R_scope. -End R_scope. -]. +Axiom NRplus : R -> R. +Axiom NRmult : R -> R. diff --git a/theories/Reals/Rtopology.v b/theories/Reals/Rtopology.v index c59db60ce..17b884d45 100644 --- a/theories/Reals/Rtopology.v +++ b/theories/Reals/Rtopology.v @@ -8,879 +8,1263 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Ranalysis1. -Require RList. -Require Classical_Prop. -Require Classical_Pred_Type. -V7only [Import R_scope.]. Open Local Scope R_scope. - -Definition included [D1,D2:R->Prop] : Prop := (x:R)(D1 x)->(D2 x). -Definition disc [x:R;delta:posreal] : R->Prop := [y:R]``(Rabsolu (y-x))<delta``. -Definition neighbourhood [V:R->Prop;x:R] : Prop := (EXT delta:posreal | (included (disc x delta) V)). -Definition open_set [D:R->Prop] : Prop := (x:R) (D x)->(neighbourhood D x). -Definition complementary [D:R->Prop] : R->Prop := [c:R]~(D c). -Definition closed_set [D:R->Prop] : Prop := (open_set (complementary D)). -Definition intersection_domain [D1,D2:R->Prop] : R->Prop := [c:R](D1 c)/\(D2 c). -Definition union_domain [D1,D2:R->Prop] : R->Prop := [c:R](D1 c)\/(D2 c). -Definition interior [D:R->Prop] : R->Prop := [x:R](neighbourhood D x). - -Lemma interior_P1 : (D:R->Prop) (included (interior D) D). -Intros; Unfold included; Unfold interior; Intros; Unfold neighbourhood in H; Elim H; Intros; Unfold included in H0; Apply H0; Unfold disc; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (cond_pos x0). +Require Import Rbase. +Require Import Rfunctions. +Require Import Ranalysis1. +Require Import RList. +Require Import Classical_Prop. +Require Import Classical_Pred_Type. Open Local Scope R_scope. + +Definition included (D1 D2:R -> Prop) : Prop := forall x:R, D1 x -> D2 x. +Definition disc (x:R) (delta:posreal) (y:R) : Prop := Rabs (y - x) < delta. +Definition neighbourhood (V:R -> Prop) (x:R) : Prop := + exists delta : posreal | included (disc x delta) V. +Definition open_set (D:R -> Prop) : Prop := + forall x:R, D x -> neighbourhood D x. +Definition complementary (D:R -> Prop) (c:R) : Prop := ~ D c. +Definition closed_set (D:R -> Prop) : Prop := open_set (complementary D). +Definition intersection_domain (D1 D2:R -> Prop) (c:R) : Prop := D1 c /\ D2 c. +Definition union_domain (D1 D2:R -> Prop) (c:R) : Prop := D1 c \/ D2 c. +Definition interior (D:R -> Prop) (x:R) : Prop := neighbourhood D x. + +Lemma interior_P1 : forall D:R -> Prop, included (interior D) D. +intros; unfold included in |- *; unfold interior in |- *; intros; + unfold neighbourhood in H; elim H; intros; unfold included in H0; + apply H0; unfold disc in |- *; unfold Rminus in |- *; + rewrite Rplus_opp_r; rewrite Rabs_R0; apply (cond_pos x0). Qed. -Lemma interior_P2 : (D:R->Prop) (open_set D) -> (included D (interior D)). -Intros; Unfold open_set in H; Unfold included; Intros; Assert H1 := (H ? H0); Unfold interior; Apply H1. +Lemma interior_P2 : forall D:R -> Prop, open_set D -> included D (interior D). +intros; unfold open_set in H; unfold included in |- *; intros; + assert (H1 := H _ H0); unfold interior in |- *; apply H1. Qed. -Definition point_adherent [D:R->Prop;x:R] : Prop := (V:R->Prop) (neighbourhood V x) -> (EXT y:R | (intersection_domain V D y)). -Definition adherence [D:R->Prop] : R->Prop := [x:R](point_adherent D x). - -Lemma adherence_P1 : (D:R->Prop) (included D (adherence D)). -Intro; Unfold included; Intros; Unfold adherence; Unfold point_adherent; Intros; Exists x; Unfold intersection_domain; Split. -Unfold neighbourhood in H0; Elim H0; Intros; Unfold included in H1; Apply H1; Unfold disc; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (cond_pos x0). -Apply H. +Definition point_adherent (D:R -> Prop) (x:R) : Prop := + forall V:R -> Prop, + neighbourhood V x -> exists y : R | intersection_domain V D y. +Definition adherence (D:R -> Prop) (x:R) : Prop := point_adherent D x. + +Lemma adherence_P1 : forall D:R -> Prop, included D (adherence D). +intro; unfold included in |- *; intros; unfold adherence in |- *; + unfold point_adherent in |- *; intros; exists x; + unfold intersection_domain in |- *; split. +unfold neighbourhood in H0; elim H0; intros; unfold included in H1; apply H1; + unfold disc in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; apply (cond_pos x0). +apply H. Qed. -Lemma included_trans : (D1,D2,D3:R->Prop) (included D1 D2) -> (included D2 D3) -> (included D1 D3). -Unfold included; Intros; Apply H0; Apply H; Apply H1. +Lemma included_trans : + forall D1 D2 D3:R -> Prop, + included D1 D2 -> included D2 D3 -> included D1 D3. +unfold included in |- *; intros; apply H0; apply H; apply H1. Qed. -Lemma interior_P3 : (D:R->Prop) (open_set (interior D)). -Intro; Unfold open_set interior; Unfold neighbourhood; Intros; Elim H; Intros. -Exists x0; Unfold included; Intros. -Pose del := ``x0-(Rabsolu (x-x1))``. -Cut ``0<del``. -Intro; Exists (mkposreal del H2); Intros. -Cut (included (disc x1 (mkposreal del H2)) (disc x x0)). -Intro; Assert H5 := (included_trans ? ? ? H4 H0). -Apply H5; Apply H3. -Unfold included; Unfold disc; Intros. -Apply Rle_lt_trans with ``(Rabsolu (x3-x1))+(Rabsolu (x1-x))``. -Replace ``x3-x`` with ``(x3-x1)+(x1-x)``; [Apply Rabsolu_triang | Ring]. -Replace (pos x0) with ``del+(Rabsolu (x1-x))``. -Do 2 Rewrite <- (Rplus_sym (Rabsolu ``x1-x``)); Apply Rlt_compatibility; Apply H4. -Unfold del; Rewrite <- (Rabsolu_Ropp ``x-x1``); Rewrite Ropp_distr2; Ring. -Unfold del; Apply Rlt_anti_compatibility with ``(Rabsolu (x-x1))``; Rewrite Rplus_Or; Replace ``(Rabsolu (x-x1))+(x0-(Rabsolu (x-x1)))`` with (pos x0); [Idtac | Ring]. -Unfold disc in H1; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H1. +Lemma interior_P3 : forall D:R -> Prop, open_set (interior D). +intro; unfold open_set, interior in |- *; unfold neighbourhood in |- *; + intros; elim H; intros. +exists x0; unfold included in |- *; intros. +pose (del := x0 - Rabs (x - x1)). +cut (0 < del). +intro; exists (mkposreal del H2); intros. +cut (included (disc x1 (mkposreal del H2)) (disc x x0)). +intro; assert (H5 := included_trans _ _ _ H4 H0). +apply H5; apply H3. +unfold included in |- *; unfold disc in |- *; intros. +apply Rle_lt_trans with (Rabs (x3 - x1) + Rabs (x1 - x)). +replace (x3 - x) with (x3 - x1 + (x1 - x)); [ apply Rabs_triang | ring ]. +replace (pos x0) with (del + Rabs (x1 - x)). +do 2 rewrite <- (Rplus_comm (Rabs (x1 - x))); apply Rplus_lt_compat_l; + apply H4. +unfold del in |- *; rewrite <- (Rabs_Ropp (x - x1)); rewrite Ropp_minus_distr; + ring. +unfold del in |- *; apply Rplus_lt_reg_r with (Rabs (x - x1)); + rewrite Rplus_0_r; + replace (Rabs (x - x1) + (x0 - Rabs (x - x1))) with (pos x0); + [ idtac | ring ]. +unfold disc in H1; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H1. Qed. -Lemma complementary_P1 : (D:R->Prop) ~(EXT y:R | (intersection_domain D (complementary D) y)). -Intro; Red; Intro; Elim H; Intros; Unfold intersection_domain complementary in H0; Elim H0; Intros; Elim H2; Assumption. +Lemma complementary_P1 : + forall D:R -> Prop, + ~ ( exists y : R | intersection_domain D (complementary D) y). +intro; red in |- *; intro; elim H; intros; + unfold intersection_domain, complementary in H0; elim H0; + intros; elim H2; assumption. Qed. -Lemma adherence_P2 : (D:R->Prop) (closed_set D) -> (included (adherence D) D). -Unfold closed_set; Unfold open_set complementary; Intros; Unfold included adherence; Intros; Assert H1 := (classic (D x)); Elim H1; Intro. -Assumption. -Assert H3 := (H ? H2); Assert H4 := (H0 ? H3); Elim H4; Intros; Unfold intersection_domain in H5; Elim H5; Intros; Elim H6; Assumption. +Lemma adherence_P2 : + forall D:R -> Prop, closed_set D -> included (adherence D) D. +unfold closed_set in |- *; unfold open_set, complementary in |- *; intros; + unfold included, adherence in |- *; intros; assert (H1 := classic (D x)); + elim H1; intro. +assumption. +assert (H3 := H _ H2); assert (H4 := H0 _ H3); elim H4; intros; + unfold intersection_domain in H5; elim H5; intros; + elim H6; assumption. Qed. -Lemma adherence_P3 : (D:R->Prop) (closed_set (adherence D)). -Intro; Unfold closed_set adherence; Unfold open_set complementary point_adherent; Intros; Pose P := [V:R->Prop](neighbourhood V x)->(EXT y:R | (intersection_domain V D y)); Assert H0 := (not_all_ex_not ? P H); Elim H0; Intros V0 H1; Unfold P in H1; Assert H2 := (imply_to_and ? ? H1); Unfold neighbourhood; Elim H2; Intros; Unfold neighbourhood in H3; Elim H3; Intros; Exists x0; Unfold included; Intros; Red; Intro. -Assert H8 := (H7 V0); Cut (EXT delta:posreal | (x:R)(disc x1 delta x)->(V0 x)). -Intro; Assert H10 := (H8 H9); Elim H4; Assumption. -Cut ``0<x0-(Rabsolu (x-x1))``. -Intro; Pose del := (mkposreal ? H9); Exists del; Intros; Unfold included in H5; Apply H5; Unfold disc; Apply Rle_lt_trans with ``(Rabsolu (x2-x1))+(Rabsolu (x1-x))``. -Replace ``x2-x`` with ``(x2-x1)+(x1-x)``; [Apply Rabsolu_triang | Ring]. -Replace (pos x0) with ``del+(Rabsolu (x1-x))``. -Do 2 Rewrite <- (Rplus_sym ``(Rabsolu (x1-x))``); Apply Rlt_compatibility; Apply H10. -Unfold del; Simpl; Rewrite <- (Rabsolu_Ropp ``x-x1``); Rewrite Ropp_distr2; Ring. -Apply Rlt_anti_compatibility with ``(Rabsolu (x-x1))``; Rewrite Rplus_Or; Replace ``(Rabsolu (x-x1))+(x0-(Rabsolu (x-x1)))`` with (pos x0); [Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H6 | Ring]. +Lemma adherence_P3 : forall D:R -> Prop, closed_set (adherence D). +intro; unfold closed_set, adherence in |- *; + unfold open_set, complementary, point_adherent in |- *; + intros; + pose + (P := + fun V:R -> Prop => + neighbourhood V x -> exists y : R | intersection_domain V D y); + assert (H0 := not_all_ex_not _ P H); elim H0; intros V0 H1; + unfold P in H1; assert (H2 := imply_to_and _ _ H1); + unfold neighbourhood in |- *; elim H2; intros; unfold neighbourhood in H3; + elim H3; intros; exists x0; unfold included in |- *; + intros; red in |- *; intro. +assert (H8 := H7 V0); + cut ( exists delta : posreal | (forall x:R, disc x1 delta x -> V0 x)). +intro; assert (H10 := H8 H9); elim H4; assumption. +cut (0 < x0 - Rabs (x - x1)). +intro; pose (del := mkposreal _ H9); exists del; intros; + unfold included in H5; apply H5; unfold disc in |- *; + apply Rle_lt_trans with (Rabs (x2 - x1) + Rabs (x1 - x)). +replace (x2 - x) with (x2 - x1 + (x1 - x)); [ apply Rabs_triang | ring ]. +replace (pos x0) with (del + Rabs (x1 - x)). +do 2 rewrite <- (Rplus_comm (Rabs (x1 - x))); apply Rplus_lt_compat_l; + apply H10. +unfold del in |- *; simpl in |- *; rewrite <- (Rabs_Ropp (x - x1)); + rewrite Ropp_minus_distr; ring. +apply Rplus_lt_reg_r with (Rabs (x - x1)); rewrite Rplus_0_r; + replace (Rabs (x - x1) + (x0 - Rabs (x - x1))) with (pos x0); + [ rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H6 | ring ]. Qed. -Definition eq_Dom [D1,D2:R->Prop] : Prop := (included D1 D2)/\(included D2 D1). +Definition eq_Dom (D1 D2:R -> Prop) : Prop := + included D1 D2 /\ included D2 D1. -Infix "=_D" eq_Dom (at level 5, no associativity). +Infix "=_D" := eq_Dom (at level 70, no associativity). -Lemma open_set_P1 : (D:R->Prop) (open_set D) <-> D =_D (interior D). -Intro; Split. -Intro; Unfold eq_Dom; Split. -Apply interior_P2; Assumption. -Apply interior_P1. -Intro; Unfold eq_Dom in H; Elim H; Clear H; Intros; Unfold open_set; Intros; Unfold included interior in H; Unfold included in H0; Apply (H ? H1). +Lemma open_set_P1 : forall D:R -> Prop, open_set D <-> D =_D interior D. +intro; split. +intro; unfold eq_Dom in |- *; split. +apply interior_P2; assumption. +apply interior_P1. +intro; unfold eq_Dom in H; elim H; clear H; intros; unfold open_set in |- *; + intros; unfold included, interior in H; unfold included in H0; + apply (H _ H1). Qed. -Lemma closed_set_P1 : (D:R->Prop) (closed_set D) <-> D =_D (adherence D). -Intro; Split. -Intro; Unfold eq_Dom; Split. -Apply adherence_P1. -Apply adherence_P2; Assumption. -Unfold eq_Dom; Unfold included; Intros; Assert H0 := (adherence_P3 D); Unfold closed_set in H0; Unfold closed_set; Unfold open_set; Unfold open_set in H0; Intros; Assert H2 : (complementary (adherence D) x). -Unfold complementary; Unfold complementary in H1; Red; Intro; Elim H; Clear H; Intros _ H; Elim H1; Apply (H ? H2). -Assert H3 := (H0 ? H2); Unfold neighbourhood; Unfold neighbourhood in H3; Elim H3; Intros; Exists x0; Unfold included; Unfold included in H4; Intros; Assert H6 := (H4 ? H5); Unfold complementary in H6; Unfold complementary; Red; Intro; Elim H; Clear H; Intros H _; Elim H6; Apply (H ? H7). +Lemma closed_set_P1 : forall D:R -> Prop, closed_set D <-> D =_D adherence D. +intro; split. +intro; unfold eq_Dom in |- *; split. +apply adherence_P1. +apply adherence_P2; assumption. +unfold eq_Dom in |- *; unfold included in |- *; intros; + assert (H0 := adherence_P3 D); unfold closed_set in H0; + unfold closed_set in |- *; unfold open_set in |- *; + unfold open_set in H0; intros; assert (H2 : complementary (adherence D) x). +unfold complementary in |- *; unfold complementary in H1; red in |- *; intro; + elim H; clear H; intros _ H; elim H1; apply (H _ H2). +assert (H3 := H0 _ H2); unfold neighbourhood in |- *; + unfold neighbourhood in H3; elim H3; intros; exists x0; + unfold included in |- *; unfold included in H4; intros; + assert (H6 := H4 _ H5); unfold complementary in H6; + unfold complementary in |- *; red in |- *; intro; + elim H; clear H; intros H _; elim H6; apply (H _ H7). Qed. -Lemma neighbourhood_P1 : (D1,D2:R->Prop;x:R) (included D1 D2) -> (neighbourhood D1 x) -> (neighbourhood D2 x). -Unfold included neighbourhood; Intros; Elim H0; Intros; Exists x0; Intros; Unfold included; Unfold included in H1; Intros; Apply (H ? (H1 ? H2)). +Lemma neighbourhood_P1 : + forall (D1 D2:R -> Prop) (x:R), + included D1 D2 -> neighbourhood D1 x -> neighbourhood D2 x. +unfold included, neighbourhood in |- *; intros; elim H0; intros; exists x0; + intros; unfold included in |- *; unfold included in H1; + intros; apply (H _ (H1 _ H2)). Qed. -Lemma open_set_P2 : (D1,D2:R->Prop) (open_set D1) -> (open_set D2) -> (open_set (union_domain D1 D2)). -Unfold open_set; Intros; Unfold union_domain in H1; Elim H1; Intro. -Apply neighbourhood_P1 with D1. -Unfold included union_domain; Tauto. -Apply H; Assumption. -Apply neighbourhood_P1 with D2. -Unfold included union_domain; Tauto. -Apply H0; Assumption. +Lemma open_set_P2 : + forall D1 D2:R -> Prop, + open_set D1 -> open_set D2 -> open_set (union_domain D1 D2). +unfold open_set in |- *; intros; unfold union_domain in H1; elim H1; intro. +apply neighbourhood_P1 with D1. +unfold included, union_domain in |- *; tauto. +apply H; assumption. +apply neighbourhood_P1 with D2. +unfold included, union_domain in |- *; tauto. +apply H0; assumption. Qed. -Lemma open_set_P3 : (D1,D2:R->Prop) (open_set D1) -> (open_set D2) -> (open_set (intersection_domain D1 D2)). -Unfold open_set; Intros; Unfold intersection_domain in H1; Elim H1; Intros. -Assert H4 := (H ? H2); Assert H5 := (H0 ? H3); Unfold intersection_domain; Unfold neighbourhood in H4 H5; Elim H4; Clear H; Intros del1 H; Elim H5; Clear H0; Intros del2 H0; Cut ``0<(Rmin del1 del2)``. -Intro; Pose del := (mkposreal ? H6). -Exists del; Unfold included; Intros; Unfold included in H H0; Unfold disc in H H0 H7. -Split. -Apply H; Apply Rlt_le_trans with (pos del). -Apply H7. -Unfold del; Simpl; Apply Rmin_l. -Apply H0; Apply Rlt_le_trans with (pos del). -Apply H7. -Unfold del; Simpl; Apply Rmin_r. -Unfold Rmin; Case (total_order_Rle del1 del2); Intro. -Apply (cond_pos del1). -Apply (cond_pos del2). +Lemma open_set_P3 : + forall D1 D2:R -> Prop, + open_set D1 -> open_set D2 -> open_set (intersection_domain D1 D2). +unfold open_set in |- *; intros; unfold intersection_domain in H1; elim H1; + intros. +assert (H4 := H _ H2); assert (H5 := H0 _ H3); + unfold intersection_domain in |- *; unfold neighbourhood in H4, H5; + elim H4; clear H; intros del1 H; elim H5; clear H0; + intros del2 H0; cut (0 < Rmin del1 del2). +intro; pose (del := mkposreal _ H6). +exists del; unfold included in |- *; intros; unfold included in H, H0; + unfold disc in H, H0, H7. +split. +apply H; apply Rlt_le_trans with (pos del). +apply H7. +unfold del in |- *; simpl in |- *; apply Rmin_l. +apply H0; apply Rlt_le_trans with (pos del). +apply H7. +unfold del in |- *; simpl in |- *; apply Rmin_r. +unfold Rmin in |- *; case (Rle_dec del1 del2); intro. +apply (cond_pos del1). +apply (cond_pos del2). Qed. -Lemma open_set_P4 : (open_set [x:R]False). -Unfold open_set; Intros; Elim H. +Lemma open_set_P4 : open_set (fun x:R => False). +unfold open_set in |- *; intros; elim H. Qed. -Lemma open_set_P5 : (open_set [x:R]True). -Unfold open_set; Intros; Unfold neighbourhood. -Exists (mkposreal R1 Rlt_R0_R1); Unfold included; Intros; Trivial. +Lemma open_set_P5 : open_set (fun x:R => True). +unfold open_set in |- *; intros; unfold neighbourhood in |- *. +exists (mkposreal 1 Rlt_0_1); unfold included in |- *; intros; trivial. Qed. -Lemma disc_P1 : (x:R;del:posreal) (open_set (disc x del)). -Intros; Assert H := (open_set_P1 (disc x del)). -Elim H; Intros; Apply H1. -Unfold eq_Dom; Split. -Unfold included interior disc; Intros; Cut ``0<del-(Rabsolu (x-x0))``. -Intro; Pose del2 := (mkposreal ? H3). -Exists del2; Unfold included; Intros. -Apply Rle_lt_trans with ``(Rabsolu (x1-x0))+(Rabsolu (x0 -x))``. -Replace ``x1-x`` with ``(x1-x0)+(x0-x)``; [Apply Rabsolu_triang | Ring]. -Replace (pos del) with ``del2 + (Rabsolu (x0-x))``. -Do 2 Rewrite <- (Rplus_sym ``(Rabsolu (x0-x))``); Apply Rlt_compatibility. -Apply H4. -Unfold del2; Simpl; Rewrite <- (Rabsolu_Ropp ``x-x0``); Rewrite Ropp_distr2; Ring. -Apply Rlt_anti_compatibility with ``(Rabsolu (x-x0))``; Rewrite Rplus_Or; Replace ``(Rabsolu (x-x0))+(del-(Rabsolu (x-x0)))`` with (pos del); [Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H2 | Ring]. -Apply interior_P1. +Lemma disc_P1 : forall (x:R) (del:posreal), open_set (disc x del). +intros; assert (H := open_set_P1 (disc x del)). +elim H; intros; apply H1. +unfold eq_Dom in |- *; split. +unfold included, interior, disc in |- *; intros; + cut (0 < del - Rabs (x - x0)). +intro; pose (del2 := mkposreal _ H3). +exists del2; unfold included in |- *; intros. +apply Rle_lt_trans with (Rabs (x1 - x0) + Rabs (x0 - x)). +replace (x1 - x) with (x1 - x0 + (x0 - x)); [ apply Rabs_triang | ring ]. +replace (pos del) with (del2 + Rabs (x0 - x)). +do 2 rewrite <- (Rplus_comm (Rabs (x0 - x))); apply Rplus_lt_compat_l. +apply H4. +unfold del2 in |- *; simpl in |- *; rewrite <- (Rabs_Ropp (x - x0)); + rewrite Ropp_minus_distr; ring. +apply Rplus_lt_reg_r with (Rabs (x - x0)); rewrite Rplus_0_r; + replace (Rabs (x - x0) + (del - Rabs (x - x0))) with (pos del); + [ rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H2 | ring ]. +apply interior_P1. Qed. -Lemma continuity_P1 : (f:R->R;x:R) (continuity_pt f x) <-> (W:R->Prop)(neighbourhood W (f x)) -> (EXT V:R->Prop | (neighbourhood V x) /\ ((y:R)(V y)->(W (f y)))). -Intros; Split. -Intros; Unfold neighbourhood in H0. -Elim H0; Intros del1 H1. -Unfold continuity_pt in H; Unfold continue_in in H; Unfold limit1_in in H; Unfold limit_in in H; Simpl in H; Unfold R_dist in H. -Assert H2 := (H del1 (cond_pos del1)). -Elim H2; Intros del2 H3. -Elim H3; Intros. -Exists (disc x (mkposreal del2 H4)). -Intros; Unfold included in H1; Split. -Unfold neighbourhood disc. -Exists (mkposreal del2 H4). -Unfold included; Intros; Assumption. -Intros; Apply H1; Unfold disc; Case (Req_EM y x); Intro. -Rewrite H7; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (cond_pos del1). -Apply H5; Split. -Unfold D_x no_cond; Split. -Trivial. -Apply not_sym; Apply H7. -Unfold disc in H6; Apply H6. -Intros; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Intros. -Assert H1 := (H (disc (f x) (mkposreal eps H0))). -Cut (neighbourhood (disc (f x) (mkposreal eps H0)) (f x)). -Intro; Assert H3 := (H1 H2). -Elim H3; Intros D H4; Elim H4; Intros; Unfold neighbourhood in H5; Elim H5; Intros del1 H7. -Exists (pos del1); Split. -Apply (cond_pos del1). -Intros; Elim H8; Intros; Simpl in H10; Unfold R_dist in H10; Simpl; Unfold R_dist; Apply (H6 ? (H7 ? H10)). -Unfold neighbourhood disc; Exists (mkposreal eps H0); Unfold included; Intros; Assumption. +Lemma continuity_P1 : + forall (f:R -> R) (x:R), + continuity_pt f x <-> + (forall W:R -> Prop, + neighbourhood W (f x) -> + exists V : R -> Prop + | neighbourhood V x /\ (forall y:R, V y -> W (f y))). +intros; split. +intros; unfold neighbourhood in H0. +elim H0; intros del1 H1. +unfold continuity_pt in H; unfold continue_in in H; unfold limit1_in in H; + unfold limit_in in H; simpl in H; unfold R_dist in H. +assert (H2 := H del1 (cond_pos del1)). +elim H2; intros del2 H3. +elim H3; intros. +exists (disc x (mkposreal del2 H4)). +intros; unfold included in H1; split. +unfold neighbourhood, disc in |- *. +exists (mkposreal del2 H4). +unfold included in |- *; intros; assumption. +intros; apply H1; unfold disc in |- *; case (Req_dec y x); intro. +rewrite H7; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + apply (cond_pos del1). +apply H5; split. +unfold D_x, no_cond in |- *; split. +trivial. +apply (sym_not_eq (A:=R)); apply H7. +unfold disc in H6; apply H6. +intros; unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + intros. +assert (H1 := H (disc (f x) (mkposreal eps H0))). +cut (neighbourhood (disc (f x) (mkposreal eps H0)) (f x)). +intro; assert (H3 := H1 H2). +elim H3; intros D H4; elim H4; intros; unfold neighbourhood in H5; elim H5; + intros del1 H7. +exists (pos del1); split. +apply (cond_pos del1). +intros; elim H8; intros; simpl in H10; unfold R_dist in H10; simpl in |- *; + unfold R_dist in |- *; apply (H6 _ (H7 _ H10)). +unfold neighbourhood, disc in |- *; exists (mkposreal eps H0); + unfold included in |- *; intros; assumption. Qed. -Definition image_rec [f:R->R;D:R->Prop] : R->Prop := [x:R](D (f x)). +Definition image_rec (f:R -> R) (D:R -> Prop) (x:R) : Prop := D (f x). (**********) -Lemma continuity_P2 : (f:R->R;D:R->Prop) (continuity f) -> (open_set D) -> (open_set (image_rec f D)). -Intros; Unfold open_set in H0; Unfold open_set; Intros; Assert H2 := (continuity_P1 f x); Elim H2; Intros H3 _; Assert H4 := (H3 (H x)); Unfold neighbourhood image_rec; Unfold image_rec in H1; Assert H5 := (H4 D (H0 (f x) H1)); Elim H5; Intros V0 H6; Elim H6; Intros; Unfold neighbourhood in H7; Elim H7; Intros del H9; Exists del; Unfold included in H9; Unfold included; Intros; Apply (H8 ? (H9 ? H10)). +Lemma continuity_P2 : + forall (f:R -> R) (D:R -> Prop), + continuity f -> open_set D -> open_set (image_rec f D). +intros; unfold open_set in H0; unfold open_set in |- *; intros; + assert (H2 := continuity_P1 f x); elim H2; intros H3 _; + assert (H4 := H3 (H x)); unfold neighbourhood, image_rec in |- *; + unfold image_rec in H1; assert (H5 := H4 D (H0 (f x) H1)); + elim H5; intros V0 H6; elim H6; intros; unfold neighbourhood in H7; + elim H7; intros del H9; exists del; unfold included in H9; + unfold included in |- *; intros; apply (H8 _ (H9 _ H10)). Qed. (**********) -Lemma continuity_P3 : (f:R->R) (continuity f) <-> (D:R->Prop) (open_set D)->(open_set (image_rec f D)). -Intros; Split. -Intros; Apply continuity_P2; Assumption. -Intros; Unfold continuity; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Cut (open_set (disc (f x) (mkposreal ? H0))). -Intro; Assert H2 := (H ? H1). -Unfold open_set image_rec in H2; Cut (disc (f x) (mkposreal ? H0) (f x)). -Intro; Assert H4 := (H2 ? H3). -Unfold neighbourhood in H4; Elim H4; Intros del H5. -Exists (pos del); Split. -Apply (cond_pos del). -Intros; Unfold included in H5; Apply H5; Elim H6; Intros; Apply H8. -Unfold disc; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply H0. -Apply disc_P1. +Lemma continuity_P3 : + forall f:R -> R, + continuity f <-> + (forall D:R -> Prop, open_set D -> open_set (image_rec f D)). +intros; split. +intros; apply continuity_P2; assumption. +intros; unfold continuity in |- *; unfold continuity_pt in |- *; + unfold continue_in in |- *; unfold limit1_in in |- *; + unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *; + intros; cut (open_set (disc (f x) (mkposreal _ H0))). +intro; assert (H2 := H _ H1). +unfold open_set, image_rec in H2; cut (disc (f x) (mkposreal _ H0) (f x)). +intro; assert (H4 := H2 _ H3). +unfold neighbourhood in H4; elim H4; intros del H5. +exists (pos del); split. +apply (cond_pos del). +intros; unfold included in H5; apply H5; elim H6; intros; apply H8. +unfold disc in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; apply H0. +apply disc_P1. Qed. (**********) -Theorem Rsepare : (x,y:R) ``x<>y``->(EXT V:R->Prop | (EXT W:R->Prop | (neighbourhood V x)/\(neighbourhood W y)/\~(EXT y:R | (intersection_domain V W y)))). -Intros x y Hsep; Pose D := ``(Rabsolu (x-y))``. -Cut ``0<D/2``. -Intro; Exists (disc x (mkposreal ? H)). -Exists (disc y (mkposreal ? H)); Split. -Unfold neighbourhood; Exists (mkposreal ? H); Unfold included; Tauto. -Split. -Unfold neighbourhood; Exists (mkposreal ? H); Unfold included; Tauto. -Red; Intro; Elim H0; Intros; Unfold intersection_domain in H1; Elim H1; Intros. -Cut ``D<D``. -Intro; Elim (Rlt_antirefl ? H4). -Change ``(Rabsolu (x-y))<D``; Apply Rle_lt_trans with ``(Rabsolu (x-x0))+(Rabsolu (x0-y))``. -Replace ``x-y`` with ``(x-x0)+(x0-y)``; [Apply Rabsolu_triang | Ring]. -Rewrite (double_var D); Apply Rplus_lt. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H2. -Apply H3. -Unfold Rdiv; Apply Rmult_lt_pos. -Unfold D; Apply Rabsolu_pos_lt; Apply (Rminus_eq_contra ? ? Hsep). -Apply Rlt_Rinv; Sup0. +Theorem Rsepare : + forall x y:R, + x <> y -> + exists V : R -> Prop + | ( exists W : R -> Prop + | neighbourhood V x /\ + neighbourhood W y /\ ~ ( exists y : R | intersection_domain V W y)). +intros x y Hsep; pose (D := Rabs (x - y)). +cut (0 < D / 2). +intro; exists (disc x (mkposreal _ H)). +exists (disc y (mkposreal _ H)); split. +unfold neighbourhood in |- *; exists (mkposreal _ H); unfold included in |- *; + tauto. +split. +unfold neighbourhood in |- *; exists (mkposreal _ H); unfold included in |- *; + tauto. +red in |- *; intro; elim H0; intros; unfold intersection_domain in H1; + elim H1; intros. +cut (D < D). +intro; elim (Rlt_irrefl _ H4). +change (Rabs (x - y) < D) in |- *; + apply Rle_lt_trans with (Rabs (x - x0) + Rabs (x0 - y)). +replace (x - y) with (x - x0 + (x0 - y)); [ apply Rabs_triang | ring ]. +rewrite (double_var D); apply Rplus_lt_compat. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H2. +apply H3. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +unfold D in |- *; apply Rabs_pos_lt; apply (Rminus_eq_contra _ _ Hsep). +apply Rinv_0_lt_compat; prove_sup0. Qed. -Record family : Type := mkfamily { - ind : R->Prop; - f :> R->R->Prop; - cond_fam : (x:R)(EXT y:R|(f x y))->(ind x) }. +Record family : Type := mkfamily + {ind : R -> Prop; + f :> R -> R -> Prop; + cond_fam : forall x:R, ( exists y : R | f x y) -> ind x}. -Definition family_open_set [f:family] : Prop := (x:R) (open_set (f x)). +Definition family_open_set (f:family) : Prop := forall x:R, open_set (f x). -Definition domain_finite [D:R->Prop] : Prop := (EXT l:Rlist | (x:R)(D x)<->(In x l)). +Definition domain_finite (D:R -> Prop) : Prop := + exists l : Rlist | (forall x:R, D x <-> In x l). -Definition family_finite [f:family] : Prop := (domain_finite (ind f)). +Definition family_finite (f:family) : Prop := domain_finite (ind f). -Definition covering [D:R->Prop;f:family] : Prop := (x:R) (D x)->(EXT y:R | (f y x)). +Definition covering (D:R -> Prop) (f:family) : Prop := + forall x:R, D x -> exists y : R | f y x. -Definition covering_open_set [D:R->Prop;f:family] : Prop := (covering D f)/\(family_open_set f). +Definition covering_open_set (D:R -> Prop) (f:family) : Prop := + covering D f /\ family_open_set f. -Definition covering_finite [D:R->Prop;f:family] : Prop := (covering D f)/\(family_finite f). +Definition covering_finite (D:R -> Prop) (f:family) : Prop := + covering D f /\ family_finite f. -Lemma restriction_family : (f:family;D:R->Prop) (x:R)(EXT y:R|([z1:R][z2:R](f z1 z2)/\(D z1) x y))->(intersection_domain (ind f) D x). -Intros; Elim H; Intros; Unfold intersection_domain; Elim H0; Intros; Split. -Apply (cond_fam f0); Exists x0; Assumption. -Assumption. +Lemma restriction_family : + forall (f:family) (D:R -> Prop) (x:R), + ( exists y : R | (fun z1 z2:R => f z1 z2 /\ D z1) x y) -> + intersection_domain (ind f) D x. +intros; elim H; intros; unfold intersection_domain in |- *; elim H0; intros; + split. +apply (cond_fam f0); exists x0; assumption. +assumption. Qed. -Definition subfamily [f:family;D:R->Prop] : family := (mkfamily (intersection_domain (ind f) D) [x:R][y:R](f x y)/\(D x) (restriction_family f D)). +Definition subfamily (f:family) (D:R -> Prop) : family := + mkfamily (intersection_domain (ind f) D) (fun x y:R => f x y /\ D x) + (restriction_family f D). -Definition compact [X:R->Prop] : Prop := (f:family) (covering_open_set X f) -> (EXT D:R->Prop | (covering_finite X (subfamily f D))). +Definition compact (X:R -> Prop) : Prop := + forall f:family, + covering_open_set X f -> + exists D : R -> Prop | covering_finite X (subfamily f D). (**********) -Lemma family_P1 : (f:family;D:R->Prop) (family_open_set f) -> (family_open_set (subfamily f D)). -Unfold family_open_set; Intros; Unfold subfamily; Simpl; Assert H0 := (classic (D x)). -Elim H0; Intro. -Cut (open_set (f0 x))->(open_set [y:R](f0 x y)/\(D x)). -Intro; Apply H2; Apply H. -Unfold open_set; Unfold neighbourhood; Intros; Elim H3; Intros; Assert H6 := (H2 ? H4); Elim H6; Intros; Exists x1; Unfold included; Intros; Split. -Apply (H7 ? H8). -Assumption. -Cut (open_set [y:R]False) -> (open_set [y:R](f0 x y)/\(D x)). -Intro; Apply H2; Apply open_set_P4. -Unfold open_set; Unfold neighbourhood; Intros; Elim H3; Intros; Elim H1; Assumption. +Lemma family_P1 : + forall (f:family) (D:R -> Prop), + family_open_set f -> family_open_set (subfamily f D). +unfold family_open_set in |- *; intros; unfold subfamily in |- *; + simpl in |- *; assert (H0 := classic (D x)). +elim H0; intro. +cut (open_set (f0 x) -> open_set (fun y:R => f0 x y /\ D x)). +intro; apply H2; apply H. +unfold open_set in |- *; unfold neighbourhood in |- *; intros; elim H3; + intros; assert (H6 := H2 _ H4); elim H6; intros; exists x1; + unfold included in |- *; intros; split. +apply (H7 _ H8). +assumption. +cut (open_set (fun y:R => False) -> open_set (fun y:R => f0 x y /\ D x)). +intro; apply H2; apply open_set_P4. +unfold open_set in |- *; unfold neighbourhood in |- *; intros; elim H3; + intros; elim H1; assumption. Qed. -Definition bounded [D:R->Prop] : Prop := (EXT m:R | (EXT M:R | (x:R)(D x)->``m<=x<=M``)). +Definition bounded (D:R -> Prop) : Prop := + exists m : R | ( exists M : R | (forall x:R, D x -> m <= x <= M)). -Lemma open_set_P6 : (D1,D2:R->Prop) (open_set D1) -> D1 =_D D2 -> (open_set D2). -Unfold open_set; Unfold neighbourhood; Intros. -Unfold eq_Dom in H0; Elim H0; Intros. -Assert H4 := (H ? (H3 ? H1)). -Elim H4; Intros. -Exists x0; Apply included_trans with D1; Assumption. +Lemma open_set_P6 : + forall D1 D2:R -> Prop, open_set D1 -> D1 =_D D2 -> open_set D2. +unfold open_set in |- *; unfold neighbourhood in |- *; intros. +unfold eq_Dom in H0; elim H0; intros. +assert (H4 := H _ (H3 _ H1)). +elim H4; intros. +exists x0; apply included_trans with D1; assumption. Qed. (**********) -Lemma compact_P1 : (X:R->Prop) (compact X) -> (bounded X). -Intros; Unfold compact in H; Pose D := [x:R]True; Pose g := [x:R][y:R]``(Rabsolu y)<x``; Cut (x:R)(EXT y|(g x y))->True; [Intro | Intro; Trivial]. -Pose f0 := (mkfamily D g H0); Assert H1 := (H f0); Cut (covering_open_set X f0). -Intro; Assert H3 := (H1 H2); Elim H3; Intros D' H4; Unfold covering_finite in H4; Elim H4; Intros; Unfold family_finite in H6; Unfold domain_finite in H6; Elim H6; Intros l H7; Unfold bounded; Pose r := (MaxRlist l). -Exists ``-r``; Exists r; Intros. -Unfold covering in H5; Assert H9 := (H5 ? H8); Elim H9; Intros; Unfold subfamily in H10; Simpl in H10; Elim H10; Intros; Assert H13 := (H7 x0); Simpl in H13; Cut (intersection_domain D D' x0). -Elim H13; Clear H13; Intros. -Assert H16 := (H13 H15); Unfold g in H11; Split. -Cut ``x0<=r``. -Intro; Cut ``(Rabsolu x)<r``. -Intro; Assert H19 := (Rabsolu_def2 x r H18); Elim H19; Intros; Left; Assumption. -Apply Rlt_le_trans with x0; Assumption. -Apply (MaxRlist_P1 l x0 H16). -Cut ``x0<=r``. -Intro; Apply Rle_trans with (Rabsolu x). -Apply Rle_Rabsolu. -Apply Rle_trans with x0. -Left; Apply H11. -Assumption. -Apply (MaxRlist_P1 l x0 H16). -Unfold intersection_domain D; Tauto. -Unfold covering_open_set; Split. -Unfold covering; Intros; Simpl; Exists ``(Rabsolu x)+1``; Unfold g; Pattern 1 (Rabsolu x); Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1. -Unfold family_open_set; Intro; Case (total_order R0 x); Intro. -Apply open_set_P6 with (disc R0 (mkposreal ? H2)). -Apply disc_P1. -Unfold eq_Dom; Unfold f0; Simpl; Unfold g disc; Split. -Unfold included; Intros; Unfold Rminus in H3; Rewrite Ropp_O in H3; Rewrite Rplus_Or in H3; Apply H3. -Unfold included; Intros; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply H3. -Apply open_set_P6 with [x:R]False. -Apply open_set_P4. -Unfold eq_Dom; Split. -Unfold included; Intros; Elim H3. -Unfold included f0; Simpl; Unfold g; Intros; Elim H2; Intro; [Rewrite <- H4 in H3; Assert H5 := (Rabsolu_pos x0); Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H5 H3)) | Assert H6 := (Rabsolu_pos x0); Assert H7 := (Rlt_trans ? ? ? H3 H4); Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H6 H7))]. +Lemma compact_P1 : forall X:R -> Prop, compact X -> bounded X. +intros; unfold compact in H; pose (D := fun x:R => True); + pose (g := fun x y:R => Rabs y < x); + cut (forall x:R, ( exists y : _ | g x y) -> True); + [ intro | intro; trivial ]. +pose (f0 := mkfamily D g H0); assert (H1 := H f0); + cut (covering_open_set X f0). +intro; assert (H3 := H1 H2); elim H3; intros D' H4; + unfold covering_finite in H4; elim H4; intros; unfold family_finite in H6; + unfold domain_finite in H6; elim H6; intros l H7; + unfold bounded in |- *; pose (r := MaxRlist l). +exists (- r); exists r; intros. +unfold covering in H5; assert (H9 := H5 _ H8); elim H9; intros; + unfold subfamily in H10; simpl in H10; elim H10; intros; + assert (H13 := H7 x0); simpl in H13; cut (intersection_domain D D' x0). +elim H13; clear H13; intros. +assert (H16 := H13 H15); unfold g in H11; split. +cut (x0 <= r). +intro; cut (Rabs x < r). +intro; assert (H19 := Rabs_def2 x r H18); elim H19; intros; left; assumption. +apply Rlt_le_trans with x0; assumption. +apply (MaxRlist_P1 l x0 H16). +cut (x0 <= r). +intro; apply Rle_trans with (Rabs x). +apply RRle_abs. +apply Rle_trans with x0. +left; apply H11. +assumption. +apply (MaxRlist_P1 l x0 H16). +unfold intersection_domain, D in |- *; tauto. +unfold covering_open_set in |- *; split. +unfold covering in |- *; intros; simpl in |- *; exists (Rabs x + 1); + unfold g in |- *; pattern (Rabs x) at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_lt_compat_l; apply Rlt_0_1. +unfold family_open_set in |- *; intro; case (Rtotal_order 0 x); intro. +apply open_set_P6 with (disc 0 (mkposreal _ H2)). +apply disc_P1. +unfold eq_Dom in |- *; unfold f0 in |- *; simpl in |- *; + unfold g, disc in |- *; split. +unfold included in |- *; intros; unfold Rminus in H3; rewrite Ropp_0 in H3; + rewrite Rplus_0_r in H3; apply H3. +unfold included in |- *; intros; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; apply H3. +apply open_set_P6 with (fun x:R => False). +apply open_set_P4. +unfold eq_Dom in |- *; split. +unfold included in |- *; intros; elim H3. +unfold included, f0 in |- *; simpl in |- *; unfold g in |- *; intros; elim H2; + intro; + [ rewrite <- H4 in H3; assert (H5 := Rabs_pos x0); + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H5 H3)) + | assert (H6 := Rabs_pos x0); assert (H7 := Rlt_trans _ _ _ H3 H4); + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H6 H7)) ]. Qed. (**********) -Lemma compact_P2 : (X:R->Prop) (compact X) -> (closed_set X). -Intros; Assert H0 := (closed_set_P1 X); Elim H0; Clear H0; Intros _ H0; Apply H0; Clear H0. -Unfold eq_Dom; Split. -Apply adherence_P1. -Unfold included; Unfold adherence; Unfold point_adherent; Intros; Unfold compact in H; Assert H1 := (classic (X x)); Elim H1; Clear H1; Intro. -Assumption. -Cut (y:R)(X y)->``0<(Rabsolu (y-x))/2``. -Intro; Pose D := X; Pose g := [y:R][z:R]``(Rabsolu (y-z))<(Rabsolu (y-x))/2``/\(D y); Cut (x:R)(EXT y|(g x y))->(D x). -Intro; Pose f0 := (mkfamily D g H3); Assert H4 := (H f0); Cut (covering_open_set X f0). -Intro; Assert H6 := (H4 H5); Elim H6; Clear H6; Intros D' H6. -Unfold covering_finite in H6; Decompose [and] H6; Unfold covering subfamily in H7; Simpl in H7; Unfold family_finite subfamily in H8; Simpl in H8; Unfold domain_finite in H8; Elim H8; Clear H8; Intros l H8; Pose alp := (MinRlist (AbsList l x)); Cut ``0<alp``. -Intro; Assert H10 := (H0 (disc x (mkposreal ? H9))); Cut (neighbourhood (disc x (mkposreal alp H9)) x). -Intro; Assert H12 := (H10 H11); Elim H12; Clear H12; Intros y H12; Unfold intersection_domain in H12; Elim H12; Clear H12; Intros; Assert H14 := (H7 ? H13); Elim H14; Clear H14; Intros y0 H14; Elim H14; Clear H14; Intros; Unfold g in H14; Elim H14; Clear H14; Intros; Unfold disc in H12; Simpl in H12; Cut ``alp<=(Rabsolu (y0-x))/2``. -Intro; Assert H18 := (Rlt_le_trans ? ? ? H12 H17); Cut ``(Rabsolu (y0-x))<(Rabsolu (y0-x))``. -Intro; Elim (Rlt_antirefl ? H19). -Apply Rle_lt_trans with ``(Rabsolu (y0-y))+(Rabsolu (y-x))``. -Replace ``y0-x`` with ``(y0-y)+(y-x)``; [Apply Rabsolu_triang | Ring]. -Rewrite (double_var ``(Rabsolu (y0-x))``); Apply Rplus_lt; Assumption. -Apply (MinRlist_P1 (AbsList l x) ``(Rabsolu (y0-x))/2``); Apply AbsList_P1; Elim (H8 y0); Clear H8; Intros; Apply H8; Unfold intersection_domain; Split; Assumption. -Assert H11 := (disc_P1 x (mkposreal alp H9)); Unfold open_set in H11; Apply H11. -Unfold disc; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply H9. -Unfold alp; Apply MinRlist_P2; Intros; Assert H10 := (AbsList_P2 ? ? ? H9); Elim H10; Clear H10; Intros z H10; Elim H10; Clear H10; Intros; Rewrite H11; Apply H2; Elim (H8 z); Clear H8; Intros; Assert H13 := (H12 H10); Unfold intersection_domain D in H13; Elim H13; Clear H13; Intros; Assumption. -Unfold covering_open_set; Split. -Unfold covering; Intros; Exists x0; Simpl; Unfold g; Split. -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Unfold Rminus in H2; Apply (H2 ? H5). -Apply H5. -Unfold family_open_set; Intro; Simpl; Unfold g; Elim (classic (D x0)); Intro. -Apply open_set_P6 with (disc x0 (mkposreal ? (H2 ? H5))). -Apply disc_P1. -Unfold eq_Dom; Split. -Unfold included disc; Simpl; Intros; Split. -Rewrite <- (Rabsolu_Ropp ``x0-x1``); Rewrite Ropp_distr2; Apply H6. -Apply H5. -Unfold included disc; Simpl; Intros; Elim H6; Intros; Rewrite <- (Rabsolu_Ropp ``x1-x0``); Rewrite Ropp_distr2; Apply H7. -Apply open_set_P6 with [z:R]False. -Apply open_set_P4. -Unfold eq_Dom; Split. -Unfold included; Intros; Elim H6. -Unfold included; Intros; Elim H6; Intros; Elim H5; Assumption. -Intros; Elim H3; Intros; Unfold g in H4; Elim H4; Clear H4; Intros _ H4; Apply H4. -Intros; Unfold Rdiv; Apply Rmult_lt_pos. -Apply Rabsolu_pos_lt; Apply Rminus_eq_contra; Red; Intro; Rewrite H3 in H2; Elim H1; Apply H2. -Apply Rlt_Rinv; Sup0. +Lemma compact_P2 : forall X:R -> Prop, compact X -> closed_set X. +intros; assert (H0 := closed_set_P1 X); elim H0; clear H0; intros _ H0; + apply H0; clear H0. +unfold eq_Dom in |- *; split. +apply adherence_P1. +unfold included in |- *; unfold adherence in |- *; + unfold point_adherent in |- *; intros; unfold compact in H; + assert (H1 := classic (X x)); elim H1; clear H1; intro. +assumption. +cut (forall y:R, X y -> 0 < Rabs (y - x) / 2). +intro; pose (D := X); + pose (g := fun y z:R => Rabs (y - z) < Rabs (y - x) / 2 /\ D y); + cut (forall x:R, ( exists y : _ | g x y) -> D x). +intro; pose (f0 := mkfamily D g H3); assert (H4 := H f0); + cut (covering_open_set X f0). +intro; assert (H6 := H4 H5); elim H6; clear H6; intros D' H6. +unfold covering_finite in H6; decompose [and] H6; + unfold covering, subfamily in H7; simpl in H7; + unfold family_finite, subfamily in H8; simpl in H8; + unfold domain_finite in H8; elim H8; clear H8; intros l H8; + pose (alp := MinRlist (AbsList l x)); cut (0 < alp). +intro; assert (H10 := H0 (disc x (mkposreal _ H9))); + cut (neighbourhood (disc x (mkposreal alp H9)) x). +intro; assert (H12 := H10 H11); elim H12; clear H12; intros y H12; + unfold intersection_domain in H12; elim H12; clear H12; + intros; assert (H14 := H7 _ H13); elim H14; clear H14; + intros y0 H14; elim H14; clear H14; intros; unfold g in H14; + elim H14; clear H14; intros; unfold disc in H12; simpl in H12; + cut (alp <= Rabs (y0 - x) / 2). +intro; assert (H18 := Rlt_le_trans _ _ _ H12 H17); + cut (Rabs (y0 - x) < Rabs (y0 - x)). +intro; elim (Rlt_irrefl _ H19). +apply Rle_lt_trans with (Rabs (y0 - y) + Rabs (y - x)). +replace (y0 - x) with (y0 - y + (y - x)); [ apply Rabs_triang | ring ]. +rewrite (double_var (Rabs (y0 - x))); apply Rplus_lt_compat; assumption. +apply (MinRlist_P1 (AbsList l x) (Rabs (y0 - x) / 2)); apply AbsList_P1; + elim (H8 y0); clear H8; intros; apply H8; unfold intersection_domain in |- *; + split; assumption. +assert (H11 := disc_P1 x (mkposreal alp H9)); unfold open_set in H11; + apply H11. +unfold disc in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; apply H9. +unfold alp in |- *; apply MinRlist_P2; intros; + assert (H10 := AbsList_P2 _ _ _ H9); elim H10; clear H10; + intros z H10; elim H10; clear H10; intros; rewrite H11; + apply H2; elim (H8 z); clear H8; intros; assert (H13 := H12 H10); + unfold intersection_domain, D in H13; elim H13; clear H13; + intros; assumption. +unfold covering_open_set in |- *; split. +unfold covering in |- *; intros; exists x0; simpl in |- *; unfold g in |- *; + split. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + unfold Rminus in H2; apply (H2 _ H5). +apply H5. +unfold family_open_set in |- *; intro; simpl in |- *; unfold g in |- *; + elim (classic (D x0)); intro. +apply open_set_P6 with (disc x0 (mkposreal _ (H2 _ H5))). +apply disc_P1. +unfold eq_Dom in |- *; split. +unfold included, disc in |- *; simpl in |- *; intros; split. +rewrite <- (Rabs_Ropp (x0 - x1)); rewrite Ropp_minus_distr; apply H6. +apply H5. +unfold included, disc in |- *; simpl in |- *; intros; elim H6; intros; + rewrite <- (Rabs_Ropp (x1 - x0)); rewrite Ropp_minus_distr; + apply H7. +apply open_set_P6 with (fun z:R => False). +apply open_set_P4. +unfold eq_Dom in |- *; split. +unfold included in |- *; intros; elim H6. +unfold included in |- *; intros; elim H6; intros; elim H5; assumption. +intros; elim H3; intros; unfold g in H4; elim H4; clear H4; intros _ H4; + apply H4. +intros; unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply Rabs_pos_lt; apply Rminus_eq_contra; red in |- *; intro; + rewrite H3 in H2; elim H1; apply H2. +apply Rinv_0_lt_compat; prove_sup0. Qed. (**********) -Lemma compact_EMP : (compact [_:R]False). -Unfold compact; Intros; Exists [x:R]False; Unfold covering_finite; Split. -Unfold covering; Intros; Elim H0. -Unfold family_finite; Unfold domain_finite; Exists nil; Intro. -Split. -Simpl; Unfold intersection_domain; Intros; Elim H0. -Elim H0; Clear H0; Intros _ H0; Elim H0. -Simpl; Intro; Elim H0. +Lemma compact_EMP : compact (fun _:R => False). +unfold compact in |- *; intros; exists (fun x:R => False); + unfold covering_finite in |- *; split. +unfold covering in |- *; intros; elim H0. +unfold family_finite in |- *; unfold domain_finite in |- *; exists nil; intro. +split. +simpl in |- *; unfold intersection_domain in |- *; intros; elim H0. +elim H0; clear H0; intros _ H0; elim H0. +simpl in |- *; intro; elim H0. Qed. -Lemma compact_eqDom : (X1,X2:R->Prop) (compact X1) -> X1 =_D X2 -> (compact X2). -Unfold compact; Intros; Unfold eq_Dom in H0; Elim H0; Clear H0; Unfold included; Intros; Assert H3 : (covering_open_set X1 f0). -Unfold covering_open_set; Unfold covering_open_set in H1; Elim H1; Clear H1; Intros; Split. -Unfold covering in H1; Unfold covering; Intros; Apply (H1 ? (H0 ? H4)). -Apply H3. -Elim (H ? H3); Intros D H4; Exists D; Unfold covering_finite; Unfold covering_finite in H4; Elim H4; Intros; Split. -Unfold covering in H5; Unfold covering; Intros; Apply (H5 ? (H2 ? H7)). -Apply H6. +Lemma compact_eqDom : + forall X1 X2:R -> Prop, compact X1 -> X1 =_D X2 -> compact X2. +unfold compact in |- *; intros; unfold eq_Dom in H0; elim H0; clear H0; + unfold included in |- *; intros; assert (H3 : covering_open_set X1 f0). +unfold covering_open_set in |- *; unfold covering_open_set in H1; elim H1; + clear H1; intros; split. +unfold covering in H1; unfold covering in |- *; intros; + apply (H1 _ (H0 _ H4)). +apply H3. +elim (H _ H3); intros D H4; exists D; unfold covering_finite in |- *; + unfold covering_finite in H4; elim H4; intros; split. +unfold covering in H5; unfold covering in |- *; intros; + apply (H5 _ (H2 _ H7)). +apply H6. Qed. (* Borel-Lebesgue's lemma *) -Lemma compact_P3 : (a,b:R) (compact [c:R]``a<=c<=b``). -Intros; Case (total_order_Rle a b); Intro. -Unfold compact; Intros; Pose A := [x:R]``a<=x<=b``/\(EXT D:R->Prop | (covering_finite [c:R]``a <= c <= x`` (subfamily f0 D))); Cut (A a). -Intro; Cut (bound A). -Intro; Cut (EXT a0:R | (A a0)). -Intro; Assert H3 := (complet A H1 H2); Elim H3; Clear H3; Intros m H3; Unfold is_lub in H3; Cut ``a<=m<=b``. -Intro; Unfold covering_open_set in H; Elim H; Clear H; Intros; Unfold covering in H; Assert H6 := (H m H4); Elim H6; Clear H6; Intros y0 H6; Unfold family_open_set in H5; Assert H7 := (H5 y0); Unfold open_set in H7; Assert H8 := (H7 m H6); Unfold neighbourhood in H8; Elim H8; Clear H8; Intros eps H8; Cut (EXT x:R | (A x)/\``m-eps<x<=m``). -Intro; Elim H9; Clear H9; Intros x H9; Elim H9; Clear H9; Intros; Case (Req_EM m b); Intro. -Rewrite H11 in H10; Rewrite H11 in H8; Unfold A in H9; Elim H9; Clear H9; Intros; Elim H12; Clear H12; Intros Dx H12; Pose Db := [x:R](Dx x)\/x==y0; Exists Db; Unfold covering_finite; Split. -Unfold covering; Unfold covering_finite in H12; Elim H12; Clear H12; Intros; Unfold covering in H12; Case (total_order_Rle x0 x); Intro. -Cut ``a<=x0<=x``. -Intro; Assert H16 := (H12 x0 H15); Elim H16; Clear H16; Intros; Exists x1; Simpl in H16; Simpl; Unfold Db; Elim H16; Clear H16; Intros; Split; [Apply H16 | Left; Apply H17]. -Split. -Elim H14; Intros; Assumption. -Assumption. -Exists y0; Simpl; Split. -Apply H8; Unfold disc; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Rewrite Rabsolu_right. -Apply Rlt_trans with ``b-x``. -Unfold Rminus; Apply Rlt_compatibility; Apply Rlt_Ropp; Auto with real. -Elim H10; Intros H15 _; Apply Rlt_anti_compatibility with ``x-eps``; Replace ``x-eps+(b-x)`` with ``b-eps``; [Replace ``x-eps+eps`` with x; [Apply H15 | Ring] | Ring]. -Apply Rge_minus; Apply Rle_sym1; Elim H14; Intros _ H15; Apply H15. -Unfold Db; Right; Reflexivity. -Unfold family_finite; Unfold domain_finite; Unfold covering_finite in H12; Elim H12; Clear H12; Intros; Unfold family_finite in H13; Unfold domain_finite in H13; Elim H13; Clear H13; Intros l H13; Exists (cons y0 l); Intro; Split. -Intro; Simpl in H14; Unfold intersection_domain in H14; Elim (H13 x0); Clear H13; Intros; Case (Req_EM x0 y0); Intro. -Simpl; Left; Apply H16. -Simpl; Right; Apply H13. -Simpl; Unfold intersection_domain; Unfold Db in H14; Decompose [and or] H14. -Split; Assumption. -Elim H16; Assumption. -Intro; Simpl in H14; Elim H14; Intro; Simpl; Unfold intersection_domain. -Split. -Apply (cond_fam f0); Rewrite H15; Exists m; Apply H6. -Unfold Db; Right; Assumption. -Simpl; Unfold intersection_domain; Elim (H13 x0). -Intros _ H16; Assert H17 := (H16 H15); Simpl in H17; Unfold intersection_domain in H17; Split. -Elim H17; Intros; Assumption. -Unfold Db; Left; Elim H17; Intros; Assumption. -Pose m' := (Rmin ``m+eps/2`` b); Cut (A m'). -Intro; Elim H3; Intros; Unfold is_upper_bound in H13; Assert H15 := (H13 m' H12); Cut ``m<m'``. -Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H15 H16)). -Unfold m'; Unfold Rmin; Case (total_order_Rle ``m+eps/2`` b); Intro. -Pattern 1 m; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos eps) | Apply Rlt_Rinv; Sup0]. -Elim H4; Intros. -Elim H17; Intro. -Assumption. -Elim H11; Assumption. -Unfold A; Split. -Split. -Apply Rle_trans with m. -Elim H4; Intros; Assumption. -Unfold m'; Unfold Rmin; Case (total_order_Rle ``m+eps/2`` b); Intro. -Pattern 1 m; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos eps) | Apply Rlt_Rinv; Sup0]. -Elim H4; Intros. -Elim H13; Intro. -Assumption. -Elim H11; Assumption. -Unfold m'; Apply Rmin_r. -Unfold A in H9; Elim H9; Clear H9; Intros; Elim H12; Clear H12; Intros Dx H12; Pose Db := [x:R](Dx x)\/x==y0; Exists Db; Unfold covering_finite; Split. -Unfold covering; Unfold covering_finite in H12; Elim H12; Clear H12; Intros; Unfold covering in H12; Case (total_order_Rle x0 x); Intro. -Cut ``a<=x0<=x``. -Intro; Assert H16 := (H12 x0 H15); Elim H16; Clear H16; Intros; Exists x1; Simpl in H16; Simpl; Unfold Db. -Elim H16; Clear H16; Intros; Split; [Apply H16 | Left; Apply H17]. -Elim H14; Intros; Split; Assumption. -Exists y0; Simpl; Split. -Apply H8; Unfold disc; Unfold Rabsolu; Case (case_Rabsolu ``x0-m``); Intro. -Rewrite Ropp_distr2; Apply Rlt_trans with ``m-x``. -Unfold Rminus; Apply Rlt_compatibility; Apply Rlt_Ropp; Auto with real. -Apply Rlt_anti_compatibility with ``x-eps``; Replace ``x-eps+(m-x)`` with ``m-eps``. -Replace ``x-eps+eps`` with x. -Elim H10; Intros; Assumption. -Ring. -Ring. -Apply Rle_lt_trans with ``m'-m``. -Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-m``); Apply Rle_compatibility; Elim H14; Intros; Assumption. -Apply Rlt_anti_compatibility with m; Replace ``m+(m'-m)`` with m'. -Apply Rle_lt_trans with ``m+eps/2``. -Unfold m'; Apply Rmin_l. -Apply Rlt_compatibility; Apply Rlt_monotony_contra with ``2``. -Sup0. -Unfold Rdiv; Rewrite <- (Rmult_sym ``/2``); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Pattern 1 (pos eps); Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Apply (cond_pos eps). -DiscrR. -Ring. -Unfold Db; Right; Reflexivity. -Unfold family_finite; Unfold domain_finite; Unfold covering_finite in H12; Elim H12; Clear H12; Intros; Unfold family_finite in H13; Unfold domain_finite in H13; Elim H13; Clear H13; Intros l H13; Exists (cons y0 l); Intro; Split. -Intro; Simpl in H14; Unfold intersection_domain in H14; Elim (H13 x0); Clear H13; Intros; Case (Req_EM x0 y0); Intro. -Simpl; Left; Apply H16. -Simpl; Right; Apply H13; Simpl; Unfold intersection_domain; Unfold Db in H14; Decompose [and or] H14. -Split; Assumption. -Elim H16; Assumption. -Intro; Simpl in H14; Elim H14; Intro; Simpl; Unfold intersection_domain. -Split. -Apply (cond_fam f0); Rewrite H15; Exists m; Apply H6. -Unfold Db; Right; Assumption. -Elim (H13 x0); Intros _ H16. -Assert H17 := (H16 H15). -Simpl in H17. -Unfold intersection_domain in H17. -Split. -Elim H17; Intros; Assumption. -Unfold Db; Left; Elim H17; Intros; Assumption. -Elim (classic (EXT x:R | (A x)/\``m-eps < x <= m``)); Intro. -Assumption. -Elim H3; Intros; Cut (is_upper_bound A ``m-eps``). -Intro; Assert H13 := (H11 ? H12); Cut ``m-eps<m``. -Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H13 H14)). -Pattern 2 m; Rewrite <- Rplus_Or; Unfold Rminus; Apply Rlt_compatibility; Apply Ropp_Rlt; Rewrite Ropp_Ropp; Rewrite Ropp_O; Apply (cond_pos eps). -Pose P := [n:R](A n)/\``m-eps<n<=m``; Assert H12 := (not_ex_all_not ? P H9); Unfold P in H12; Unfold is_upper_bound; Intros; Assert H14 := (not_and_or ? ? (H12 x)); Elim H14; Intro. -Elim H15; Apply H13. -Elim (not_and_or ? ? H15); Intro. -Case (total_order_Rle x ``m-eps``); Intro. -Assumption. -Elim H16; Auto with real. -Unfold is_upper_bound in H10; Assert H17 := (H10 x H13); Elim H16; Apply H17. -Elim H3; Clear H3; Intros. -Unfold is_upper_bound in H3. -Split. -Apply (H3 ? H0). -Apply (H4 b); Unfold is_upper_bound; Intros; Unfold A in H5; Elim H5; Clear H5; Intros H5 _; Elim H5; Clear H5; Intros _ H5; Apply H5. -Exists a; Apply H0. -Unfold bound; Exists b; Unfold is_upper_bound; Intros; Unfold A in H1; Elim H1; Clear H1; Intros H1 _; Elim H1; Clear H1; Intros _ H1; Apply H1. -Unfold A; Split. -Split; [Right; Reflexivity | Apply r]. -Unfold covering_open_set in H; Elim H; Clear H; Intros; Unfold covering in H; Cut ``a<=a<=b``. -Intro; Elim (H ? H1); Intros y0 H2; Pose D':=[x:R]x==y0; Exists D'; Unfold covering_finite; Split. -Unfold covering; Simpl; Intros; Cut x==a. -Intro; Exists y0; Split. -Rewrite H4; Apply H2. -Unfold D'; Reflexivity. -Elim H3; Intros; Apply Rle_antisym; Assumption. -Unfold family_finite; Unfold domain_finite; Exists (cons y0 nil); Intro; Split. -Simpl; Unfold intersection_domain; Intro; Elim H3; Clear H3; Intros; Unfold D' in H4; Left; Apply H4. -Simpl; Unfold intersection_domain; Intro; Elim H3; Intro. -Split; [Rewrite H4; Apply (cond_fam f0); Exists a; Apply H2 | Apply H4]. -Elim H4. -Split; [Right; Reflexivity | Apply r]. -Apply compact_eqDom with [c:R]False. -Apply compact_EMP. -Unfold eq_Dom; Split. -Unfold included; Intros; Elim H. -Unfold included; Intros; Elim H; Clear H; Intros; Assert H1 := (Rle_trans ? ? ? H H0); Elim n; Apply H1. +Lemma compact_P3 : forall a b:R, compact (fun c:R => a <= c <= b). +intros; case (Rle_dec a b); intro. +unfold compact in |- *; intros; + pose + (A := + fun x:R => + a <= x <= b /\ + ( exists D : R -> Prop + | covering_finite (fun c:R => a <= c <= x) (subfamily f0 D))); + cut (A a). +intro; cut (bound A). +intro; cut ( exists a0 : R | A a0). +intro; assert (H3 := completeness A H1 H2); elim H3; clear H3; intros m H3; + unfold is_lub in H3; cut (a <= m <= b). +intro; unfold covering_open_set in H; elim H; clear H; intros; + unfold covering in H; assert (H6 := H m H4); elim H6; + clear H6; intros y0 H6; unfold family_open_set in H5; + assert (H7 := H5 y0); unfold open_set in H7; assert (H8 := H7 m H6); + unfold neighbourhood in H8; elim H8; clear H8; intros eps H8; + cut ( exists x : R | A x /\ m - eps < x <= m). +intro; elim H9; clear H9; intros x H9; elim H9; clear H9; intros; + case (Req_dec m b); intro. +rewrite H11 in H10; rewrite H11 in H8; unfold A in H9; elim H9; clear H9; + intros; elim H12; clear H12; intros Dx H12; + pose (Db := fun x:R => Dx x \/ x = y0); exists Db; + unfold covering_finite in |- *; split. +unfold covering in |- *; unfold covering_finite in H12; elim H12; clear H12; + intros; unfold covering in H12; case (Rle_dec x0 x); + intro. +cut (a <= x0 <= x). +intro; assert (H16 := H12 x0 H15); elim H16; clear H16; intros; exists x1; + simpl in H16; simpl in |- *; unfold Db in |- *; elim H16; + clear H16; intros; split; [ apply H16 | left; apply H17 ]. +split. +elim H14; intros; assumption. +assumption. +exists y0; simpl in |- *; split. +apply H8; unfold disc in |- *; rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; + rewrite Rabs_right. +apply Rlt_trans with (b - x). +unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_gt_contravar; + auto with real. +elim H10; intros H15 _; apply Rplus_lt_reg_r with (x - eps); + replace (x - eps + (b - x)) with (b - eps); + [ replace (x - eps + eps) with x; [ apply H15 | ring ] | ring ]. +apply Rge_minus; apply Rle_ge; elim H14; intros _ H15; apply H15. +unfold Db in |- *; right; reflexivity. +unfold family_finite in |- *; unfold domain_finite in |- *; + unfold covering_finite in H12; elim H12; clear H12; + intros; unfold family_finite in H13; unfold domain_finite in H13; + elim H13; clear H13; intros l H13; exists (cons y0 l); + intro; split. +intro; simpl in H14; unfold intersection_domain in H14; elim (H13 x0); + clear H13; intros; case (Req_dec x0 y0); intro. +simpl in |- *; left; apply H16. +simpl in |- *; right; apply H13. +simpl in |- *; unfold intersection_domain in |- *; unfold Db in H14; + decompose [and or] H14. +split; assumption. +elim H16; assumption. +intro; simpl in H14; elim H14; intro; simpl in |- *; + unfold intersection_domain in |- *. +split. +apply (cond_fam f0); rewrite H15; exists m; apply H6. +unfold Db in |- *; right; assumption. +simpl in |- *; unfold intersection_domain in |- *; elim (H13 x0). +intros _ H16; assert (H17 := H16 H15); simpl in H17; + unfold intersection_domain in H17; split. +elim H17; intros; assumption. +unfold Db in |- *; left; elim H17; intros; assumption. +pose (m' := Rmin (m + eps / 2) b); cut (A m'). +intro; elim H3; intros; unfold is_upper_bound in H13; + assert (H15 := H13 m' H12); cut (m < m'). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H15 H16)). +unfold m' in |- *; unfold Rmin in |- *; case (Rle_dec (m + eps / 2) b); intro. +pattern m at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ]. +elim H4; intros. +elim H17; intro. +assumption. +elim H11; assumption. +unfold A in |- *; split. +split. +apply Rle_trans with m. +elim H4; intros; assumption. +unfold m' in |- *; unfold Rmin in |- *; case (Rle_dec (m + eps / 2) b); intro. +pattern m at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos eps) | apply Rinv_0_lt_compat; prove_sup0 ]. +elim H4; intros. +elim H13; intro. +assumption. +elim H11; assumption. +unfold m' in |- *; apply Rmin_r. +unfold A in H9; elim H9; clear H9; intros; elim H12; clear H12; intros Dx H12; + pose (Db := fun x:R => Dx x \/ x = y0); exists Db; + unfold covering_finite in |- *; split. +unfold covering in |- *; unfold covering_finite in H12; elim H12; clear H12; + intros; unfold covering in H12; case (Rle_dec x0 x); + intro. +cut (a <= x0 <= x). +intro; assert (H16 := H12 x0 H15); elim H16; clear H16; intros; exists x1; + simpl in H16; simpl in |- *; unfold Db in |- *. +elim H16; clear H16; intros; split; [ apply H16 | left; apply H17 ]. +elim H14; intros; split; assumption. +exists y0; simpl in |- *; split. +apply H8; unfold disc in |- *; unfold Rabs in |- *; case (Rcase_abs (x0 - m)); + intro. +rewrite Ropp_minus_distr; apply Rlt_trans with (m - x). +unfold Rminus in |- *; apply Rplus_lt_compat_l; apply Ropp_lt_gt_contravar; + auto with real. +apply Rplus_lt_reg_r with (x - eps); + replace (x - eps + (m - x)) with (m - eps). +replace (x - eps + eps) with x. +elim H10; intros; assumption. +ring. +ring. +apply Rle_lt_trans with (m' - m). +unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- m)); + apply Rplus_le_compat_l; elim H14; intros; assumption. +apply Rplus_lt_reg_r with m; replace (m + (m' - m)) with m'. +apply Rle_lt_trans with (m + eps / 2). +unfold m' in |- *; apply Rmin_l. +apply Rplus_lt_compat_l; apply Rmult_lt_reg_l with 2. +prove_sup0. +unfold Rdiv in |- *; rewrite <- (Rmult_comm (/ 2)); rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; pattern (pos eps) at 1 in |- *; rewrite <- Rplus_0_r; + rewrite double; apply Rplus_lt_compat_l; apply (cond_pos eps). +discrR. +ring. +unfold Db in |- *; right; reflexivity. +unfold family_finite in |- *; unfold domain_finite in |- *; + unfold covering_finite in H12; elim H12; clear H12; + intros; unfold family_finite in H13; unfold domain_finite in H13; + elim H13; clear H13; intros l H13; exists (cons y0 l); + intro; split. +intro; simpl in H14; unfold intersection_domain in H14; elim (H13 x0); + clear H13; intros; case (Req_dec x0 y0); intro. +simpl in |- *; left; apply H16. +simpl in |- *; right; apply H13; simpl in |- *; + unfold intersection_domain in |- *; unfold Db in H14; + decompose [and or] H14. +split; assumption. +elim H16; assumption. +intro; simpl in H14; elim H14; intro; simpl in |- *; + unfold intersection_domain in |- *. +split. +apply (cond_fam f0); rewrite H15; exists m; apply H6. +unfold Db in |- *; right; assumption. +elim (H13 x0); intros _ H16. +assert (H17 := H16 H15). +simpl in H17. +unfold intersection_domain in H17. +split. +elim H17; intros; assumption. +unfold Db in |- *; left; elim H17; intros; assumption. +elim (classic ( exists x : R | A x /\ m - eps < x <= m)); intro. +assumption. +elim H3; intros; cut (is_upper_bound A (m - eps)). +intro; assert (H13 := H11 _ H12); cut (m - eps < m). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H13 H14)). +pattern m at 2 in |- *; rewrite <- Rplus_0_r; unfold Rminus in |- *; + apply Rplus_lt_compat_l; apply Ropp_lt_cancel; rewrite Ropp_involutive; + rewrite Ropp_0; apply (cond_pos eps). +pose (P := fun n:R => A n /\ m - eps < n <= m); + assert (H12 := not_ex_all_not _ P H9); unfold P in H12; + unfold is_upper_bound in |- *; intros; + assert (H14 := not_and_or _ _ (H12 x)); elim H14; + intro. +elim H15; apply H13. +elim (not_and_or _ _ H15); intro. +case (Rle_dec x (m - eps)); intro. +assumption. +elim H16; auto with real. +unfold is_upper_bound in H10; assert (H17 := H10 x H13); elim H16; apply H17. +elim H3; clear H3; intros. +unfold is_upper_bound in H3. +split. +apply (H3 _ H0). +apply (H4 b); unfold is_upper_bound in |- *; intros; unfold A in H5; elim H5; + clear H5; intros H5 _; elim H5; clear H5; intros _ H5; + apply H5. +exists a; apply H0. +unfold bound in |- *; exists b; unfold is_upper_bound in |- *; intros; + unfold A in H1; elim H1; clear H1; intros H1 _; elim H1; + clear H1; intros _ H1; apply H1. +unfold A in |- *; split. +split; [ right; reflexivity | apply r ]. +unfold covering_open_set in H; elim H; clear H; intros; unfold covering in H; + cut (a <= a <= b). +intro; elim (H _ H1); intros y0 H2; pose (D' := fun x:R => x = y0); exists D'; + unfold covering_finite in |- *; split. +unfold covering in |- *; simpl in |- *; intros; cut (x = a). +intro; exists y0; split. +rewrite H4; apply H2. +unfold D' in |- *; reflexivity. +elim H3; intros; apply Rle_antisym; assumption. +unfold family_finite in |- *; unfold domain_finite in |- *; + exists (cons y0 nil); intro; split. +simpl in |- *; unfold intersection_domain in |- *; intro; elim H3; clear H3; + intros; unfold D' in H4; left; apply H4. +simpl in |- *; unfold intersection_domain in |- *; intro; elim H3; intro. +split; [ rewrite H4; apply (cond_fam f0); exists a; apply H2 | apply H4 ]. +elim H4. +split; [ right; reflexivity | apply r ]. +apply compact_eqDom with (fun c:R => False). +apply compact_EMP. +unfold eq_Dom in |- *; split. +unfold included in |- *; intros; elim H. +unfold included in |- *; intros; elim H; clear H; intros; + assert (H1 := Rle_trans _ _ _ H H0); elim n; apply H1. Qed. -Lemma compact_P4 : (X,F:R->Prop) (compact X) -> (closed_set F) -> (included F X) -> (compact F). -Unfold compact; Intros; Elim (classic (EXT z:R | (F z))); Intro Hyp_F_NE. -Pose D := (ind f0); Pose g := (f f0); Unfold closed_set in H0. -Pose g' := [x:R][y:R](f0 x y)\/((complementary F y)/\(D x)). -Pose D' := D. -Cut (x:R)(EXT y:R | (g' x y))->(D' x). -Intro; Pose f' := (mkfamily D' g' H3); Cut (covering_open_set X f'). -Intro; Elim (H ? H4); Intros DX H5; Exists DX. -Unfold covering_finite; Unfold covering_finite in H5; Elim H5; Clear H5; Intros. -Split. -Unfold covering; Unfold covering in H5; Intros. -Elim (H5 ? (H1 ? H7)); Intros y0 H8; Exists y0; Simpl in H8; Simpl; Elim H8; Clear H8; Intros. -Split. -Unfold g' in H8; Elim H8; Intro. -Apply H10. -Elim H10; Intros H11 _; Unfold complementary in H11; Elim H11; Apply H7. -Apply H9. -Unfold family_finite; Unfold domain_finite; Unfold family_finite in H6; Unfold domain_finite in H6; Elim H6; Clear H6; Intros l H6; Exists l; Intro; Assert H7 := (H6 x); Elim H7; Clear H7; Intros. -Split. -Intro; Apply H7; Simpl; Unfold intersection_domain; Simpl in H9; Unfold intersection_domain in H9; Unfold D'; Apply H9. -Intro; Assert H10 := (H8 H9); Simpl in H10; Unfold intersection_domain in H10; Simpl; Unfold intersection_domain; Unfold D' in H10; Apply H10. -Unfold covering_open_set; Unfold covering_open_set in H2; Elim H2; Clear H2; Intros. -Split. -Unfold covering; Unfold covering in H2; Intros. -Elim (classic (F x)); Intro. -Elim (H2 ? H6); Intros y0 H7; Exists y0; Simpl; Unfold g'; Left; Assumption. -Cut (EXT z:R | (D z)). -Intro; Elim H7; Clear H7; Intros x0 H7; Exists x0; Simpl; Unfold g'; Right. -Split. -Unfold complementary; Apply H6. -Apply H7. -Elim Hyp_F_NE; Intros z0 H7. -Assert H8 := (H2 ? H7). -Elim H8; Clear H8; Intros t H8; Exists t; Apply (cond_fam f0); Exists z0; Apply H8. -Unfold family_open_set; Intro; Simpl; Unfold g'; Elim (classic (D x)); Intro. -Apply open_set_P6 with (union_domain (f0 x) (complementary F)). -Apply open_set_P2. -Unfold family_open_set in H4; Apply H4. -Apply H0. -Unfold eq_Dom; Split. -Unfold included union_domain complementary; Intros. -Elim H6; Intro; [Left; Apply H7 | Right; Split; Assumption]. -Unfold included union_domain complementary; Intros. -Elim H6; Intro; [Left; Apply H7 | Right; Elim H7; Intros; Apply H8]. -Apply open_set_P6 with (f0 x). -Unfold family_open_set in H4; Apply H4. -Unfold eq_Dom; Split. -Unfold included complementary; Intros; Left; Apply H6. -Unfold included complementary; Intros. -Elim H6; Intro. -Apply H7. -Elim H7; Intros _ H8; Elim H5; Apply H8. -Intros; Elim H3; Intros y0 H4; Unfold g' in H4; Elim H4; Intro. -Apply (cond_fam f0); Exists y0; Apply H5. -Elim H5; Clear H5; Intros _ H5; Apply H5. +Lemma compact_P4 : + forall X F:R -> Prop, compact X -> closed_set F -> included F X -> compact F. +unfold compact in |- *; intros; elim (classic ( exists z : R | F z)); + intro Hyp_F_NE. +pose (D := ind f0); pose (g := f f0); unfold closed_set in H0. +pose (g' := fun x y:R => f0 x y \/ complementary F y /\ D x). +pose (D' := D). +cut (forall x:R, ( exists y : R | g' x y) -> D' x). +intro; pose (f' := mkfamily D' g' H3); cut (covering_open_set X f'). +intro; elim (H _ H4); intros DX H5; exists DX. +unfold covering_finite in |- *; unfold covering_finite in H5; elim H5; + clear H5; intros. +split. +unfold covering in |- *; unfold covering in H5; intros. +elim (H5 _ (H1 _ H7)); intros y0 H8; exists y0; simpl in H8; simpl in |- *; + elim H8; clear H8; intros. +split. +unfold g' in H8; elim H8; intro. +apply H10. +elim H10; intros H11 _; unfold complementary in H11; elim H11; apply H7. +apply H9. +unfold family_finite in |- *; unfold domain_finite in |- *; + unfold family_finite in H6; unfold domain_finite in H6; + elim H6; clear H6; intros l H6; exists l; intro; assert (H7 := H6 x); + elim H7; clear H7; intros. +split. +intro; apply H7; simpl in |- *; unfold intersection_domain in |- *; + simpl in H9; unfold intersection_domain in H9; unfold D' in |- *; + apply H9. +intro; assert (H10 := H8 H9); simpl in H10; unfold intersection_domain in H10; + simpl in |- *; unfold intersection_domain in |- *; + unfold D' in H10; apply H10. +unfold covering_open_set in |- *; unfold covering_open_set in H2; elim H2; + clear H2; intros. +split. +unfold covering in |- *; unfold covering in H2; intros. +elim (classic (F x)); intro. +elim (H2 _ H6); intros y0 H7; exists y0; simpl in |- *; unfold g' in |- *; + left; assumption. +cut ( exists z : R | D z). +intro; elim H7; clear H7; intros x0 H7; exists x0; simpl in |- *; + unfold g' in |- *; right. +split. +unfold complementary in |- *; apply H6. +apply H7. +elim Hyp_F_NE; intros z0 H7. +assert (H8 := H2 _ H7). +elim H8; clear H8; intros t H8; exists t; apply (cond_fam f0); exists z0; + apply H8. +unfold family_open_set in |- *; intro; simpl in |- *; unfold g' in |- *; + elim (classic (D x)); intro. +apply open_set_P6 with (union_domain (f0 x) (complementary F)). +apply open_set_P2. +unfold family_open_set in H4; apply H4. +apply H0. +unfold eq_Dom in |- *; split. +unfold included, union_domain, complementary in |- *; intros. +elim H6; intro; [ left; apply H7 | right; split; assumption ]. +unfold included, union_domain, complementary in |- *; intros. +elim H6; intro; [ left; apply H7 | right; elim H7; intros; apply H8 ]. +apply open_set_P6 with (f0 x). +unfold family_open_set in H4; apply H4. +unfold eq_Dom in |- *; split. +unfold included, complementary in |- *; intros; left; apply H6. +unfold included, complementary in |- *; intros. +elim H6; intro. +apply H7. +elim H7; intros _ H8; elim H5; apply H8. +intros; elim H3; intros y0 H4; unfold g' in H4; elim H4; intro. +apply (cond_fam f0); exists y0; apply H5. +elim H5; clear H5; intros _ H5; apply H5. (* Cas ou F est l'ensemble vide *) -Cut (compact F). -Intro; Apply (H3 f0 H2). -Apply compact_eqDom with [_:R]False. -Apply compact_EMP. -Unfold eq_Dom; Split. -Unfold included; Intros; Elim H3. -Assert H3 := (not_ex_all_not ? ? Hyp_F_NE); Unfold included; Intros; Elim (H3 x); Apply H4. +cut (compact F). +intro; apply (H3 f0 H2). +apply compact_eqDom with (fun _:R => False). +apply compact_EMP. +unfold eq_Dom in |- *; split. +unfold included in |- *; intros; elim H3. +assert (H3 := not_ex_all_not _ _ Hyp_F_NE); unfold included in |- *; intros; + elim (H3 x); apply H4. Qed. (**********) -Lemma compact_P5 : (X:R->Prop) (closed_set X)->(bounded X)->(compact X). -Intros; Unfold bounded in H0. -Elim H0; Clear H0; Intros m H0. -Elim H0; Clear H0; Intros M H0. -Assert H1 := (compact_P3 m M). -Apply (compact_P4 [c:R]``m<=c<=M`` X H1 H H0). +Lemma compact_P5 : forall X:R -> Prop, closed_set X -> bounded X -> compact X. +intros; unfold bounded in H0. +elim H0; clear H0; intros m H0. +elim H0; clear H0; intros M H0. +assert (H1 := compact_P3 m M). +apply (compact_P4 (fun c:R => m <= c <= M) X H1 H H0). Qed. (**********) -Lemma compact_carac : (X:R->Prop) (compact X)<->(closed_set X)/\(bounded X). -Intro; Split. -Intro; Split; [Apply (compact_P2 ? H) | Apply (compact_P1 ? H)]. -Intro; Elim H; Clear H; Intros; Apply (compact_P5 ? H H0). +Lemma compact_carac : + forall X:R -> Prop, compact X <-> closed_set X /\ bounded X. +intro; split. +intro; split; [ apply (compact_P2 _ H) | apply (compact_P1 _ H) ]. +intro; elim H; clear H; intros; apply (compact_P5 _ H H0). Qed. -Definition image_dir [f:R->R;D:R->Prop] : R->Prop := [x:R](EXT y:R | x==(f y)/\(D y)). +Definition image_dir (f:R -> R) (D:R -> Prop) (x:R) : Prop := + exists y : R | x = f y /\ D y. (**********) -Lemma continuity_compact : (f:R->R;X:R->Prop) ((x:R)(continuity_pt f x)) -> (compact X) -> (compact (image_dir f X)). -Unfold compact; Intros; Unfold covering_open_set in H1. -Elim H1; Clear H1; Intros. -Pose D := (ind f1). -Pose g := [x:R][y:R](image_rec f0 (f1 x) y). -Cut (x:R)(EXT y:R | (g x y))->(D x). -Intro; Pose f' := (mkfamily D g H3). -Cut (covering_open_set X f'). -Intro; Elim (H0 f' H4); Intros D' H5; Exists D'. -Unfold covering_finite in H5; Elim H5; Clear H5; Intros; Unfold covering_finite; Split. -Unfold covering image_dir; Simpl; Unfold covering in H5; Intros; Elim H7; Intros y H8; Elim H8; Intros; Assert H11 := (H5 ? H10); Simpl in H11; Elim H11; Intros z H12; Exists z; Unfold g in H12; Unfold image_rec in H12; Rewrite H9; Apply H12. -Unfold family_finite in H6; Unfold domain_finite in H6; Unfold family_finite; Unfold domain_finite; Elim H6; Intros l H7; Exists l; Intro; Elim (H7 x); Intros; Split; Intro. -Apply H8; Simpl in H10; Simpl; Apply H10. -Apply (H9 H10). -Unfold covering_open_set; Split. -Unfold covering; Intros; Simpl; Unfold covering in H1; Unfold image_dir in H1; Unfold g; Unfold image_rec; Apply H1. -Exists x; Split; [Reflexivity | Apply H4]. -Unfold family_open_set; Unfold family_open_set in H2; Intro; Simpl; Unfold g; Cut ([y:R](image_rec f0 (f1 x) y))==(image_rec f0 (f1 x)). -Intro; Rewrite H4. -Apply (continuity_P2 f0 (f1 x) H (H2 x)). -Reflexivity. -Intros; Apply (cond_fam f1); Unfold g in H3; Unfold image_rec in H3; Elim H3; Intros; Exists (f0 x0); Apply H4. +Lemma continuity_compact : + forall (f:R -> R) (X:R -> Prop), + (forall x:R, continuity_pt f x) -> compact X -> compact (image_dir f X). +unfold compact in |- *; intros; unfold covering_open_set in H1. +elim H1; clear H1; intros. +pose (D := ind f1). +pose (g := fun x y:R => image_rec f0 (f1 x) y). +cut (forall x:R, ( exists y : R | g x y) -> D x). +intro; pose (f' := mkfamily D g H3). +cut (covering_open_set X f'). +intro; elim (H0 f' H4); intros D' H5; exists D'. +unfold covering_finite in H5; elim H5; clear H5; intros; + unfold covering_finite in |- *; split. +unfold covering, image_dir in |- *; simpl in |- *; unfold covering in H5; + intros; elim H7; intros y H8; elim H8; intros; assert (H11 := H5 _ H10); + simpl in H11; elim H11; intros z H12; exists z; unfold g in H12; + unfold image_rec in H12; rewrite H9; apply H12. +unfold family_finite in H6; unfold domain_finite in H6; + unfold family_finite in |- *; unfold domain_finite in |- *; + elim H6; intros l H7; exists l; intro; elim (H7 x); + intros; split; intro. +apply H8; simpl in H10; simpl in |- *; apply H10. +apply (H9 H10). +unfold covering_open_set in |- *; split. +unfold covering in |- *; intros; simpl in |- *; unfold covering in H1; + unfold image_dir in H1; unfold g in |- *; unfold image_rec in |- *; + apply H1. +exists x; split; [ reflexivity | apply H4 ]. +unfold family_open_set in |- *; unfold family_open_set in H2; intro; + simpl in |- *; unfold g in |- *; + cut ((fun y:R => image_rec f0 (f1 x) y) = image_rec f0 (f1 x)). +intro; rewrite H4. +apply (continuity_P2 f0 (f1 x) H (H2 x)). +reflexivity. +intros; apply (cond_fam f1); unfold g in H3; unfold image_rec in H3; elim H3; + intros; exists (f0 x0); apply H4. Qed. -Lemma Rlt_Rminus : (a,b:R) ``a<b`` -> ``0<b-a``. -Intros; Apply Rlt_anti_compatibility with a; Rewrite Rplus_Or; Replace ``a+(b-a)`` with b; [Assumption | Ring]. +Lemma Rlt_Rminus : forall a b:R, a < b -> 0 < b - a. +intros; apply Rplus_lt_reg_r with a; rewrite Rplus_0_r; + replace (a + (b - a)) with b; [ assumption | ring ]. Qed. -Lemma prolongement_C0 : (f:R->R;a,b:R) ``a<=b`` -> ((c:R)``a<=c<=b``->(continuity_pt f c)) -> (EXT g:R->R | (continuity g)/\((c:R)``a<=c<=b``->(g c)==(f c))). -Intros; Elim H; Intro. -Pose h := [x:R](Cases (total_order_Rle x a) of - (leftT _) => (f0 a) -| (rightT _) => (Cases (total_order_Rle x b) of - (leftT _) => (f0 x) - | (rightT _) => (f0 b) end) end). -Assert H2 : ``0<b-a``. -Apply Rlt_Rminus; Assumption. -Exists h; Split. -Unfold continuity; Intro; Case (total_order x a); Intro. -Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Exists ``a-x``; Split. -Change ``0<a-x``; Apply Rlt_Rminus; Assumption. -Intros; Elim H5; Clear H5; Intros _ H5; Unfold h. -Case (total_order_Rle x a); Intro. -Case (total_order_Rle x0 a); Intro. -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Elim n; Left; Apply Rlt_anti_compatibility with ``-x``; Do 2 Rewrite (Rplus_sym ``-x``); Apply Rle_lt_trans with ``(Rabsolu (x0-x))``. -Apply Rle_Rabsolu. -Assumption. -Elim n; Left; Assumption. -Elim H3; Intro. -Assert H5 : ``a<=a<=b``. -Split; [Right; Reflexivity | Left; Assumption]. -Assert H6 := (H0 ? H5); Unfold continuity_pt in H6; Unfold continue_in in H6; Unfold limit1_in in H6; Unfold limit_in in H6; Simpl in H6; Unfold R_dist in H6; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Elim (H6 ? H7); Intros; Exists (Rmin x0 ``b-a``); Split. -Unfold Rmin; Case (total_order_Rle x0 ``b-a``); Intro. -Elim H8; Intros; Assumption. -Change ``0<b-a``; Apply Rlt_Rminus; Assumption. -Intros; Elim H9; Clear H9; Intros _ H9; Cut ``x1<b``. -Intro; Unfold h; Case (total_order_Rle x a); Intro. -Case (total_order_Rle x1 a); Intro. -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Case (total_order_Rle x1 b); Intro. -Elim H8; Intros; Apply H12; Split. -Unfold D_x no_cond; Split. -Trivial. -Red; Intro; Elim n; Right; Symmetry; Assumption. -Apply Rlt_le_trans with (Rmin x0 ``b-a``). -Rewrite H4 in H9; Apply H9. -Apply Rmin_l. -Elim n0; Left; Assumption. -Elim n; Right; Assumption. -Apply Rlt_anti_compatibility with ``-a``; Do 2 Rewrite (Rplus_sym ``-a``); Rewrite H4 in H9; Apply Rle_lt_trans with ``(Rabsolu (x1-a))``. -Apply Rle_Rabsolu. -Apply Rlt_le_trans with ``(Rmin x0 (b-a))``. -Assumption. -Apply Rmin_r. -Case (total_order x b); Intro. -Assert H6 : ``a<=x<=b``. -Split; Left; Assumption. -Assert H7 := (H0 ? H6); Unfold continuity_pt in H7; Unfold continue_in in H7; Unfold limit1_in in H7; Unfold limit_in in H7; Simpl in H7; Unfold R_dist in H7; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Elim (H7 ? H8); Intros; Elim H9; Clear H9; Intros. -Assert H11 : ``0<x-a``. -Apply Rlt_Rminus; Assumption. -Assert H12 : ``0<b-x``. -Apply Rlt_Rminus; Assumption. -Exists (Rmin x0 (Rmin ``x-a`` ``b-x``)); Split. -Unfold Rmin; Case (total_order_Rle ``x-a`` ``b-x``); Intro. -Case (total_order_Rle x0 ``x-a``); Intro. -Assumption. -Assumption. -Case (total_order_Rle x0 ``b-x``); Intro. -Assumption. -Assumption. -Intros; Elim H13; Clear H13; Intros; Cut ``a<x1<b``. -Intro; Elim H15; Clear H15; Intros; Unfold h; Case (total_order_Rle x a); Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H4)). -Case (total_order_Rle x b); Intro. -Case (total_order_Rle x1 a); Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r0 H15)). -Case (total_order_Rle x1 b); Intro. -Apply H10; Split. -Assumption. -Apply Rlt_le_trans with ``(Rmin x0 (Rmin (x-a) (b-x)))``. -Assumption. -Apply Rmin_l. -Elim n1; Left; Assumption. -Elim n0; Left; Assumption. -Split. -Apply Ropp_Rlt; Apply Rlt_anti_compatibility with x; Apply Rle_lt_trans with ``(Rabsolu (x1-x))``. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply Rle_Rabsolu. -Apply Rlt_le_trans with ``(Rmin x0 (Rmin (x-a) (b-x)))``. -Assumption. -Apply Rle_trans with ``(Rmin (x-a) (b-x))``. -Apply Rmin_r. -Apply Rmin_l. -Apply Rlt_anti_compatibility with ``-x``; Do 2 Rewrite (Rplus_sym ``-x``); Apply Rle_lt_trans with ``(Rabsolu (x1-x))``. -Apply Rle_Rabsolu. -Apply Rlt_le_trans with ``(Rmin x0 (Rmin (x-a) (b-x)))``. -Assumption. -Apply Rle_trans with ``(Rmin (x-a) (b-x))``; Apply Rmin_r. -Elim H5; Intro. -Assert H7 : ``a<=b<=b``. -Split; [Left; Assumption | Right; Reflexivity]. -Assert H8 := (H0 ? H7); Unfold continuity_pt in H8; Unfold continue_in in H8; Unfold limit1_in in H8; Unfold limit_in in H8; Simpl in H8; Unfold R_dist in H8; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Elim (H8 ? H9); Intros; Exists (Rmin x0 ``b-a``); Split. -Unfold Rmin; Case (total_order_Rle x0 ``b-a``); Intro. -Elim H10; Intros; Assumption. -Change ``0<b-a``; Apply Rlt_Rminus; Assumption. -Intros; Elim H11; Clear H11; Intros _ H11; Cut ``a<x1``. -Intro; Unfold h; Case (total_order_Rle x a); Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H4)). -Case (total_order_Rle x1 a); Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H12)). -Case (total_order_Rle x b); Intro. -Case (total_order_Rle x1 b); Intro. -Rewrite H6; Elim H10; Intros; Elim r0; Intro. -Apply H14; Split. -Unfold D_x no_cond; Split. -Trivial. -Red; Intro; Rewrite <- H16 in H15; Elim (Rlt_antirefl ? H15). -Rewrite H6 in H11; Apply Rlt_le_trans with ``(Rmin x0 (b-a))``. -Apply H11. -Apply Rmin_l. -Rewrite H15; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Rewrite H6; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Elim n1; Right; Assumption. -Rewrite H6 in H11; Apply Ropp_Rlt; Apply Rlt_anti_compatibility with b; Apply Rle_lt_trans with ``(Rabsolu (x1-b))``. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply Rle_Rabsolu. -Apply Rlt_le_trans with ``(Rmin x0 (b-a))``. -Assumption. -Apply Rmin_r. -Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Exists ``x-b``; Split. -Change ``0<x-b``; Apply Rlt_Rminus; Assumption. -Intros; Elim H8; Clear H8; Intros. -Assert H10 : ``b<x0``. -Apply Ropp_Rlt; Apply Rlt_anti_compatibility with x; Apply Rle_lt_trans with ``(Rabsolu (x0-x))``. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply Rle_Rabsolu. -Assumption. -Unfold h; Case (total_order_Rle x a); Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H4)). -Case (total_order_Rle x b); Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H6)). -Case (total_order_Rle x0 a); Intro. -Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H1 (Rlt_le_trans ? ? ? H10 r))). -Case (total_order_Rle x0 b); Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? r H10)). -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Intros; Elim H3; Intros; Unfold h; Case (total_order_Rle c a); Intro. -Elim r; Intro. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H4 H6)). -Rewrite H6; Reflexivity. -Case (total_order_Rle c b); Intro. -Reflexivity. -Elim n0; Assumption. -Exists [_:R](f0 a); Split. -Apply derivable_continuous; Apply (derivable_const (f0 a)). -Intros; Elim H2; Intros; Rewrite H1 in H3; Cut b==c. -Intro; Rewrite <- H5; Rewrite H1; Reflexivity. -Apply Rle_antisym; Assumption. +Lemma prolongement_C0 : + forall (f:R -> R) (a b:R), + a <= b -> + (forall c:R, a <= c <= b -> continuity_pt f c) -> + exists g : R -> R + | continuity g /\ (forall c:R, a <= c <= b -> g c = f c). +intros; elim H; intro. +pose + (h := + fun x:R => + match Rle_dec x a with + | left _ => f0 a + | right _ => + match Rle_dec x b with + | left _ => f0 x + | right _ => f0 b + end + end). +assert (H2 : 0 < b - a). +apply Rlt_Rminus; assumption. +exists h; split. +unfold continuity in |- *; intro; case (Rtotal_order x a); intro. +unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros; exists (a - x); + split. +change (0 < a - x) in |- *; apply Rlt_Rminus; assumption. +intros; elim H5; clear H5; intros _ H5; unfold h in |- *. +case (Rle_dec x a); intro. +case (Rle_dec x0 a); intro. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. +elim n; left; apply Rplus_lt_reg_r with (- x); + do 2 rewrite (Rplus_comm (- x)); apply Rle_lt_trans with (Rabs (x0 - x)). +apply RRle_abs. +assumption. +elim n; left; assumption. +elim H3; intro. +assert (H5 : a <= a <= b). +split; [ right; reflexivity | left; assumption ]. +assert (H6 := H0 _ H5); unfold continuity_pt in H6; unfold continue_in in H6; + unfold limit1_in in H6; unfold limit_in in H6; simpl in H6; + unfold R_dist in H6; unfold continuity_pt in |- *; + unfold continue_in in |- *; unfold limit1_in in |- *; + unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *; + intros; elim (H6 _ H7); intros; exists (Rmin x0 (b - a)); + split. +unfold Rmin in |- *; case (Rle_dec x0 (b - a)); intro. +elim H8; intros; assumption. +change (0 < b - a) in |- *; apply Rlt_Rminus; assumption. +intros; elim H9; clear H9; intros _ H9; cut (x1 < b). +intro; unfold h in |- *; case (Rle_dec x a); intro. +case (Rle_dec x1 a); intro. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. +case (Rle_dec x1 b); intro. +elim H8; intros; apply H12; split. +unfold D_x, no_cond in |- *; split. +trivial. +red in |- *; intro; elim n; right; symmetry in |- *; assumption. +apply Rlt_le_trans with (Rmin x0 (b - a)). +rewrite H4 in H9; apply H9. +apply Rmin_l. +elim n0; left; assumption. +elim n; right; assumption. +apply Rplus_lt_reg_r with (- a); do 2 rewrite (Rplus_comm (- a)); + rewrite H4 in H9; apply Rle_lt_trans with (Rabs (x1 - a)). +apply RRle_abs. +apply Rlt_le_trans with (Rmin x0 (b - a)). +assumption. +apply Rmin_r. +case (Rtotal_order x b); intro. +assert (H6 : a <= x <= b). +split; left; assumption. +assert (H7 := H0 _ H6); unfold continuity_pt in H7; unfold continue_in in H7; + unfold limit1_in in H7; unfold limit_in in H7; simpl in H7; + unfold R_dist in H7; unfold continuity_pt in |- *; + unfold continue_in in |- *; unfold limit1_in in |- *; + unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *; + intros; elim (H7 _ H8); intros; elim H9; clear H9; + intros. +assert (H11 : 0 < x - a). +apply Rlt_Rminus; assumption. +assert (H12 : 0 < b - x). +apply Rlt_Rminus; assumption. +exists (Rmin x0 (Rmin (x - a) (b - x))); split. +unfold Rmin in |- *; case (Rle_dec (x - a) (b - x)); intro. +case (Rle_dec x0 (x - a)); intro. +assumption. +assumption. +case (Rle_dec x0 (b - x)); intro. +assumption. +assumption. +intros; elim H13; clear H13; intros; cut (a < x1 < b). +intro; elim H15; clear H15; intros; unfold h in |- *; case (Rle_dec x a); + intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H4)). +case (Rle_dec x b); intro. +case (Rle_dec x1 a); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 H15)). +case (Rle_dec x1 b); intro. +apply H10; split. +assumption. +apply Rlt_le_trans with (Rmin x0 (Rmin (x - a) (b - x))). +assumption. +apply Rmin_l. +elim n1; left; assumption. +elim n0; left; assumption. +split. +apply Ropp_lt_cancel; apply Rplus_lt_reg_r with x; + apply Rle_lt_trans with (Rabs (x1 - x)). +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply RRle_abs. +apply Rlt_le_trans with (Rmin x0 (Rmin (x - a) (b - x))). +assumption. +apply Rle_trans with (Rmin (x - a) (b - x)). +apply Rmin_r. +apply Rmin_l. +apply Rplus_lt_reg_r with (- x); do 2 rewrite (Rplus_comm (- x)); + apply Rle_lt_trans with (Rabs (x1 - x)). +apply RRle_abs. +apply Rlt_le_trans with (Rmin x0 (Rmin (x - a) (b - x))). +assumption. +apply Rle_trans with (Rmin (x - a) (b - x)); apply Rmin_r. +elim H5; intro. +assert (H7 : a <= b <= b). +split; [ left; assumption | right; reflexivity ]. +assert (H8 := H0 _ H7); unfold continuity_pt in H8; unfold continue_in in H8; + unfold limit1_in in H8; unfold limit_in in H8; simpl in H8; + unfold R_dist in H8; unfold continuity_pt in |- *; + unfold continue_in in |- *; unfold limit1_in in |- *; + unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *; + intros; elim (H8 _ H9); intros; exists (Rmin x0 (b - a)); + split. +unfold Rmin in |- *; case (Rle_dec x0 (b - a)); intro. +elim H10; intros; assumption. +change (0 < b - a) in |- *; apply Rlt_Rminus; assumption. +intros; elim H11; clear H11; intros _ H11; cut (a < x1). +intro; unfold h in |- *; case (Rle_dec x a); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H4)). +case (Rle_dec x1 a); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H12)). +case (Rle_dec x b); intro. +case (Rle_dec x1 b); intro. +rewrite H6; elim H10; intros; elim r0; intro. +apply H14; split. +unfold D_x, no_cond in |- *; split. +trivial. +red in |- *; intro; rewrite <- H16 in H15; elim (Rlt_irrefl _ H15). +rewrite H6 in H11; apply Rlt_le_trans with (Rmin x0 (b - a)). +apply H11. +apply Rmin_l. +rewrite H15; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + assumption. +rewrite H6; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + assumption. +elim n1; right; assumption. +rewrite H6 in H11; apply Ropp_lt_cancel; apply Rplus_lt_reg_r with b; + apply Rle_lt_trans with (Rabs (x1 - b)). +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply RRle_abs. +apply Rlt_le_trans with (Rmin x0 (b - a)). +assumption. +apply Rmin_r. +unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros; exists (x - b); + split. +change (0 < x - b) in |- *; apply Rlt_Rminus; assumption. +intros; elim H8; clear H8; intros. +assert (H10 : b < x0). +apply Ropp_lt_cancel; apply Rplus_lt_reg_r with x; + apply Rle_lt_trans with (Rabs (x0 - x)). +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply RRle_abs. +assumption. +unfold h in |- *; case (Rle_dec x a); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H4)). +case (Rle_dec x b); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H6)). +case (Rle_dec x0 a); intro. +elim (Rlt_irrefl _ (Rlt_trans _ _ _ H1 (Rlt_le_trans _ _ _ H10 r))). +case (Rle_dec x0 b); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r H10)). +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. +intros; elim H3; intros; unfold h in |- *; case (Rle_dec c a); intro. +elim r; intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 H6)). +rewrite H6; reflexivity. +case (Rle_dec c b); intro. +reflexivity. +elim n0; assumption. +exists (fun _:R => f0 a); split. +apply derivable_continuous; apply (derivable_const (f0 a)). +intros; elim H2; intros; rewrite H1 in H3; cut (b = c). +intro; rewrite <- H5; rewrite H1; reflexivity. +apply Rle_antisym; assumption. Qed. (**********) -Lemma continuity_ab_maj : (f:R->R;a,b:R) ``a<=b`` -> ((c:R)``a<=c<=b``->(continuity_pt f c)) -> (EXT Mx : R | ((c:R)``a<=c<=b``->``(f c)<=(f Mx)``)/\``a<=Mx<=b``). -Intros; Cut (EXT g:R->R | (continuity g)/\((c:R)``a<=c<=b``->(g c)==(f0 c))). -Intro HypProl. -Elim HypProl; Intros g Hcont_eq. -Elim Hcont_eq; Clear Hcont_eq; Intros Hcont Heq. -Assert H1 := (compact_P3 a b). -Assert H2 := (continuity_compact g [c:R]``a<=c<=b`` Hcont H1). -Assert H3 := (compact_P2 ? H2). -Assert H4 := (compact_P1 ? H2). -Cut (bound (image_dir g [c:R]``a <= c <= b``)). -Cut (ExT [x:R] ((image_dir g [c:R]``a <= c <= b``) x)). -Intros; Assert H7 := (complet ? H6 H5). -Elim H7; Clear H7; Intros M H7; Cut (image_dir g [c:R]``a <= c <= b`` M). -Intro; Unfold image_dir in H8; Elim H8; Clear H8; Intros Mxx H8; Elim H8; Clear H8; Intros; Exists Mxx; Split. -Intros; Rewrite <- (Heq c H10); Rewrite <- (Heq Mxx H9); Intros; Rewrite <- H8; Unfold is_lub in H7; Elim H7; Clear H7; Intros H7 _; Unfold is_upper_bound in H7; Apply H7; Unfold image_dir; Exists c; Split; [Reflexivity | Apply H10]. -Apply H9. -Elim (classic (image_dir g [c:R]``a <= c <= b`` M)); Intro. -Assumption. -Cut (EXT eps:posreal | (y:R)~(intersection_domain (disc M eps) (image_dir g [c:R]``a <= c <= b``) y)). -Intro; Elim H9; Clear H9; Intros eps H9; Unfold is_lub in H7; Elim H7; Clear H7; Intros; Cut (is_upper_bound (image_dir g [c:R]``a <= c <= b``) ``M-eps``). -Intro; Assert H12 := (H10 ? H11); Cut ``M-eps<M``. -Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H12 H13)). -Pattern 2 M; Rewrite <- Rplus_Or; Unfold Rminus; Apply Rlt_compatibility; Apply Ropp_Rlt; Rewrite Ropp_O; Rewrite Ropp_Ropp; Apply (cond_pos eps). -Unfold is_upper_bound image_dir; Intros; Cut ``x<=M``. -Intro; Case (total_order_Rle x ``M-eps``); Intro. -Apply r. -Elim (H9 x); Unfold intersection_domain disc image_dir; Split. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Rewrite Rabsolu_right. -Apply Rlt_anti_compatibility with ``x-eps``; Replace ``x-eps+(M-x)`` with ``M-eps``. -Replace ``x-eps+eps`` with x. -Auto with real. -Ring. -Ring. -Apply Rge_minus; Apply Rle_sym1; Apply H12. -Apply H11. -Apply H7; Apply H11. -Cut (EXT V:R->Prop | (neighbourhood V M)/\((y:R)~(intersection_domain V (image_dir g [c:R]``a <= c <= b``) y))). -Intro; Elim H9; Intros V H10; Elim H10; Clear H10; Intros. -Unfold neighbourhood in H10; Elim H10; Intros del H12; Exists del; Intros; Red; Intro; Elim (H11 y). -Unfold intersection_domain; Unfold intersection_domain in H13; Elim H13; Clear H13; Intros; Split. -Apply (H12 ? H13). -Apply H14. -Cut ~(point_adherent (image_dir g [c:R]``a <= c <= b``) M). -Intro; Unfold point_adherent in H9. -Assert H10 := (not_all_ex_not ? [V:R->Prop](neighbourhood V M) - ->(EXT y:R | - (intersection_domain V - (image_dir g [c:R]``a <= c <= b``) y)) H9). -Elim H10; Intros V0 H11; Exists V0; Assert H12 := (imply_to_and ? ? H11); Elim H12; Clear H12; Intros. -Split. -Apply H12. -Apply (not_ex_all_not ? ? H13). -Red; Intro; Cut (adherence (image_dir g [c:R]``a <= c <= b``) M). -Intro; Elim (closed_set_P1 (image_dir g [c:R]``a <= c <= b``)); Intros H11 _; Assert H12 := (H11 H3). -Elim H8. -Unfold eq_Dom in H12; Elim H12; Clear H12; Intros. -Apply (H13 ? H10). -Apply H9. -Exists (g a); Unfold image_dir; Exists a; Split. -Reflexivity. -Split; [Right; Reflexivity | Apply H]. -Unfold bound; Unfold bounded in H4; Elim H4; Clear H4; Intros m H4; Elim H4; Clear H4; Intros M H4; Exists M; Unfold is_upper_bound; Intros; Elim (H4 ? H5); Intros _ H6; Apply H6. -Apply prolongement_C0; Assumption. +Lemma continuity_ab_maj : + forall (f:R -> R) (a b:R), + a <= b -> + (forall c:R, a <= c <= b -> continuity_pt f c) -> + exists Mx : R | (forall c:R, a <= c <= b -> f c <= f Mx) /\ a <= Mx <= b. +intros; + cut + ( exists g : R -> R + | continuity g /\ (forall c:R, a <= c <= b -> g c = f0 c)). +intro HypProl. +elim HypProl; intros g Hcont_eq. +elim Hcont_eq; clear Hcont_eq; intros Hcont Heq. +assert (H1 := compact_P3 a b). +assert (H2 := continuity_compact g (fun c:R => a <= c <= b) Hcont H1). +assert (H3 := compact_P2 _ H2). +assert (H4 := compact_P1 _ H2). +cut (bound (image_dir g (fun c:R => a <= c <= b))). +cut ( exists x : R | image_dir g (fun c:R => a <= c <= b) x). +intros; assert (H7 := completeness _ H6 H5). +elim H7; clear H7; intros M H7; cut (image_dir g (fun c:R => a <= c <= b) M). +intro; unfold image_dir in H8; elim H8; clear H8; intros Mxx H8; elim H8; + clear H8; intros; exists Mxx; split. +intros; rewrite <- (Heq c H10); rewrite <- (Heq Mxx H9); intros; + rewrite <- H8; unfold is_lub in H7; elim H7; clear H7; + intros H7 _; unfold is_upper_bound in H7; apply H7; + unfold image_dir in |- *; exists c; split; [ reflexivity | apply H10 ]. +apply H9. +elim (classic (image_dir g (fun c:R => a <= c <= b) M)); intro. +assumption. +cut + ( exists eps : posreal + | (forall y:R, + ~ + intersection_domain (disc M eps) + (image_dir g (fun c:R => a <= c <= b)) y)). +intro; elim H9; clear H9; intros eps H9; unfold is_lub in H7; elim H7; + clear H7; intros; + cut (is_upper_bound (image_dir g (fun c:R => a <= c <= b)) (M - eps)). +intro; assert (H12 := H10 _ H11); cut (M - eps < M). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H12 H13)). +pattern M at 2 in |- *; rewrite <- Rplus_0_r; unfold Rminus in |- *; + apply Rplus_lt_compat_l; apply Ropp_lt_cancel; rewrite Ropp_0; + rewrite Ropp_involutive; apply (cond_pos eps). +unfold is_upper_bound, image_dir in |- *; intros; cut (x <= M). +intro; case (Rle_dec x (M - eps)); intro. +apply r. +elim (H9 x); unfold intersection_domain, disc, image_dir in |- *; split. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; rewrite Rabs_right. +apply Rplus_lt_reg_r with (x - eps); + replace (x - eps + (M - x)) with (M - eps). +replace (x - eps + eps) with x. +auto with real. +ring. +ring. +apply Rge_minus; apply Rle_ge; apply H12. +apply H11. +apply H7; apply H11. +cut + ( exists V : R -> Prop + | neighbourhood V M /\ + (forall y:R, + ~ intersection_domain V (image_dir g (fun c:R => a <= c <= b)) y)). +intro; elim H9; intros V H10; elim H10; clear H10; intros. +unfold neighbourhood in H10; elim H10; intros del H12; exists del; intros; + red in |- *; intro; elim (H11 y). +unfold intersection_domain in |- *; unfold intersection_domain in H13; + elim H13; clear H13; intros; split. +apply (H12 _ H13). +apply H14. +cut (~ point_adherent (image_dir g (fun c:R => a <= c <= b)) M). +intro; unfold point_adherent in H9. +assert + (H10 := + not_all_ex_not _ + (fun V:R -> Prop => + neighbourhood V M -> + exists y : R + | intersection_domain V (image_dir g (fun c:R => a <= c <= b)) y) H9). +elim H10; intros V0 H11; exists V0; assert (H12 := imply_to_and _ _ H11); + elim H12; clear H12; intros. +split. +apply H12. +apply (not_ex_all_not _ _ H13). +red in |- *; intro; cut (adherence (image_dir g (fun c:R => a <= c <= b)) M). +intro; elim (closed_set_P1 (image_dir g (fun c:R => a <= c <= b))); + intros H11 _; assert (H12 := H11 H3). +elim H8. +unfold eq_Dom in H12; elim H12; clear H12; intros. +apply (H13 _ H10). +apply H9. +exists (g a); unfold image_dir in |- *; exists a; split. +reflexivity. +split; [ right; reflexivity | apply H ]. +unfold bound in |- *; unfold bounded in H4; elim H4; clear H4; intros m H4; + elim H4; clear H4; intros M H4; exists M; unfold is_upper_bound in |- *; + intros; elim (H4 _ H5); intros _ H6; apply H6. +apply prolongement_C0; assumption. Qed. (**********) -Lemma continuity_ab_min : (f:(R->R); a,b:R) ``a <= b``->((c:R)``a<=c<=b``->(continuity_pt f c))->(EXT mx:R | ((c:R)``a <= c <= b``->``(f mx) <= (f c)``)/\``a <= mx <= b``). -Intros. -Cut ((c:R)``a<=c<=b``->(continuity_pt (opp_fct f0) c)). -Intro; Assert H2 := (continuity_ab_maj (opp_fct f0) a b H H1); Elim H2; Intros x0 H3; Exists x0; Intros; Split. -Intros; Rewrite <- (Ropp_Ropp (f0 x0)); Rewrite <- (Ropp_Ropp (f0 c)); Apply Rle_Ropp1; Elim H3; Intros; Unfold opp_fct in H5; Apply H5; Apply H4. -Elim H3; Intros; Assumption. -Intros. -Assert H2 := (H0 ? H1). -Apply (continuity_pt_opp ? ? H2). +Lemma continuity_ab_min : + forall (f:R -> R) (a b:R), + a <= b -> + (forall c:R, a <= c <= b -> continuity_pt f c) -> + exists mx : R | (forall c:R, a <= c <= b -> f mx <= f c) /\ a <= mx <= b. +intros. +cut (forall c:R, a <= c <= b -> continuity_pt (- f0) c). +intro; assert (H2 := continuity_ab_maj (- f0)%F a b H H1); elim H2; + intros x0 H3; exists x0; intros; split. +intros; rewrite <- (Ropp_involutive (f0 x0)); + rewrite <- (Ropp_involutive (f0 c)); apply Ropp_le_contravar; + elim H3; intros; unfold opp_fct in H5; apply H5; apply H4. +elim H3; intros; assumption. +intros. +assert (H2 := H0 _ H1). +apply (continuity_pt_opp _ _ H2). Qed. @@ -888,291 +1272,554 @@ Qed. (* Proof of Bolzano-Weierstrass theorem *) (********************************************************) -Definition ValAdh [un:nat->R;x:R] : Prop := (V:R->Prop;N:nat) (neighbourhood V x) -> (EX p:nat | (le N p)/\(V (un p))). - -Definition intersection_family [f:family] : R->Prop := [x:R](y:R)(ind f y)->(f y x). - -Lemma ValAdh_un_exists : (un:nat->R) let D=[x:R](EX n:nat | x==(INR n)) in let f=[x:R](adherence [y:R](EX p:nat | y==(un p)/\``x<=(INR p)``)/\(D x)) in ((x:R)(EXT y:R | (f x y))->(D x)). -Intros; Elim H; Intros; Unfold f in H0; Unfold adherence in H0; Unfold point_adherent in H0; Assert H1 : (neighbourhood (disc x0 (mkposreal ? Rlt_R0_R1)) x0). -Unfold neighbourhood disc; Exists (mkposreal ? Rlt_R0_R1); Unfold included; Trivial. -Elim (H0 ? H1); Intros; Unfold intersection_domain in H2; Elim H2; Intros; Elim H4; Intros; Apply H6. +Definition ValAdh (un:nat -> R) (x:R) : Prop := + forall (V:R -> Prop) (N:nat), + neighbourhood V x -> exists p : nat | (N <= p)%nat /\ V (un p). + +Definition intersection_family (f:family) (x:R) : Prop := + forall y:R, ind f y -> f y x. + +Lemma ValAdh_un_exists : + forall (un:nat -> R) (D:=fun x:R => exists n : nat | x = INR n) + (f:= + fun x:R => + adherence + (fun y:R => ( exists p : nat | y = un p /\ x <= INR p) /\ D x)) + (x:R), ( exists y : R | f x y) -> D x. +intros; elim H; intros; unfold f in H0; unfold adherence in H0; + unfold point_adherent in H0; + assert (H1 : neighbourhood (disc x0 (mkposreal _ Rlt_0_1)) x0). +unfold neighbourhood, disc in |- *; exists (mkposreal _ Rlt_0_1); + unfold included in |- *; trivial. +elim (H0 _ H1); intros; unfold intersection_domain in H2; elim H2; intros; + elim H4; intros; apply H6. Qed. -Definition ValAdh_un [un:nat->R] : R->Prop := let D=[x:R](EX n:nat | x==(INR n)) in let f=[x:R](adherence [y:R](EX p:nat | y==(un p)/\``x<=(INR p)``)/\(D x)) in (intersection_family (mkfamily D f (ValAdh_un_exists un))). - -Lemma ValAdh_un_prop : (un:nat->R;x:R) (ValAdh un x) <-> (ValAdh_un un x). -Intros; Split; Intro. -Unfold ValAdh in H; Unfold ValAdh_un; Unfold intersection_family; Simpl; Intros; Elim H0; Intros N H1; Unfold adherence; Unfold point_adherent; Intros; Elim (H V N H2); Intros; Exists (un x0); Unfold intersection_domain; Elim H3; Clear H3; Intros; Split. -Assumption. -Split. -Exists x0; Split; [Reflexivity | Rewrite H1; Apply (le_INR ? ? H3)]. -Exists N; Assumption. -Unfold ValAdh; Intros; Unfold ValAdh_un in H; Unfold intersection_family in H; Simpl in H; Assert H1 : (adherence [y0:R](EX p:nat | ``y0 == (un p)``/\``(INR N) <= (INR p)``)/\(EX n:nat | ``(INR N) == (INR n)``) x). -Apply H; Exists N; Reflexivity. -Unfold adherence in H1; Unfold point_adherent in H1; Assert H2 := (H1 ? H0); Elim H2; Intros; Unfold intersection_domain in H3; Elim H3; Clear H3; Intros; Elim H4; Clear H4; Intros; Elim H4; Clear H4; Intros; Elim H4; Clear H4; Intros; Exists x1; Split. -Apply (INR_le ? ? H6). -Rewrite H4 in H3; Apply H3. +Definition ValAdh_un (un:nat -> R) : R -> Prop := + let D := fun x:R => exists n : nat | x = INR n in + let f := + fun x:R => + adherence + (fun y:R => ( exists p : nat | y = un p /\ x <= INR p) /\ D x) in + intersection_family (mkfamily D f (ValAdh_un_exists un)). + +Lemma ValAdh_un_prop : + forall (un:nat -> R) (x:R), ValAdh un x <-> ValAdh_un un x. +intros; split; intro. +unfold ValAdh in H; unfold ValAdh_un in |- *; + unfold intersection_family in |- *; simpl in |- *; + intros; elim H0; intros N H1; unfold adherence in |- *; + unfold point_adherent in |- *; intros; elim (H V N H2); + intros; exists (un x0); unfold intersection_domain in |- *; + elim H3; clear H3; intros; split. +assumption. +split. +exists x0; split; [ reflexivity | rewrite H1; apply (le_INR _ _ H3) ]. +exists N; assumption. +unfold ValAdh in |- *; intros; unfold ValAdh_un in H; + unfold intersection_family in H; simpl in H; + assert + (H1 : + adherence + (fun y0:R => + ( exists p : nat | y0 = un p /\ INR N <= INR p) /\ + ( exists n : nat | INR N = INR n)) x). +apply H; exists N; reflexivity. +unfold adherence in H1; unfold point_adherent in H1; assert (H2 := H1 _ H0); + elim H2; intros; unfold intersection_domain in H3; + elim H3; clear H3; intros; elim H4; clear H4; intros; + elim H4; clear H4; intros; elim H4; clear H4; intros; + exists x1; split. +apply (INR_le _ _ H6). +rewrite H4 in H3; apply H3. Qed. -Lemma adherence_P4 : (F,G:R->Prop) (included F G) -> (included (adherence F) (adherence G)). -Unfold adherence included; Unfold point_adherent; Intros; Elim (H0 ? H1); Unfold intersection_domain; Intros; Elim H2; Clear H2; Intros; Exists x0; Split; [Assumption | Apply (H ? H3)]. +Lemma adherence_P4 : + forall F G:R -> Prop, included F G -> included (adherence F) (adherence G). +unfold adherence, included in |- *; unfold point_adherent in |- *; intros; + elim (H0 _ H1); unfold intersection_domain in |- *; + intros; elim H2; clear H2; intros; exists x0; split; + [ assumption | apply (H _ H3) ]. Qed. -Definition family_closed_set [f:family] : Prop := (x:R) (closed_set (f x)). +Definition family_closed_set (f:family) : Prop := + forall x:R, closed_set (f x). -Definition intersection_vide_in [D:R->Prop;f:family] : Prop := ((x:R)((ind f x)->(included (f x) D))/\~(EXT y:R | (intersection_family f y))). +Definition intersection_vide_in (D:R -> Prop) (f:family) : Prop := + forall x:R, + (ind f x -> included (f x) D) /\ + ~ ( exists y : R | intersection_family f y). -Definition intersection_vide_finite_in [D:R->Prop;f:family] : Prop := (intersection_vide_in D f)/\(family_finite f). +Definition intersection_vide_finite_in (D:R -> Prop) + (f:family) : Prop := intersection_vide_in D f /\ family_finite f. (**********) -Lemma compact_P6 : (X:R->Prop) (compact X) -> (EXT z:R | (X z)) -> ((g:family) (family_closed_set g) -> (intersection_vide_in X g) -> (EXT D:R->Prop | (intersection_vide_finite_in X (subfamily g D)))). -Intros X H Hyp g H0 H1. -Pose D' := (ind g). -Pose f' := [x:R][y:R](complementary (g x) y)/\(D' x). -Assert H2 : (x:R)(EXT y:R|(f' x y))->(D' x). -Intros; Elim H2; Intros; Unfold f' in H3; Elim H3; Intros; Assumption. -Pose f0 := (mkfamily D' f' H2). -Unfold compact in H; Assert H3 : (covering_open_set X f0). -Unfold covering_open_set; Split. -Unfold covering; Intros; Unfold intersection_vide_in in H1; Elim (H1 x); Intros; Unfold intersection_family in H5; Assert H6 := (not_ex_all_not ? [y:R](y0:R)(ind g y0)->(g y0 y) H5 x); Assert H7 := (not_all_ex_not ? [y0:R](ind g y0)->(g y0 x) H6); Elim H7; Intros; Exists x0; Elim (imply_to_and ? ? H8); Intros; Unfold f0; Simpl; Unfold f'; Split; [Apply H10 | Apply H9]. -Unfold family_open_set; Intro; Elim (classic (D' x)); Intro. -Apply open_set_P6 with (complementary (g x)). -Unfold family_closed_set in H0; Unfold closed_set in H0; Apply H0. -Unfold f0; Simpl; Unfold f'; Unfold eq_Dom; Split. -Unfold included; Intros; Split; [Apply H4 | Apply H3]. -Unfold included; Intros; Elim H4; Intros; Assumption. -Apply open_set_P6 with [_:R]False. -Apply open_set_P4. -Unfold eq_Dom; Unfold included; Split; Intros; [Elim H4 | Simpl in H4; Unfold f' in H4; Elim H4; Intros; Elim H3; Assumption]. -Elim (H ? H3); Intros SF H4; Exists SF; Unfold intersection_vide_finite_in; Split. -Unfold intersection_vide_in; Simpl; Intros; Split. -Intros; Unfold included; Intros; Unfold intersection_vide_in in H1; Elim (H1 x); Intros; Elim H6; Intros; Apply H7. -Unfold intersection_domain in H5; Elim H5; Intros; Assumption. -Assumption. -Elim (classic (EXT y:R | (intersection_domain (ind g) SF y))); Intro Hyp'. -Red; Intro; Elim H5; Intros; Unfold intersection_family in H6; Simpl in H6. -Cut (X x0). -Intro; Unfold covering_finite in H4; Elim H4; Clear H4; Intros H4 _; Unfold covering in H4; Elim (H4 x0 H7); Intros; Simpl in H8; Unfold intersection_domain in H6; Cut (ind g x1)/\(SF x1). -Intro; Assert H10 := (H6 x1 H9); Elim H10; Clear H10; Intros H10 _; Elim H8; Clear H8; Intros H8 _; Unfold f' in H8; Unfold complementary in H8; Elim H8; Clear H8; Intros H8 _; Elim H8; Assumption. -Split. -Apply (cond_fam f0). -Exists x0; Elim H8; Intros; Assumption. -Elim H8; Intros; Assumption. -Unfold intersection_vide_in in H1; Elim Hyp'; Intros; Assert H8 := (H6 ? H7); Elim H8; Intros; Cut (ind g x1). -Intro; Elim (H1 x1); Intros; Apply H12. -Apply H11. -Apply H9. -Apply (cond_fam g); Exists x0; Assumption. -Unfold covering_finite in H4; Elim H4; Clear H4; Intros H4 _; Cut (EXT z:R | (X z)). -Intro; Elim H5; Clear H5; Intros; Unfold covering in H4; Elim (H4 x0 H5); Intros; Simpl in H6; Elim Hyp'; Exists x1; Elim H6; Intros; Unfold intersection_domain; Split. -Apply (cond_fam f0); Exists x0; Apply H7. -Apply H8. -Apply Hyp. -Unfold covering_finite in H4; Elim H4; Clear H4; Intros; Unfold family_finite in H5; Unfold domain_finite in H5; Unfold family_finite; Unfold domain_finite; Elim H5; Clear H5; Intros l H5; Exists l; Intro; Elim (H5 x); Intros; Split; Intro; [Apply H6; Simpl; Simpl in H8; Apply H8 | Apply (H7 H8)]. +Lemma compact_P6 : + forall X:R -> Prop, + compact X -> + ( exists z : R | X z) -> + forall g:family, + family_closed_set g -> + intersection_vide_in X g -> + exists D : R -> Prop | intersection_vide_finite_in X (subfamily g D). +intros X H Hyp g H0 H1. +pose (D' := ind g). +pose (f' := fun x y:R => complementary (g x) y /\ D' x). +assert (H2 : forall x:R, ( exists y : R | f' x y) -> D' x). +intros; elim H2; intros; unfold f' in H3; elim H3; intros; assumption. +pose (f0 := mkfamily D' f' H2). +unfold compact in H; assert (H3 : covering_open_set X f0). +unfold covering_open_set in |- *; split. +unfold covering in |- *; intros; unfold intersection_vide_in in H1; + elim (H1 x); intros; unfold intersection_family in H5; + assert + (H6 := not_ex_all_not _ (fun y:R => forall y0:R, ind g y0 -> g y0 y) H5 x); + assert (H7 := not_all_ex_not _ (fun y0:R => ind g y0 -> g y0 x) H6); + elim H7; intros; exists x0; elim (imply_to_and _ _ H8); + intros; unfold f0 in |- *; simpl in |- *; unfold f' in |- *; + split; [ apply H10 | apply H9 ]. +unfold family_open_set in |- *; intro; elim (classic (D' x)); intro. +apply open_set_P6 with (complementary (g x)). +unfold family_closed_set in H0; unfold closed_set in H0; apply H0. +unfold f0 in |- *; simpl in |- *; unfold f' in |- *; unfold eq_Dom in |- *; + split. +unfold included in |- *; intros; split; [ apply H4 | apply H3 ]. +unfold included in |- *; intros; elim H4; intros; assumption. +apply open_set_P6 with (fun _:R => False). +apply open_set_P4. +unfold eq_Dom in |- *; unfold included in |- *; split; intros; + [ elim H4 + | simpl in H4; unfold f' in H4; elim H4; intros; elim H3; assumption ]. +elim (H _ H3); intros SF H4; exists SF; + unfold intersection_vide_finite_in in |- *; split. +unfold intersection_vide_in in |- *; simpl in |- *; intros; split. +intros; unfold included in |- *; intros; unfold intersection_vide_in in H1; + elim (H1 x); intros; elim H6; intros; apply H7. +unfold intersection_domain in H5; elim H5; intros; assumption. +assumption. +elim (classic ( exists y : R | intersection_domain (ind g) SF y)); intro Hyp'. +red in |- *; intro; elim H5; intros; unfold intersection_family in H6; + simpl in H6. +cut (X x0). +intro; unfold covering_finite in H4; elim H4; clear H4; intros H4 _; + unfold covering in H4; elim (H4 x0 H7); intros; simpl in H8; + unfold intersection_domain in H6; cut (ind g x1 /\ SF x1). +intro; assert (H10 := H6 x1 H9); elim H10; clear H10; intros H10 _; elim H8; + clear H8; intros H8 _; unfold f' in H8; unfold complementary in H8; + elim H8; clear H8; intros H8 _; elim H8; assumption. +split. +apply (cond_fam f0). +exists x0; elim H8; intros; assumption. +elim H8; intros; assumption. +unfold intersection_vide_in in H1; elim Hyp'; intros; assert (H8 := H6 _ H7); + elim H8; intros; cut (ind g x1). +intro; elim (H1 x1); intros; apply H12. +apply H11. +apply H9. +apply (cond_fam g); exists x0; assumption. +unfold covering_finite in H4; elim H4; clear H4; intros H4 _; + cut ( exists z : R | X z). +intro; elim H5; clear H5; intros; unfold covering in H4; elim (H4 x0 H5); + intros; simpl in H6; elim Hyp'; exists x1; elim H6; + intros; unfold intersection_domain in |- *; split. +apply (cond_fam f0); exists x0; apply H7. +apply H8. +apply Hyp. +unfold covering_finite in H4; elim H4; clear H4; intros; + unfold family_finite in H5; unfold domain_finite in H5; + unfold family_finite in |- *; unfold domain_finite in |- *; + elim H5; clear H5; intros l H5; exists l; intro; elim (H5 x); + intros; split; intro; + [ apply H6; simpl in |- *; simpl in H8; apply H8 | apply (H7 H8) ]. Qed. -Theorem Bolzano_Weierstrass : (un:nat->R;X:R->Prop) (compact X) -> ((n:nat)(X (un n))) -> (EXT l:R | (ValAdh un l)). -Intros; Cut (EXT l:R | (ValAdh_un un l)). -Intro; Elim H1; Intros; Exists x; Elim (ValAdh_un_prop un x); Intros; Apply (H4 H2). -Assert H1 : (EXT z:R | (X z)). -Exists (un O); Apply H0. -Pose D:=[x:R](EX n:nat | x==(INR n)). -Pose g:=[x:R](adherence [y:R](EX p:nat | y==(un p)/\``x<=(INR p)``)/\(D x)). -Assert H2 : (x:R)(EXT y:R | (g x y))->(D x). -Intros; Elim H2; Intros; Unfold g in H3; Unfold adherence in H3; Unfold point_adherent in H3. -Assert H4 : (neighbourhood (disc x0 (mkposreal ? Rlt_R0_R1)) x0). -Unfold neighbourhood; Exists (mkposreal ? Rlt_R0_R1); Unfold included; Trivial. -Elim (H3 ? H4); Intros; Unfold intersection_domain in H5; Decompose [and] H5; Assumption. -Pose f0 := (mkfamily D g H2). -Assert H3 := (compact_P6 X H H1 f0). -Elim (classic (EXT l:R | (ValAdh_un un l))); Intro. -Assumption. -Cut (family_closed_set f0). -Intro; Cut (intersection_vide_in X f0). -Intro; Assert H7 := (H3 H5 H6). -Elim H7; Intros SF H8; Unfold intersection_vide_finite_in in H8; Elim H8; Clear H8; Intros; Unfold intersection_vide_in in H8; Elim (H8 R0); Intros _ H10; Elim H10; Unfold family_finite in H9; Unfold domain_finite in H9; Elim H9; Clear H9; Intros l H9; Pose r := (MaxRlist l); Cut (D r). -Intro; Unfold D in H11; Elim H11; Intros; Exists (un x); Unfold intersection_family; Simpl; Unfold intersection_domain; Intros; Split. -Unfold g; Apply adherence_P1; Split. -Exists x; Split; [Reflexivity | Rewrite <- H12; Unfold r; Apply MaxRlist_P1; Elim (H9 y); Intros; Apply H14; Simpl; Apply H13]. -Elim H13; Intros; Assumption. -Elim H13; Intros; Assumption. -Elim (H9 r); Intros. -Simpl in H12; Unfold intersection_domain in H12; Cut (In r l). -Intro; Elim (H12 H13); Intros; Assumption. -Unfold r; Apply MaxRlist_P2; Cut (EXT z:R | (intersection_domain (ind f0) SF z)). -Intro; Elim H13; Intros; Elim (H9 x); Intros; Simpl in H15; Assert H17 := (H15 H14); Exists x; Apply H17. -Elim (classic (EXT z:R | (intersection_domain (ind f0) SF z))); Intro. -Assumption. -Elim (H8 R0); Intros _ H14; Elim H1; Intros; Assert H16 := (not_ex_all_not ? [y:R](intersection_family (subfamily f0 SF) y) H14); Assert H17 := (not_ex_all_not ? [z:R](intersection_domain (ind f0) SF z) H13); Assert H18 := (H16 x); Unfold intersection_family in H18; Simpl in H18; Assert H19 := (not_all_ex_not ? [y:R](intersection_domain D SF y)->(g y x)/\(SF y) H18); Elim H19; Intros; Assert H21 := (imply_to_and ? ? H20); Elim (H17 x0); Elim H21; Intros; Assumption. -Unfold intersection_vide_in; Intros; Split. -Intro; Simpl in H6; Unfold f0; Simpl; Unfold g; Apply included_trans with (adherence X). -Apply adherence_P4. -Unfold included; Intros; Elim H7; Intros; Elim H8; Intros; Elim H10; Intros; Rewrite H11; Apply H0. -Apply adherence_P2; Apply compact_P2; Assumption. -Apply H4. -Unfold family_closed_set; Unfold f0; Simpl; Unfold g; Intro; Apply adherence_P3. +Theorem Bolzano_Weierstrass : + forall (un:nat -> R) (X:R -> Prop), + compact X -> (forall n:nat, X (un n)) -> exists l : R | ValAdh un l. +intros; cut ( exists l : R | ValAdh_un un l). +intro; elim H1; intros; exists x; elim (ValAdh_un_prop un x); intros; + apply (H4 H2). +assert (H1 : exists z : R | X z). +exists (un 0%nat); apply H0. +pose (D := fun x:R => exists n : nat | x = INR n). +pose + (g := + fun x:R => + adherence (fun y:R => ( exists p : nat | y = un p /\ x <= INR p) /\ D x)). +assert (H2 : forall x:R, ( exists y : R | g x y) -> D x). +intros; elim H2; intros; unfold g in H3; unfold adherence in H3; + unfold point_adherent in H3. +assert (H4 : neighbourhood (disc x0 (mkposreal _ Rlt_0_1)) x0). +unfold neighbourhood in |- *; exists (mkposreal _ Rlt_0_1); + unfold included in |- *; trivial. +elim (H3 _ H4); intros; unfold intersection_domain in H5; decompose [and] H5; + assumption. +pose (f0 := mkfamily D g H2). +assert (H3 := compact_P6 X H H1 f0). +elim (classic ( exists l : R | ValAdh_un un l)); intro. +assumption. +cut (family_closed_set f0). +intro; cut (intersection_vide_in X f0). +intro; assert (H7 := H3 H5 H6). +elim H7; intros SF H8; unfold intersection_vide_finite_in in H8; elim H8; + clear H8; intros; unfold intersection_vide_in in H8; + elim (H8 0); intros _ H10; elim H10; unfold family_finite in H9; + unfold domain_finite in H9; elim H9; clear H9; intros l H9; + pose (r := MaxRlist l); cut (D r). +intro; unfold D in H11; elim H11; intros; exists (un x); + unfold intersection_family in |- *; simpl in |- *; + unfold intersection_domain in |- *; intros; split. +unfold g in |- *; apply adherence_P1; split. +exists x; split; + [ reflexivity + | rewrite <- H12; unfold r in |- *; apply MaxRlist_P1; elim (H9 y); intros; + apply H14; simpl in |- *; apply H13 ]. +elim H13; intros; assumption. +elim H13; intros; assumption. +elim (H9 r); intros. +simpl in H12; unfold intersection_domain in H12; cut (In r l). +intro; elim (H12 H13); intros; assumption. +unfold r in |- *; apply MaxRlist_P2; + cut ( exists z : R | intersection_domain (ind f0) SF z). +intro; elim H13; intros; elim (H9 x); intros; simpl in H15; + assert (H17 := H15 H14); exists x; apply H17. +elim (classic ( exists z : R | intersection_domain (ind f0) SF z)); intro. +assumption. +elim (H8 0); intros _ H14; elim H1; intros; + assert + (H16 := + not_ex_all_not _ (fun y:R => intersection_family (subfamily f0 SF) y) H14); + assert + (H17 := + not_ex_all_not _ (fun z:R => intersection_domain (ind f0) SF z) H13); + assert (H18 := H16 x); unfold intersection_family in H18; + simpl in H18; + assert + (H19 := + not_all_ex_not _ (fun y:R => intersection_domain D SF y -> g y x /\ SF y) + H18); elim H19; intros; assert (H21 := imply_to_and _ _ H20); + elim (H17 x0); elim H21; intros; assumption. +unfold intersection_vide_in in |- *; intros; split. +intro; simpl in H6; unfold f0 in |- *; simpl in |- *; unfold g in |- *; + apply included_trans with (adherence X). +apply adherence_P4. +unfold included in |- *; intros; elim H7; intros; elim H8; intros; elim H10; + intros; rewrite H11; apply H0. +apply adherence_P2; apply compact_P2; assumption. +apply H4. +unfold family_closed_set in |- *; unfold f0 in |- *; simpl in |- *; + unfold g in |- *; intro; apply adherence_P3. Qed. (********************************************************) (* Proof of Heine's theorem *) (********************************************************) -Definition uniform_continuity [f:R->R;X:R->Prop] : Prop := (eps:posreal)(EXT delta:posreal | (x,y:R) (X x)->(X y)->``(Rabsolu (x-y))<delta`` ->``(Rabsolu ((f x)-(f y)))<eps``). +Definition uniform_continuity (f:R -> R) (X:R -> Prop) : Prop := + forall eps:posreal, + exists delta : posreal + | (forall x y:R, + X x -> X y -> Rabs (x - y) < delta -> Rabs (f x - f y) < eps). -Lemma is_lub_u : (E:R->Prop;x,y:R) (is_lub E x) -> (is_lub E y) -> x==y. -Unfold is_lub; Intros; Elim H; Elim H0; Intros; Apply Rle_antisym; [Apply (H4 ? H1) | Apply (H2 ? H3)]. +Lemma is_lub_u : + forall (E:R -> Prop) (x y:R), is_lub E x -> is_lub E y -> x = y. +unfold is_lub in |- *; intros; elim H; elim H0; intros; apply Rle_antisym; + [ apply (H4 _ H1) | apply (H2 _ H3) ]. Qed. -Lemma domain_P1 : (X:R->Prop) ~(EXT y:R | (X y))\/(EXT y:R | (X y)/\((x:R)(X x)->x==y))\/(EXT x:R | (EXT y:R | (X x)/\(X y)/\``x<>y``)). -Intro; Elim (classic (EXT y:R | (X y))); Intro. -Right; Elim H; Intros; Elim (classic (EXT y:R | (X y)/\``y<>x``)); Intro. -Right; Elim H1; Intros; Elim H2; Intros; Exists x; Exists x0; Intros. -Split; [Assumption | Split; [Assumption | Apply not_sym; Assumption]]. -Left; Exists x; Split. -Assumption. -Intros; Case (Req_EM x0 x); Intro. -Assumption. -Elim H1; Exists x0; Split; Assumption. -Left; Assumption. +Lemma domain_P1 : + forall X:R -> Prop, + ~ ( exists y : R | X y) \/ + ( exists y : R | X y /\ (forall x:R, X x -> x = y)) \/ + ( exists x : R | ( exists y : R | X x /\ X y /\ x <> y)). +intro; elim (classic ( exists y : R | X y)); intro. +right; elim H; intros; elim (classic ( exists y : R | X y /\ y <> x)); intro. +right; elim H1; intros; elim H2; intros; exists x; exists x0; intros. +split; + [ assumption + | split; [ assumption | apply (sym_not_eq (A:=R)); assumption ] ]. +left; exists x; split. +assumption. +intros; case (Req_dec x0 x); intro. +assumption. +elim H1; exists x0; split; assumption. +left; assumption. Qed. -Theorem Heine : (f:R->R;X:R->Prop) (compact X) -> ((x:R)(X x)->(continuity_pt f x)) -> (uniform_continuity f X). -Intros f0 X H0 H; Elim (domain_P1 X); Intro Hyp. +Theorem Heine : + forall (f:R -> R) (X:R -> Prop), + compact X -> + (forall x:R, X x -> continuity_pt f x) -> uniform_continuity f X. +intros f0 X H0 H; elim (domain_P1 X); intro Hyp. (* X est vide *) -Unfold uniform_continuity; Intros; Exists (mkposreal ? Rlt_R0_R1); Intros; Elim Hyp; Exists x; Assumption. -Elim Hyp; Clear Hyp; Intro Hyp. +unfold uniform_continuity in |- *; intros; exists (mkposreal _ Rlt_0_1); + intros; elim Hyp; exists x; assumption. +elim Hyp; clear Hyp; intro Hyp. (* X possède un seul élément *) -Unfold uniform_continuity; Intros; Exists (mkposreal ? Rlt_R0_R1); Intros; Elim Hyp; Clear Hyp; Intros; Elim H4; Clear H4; Intros; Assert H6 := (H5 ? H1); Assert H7 := (H5 ? H2); Rewrite H6; Rewrite H7; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (cond_pos eps). +unfold uniform_continuity in |- *; intros; exists (mkposreal _ Rlt_0_1); + intros; elim Hyp; clear Hyp; intros; elim H4; clear H4; + intros; assert (H6 := H5 _ H1); assert (H7 := H5 _ H2); + rewrite H6; rewrite H7; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; apply (cond_pos eps). (* X possède au moins deux éléments distincts *) -Assert X_enc : (EXT m:R | (EXT M:R | ((x:R)(X x)->``m<=x<=M``)/\``m<M``)). -Assert H1 := (compact_P1 X H0); Unfold bounded in H1; Elim H1; Intros; Elim H2; Intros; Exists x; Exists x0; Split. -Apply H3. -Elim Hyp; Intros; Elim H4; Intros; Decompose [and] H5; Assert H10 := (H3 ? H6); Assert H11 := (H3 ? H8); Elim H10; Intros; Elim H11; Intros; Case (total_order_T x x0); Intro. -Elim s; Intro. -Assumption. -Rewrite b in H13; Rewrite b in H7; Elim H9; Apply Rle_antisym; Apply Rle_trans with x0; Assumption. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? (Rle_trans ? ? ? H13 H14) r)). -Elim X_enc; Clear X_enc; Intros m X_enc; Elim X_enc; Clear X_enc; Intros M X_enc; Elim X_enc; Clear X_enc Hyp; Intros X_enc Hyp; Unfold uniform_continuity; Intro; Assert H1 : (t:posreal)``0<t/2``. -Intro; Unfold Rdiv; Apply Rmult_lt_pos; [Apply (cond_pos t) | Apply Rlt_Rinv; Sup0]. -Pose g := [x:R][y:R](X x)/\(EXT del:posreal | ((z:R) ``(Rabsolu (z-x))<del``->``(Rabsolu ((f0 z)-(f0 x)))<eps/2``)/\(is_lub [zeta:R]``0<zeta<=M-m``/\((z:R) ``(Rabsolu (z-x))<zeta``->``(Rabsolu ((f0 z)-(f0 x)))<eps/2``) del)/\(disc x (mkposreal ``del/2`` (H1 del)) y)). -Assert H2 : (x:R)(EXT y:R | (g x y))->(X x). -Intros; Elim H2; Intros; Unfold g in H3; Elim H3; Clear H3; Intros H3 _; Apply H3. -Pose f' := (mkfamily X g H2); Unfold compact in H0; Assert H3 : (covering_open_set X f'). -Unfold covering_open_set; Split. -Unfold covering; Intros; Exists x; Simpl; Unfold g; Split. -Assumption. -Assert H4 := (H ? H3); Unfold continuity_pt in H4; Unfold continue_in in H4; Unfold limit1_in in H4; Unfold limit_in in H4; Simpl in H4; Unfold R_dist in H4; Elim (H4 ``eps/2`` (H1 eps)); Intros; Pose E:=[zeta:R]``0<zeta <= M-m``/\((z:R)``(Rabsolu (z-x)) < zeta``->``(Rabsolu ((f0 z)-(f0 x))) < eps/2``); Assert H6 : (bound E). -Unfold bound; Exists ``M-m``; Unfold is_upper_bound; Unfold E; Intros; Elim H6; Clear H6; Intros H6 _; Elim H6; Clear H6; Intros _ H6; Apply H6. -Assert H7 : (EXT x:R | (E x)). -Elim H5; Clear H5; Intros; Exists (Rmin x0 ``M-m``); Unfold E; Intros; Split. -Split. -Unfold Rmin; Case (total_order_Rle x0 ``M-m``); Intro. -Apply H5. -Apply Rlt_Rminus; Apply Hyp. -Apply Rmin_r. -Intros; Case (Req_EM x z); Intro. -Rewrite H9; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (H1 eps). -Apply H7; Split. -Unfold D_x no_cond; Split; [Trivial | Assumption]. -Apply Rlt_le_trans with (Rmin x0 ``M-m``); [Apply H8 | Apply Rmin_l]. -Assert H8 := (complet ? H6 H7); Elim H8; Clear H8; Intros; Cut ``0<x1<=(M-m)``. -Intro; Elim H8; Clear H8; Intros; Exists (mkposreal ? H8); Split. -Intros; Cut (EXT alp:R | ``(Rabsolu (z-x))<alp<=x1``/\(E alp)). -Intros; Elim H11; Intros; Elim H12; Clear H12; Intros; Unfold E in H13; Elim H13; Intros; Apply H15. -Elim H12; Intros; Assumption. -Elim (classic (EXT alp:R | ``(Rabsolu (z-x)) < alp <= x1``/\(E alp))); Intro. -Assumption. -Assert H12 := (not_ex_all_not ? [alp:R]``(Rabsolu (z-x)) < alp <= x1``/\(E alp) H11); Unfold is_lub in p; Elim p; Intros; Cut (is_upper_bound E ``(Rabsolu (z-x))``). -Intro; Assert H16 := (H14 ? H15); Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H10 H16)). -Unfold is_upper_bound; Intros; Unfold is_upper_bound in H13; Assert H16 := (H13 ? H15); Case (total_order_Rle x2 ``(Rabsolu (z-x))``); Intro. -Assumption. -Elim (H12 x2); Split; [Split; [Auto with real | Assumption] | Assumption]. -Split. -Apply p. -Unfold disc; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Simpl; Unfold Rdiv; Apply Rmult_lt_pos; [Apply H8 | Apply Rlt_Rinv; Sup0]. -Elim H7; Intros; Unfold E in H8; Elim H8; Intros H9 _; Elim H9; Intros H10 _; Unfold is_lub in p; Elim p; Intros; Unfold is_upper_bound in H12; Unfold is_upper_bound in H11; Split. -Apply Rlt_le_trans with x2; [Assumption | Apply (H11 ? H8)]. -Apply H12; Intros; Unfold E in H13; Elim H13; Intros; Elim H14; Intros; Assumption. -Unfold family_open_set; Intro; Simpl; Elim (classic (X x)); Intro. -Unfold g; Unfold open_set; Intros; Elim H4; Clear H4; Intros _ H4; Elim H4; Clear H4; Intros; Elim H4; Clear H4; Intros; Unfold neighbourhood; Case (Req_EM x x0); Intro. -Exists (mkposreal ? (H1 x1)); Rewrite <- H6; Unfold included; Intros; Split. -Assumption. -Exists x1; Split. -Apply H4. -Split. -Elim H5; Intros; Apply H8. -Apply H7. -Pose d := ``x1/2-(Rabsolu (x0-x))``; Assert H7 : ``0<d``. -Unfold d; Apply Rlt_Rminus; Elim H5; Clear H5; Intros; Unfold disc in H7; Apply H7. -Exists (mkposreal ? H7); Unfold included; Intros; Split. -Assumption. -Exists x1; Split. -Apply H4. -Elim H5; Intros; Split. -Assumption. -Unfold disc in H8; Simpl in H8; Unfold disc; Simpl; Unfold disc in H10; Simpl in H10; Apply Rle_lt_trans with ``(Rabsolu (x2-x0))+(Rabsolu (x0-x))``. -Replace ``x2-x`` with ``(x2-x0)+(x0-x)``; [Apply Rabsolu_triang | Ring]. -Replace ``x1/2`` with ``d+(Rabsolu (x0-x))``; [Idtac | Unfold d; Ring]. -Do 2 Rewrite <- (Rplus_sym ``(Rabsolu (x0-x))``); Apply Rlt_compatibility; Apply H8. -Apply open_set_P6 with [_:R]False. -Apply open_set_P4. -Unfold eq_Dom; Unfold included; Intros; Split. -Intros; Elim H4. -Intros; Unfold g in H4; Elim H4; Clear H4; Intros H4 _; Elim H3; Apply H4. -Elim (H0 ? H3); Intros DF H4; Unfold covering_finite in H4; Elim H4; Clear H4; Intros; Unfold family_finite in H5; Unfold domain_finite in H5; Unfold covering in H4; Simpl in H4; Simpl in H5; Elim H5; Clear H5; Intros l H5; Unfold intersection_domain in H5; Cut (x:R)(In x l)->(EXT del:R | ``0<del``/\((z:R)``(Rabsolu (z-x)) < del``->``(Rabsolu ((f0 z)-(f0 x))) < eps/2``)/\(included (g x) [z:R]``(Rabsolu (z-x))<del/2``)). -Intros; Assert H7 := (Rlist_P1 l [x:R][del:R]``0<del``/\((z:R)``(Rabsolu (z-x)) < del``->``(Rabsolu ((f0 z)-(f0 x))) < eps/2``)/\(included (g x) [z:R]``(Rabsolu (z-x))<del/2``) H6); Elim H7; Clear H7; Intros l' H7; Elim H7; Clear H7; Intros; Pose D := (MinRlist l'); Cut ``0<D/2``. -Intro; Exists (mkposreal ? H9); Intros; Assert H13 := (H4 ? H10); Elim H13; Clear H13; Intros xi H13; Assert H14 : (In xi l). -Unfold g in H13; Decompose [and] H13; Elim (H5 xi); Intros; Apply H14; Split; Assumption. -Elim (pos_Rl_P2 l xi); Intros H15 _; Elim (H15 H14); Intros i H16; Elim H16; Intros; Apply Rle_lt_trans with ``(Rabsolu ((f0 x)-(f0 xi)))+(Rabsolu ((f0 xi)-(f0 y)))``. -Replace ``(f0 x)-(f0 y)`` with ``((f0 x)-(f0 xi))+((f0 xi)-(f0 y))``; [Apply Rabsolu_triang | Ring]. -Rewrite (double_var eps); Apply Rplus_lt. -Assert H19 := (H8 i H17); Elim H19; Clear H19; Intros; Rewrite <- H18 in H20; Elim H20; Clear H20; Intros; Apply H20; Unfold included in H21; Apply Rlt_trans with ``(pos_Rl l' i)/2``. -Apply H21. -Elim H13; Clear H13; Intros; Assumption. -Unfold Rdiv; Apply Rlt_monotony_contra with ``2``. -Sup0. -Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Pattern 1 (pos_Rl l' i); Rewrite <- Rplus_Or; Rewrite double; Apply Rlt_compatibility; Apply H19. -DiscrR. -Assert H19 := (H8 i H17); Elim H19; Clear H19; Intros; Rewrite <- H18 in H20; Elim H20; Clear H20; Intros; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H20; Unfold included in H21; Elim H13; Intros; Assert H24 := (H21 x H22); Apply Rle_lt_trans with ``(Rabsolu (y-x))+(Rabsolu (x-xi))``. -Replace ``y-xi`` with ``(y-x)+(x-xi)``; [Apply Rabsolu_triang | Ring]. -Rewrite (double_var (pos_Rl l' i)); Apply Rplus_lt. -Apply Rlt_le_trans with ``D/2``. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H12. -Unfold Rdiv; Do 2 Rewrite <- (Rmult_sym ``/2``); Apply Rle_monotony. -Left; Apply Rlt_Rinv; Sup0. -Unfold D; Apply MinRlist_P1; Elim (pos_Rl_P2 l' (pos_Rl l' i)); Intros; Apply H26; Exists i; Split; [Rewrite <- H7; Assumption | Reflexivity]. -Assumption. -Unfold Rdiv; Apply Rmult_lt_pos; [Unfold D; Apply MinRlist_P2; Intros; Elim (pos_Rl_P2 l' y); Intros; Elim (H10 H9); Intros; Elim H12; Intros; Rewrite H14; Rewrite <- H7 in H13; Elim (H8 x H13); Intros; Apply H15 | Apply Rlt_Rinv; Sup0]. -Intros; Elim (H5 x); Intros; Elim (H8 H6); Intros; Pose E:=[zeta:R]``0<zeta <= M-m``/\((z:R)``(Rabsolu (z-x)) < zeta``->``(Rabsolu ((f0 z)-(f0 x))) < eps/2``); Assert H11 : (bound E). -Unfold bound; Exists ``M-m``; Unfold is_upper_bound; Unfold E; Intros; Elim H11; Clear H11; Intros H11 _; Elim H11; Clear H11; Intros _ H11; Apply H11. -Assert H12 : (EXT x:R | (E x)). -Assert H13 := (H ? H9); Unfold continuity_pt in H13; Unfold continue_in in H13; Unfold limit1_in in H13; Unfold limit_in in H13; Simpl in H13; Unfold R_dist in H13; Elim (H13 ? (H1 eps)); Intros; Elim H12; Clear H12; Intros; Exists (Rmin x0 ``M-m``); Unfold E; Intros; Split. -Split; [Unfold Rmin; Case (total_order_Rle x0 ``M-m``); Intro; [Apply H12 | Apply Rlt_Rminus; Apply Hyp] | Apply Rmin_r]. -Intros; Case (Req_EM x z); Intro. -Rewrite H16; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Apply (H1 eps). -Apply H14; Split; [Unfold D_x no_cond; Split; [Trivial | Assumption] | Apply Rlt_le_trans with (Rmin x0 ``M-m``); [Apply H15 | Apply Rmin_l]]. -Assert H13 := (complet ? H11 H12); Elim H13; Clear H13; Intros; Cut ``0<x0<=M-m``. -Intro; Elim H13; Clear H13; Intros; Exists x0; Split. -Assumption. -Split. -Intros; Cut (EXT alp:R | ``(Rabsolu (z-x))<alp<=x0``/\(E alp)). -Intros; Elim H16; Intros; Elim H17; Clear H17; Intros; Unfold E in H18; Elim H18; Intros; Apply H20; Elim H17; Intros; Assumption. -Elim (classic (EXT alp:R | ``(Rabsolu (z-x)) < alp <= x0``/\(E alp))); Intro. -Assumption. -Assert H17 := (not_ex_all_not ? [alp:R]``(Rabsolu (z-x)) < alp <= x0``/\(E alp) H16); Unfold is_lub in p; Elim p; Intros; Cut (is_upper_bound E ``(Rabsolu (z-x))``). -Intro; Assert H21 := (H19 ? H20); Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H15 H21)). -Unfold is_upper_bound; Intros; Unfold is_upper_bound in H18; Assert H21 := (H18 ? H20); Case (total_order_Rle x1 ``(Rabsolu (z-x))``); Intro. -Assumption. -Elim (H17 x1); Split. -Split; [Auto with real | Assumption]. -Assumption. -Unfold included g; Intros; Elim H15; Intros; Elim H17; Intros; Decompose [and] H18; Cut x0==x2. -Intro; Rewrite H20; Apply H22. -Unfold E in p; EApply is_lub_u. -Apply p. -Apply H21. -Elim H12; Intros; Unfold E in H13; Elim H13; Intros H14 _; Elim H14; Intros H15 _; Unfold is_lub in p; Elim p; Intros; Unfold is_upper_bound in H16; Unfold is_upper_bound in H17; Split. -Apply Rlt_le_trans with x1; [Assumption | Apply (H16 ? H13)]. -Apply H17; Intros; Unfold E in H18; Elim H18; Intros; Elim H19; Intros; Assumption. -Qed. +assert + (X_enc : + exists m : R | ( exists M : R | (forall x:R, X x -> m <= x <= M) /\ m < M)). +assert (H1 := compact_P1 X H0); unfold bounded in H1; elim H1; intros; + elim H2; intros; exists x; exists x0; split. +apply H3. +elim Hyp; intros; elim H4; intros; decompose [and] H5; + assert (H10 := H3 _ H6); assert (H11 := H3 _ H8); + elim H10; intros; elim H11; intros; case (total_order_T x x0); + intro. +elim s; intro. +assumption. +rewrite b in H13; rewrite b in H7; elim H9; apply Rle_antisym; + apply Rle_trans with x0; assumption. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ (Rle_trans _ _ _ H13 H14) r)). +elim X_enc; clear X_enc; intros m X_enc; elim X_enc; clear X_enc; + intros M X_enc; elim X_enc; clear X_enc Hyp; intros X_enc Hyp; + unfold uniform_continuity in |- *; intro; + assert (H1 : forall t:posreal, 0 < t / 2). +intro; unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply (cond_pos t) | apply Rinv_0_lt_compat; prove_sup0 ]. +pose + (g := + fun x y:R => + X x /\ + ( exists del : posreal + | (forall z:R, Rabs (z - x) < del -> Rabs (f0 z - f0 x) < eps / 2) /\ + is_lub + (fun zeta:R => + 0 < zeta <= M - m /\ + (forall z:R, Rabs (z - x) < zeta -> Rabs (f0 z - f0 x) < eps / 2)) + del /\ disc x (mkposreal (del / 2) (H1 del)) y)). +assert (H2 : forall x:R, ( exists y : R | g x y) -> X x). +intros; elim H2; intros; unfold g in H3; elim H3; clear H3; intros H3 _; + apply H3. +pose (f' := mkfamily X g H2); unfold compact in H0; + assert (H3 : covering_open_set X f'). +unfold covering_open_set in |- *; split. +unfold covering in |- *; intros; exists x; simpl in |- *; unfold g in |- *; + split. +assumption. +assert (H4 := H _ H3); unfold continuity_pt in H4; unfold continue_in in H4; + unfold limit1_in in H4; unfold limit_in in H4; simpl in H4; + unfold R_dist in H4; elim (H4 (eps / 2) (H1 eps)); + intros; + pose + (E := + fun zeta:R => + 0 < zeta <= M - m /\ + (forall z:R, Rabs (z - x) < zeta -> Rabs (f0 z - f0 x) < eps / 2)); + assert (H6 : bound E). +unfold bound in |- *; exists (M - m); unfold is_upper_bound in |- *; + unfold E in |- *; intros; elim H6; clear H6; intros H6 _; + elim H6; clear H6; intros _ H6; apply H6. +assert (H7 : exists x : R | E x). +elim H5; clear H5; intros; exists (Rmin x0 (M - m)); unfold E in |- *; intros; + split. +split. +unfold Rmin in |- *; case (Rle_dec x0 (M - m)); intro. +apply H5. +apply Rlt_Rminus; apply Hyp. +apply Rmin_r. +intros; case (Req_dec x z); intro. +rewrite H9; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + apply (H1 eps). +apply H7; split. +unfold D_x, no_cond in |- *; split; [ trivial | assumption ]. +apply Rlt_le_trans with (Rmin x0 (M - m)); [ apply H8 | apply Rmin_l ]. +assert (H8 := completeness _ H6 H7); elim H8; clear H8; intros; + cut (0 < x1 <= M - m). +intro; elim H8; clear H8; intros; exists (mkposreal _ H8); split. +intros; cut ( exists alp : R | Rabs (z - x) < alp <= x1 /\ E alp). +intros; elim H11; intros; elim H12; clear H12; intros; unfold E in H13; + elim H13; intros; apply H15. +elim H12; intros; assumption. +elim (classic ( exists alp : R | Rabs (z - x) < alp <= x1 /\ E alp)); intro. +assumption. +assert + (H12 := + not_ex_all_not _ (fun alp:R => Rabs (z - x) < alp <= x1 /\ E alp) H11); + unfold is_lub in p; elim p; intros; cut (is_upper_bound E (Rabs (z - x))). +intro; assert (H16 := H14 _ H15); + elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H10 H16)). +unfold is_upper_bound in |- *; intros; unfold is_upper_bound in H13; + assert (H16 := H13 _ H15); case (Rle_dec x2 (Rabs (z - x))); + intro. +assumption. +elim (H12 x2); split; [ split; [ auto with real | assumption ] | assumption ]. +split. +apply p. +unfold disc in |- *; unfold Rminus in |- *; rewrite Rplus_opp_r; + rewrite Rabs_R0; simpl in |- *; unfold Rdiv in |- *; + apply Rmult_lt_0_compat; [ apply H8 | apply Rinv_0_lt_compat; prove_sup0 ]. +elim H7; intros; unfold E in H8; elim H8; intros H9 _; elim H9; intros H10 _; + unfold is_lub in p; elim p; intros; unfold is_upper_bound in H12; + unfold is_upper_bound in H11; split. +apply Rlt_le_trans with x2; [ assumption | apply (H11 _ H8) ]. +apply H12; intros; unfold E in H13; elim H13; intros; elim H14; intros; + assumption. +unfold family_open_set in |- *; intro; simpl in |- *; elim (classic (X x)); + intro. +unfold g in |- *; unfold open_set in |- *; intros; elim H4; clear H4; + intros _ H4; elim H4; clear H4; intros; elim H4; clear H4; + intros; unfold neighbourhood in |- *; case (Req_dec x x0); + intro. +exists (mkposreal _ (H1 x1)); rewrite <- H6; unfold included in |- *; intros; + split. +assumption. +exists x1; split. +apply H4. +split. +elim H5; intros; apply H8. +apply H7. +pose (d := x1 / 2 - Rabs (x0 - x)); assert (H7 : 0 < d). +unfold d in |- *; apply Rlt_Rminus; elim H5; clear H5; intros; + unfold disc in H7; apply H7. +exists (mkposreal _ H7); unfold included in |- *; intros; split. +assumption. +exists x1; split. +apply H4. +elim H5; intros; split. +assumption. +unfold disc in H8; simpl in H8; unfold disc in |- *; simpl in |- *; + unfold disc in H10; simpl in H10; + apply Rle_lt_trans with (Rabs (x2 - x0) + Rabs (x0 - x)). +replace (x2 - x) with (x2 - x0 + (x0 - x)); [ apply Rabs_triang | ring ]. +replace (x1 / 2) with (d + Rabs (x0 - x)); [ idtac | unfold d in |- *; ring ]. +do 2 rewrite <- (Rplus_comm (Rabs (x0 - x))); apply Rplus_lt_compat_l; + apply H8. +apply open_set_P6 with (fun _:R => False). +apply open_set_P4. +unfold eq_Dom in |- *; unfold included in |- *; intros; split. +intros; elim H4. +intros; unfold g in H4; elim H4; clear H4; intros H4 _; elim H3; apply H4. +elim (H0 _ H3); intros DF H4; unfold covering_finite in H4; elim H4; clear H4; + intros; unfold family_finite in H5; unfold domain_finite in H5; + unfold covering in H4; simpl in H4; simpl in H5; elim H5; + clear H5; intros l H5; unfold intersection_domain in H5; + cut + (forall x:R, + In x l -> + exists del : R + | 0 < del /\ + (forall z:R, Rabs (z - x) < del -> Rabs (f0 z - f0 x) < eps / 2) /\ + included (g x) (fun z:R => Rabs (z - x) < del / 2)). +intros; + assert + (H7 := + Rlist_P1 l + (fun x del:R => + 0 < del /\ + (forall z:R, Rabs (z - x) < del -> Rabs (f0 z - f0 x) < eps / 2) /\ + included (g x) (fun z:R => Rabs (z - x) < del / 2)) H6); + elim H7; clear H7; intros l' H7; elim H7; clear H7; + intros; pose (D := MinRlist l'); cut (0 < D / 2). +intro; exists (mkposreal _ H9); intros; assert (H13 := H4 _ H10); elim H13; + clear H13; intros xi H13; assert (H14 : In xi l). +unfold g in H13; decompose [and] H13; elim (H5 xi); intros; apply H14; split; + assumption. +elim (pos_Rl_P2 l xi); intros H15 _; elim (H15 H14); intros i H16; elim H16; + intros; apply Rle_lt_trans with (Rabs (f0 x - f0 xi) + Rabs (f0 xi - f0 y)). +replace (f0 x - f0 y) with (f0 x - f0 xi + (f0 xi - f0 y)); + [ apply Rabs_triang | ring ]. +rewrite (double_var eps); apply Rplus_lt_compat. +assert (H19 := H8 i H17); elim H19; clear H19; intros; rewrite <- H18 in H20; + elim H20; clear H20; intros; apply H20; unfold included in H21; + apply Rlt_trans with (pos_Rl l' i / 2). +apply H21. +elim H13; clear H13; intros; assumption. +unfold Rdiv in |- *; apply Rmult_lt_reg_l with 2. +prove_sup0. +rewrite Rmult_comm; rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; pattern (pos_Rl l' i) at 1 in |- *; rewrite <- Rplus_0_r; + rewrite double; apply Rplus_lt_compat_l; apply H19. +discrR. +assert (H19 := H8 i H17); elim H19; clear H19; intros; rewrite <- H18 in H20; + elim H20; clear H20; intros; rewrite <- Rabs_Ropp; + rewrite Ropp_minus_distr; apply H20; unfold included in H21; + elim H13; intros; assert (H24 := H21 x H22); + apply Rle_lt_trans with (Rabs (y - x) + Rabs (x - xi)). +replace (y - xi) with (y - x + (x - xi)); [ apply Rabs_triang | ring ]. +rewrite (double_var (pos_Rl l' i)); apply Rplus_lt_compat. +apply Rlt_le_trans with (D / 2). +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H12. +unfold Rdiv in |- *; do 2 rewrite <- (Rmult_comm (/ 2)); + apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; prove_sup0. +unfold D in |- *; apply MinRlist_P1; elim (pos_Rl_P2 l' (pos_Rl l' i)); + intros; apply H26; exists i; split; + [ rewrite <- H7; assumption | reflexivity ]. +assumption. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ unfold D in |- *; apply MinRlist_P2; intros; elim (pos_Rl_P2 l' y); intros; + elim (H10 H9); intros; elim H12; intros; rewrite H14; + rewrite <- H7 in H13; elim (H8 x H13); intros; + apply H15 + | apply Rinv_0_lt_compat; prove_sup0 ]. +intros; elim (H5 x); intros; elim (H8 H6); intros; + pose + (E := + fun zeta:R => + 0 < zeta <= M - m /\ + (forall z:R, Rabs (z - x) < zeta -> Rabs (f0 z - f0 x) < eps / 2)); + assert (H11 : bound E). +unfold bound in |- *; exists (M - m); unfold is_upper_bound in |- *; + unfold E in |- *; intros; elim H11; clear H11; intros H11 _; + elim H11; clear H11; intros _ H11; apply H11. +assert (H12 : exists x : R | E x). +assert (H13 := H _ H9); unfold continuity_pt in H13; + unfold continue_in in H13; unfold limit1_in in H13; + unfold limit_in in H13; simpl in H13; unfold R_dist in H13; + elim (H13 _ (H1 eps)); intros; elim H12; clear H12; + intros; exists (Rmin x0 (M - m)); unfold E in |- *; + intros; split. +split; + [ unfold Rmin in |- *; case (Rle_dec x0 (M - m)); intro; + [ apply H12 | apply Rlt_Rminus; apply Hyp ] + | apply Rmin_r ]. +intros; case (Req_dec x z); intro. +rewrite H16; unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; + apply (H1 eps). +apply H14; split; + [ unfold D_x, no_cond in |- *; split; [ trivial | assumption ] + | apply Rlt_le_trans with (Rmin x0 (M - m)); [ apply H15 | apply Rmin_l ] ]. +assert (H13 := completeness _ H11 H12); elim H13; clear H13; intros; + cut (0 < x0 <= M - m). +intro; elim H13; clear H13; intros; exists x0; split. +assumption. +split. +intros; cut ( exists alp : R | Rabs (z - x) < alp <= x0 /\ E alp). +intros; elim H16; intros; elim H17; clear H17; intros; unfold E in H18; + elim H18; intros; apply H20; elim H17; intros; assumption. +elim (classic ( exists alp : R | Rabs (z - x) < alp <= x0 /\ E alp)); intro. +assumption. +assert + (H17 := + not_ex_all_not _ (fun alp:R => Rabs (z - x) < alp <= x0 /\ E alp) H16); + unfold is_lub in p; elim p; intros; cut (is_upper_bound E (Rabs (z - x))). +intro; assert (H21 := H19 _ H20); + elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H15 H21)). +unfold is_upper_bound in |- *; intros; unfold is_upper_bound in H18; + assert (H21 := H18 _ H20); case (Rle_dec x1 (Rabs (z - x))); + intro. +assumption. +elim (H17 x1); split. +split; [ auto with real | assumption ]. +assumption. +unfold included, g in |- *; intros; elim H15; intros; elim H17; intros; + decompose [and] H18; cut (x0 = x2). +intro; rewrite H20; apply H22. +unfold E in p; eapply is_lub_u. +apply p. +apply H21. +elim H12; intros; unfold E in H13; elim H13; intros H14 _; elim H14; + intros H15 _; unfold is_lub in p; elim p; intros; + unfold is_upper_bound in H16; unfold is_upper_bound in H17; + split. +apply Rlt_le_trans with x1; [ assumption | apply (H16 _ H13) ]. +apply H17; intros; unfold E in H18; elim H18; intros; elim H19; intros; + assumption. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Rtrigo.v b/theories/Reals/Rtrigo.v index ae23fd8a6..60f07f610 100644 --- a/theories/Reals/Rtrigo.v +++ b/theories/Reals/Rtrigo.v @@ -8,1104 +8,1700 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. Require Export Rtrigo_fun. Require Export Rtrigo_def. Require Export Rtrigo_alt. Require Export Cos_rel. Require Export Cos_plus. -Require ZArith_base. -Require Zcomplements. -Require Classical_Prop. -V7only [Import nat_scope. Import Z_scope. Import R_scope.]. +Require Import ZArith_base. +Require Import Zcomplements. +Require Import Classical_Prop. Open Local Scope nat_scope. Open Local Scope R_scope. (** sin_PI2 is the only remaining axiom **) -Axiom sin_PI2 : ``(sin (PI/2))==1``. +Axiom sin_PI2 : sin (PI / 2) = 1. (**********) -Lemma PI_neq0 : ~``PI==0``. -Red; Intro; Assert H0 := PI_RGT_0; Rewrite H in H0; Elim (Rlt_antirefl ? H0). +Lemma PI_neq0 : PI <> 0. +red in |- *; intro; assert (H0 := PI_RGT_0); rewrite H in H0; + elim (Rlt_irrefl _ H0). Qed. (**********) -Lemma cos_minus : (x,y:R) ``(cos (x-y))==(cos x)*(cos y)+(sin x)*(sin y)``. -Intros; Unfold Rminus; Rewrite cos_plus. -Rewrite <- cos_sym; Rewrite sin_antisym; Ring. +Lemma cos_minus : forall x y:R, cos (x - y) = cos x * cos y + sin x * sin y. +intros; unfold Rminus in |- *; rewrite cos_plus. +rewrite <- cos_sym; rewrite sin_antisym; ring. Qed. (**********) -Lemma sin2_cos2 : (x:R) ``(Rsqr (sin x)) + (Rsqr (cos x))==1``. -Intro; Unfold Rsqr; Rewrite Rplus_sym; Rewrite <- (cos_minus x x); Unfold Rminus; Rewrite Rplus_Ropp_r; Apply cos_0. +Lemma sin2_cos2 : forall x:R, Rsqr (sin x) + Rsqr (cos x) = 1. +intro; unfold Rsqr in |- *; rewrite Rplus_comm; rewrite <- (cos_minus x x); + unfold Rminus in |- *; rewrite Rplus_opp_r; apply cos_0. Qed. -Lemma cos2 : (x:R) ``(Rsqr (cos x))==1-(Rsqr (sin x))``. -Intro x; Generalize (sin2_cos2 x); Intro H1; Rewrite <- H1; Unfold Rminus; Rewrite <- (Rplus_sym (Rsqr (cos x))); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Symmetry; Apply Rplus_Or. +Lemma cos2 : forall x:R, Rsqr (cos x) = 1 - Rsqr (sin x). +intro x; generalize (sin2_cos2 x); intro H1; rewrite <- H1; + unfold Rminus in |- *; rewrite <- (Rplus_comm (Rsqr (cos x))); + rewrite Rplus_assoc; rewrite Rplus_opp_r; symmetry in |- *; + apply Rplus_0_r. Qed. (**********) -Lemma cos_PI2 : ``(cos (PI/2))==0``. -Apply Rsqr_eq_0; Rewrite cos2; Rewrite sin_PI2; Rewrite Rsqr_1; Unfold Rminus; Apply Rplus_Ropp_r. +Lemma cos_PI2 : cos (PI / 2) = 0. +apply Rsqr_eq_0; rewrite cos2; rewrite sin_PI2; rewrite Rsqr_1; + unfold Rminus in |- *; apply Rplus_opp_r. Qed. (**********) -Lemma cos_PI : ``(cos PI)==-1``. -Replace ``PI`` with ``PI/2+PI/2``. -Rewrite cos_plus. -Rewrite sin_PI2; Rewrite cos_PI2. -Ring. -Symmetry; Apply double_var. +Lemma cos_PI : cos PI = -1. +replace PI with (PI / 2 + PI / 2). +rewrite cos_plus. +rewrite sin_PI2; rewrite cos_PI2. +ring. +symmetry in |- *; apply double_var. Qed. -Lemma sin_PI : ``(sin PI)==0``. -Assert H := (sin2_cos2 PI). -Rewrite cos_PI in H. -Rewrite <- Rsqr_neg in H. -Rewrite Rsqr_1 in H. -Cut (Rsqr (sin PI))==R0. -Intro; Apply (Rsqr_eq_0 ? H0). -Apply r_Rplus_plus with R1. -Rewrite Rplus_Or; Rewrite Rplus_sym; Exact H. +Lemma sin_PI : sin PI = 0. +assert (H := sin2_cos2 PI). +rewrite cos_PI in H. +rewrite <- Rsqr_neg in H. +rewrite Rsqr_1 in H. +cut (Rsqr (sin PI) = 0). +intro; apply (Rsqr_eq_0 _ H0). +apply Rplus_eq_reg_l with 1. +rewrite Rplus_0_r; rewrite Rplus_comm; exact H. Qed. (**********) -Lemma neg_cos : (x:R) ``(cos (x+PI))==-(cos x)``. -Intro x; Rewrite -> cos_plus; Rewrite -> sin_PI; Rewrite -> cos_PI; Ring. +Lemma neg_cos : forall x:R, cos (x + PI) = - cos x. +intro x; rewrite cos_plus; rewrite sin_PI; rewrite cos_PI; ring. Qed. (**********) -Lemma sin_cos : (x:R) ``(sin x)==-(cos (PI/2+x))``. -Intro x; Rewrite -> cos_plus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring. +Lemma sin_cos : forall x:R, sin x = - cos (PI / 2 + x). +intro x; rewrite cos_plus; rewrite sin_PI2; rewrite cos_PI2; ring. Qed. (**********) -Lemma sin_plus : (x,y:R) ``(sin (x+y))==(sin x)*(cos y)+(cos x)*(sin y)``. -Intros. -Rewrite (sin_cos ``x+y``). -Replace ``PI/2+(x+y)`` with ``(PI/2+x)+y``; [Rewrite cos_plus | Ring]. -Rewrite (sin_cos ``PI/2+x``). -Replace ``PI/2+(PI/2+x)`` with ``x+PI``. -Rewrite neg_cos. -Replace (cos ``PI/2+x``) with ``-(sin x)``. -Ring. -Rewrite sin_cos; Rewrite Ropp_Ropp; Reflexivity. -Pattern 1 PI; Rewrite (double_var PI); Ring. +Lemma sin_plus : forall x y:R, sin (x + y) = sin x * cos y + cos x * sin y. +intros. +rewrite (sin_cos (x + y)). +replace (PI / 2 + (x + y)) with (PI / 2 + x + y); [ rewrite cos_plus | ring ]. +rewrite (sin_cos (PI / 2 + x)). +replace (PI / 2 + (PI / 2 + x)) with (x + PI). +rewrite neg_cos. +replace (cos (PI / 2 + x)) with (- sin x). +ring. +rewrite sin_cos; rewrite Ropp_involutive; reflexivity. +pattern PI at 1 in |- *; rewrite (double_var PI); ring. Qed. -Lemma sin_minus : (x,y:R) ``(sin (x-y))==(sin x)*(cos y)-(cos x)*(sin y)``. -Intros; Unfold Rminus; Rewrite sin_plus. -Rewrite <- cos_sym; Rewrite sin_antisym; Ring. +Lemma sin_minus : forall x y:R, sin (x - y) = sin x * cos y - cos x * sin y. +intros; unfold Rminus in |- *; rewrite sin_plus. +rewrite <- cos_sym; rewrite sin_antisym; ring. Qed. (**********) -Definition tan [x:R] : R := ``(sin x)/(cos x)``. - -Lemma tan_plus : (x,y:R) ~``(cos x)==0`` -> ~``(cos y)==0`` -> ~``(cos (x+y))==0`` -> ~``1-(tan x)*(tan y)==0`` -> ``(tan (x+y))==((tan x)+(tan y))/(1-(tan x)*(tan y))``. -Intros; Unfold tan; Rewrite sin_plus; Rewrite cos_plus; Unfold Rdiv; Replace ``((cos x)*(cos y)-(sin x)*(sin y))`` with ``((cos x)*(cos y))*(1-(sin x)*/(cos x)*((sin y)*/(cos y)))``. -Rewrite Rinv_Rmult. -Repeat Rewrite <- Rmult_assoc; Replace ``((sin x)*(cos y)+(cos x)*(sin y))*/((cos x)*(cos y))`` with ``((sin x)*/(cos x)+(sin y)*/(cos y))``. -Reflexivity. -Rewrite Rmult_Rplus_distrl; Rewrite Rinv_Rmult. -Repeat Rewrite Rmult_assoc; Repeat Rewrite (Rmult_sym ``(sin x)``); Repeat Rewrite <- Rmult_assoc. -Repeat Rewrite Rinv_r_simpl_m; [Reflexivity | Assumption | Assumption]. -Assumption. -Assumption. -Apply prod_neq_R0; Assumption. -Assumption. -Unfold Rminus; Rewrite Rmult_Rplus_distr; Rewrite Rmult_1r; Apply Rplus_plus_r; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym ``(sin x)``); Rewrite (Rmult_sym ``(cos y)``); Rewrite <- Ropp_mul3; Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite (Rmult_sym (sin x)); Rewrite <- Ropp_mul3; Repeat Rewrite Rmult_assoc; Apply Rmult_mult_r; Rewrite (Rmult_sym ``/(cos y)``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Apply Rmult_1r. -Assumption. -Assumption. +Definition tan (x:R) : R := sin x / cos x. + +Lemma tan_plus : + forall x y:R, + cos x <> 0 -> + cos y <> 0 -> + cos (x + y) <> 0 -> + 1 - tan x * tan y <> 0 -> + tan (x + y) = (tan x + tan y) / (1 - tan x * tan y). +intros; unfold tan in |- *; rewrite sin_plus; rewrite cos_plus; + unfold Rdiv in |- *; + replace (cos x * cos y - sin x * sin y) with + (cos x * cos y * (1 - sin x * / cos x * (sin y * / cos y))). +rewrite Rinv_mult_distr. +repeat rewrite <- Rmult_assoc; + replace ((sin x * cos y + cos x * sin y) * / (cos x * cos y)) with + (sin x * / cos x + sin y * / cos y). +reflexivity. +rewrite Rmult_plus_distr_r; rewrite Rinv_mult_distr. +repeat rewrite Rmult_assoc; repeat rewrite (Rmult_comm (sin x)); + repeat rewrite <- Rmult_assoc. +repeat rewrite Rinv_r_simpl_m; [ reflexivity | assumption | assumption ]. +assumption. +assumption. +apply prod_neq_R0; assumption. +assumption. +unfold Rminus in |- *; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r; + apply Rplus_eq_compat_l; repeat rewrite Rmult_assoc; + rewrite (Rmult_comm (sin x)); rewrite (Rmult_comm (cos y)); + rewrite <- Ropp_mult_distr_r_reverse; repeat rewrite <- Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite (Rmult_comm (sin x)); + rewrite <- Ropp_mult_distr_r_reverse; repeat rewrite Rmult_assoc; + apply Rmult_eq_compat_l; rewrite (Rmult_comm (/ cos y)); + rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +apply Rmult_1_r. +assumption. +assumption. Qed. (*******************************************************) (* Some properties of cos, sin and tan *) (*******************************************************) -Lemma sin2 : (x:R) ``(Rsqr (sin x))==1-(Rsqr (cos x))``. -Intro x; Generalize (cos2 x); Intro H1; Rewrite -> H1. -Unfold Rminus; Rewrite Ropp_distr1; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Ol; Symmetry; Apply Ropp_Ropp. +Lemma sin2 : forall x:R, Rsqr (sin x) = 1 - Rsqr (cos x). +intro x; generalize (cos2 x); intro H1; rewrite H1. +unfold Rminus in |- *; rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_l; symmetry in |- *; + apply Ropp_involutive. Qed. -Lemma sin_2a : (x:R) ``(sin (2*x))==2*(sin x)*(cos x)``. -Intro x; Rewrite double; Rewrite sin_plus. -Rewrite <- (Rmult_sym (sin x)); Symmetry; Rewrite Rmult_assoc; Apply double. +Lemma sin_2a : forall x:R, sin (2 * x) = 2 * sin x * cos x. +intro x; rewrite double; rewrite sin_plus. +rewrite <- (Rmult_comm (sin x)); symmetry in |- *; rewrite Rmult_assoc; + apply double. Qed. -Lemma cos_2a : (x:R) ``(cos (2*x))==(cos x)*(cos x)-(sin x)*(sin x)``. -Intro x; Rewrite double; Apply cos_plus. +Lemma cos_2a : forall x:R, cos (2 * x) = cos x * cos x - sin x * sin x. +intro x; rewrite double; apply cos_plus. Qed. -Lemma cos_2a_cos : (x:R) ``(cos (2*x))==2*(cos x)*(cos x)-1``. -Intro x; Rewrite double; Unfold Rminus; Rewrite Rmult_assoc; Rewrite cos_plus; Generalize (sin2_cos2 x); Rewrite double; Intro H1; Rewrite <- H1; SqRing. +Lemma cos_2a_cos : forall x:R, cos (2 * x) = 2 * cos x * cos x - 1. +intro x; rewrite double; unfold Rminus in |- *; rewrite Rmult_assoc; + rewrite cos_plus; generalize (sin2_cos2 x); rewrite double; + intro H1; rewrite <- H1; ring_Rsqr. Qed. -Lemma cos_2a_sin : (x:R) ``(cos (2*x))==1-2*(sin x)*(sin x)``. -Intro x; Rewrite Rmult_assoc; Unfold Rminus; Repeat Rewrite double. -Generalize (sin2_cos2 x); Intro H1; Rewrite <- H1; Rewrite cos_plus; SqRing. +Lemma cos_2a_sin : forall x:R, cos (2 * x) = 1 - 2 * sin x * sin x. +intro x; rewrite Rmult_assoc; unfold Rminus in |- *; repeat rewrite double. +generalize (sin2_cos2 x); intro H1; rewrite <- H1; rewrite cos_plus; + ring_Rsqr. Qed. -Lemma tan_2a : (x:R) ~``(cos x)==0`` -> ~``(cos (2*x))==0`` -> ~``1-(tan x)*(tan x)==0`` ->``(tan (2*x))==(2*(tan x))/(1-(tan x)*(tan x))``. -Repeat Rewrite double; Intros; Repeat Rewrite double; Rewrite double in H0; Apply tan_plus; Assumption. +Lemma tan_2a : + forall x:R, + cos x <> 0 -> + cos (2 * x) <> 0 -> + 1 - tan x * tan x <> 0 -> tan (2 * x) = 2 * tan x / (1 - tan x * tan x). +repeat rewrite double; intros; repeat rewrite double; rewrite double in H0; + apply tan_plus; assumption. Qed. -Lemma sin_neg : (x:R) ``(sin (-x))==-(sin x)``. -Apply sin_antisym. +Lemma sin_neg : forall x:R, sin (- x) = - sin x. +apply sin_antisym. Qed. -Lemma cos_neg : (x:R) ``(cos (-x))==(cos x)``. -Intro; Symmetry; Apply cos_sym. +Lemma cos_neg : forall x:R, cos (- x) = cos x. +intro; symmetry in |- *; apply cos_sym. Qed. -Lemma tan_0 : ``(tan 0)==0``. -Unfold tan; Rewrite -> sin_0; Rewrite -> cos_0. -Unfold Rdiv; Apply Rmult_Ol. +Lemma tan_0 : tan 0 = 0. +unfold tan in |- *; rewrite sin_0; rewrite cos_0. +unfold Rdiv in |- *; apply Rmult_0_l. Qed. -Lemma tan_neg : (x:R) ``(tan (-x))==-(tan x)``. -Intros x; Unfold tan; Rewrite sin_neg; Rewrite cos_neg; Unfold Rdiv. -Apply Ropp_mul1. +Lemma tan_neg : forall x:R, tan (- x) = - tan x. +intros x; unfold tan in |- *; rewrite sin_neg; rewrite cos_neg; + unfold Rdiv in |- *. +apply Ropp_mult_distr_l_reverse. Qed. -Lemma tan_minus : (x,y:R) ~``(cos x)==0`` -> ~``(cos y)==0`` -> ~``(cos (x-y))==0`` -> ~``1+(tan x)*(tan y)==0`` -> ``(tan (x-y))==((tan x)-(tan y))/(1+(tan x)*(tan y))``. -Intros; Unfold Rminus; Rewrite tan_plus. -Rewrite tan_neg; Unfold Rminus; Rewrite <- Ropp_mul1; Rewrite Ropp_mul2; Reflexivity. -Assumption. -Rewrite cos_neg; Assumption. -Assumption. -Rewrite tan_neg; Unfold Rminus; Rewrite <- Ropp_mul1; Rewrite Ropp_mul2; Assumption. +Lemma tan_minus : + forall x y:R, + cos x <> 0 -> + cos y <> 0 -> + cos (x - y) <> 0 -> + 1 + tan x * tan y <> 0 -> + tan (x - y) = (tan x - tan y) / (1 + tan x * tan y). +intros; unfold Rminus in |- *; rewrite tan_plus. +rewrite tan_neg; unfold Rminus in |- *; rewrite <- Ropp_mult_distr_l_reverse; + rewrite Rmult_opp_opp; reflexivity. +assumption. +rewrite cos_neg; assumption. +assumption. +rewrite tan_neg; unfold Rminus in |- *; rewrite <- Ropp_mult_distr_l_reverse; + rewrite Rmult_opp_opp; assumption. Qed. -Lemma cos_3PI2 : ``(cos (3*(PI/2)))==0``. -Replace ``3*(PI/2)`` with ``PI+(PI/2)``. -Rewrite -> cos_plus; Rewrite -> sin_PI; Rewrite -> cos_PI2; Ring. -Pattern 1 PI; Rewrite (double_var PI). -Ring. +Lemma cos_3PI2 : cos (3 * (PI / 2)) = 0. +replace (3 * (PI / 2)) with (PI + PI / 2). +rewrite cos_plus; rewrite sin_PI; rewrite cos_PI2; ring. +pattern PI at 1 in |- *; rewrite (double_var PI). +ring. Qed. -Lemma sin_2PI : ``(sin (2*PI))==0``. -Rewrite -> sin_2a; Rewrite -> sin_PI; Ring. +Lemma sin_2PI : sin (2 * PI) = 0. +rewrite sin_2a; rewrite sin_PI; ring. Qed. -Lemma cos_2PI : ``(cos (2*PI))==1``. -Rewrite -> cos_2a; Rewrite -> sin_PI; Rewrite -> cos_PI; Ring. +Lemma cos_2PI : cos (2 * PI) = 1. +rewrite cos_2a; rewrite sin_PI; rewrite cos_PI; ring. Qed. -Lemma neg_sin : (x:R) ``(sin (x+PI))==-(sin x)``. -Intro x; Rewrite -> sin_plus; Rewrite -> sin_PI; Rewrite -> cos_PI; Ring. +Lemma neg_sin : forall x:R, sin (x + PI) = - sin x. +intro x; rewrite sin_plus; rewrite sin_PI; rewrite cos_PI; ring. Qed. -Lemma sin_PI_x : (x:R) ``(sin (PI-x))==(sin x)``. -Intro x; Rewrite -> sin_minus; Rewrite -> sin_PI; Rewrite -> cos_PI; Rewrite Rmult_Ol; Unfold Rminus; Rewrite Rplus_Ol; Rewrite Ropp_mul1; Rewrite Ropp_Ropp; Apply Rmult_1l. +Lemma sin_PI_x : forall x:R, sin (PI - x) = sin x. +intro x; rewrite sin_minus; rewrite sin_PI; rewrite cos_PI; rewrite Rmult_0_l; + unfold Rminus in |- *; rewrite Rplus_0_l; rewrite Ropp_mult_distr_l_reverse; + rewrite Ropp_involutive; apply Rmult_1_l. Qed. -Lemma sin_period : (x:R)(k:nat) ``(sin (x+2*(INR k)*PI))==(sin x)``. -Intros x k; Induction k. -Cut ``x+2*(INR O)*PI==x``; [Intro; Rewrite H; Reflexivity | Ring]. -Replace ``x+2*(INR (S k))*PI`` with ``(x+2*(INR k)*PI)+(2*PI)``; [Rewrite -> sin_plus; Rewrite -> sin_2PI; Rewrite -> cos_2PI; Ring; Apply Hreck | Rewrite -> S_INR; Ring]. +Lemma sin_period : forall (x:R) (k:nat), sin (x + 2 * INR k * PI) = sin x. +intros x k; induction k as [| k Hreck]. +cut (x + 2 * INR 0 * PI = x); [ intro; rewrite H; reflexivity | ring ]. +replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI); + [ rewrite sin_plus; rewrite sin_2PI; rewrite cos_2PI; ring; apply Hreck + | rewrite S_INR; ring ]. Qed. -Lemma cos_period : (x:R)(k:nat) ``(cos (x+2*(INR k)*PI))==(cos x)``. -Intros x k; Induction k. -Cut ``x+2*(INR O)*PI==x``; [Intro; Rewrite H; Reflexivity | Ring]. -Replace ``x+2*(INR (S k))*PI`` with ``(x+2*(INR k)*PI)+(2*PI)``; [Rewrite -> cos_plus; Rewrite -> sin_2PI; Rewrite -> cos_2PI; Ring; Apply Hreck | Rewrite -> S_INR; Ring]. +Lemma cos_period : forall (x:R) (k:nat), cos (x + 2 * INR k * PI) = cos x. +intros x k; induction k as [| k Hreck]. +cut (x + 2 * INR 0 * PI = x); [ intro; rewrite H; reflexivity | ring ]. +replace (x + 2 * INR (S k) * PI) with (x + 2 * INR k * PI + 2 * PI); + [ rewrite cos_plus; rewrite sin_2PI; rewrite cos_2PI; ring; apply Hreck + | rewrite S_INR; ring ]. Qed. -Lemma sin_shift : (x:R) ``(sin (PI/2-x))==(cos x)``. -Intro x; Rewrite -> sin_minus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring. +Lemma sin_shift : forall x:R, sin (PI / 2 - x) = cos x. +intro x; rewrite sin_minus; rewrite sin_PI2; rewrite cos_PI2; ring. Qed. -Lemma cos_shift : (x:R) ``(cos (PI/2-x))==(sin x)``. -Intro x; Rewrite -> cos_minus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring. +Lemma cos_shift : forall x:R, cos (PI / 2 - x) = sin x. +intro x; rewrite cos_minus; rewrite sin_PI2; rewrite cos_PI2; ring. Qed. -Lemma cos_sin : (x:R) ``(cos x)==(sin (PI/2+x))``. -Intro x; Rewrite -> sin_plus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring. +Lemma cos_sin : forall x:R, cos x = sin (PI / 2 + x). +intro x; rewrite sin_plus; rewrite sin_PI2; rewrite cos_PI2; ring. Qed. -Lemma PI2_RGT_0 : ``0<PI/2``. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup]. +Lemma PI2_RGT_0 : 0 < PI / 2. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup ]. Qed. -Lemma SIN_bound : (x:R) ``-1<=(sin x)<=1``. -Intro; Case (total_order_Rle ``-1`` (sin x)); Intro. -Case (total_order_Rle (sin x) ``1``); Intro. -Split; Assumption. -Cut ``1<(sin x)``. -Intro; Generalize (Rsqr_incrst_1 ``1`` (sin x) H (Rlt_le ``0`` ``1`` Rlt_R0_R1) (Rlt_le ``0`` (sin x) (Rlt_trans ``0`` ``1`` (sin x) Rlt_R0_R1 H))); Rewrite Rsqr_1; Intro; Rewrite sin2 in H0; Unfold Rminus in H0; Generalize (Rlt_compatibility ``-1`` ``1`` ``1+ -(Rsqr (cos x))`` H0); Repeat Rewrite <- Rplus_assoc; Repeat Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Intro; Rewrite <- Ropp_O in H1; Generalize (Rlt_Ropp ``-0`` ``-(Rsqr (cos x))`` H1); Repeat Rewrite Ropp_Ropp; Intro; Generalize (pos_Rsqr (cos x)); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` (Rsqr (cos x)) ``0`` H3 H2)). -Auto with real. -Cut ``(sin x)< -1``. -Intro; Generalize (Rlt_Ropp (sin x) ``-1`` H); Rewrite Ropp_Ropp; Clear H; Intro; Generalize (Rsqr_incrst_1 ``1`` ``-(sin x)`` H (Rlt_le ``0`` ``1`` Rlt_R0_R1) (Rlt_le ``0`` ``-(sin x)`` (Rlt_trans ``0`` ``1`` ``-(sin x)`` Rlt_R0_R1 H))); Rewrite Rsqr_1; Intro; Rewrite <- Rsqr_neg in H0; Rewrite sin2 in H0; Unfold Rminus in H0; Generalize (Rlt_compatibility ``-1`` ``1`` ``1+ -(Rsqr (cos x))`` H0); Repeat Rewrite <- Rplus_assoc; Repeat Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Intro; Rewrite <- Ropp_O in H1; Generalize (Rlt_Ropp ``-0`` ``-(Rsqr (cos x))`` H1); Repeat Rewrite Ropp_Ropp; Intro; Generalize (pos_Rsqr (cos x)); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` (Rsqr (cos x)) ``0`` H3 H2)). -Auto with real. -Qed. - -Lemma COS_bound : (x:R) ``-1<=(cos x)<=1``. -Intro; Rewrite <- sin_shift; Apply SIN_bound. -Qed. - -Lemma cos_sin_0 : (x:R) ~(``(cos x)==0``/\``(sin x)==0``). -Intro; Red; Intro; Elim H; Intros; Generalize (sin2_cos2 x); Intro; Rewrite H0 in H2; Rewrite H1 in H2; Repeat Rewrite Rsqr_O in H2; Rewrite Rplus_Or in H2; Generalize Rlt_R0_R1; Intro; Rewrite <- H2 in H3; Elim (Rlt_antirefl ``0`` H3). +Lemma SIN_bound : forall x:R, -1 <= sin x <= 1. +intro; case (Rle_dec (-1) (sin x)); intro. +case (Rle_dec (sin x) 1); intro. +split; assumption. +cut (1 < sin x). +intro; + generalize + (Rsqr_incrst_1 1 (sin x) H (Rlt_le 0 1 Rlt_0_1) + (Rlt_le 0 (sin x) (Rlt_trans 0 1 (sin x) Rlt_0_1 H))); + rewrite Rsqr_1; intro; rewrite sin2 in H0; unfold Rminus in H0; + generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0); + repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l; + rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1; + generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1); + repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x)); + intro; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rsqr (cos x)) 0 H3 H2)). +auto with real. +cut (sin x < -1). +intro; generalize (Ropp_lt_gt_contravar (sin x) (-1) H); + rewrite Ropp_involutive; clear H; intro; + generalize + (Rsqr_incrst_1 1 (- sin x) H (Rlt_le 0 1 Rlt_0_1) + (Rlt_le 0 (- sin x) (Rlt_trans 0 1 (- sin x) Rlt_0_1 H))); + rewrite Rsqr_1; intro; rewrite <- Rsqr_neg in H0; + rewrite sin2 in H0; unfold Rminus in H0; + generalize (Rplus_lt_compat_l (-1) 1 (1 + - Rsqr (cos x)) H0); + repeat rewrite <- Rplus_assoc; repeat rewrite Rplus_opp_l; + rewrite Rplus_0_l; intro; rewrite <- Ropp_0 in H1; + generalize (Ropp_lt_gt_contravar (-0) (- Rsqr (cos x)) H1); + repeat rewrite Ropp_involutive; intro; generalize (Rle_0_sqr (cos x)); + intro; elim (Rlt_irrefl 0 (Rle_lt_trans 0 (Rsqr (cos x)) 0 H3 H2)). +auto with real. +Qed. + +Lemma COS_bound : forall x:R, -1 <= cos x <= 1. +intro; rewrite <- sin_shift; apply SIN_bound. +Qed. + +Lemma cos_sin_0 : forall x:R, ~ (cos x = 0 /\ sin x = 0). +intro; red in |- *; intro; elim H; intros; generalize (sin2_cos2 x); intro; + rewrite H0 in H2; rewrite H1 in H2; repeat rewrite Rsqr_0 in H2; + rewrite Rplus_0_r in H2; generalize Rlt_0_1; intro; + rewrite <- H2 in H3; elim (Rlt_irrefl 0 H3). Qed. -Lemma cos_sin_0_var : (x:R) ~``(cos x)==0``\/~``(sin x)==0``. -Intro; Apply not_and_or; Apply cos_sin_0. +Lemma cos_sin_0_var : forall x:R, cos x <> 0 \/ sin x <> 0. +intro; apply not_and_or; apply cos_sin_0. Qed. (*****************************************************************) (* Using series definitions of cos and sin *) (*****************************************************************) -Definition sin_lb [a:R] : R := (sin_approx a (3)). -Definition sin_ub [a:R] : R := (sin_approx a (4)). -Definition cos_lb [a:R] : R := (cos_approx a (3)). -Definition cos_ub [a:R] : R := (cos_approx a (4)). - -Lemma sin_lb_gt_0 : (a:R) ``0<a``->``a<=PI/2``->``0<(sin_lb a)``. -Intros. -Unfold sin_lb; Unfold sin_approx; Unfold sin_term. -Pose Un := [i:nat]``(pow a (plus (mult (S (S O)) i) (S O)))/(INR (fact (plus (mult (S (S O)) i) (S O))))``. -Replace (sum_f_R0 [i:nat] ``(pow ( -1) i)*(pow a (plus (mult (S (S O)) i) (S O)))/(INR (fact (plus (mult (S (S O)) i) (S O))))`` (S (S (S O)))) with (sum_f_R0 [i:nat]``(pow (-1) i)*(Un i)`` (3)); [Idtac | Apply sum_eq; Intros; Unfold Un; Reflexivity]. -Cut (n:nat)``(Un (S n))<(Un n)``. -Intro; Simpl. -Repeat Rewrite Rmult_1l; Repeat Rewrite Rmult_1r; Replace ``-1*(Un (S O))`` with ``-(Un (S O))``; [Idtac | Ring]; Replace ``-1* -1*(Un (S (S O)))`` with ``(Un (S (S O)))``; [Idtac | Ring]; Replace ``-1*( -1* -1)*(Un (S (S (S O))))`` with ``-(Un (S (S (S O))))``; [Idtac | Ring]; Replace ``(Un O)+ -(Un (S O))+(Un (S (S O)))+ -(Un (S (S (S O))))`` with ``((Un O)-(Un (S O)))+((Un (S (S O)))-(Un (S (S (S O)))))``; [Idtac | Ring]. -Apply gt0_plus_gt0_is_gt0. -Unfold Rminus; Apply Rlt_anti_compatibility with (Un (S O)); Rewrite Rplus_Or; Rewrite (Rplus_sym (Un (S O))); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply H1. -Unfold Rminus; Apply Rlt_anti_compatibility with (Un (S (S (S O)))); Rewrite Rplus_Or; Rewrite (Rplus_sym (Un (S (S (S O))))); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply H1. -Intro; Unfold Un. -Cut (plus (mult (2) (S n)) (S O)) = (plus (plus (mult (2) n) (S O)) (2)). -Intro; Rewrite H1. -Rewrite pow_add; Unfold Rdiv; Rewrite Rmult_assoc; Apply Rlt_monotony. -Apply pow_lt; Assumption. -Rewrite <- H1; Apply Rlt_monotony_contra with (INR (fact (plus (mult (S (S O)) n) (S O)))). -Apply lt_INR_0; Apply neq_O_lt. -Assert H2 := (fact_neq_0 (plus (mult (2) n) (1))). -Red; Intro; Elim H2; Symmetry; Assumption. -Rewrite <- Rinv_r_sym. -Apply Rlt_monotony_contra with (INR (fact (plus (mult (S (S O)) (S n)) (S O)))). -Apply lt_INR_0; Apply neq_O_lt. -Assert H2 := (fact_neq_0 (plus (mult (2) (S n)) (1))). -Red; Intro; Elim H2; Symmetry; Assumption. -Rewrite (Rmult_sym (INR (fact (plus (mult (S (S O)) (S n)) (S O))))); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Do 2 Rewrite Rmult_1r; Apply Rle_lt_trans with ``(INR (fact (plus (mult (S (S O)) n) (S O))))*4``. -Apply Rle_monotony. -Replace R0 with (INR O); [Idtac | Reflexivity]; Apply le_INR; Apply le_O_n. -Simpl; Rewrite Rmult_1r; Replace ``4`` with ``(Rsqr 2)``; [Idtac | SqRing]; Replace ``a*a`` with (Rsqr a); [Idtac | Reflexivity]; Apply Rsqr_incr_1. -Apply Rle_trans with ``PI/2``; [Assumption | Unfold Rdiv; Apply Rle_monotony_contra with ``2``; [ Sup0 | Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m; [Replace ``2*2`` with ``4``; [Apply PI_4 | Ring] | DiscrR]]]. -Left; Assumption. -Left; Sup0. -Rewrite H1; Replace (plus (plus (mult (S (S O)) n) (S O)) (S (S O))) with (S (S (plus (mult (S (S O)) n) (S O)))). -Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR. -Repeat Rewrite <- Rmult_assoc. -Rewrite <- (Rmult_sym (INR (fact (plus (mult (S (S O)) n) (S O))))). -Rewrite Rmult_assoc. -Apply Rlt_monotony. -Apply lt_INR_0; Apply neq_O_lt. -Assert H2 := (fact_neq_0 (plus (mult (2) n) (1))). -Red; Intro; Elim H2; Symmetry; Assumption. -Do 2 Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Pose x := (INR n); Unfold INR. -Replace ``(2*x+1+1+1)*(2*x+1+1)`` with ``4*x*x+10*x+6``; [Idtac | Ring]. -Apply Rlt_anti_compatibility with ``-4``; Rewrite Rplus_Ropp_l; Replace ``-4+(4*x*x+10*x+6)`` with ``(4*x*x+10*x)+2``; [Idtac | Ring]. -Apply ge0_plus_gt0_is_gt0. -Cut ``0<=x``. -Intro; Apply ge0_plus_ge0_is_ge0; Repeat Apply Rmult_le_pos; Assumption Orelse Left; Sup. -Unfold x; Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. -Sup0. -Apply INR_eq; Do 2 Rewrite S_INR; Do 3 Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_eq; Do 3 Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Qed. - -Lemma SIN : (a:R) ``0<=a`` -> ``a<=PI`` -> ``(sin_lb a)<=(sin a)<=(sin_ub a)``. -Intros; Unfold sin_lb sin_ub; Apply (sin_bound a (S O) H H0). -Qed. - -Lemma COS : (a:R) ``-PI/2<=a`` -> ``a<=PI/2`` -> ``(cos_lb a)<=(cos a)<=(cos_ub a)``. -Intros; Unfold cos_lb cos_ub; Apply (cos_bound a (S O) H H0). +Definition sin_lb (a:R) : R := sin_approx a 3. +Definition sin_ub (a:R) : R := sin_approx a 4. +Definition cos_lb (a:R) : R := cos_approx a 3. +Definition cos_ub (a:R) : R := cos_approx a 4. + +Lemma sin_lb_gt_0 : forall a:R, 0 < a -> a <= PI / 2 -> 0 < sin_lb a. +intros. +unfold sin_lb in |- *; unfold sin_approx in |- *; unfold sin_term in |- *. +pose (Un := fun i:nat => a ^ (2 * i + 1) / INR (fact (2 * i + 1))). +replace + (sum_f_R0 + (fun i:nat => (-1) ^ i * (a ^ (2 * i + 1) / INR (fact (2 * i + 1)))) 3) + with (sum_f_R0 (fun i:nat => (-1) ^ i * Un i) 3); + [ idtac | apply sum_eq; intros; unfold Un in |- *; reflexivity ]. +cut (forall n:nat, Un (S n) < Un n). +intro; simpl in |- *. +repeat rewrite Rmult_1_l; repeat rewrite Rmult_1_r; + replace (-1 * Un 1%nat) with (- Un 1%nat); [ idtac | ring ]; + replace (-1 * -1 * Un 2%nat) with (Un 2%nat); [ idtac | ring ]; + replace (-1 * (-1 * -1) * Un 3%nat) with (- Un 3%nat); + [ idtac | ring ]; + replace (Un 0%nat + - Un 1%nat + Un 2%nat + - Un 3%nat) with + (Un 0%nat - Un 1%nat + (Un 2%nat - Un 3%nat)); [ idtac | ring ]. +apply Rplus_lt_0_compat. +unfold Rminus in |- *; apply Rplus_lt_reg_r with (Un 1%nat); + rewrite Rplus_0_r; rewrite (Rplus_comm (Un 1%nat)); + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; + apply H1. +unfold Rminus in |- *; apply Rplus_lt_reg_r with (Un 3%nat); + rewrite Rplus_0_r; rewrite (Rplus_comm (Un 3%nat)); + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; + apply H1. +intro; unfold Un in |- *. +cut ((2 * S n + 1)%nat = (2 * n + 1 + 2)%nat). +intro; rewrite H1. +rewrite pow_add; unfold Rdiv in |- *; rewrite Rmult_assoc; + apply Rmult_lt_compat_l. +apply pow_lt; assumption. +rewrite <- H1; apply Rmult_lt_reg_l with (INR (fact (2 * n + 1))). +apply lt_INR_0; apply neq_O_lt. +assert (H2 := fact_neq_0 (2 * n + 1)). +red in |- *; intro; elim H2; symmetry in |- *; assumption. +rewrite <- Rinv_r_sym. +apply Rmult_lt_reg_l with (INR (fact (2 * S n + 1))). +apply lt_INR_0; apply neq_O_lt. +assert (H2 := fact_neq_0 (2 * S n + 1)). +red in |- *; intro; elim H2; symmetry in |- *; assumption. +rewrite (Rmult_comm (INR (fact (2 * S n + 1)))); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +do 2 rewrite Rmult_1_r; apply Rle_lt_trans with (INR (fact (2 * n + 1)) * 4). +apply Rmult_le_compat_l. +replace 0 with (INR 0); [ idtac | reflexivity ]; apply le_INR; apply le_O_n. +simpl in |- *; rewrite Rmult_1_r; replace 4 with (Rsqr 2); + [ idtac | ring_Rsqr ]; replace (a * a) with (Rsqr a); + [ idtac | reflexivity ]; apply Rsqr_incr_1. +apply Rle_trans with (PI / 2); + [ assumption + | unfold Rdiv in |- *; apply Rmult_le_reg_l with 2; + [ prove_sup0 + | rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m; + [ replace 4 with 4; [ apply PI_4 | ring ] | discrR ] ] ]. +left; assumption. +left; prove_sup0. +rewrite H1; replace (2 * n + 1 + 2)%nat with (S (S (2 * n + 1))). +do 2 rewrite fact_simpl; do 2 rewrite mult_INR. +repeat rewrite <- Rmult_assoc. +rewrite <- (Rmult_comm (INR (fact (2 * n + 1)))). +rewrite Rmult_assoc. +apply Rmult_lt_compat_l. +apply lt_INR_0; apply neq_O_lt. +assert (H2 := fact_neq_0 (2 * n + 1)). +red in |- *; intro; elim H2; symmetry in |- *; assumption. +do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR; pose (x := INR n); + unfold INR in |- *. +replace ((2 * x + 1 + 1 + 1) * (2 * x + 1 + 1)) with (4 * x * x + 10 * x + 6); + [ idtac | ring ]. +apply Rplus_lt_reg_r with (-4); rewrite Rplus_opp_l; + replace (-4 + (4 * x * x + 10 * x + 6)) with (4 * x * x + 10 * x + 2); + [ idtac | ring ]. +apply Rplus_le_lt_0_compat. +cut (0 <= x). +intro; apply Rplus_le_le_0_compat; repeat apply Rmult_le_pos; + assumption || left; prove_sup. +unfold x in |- *; replace 0 with (INR 0); + [ apply le_INR; apply le_O_n | reflexivity ]. +prove_sup0. +apply INR_eq; do 2 rewrite S_INR; do 3 rewrite plus_INR; rewrite mult_INR; + repeat rewrite S_INR; ring. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_eq; do 3 rewrite plus_INR; do 2 rewrite mult_INR; + repeat rewrite S_INR; ring. +Qed. + +Lemma SIN : forall a:R, 0 <= a -> a <= PI -> sin_lb a <= sin a <= sin_ub a. +intros; unfold sin_lb, sin_ub in |- *; apply (sin_bound a 1 H H0). +Qed. + +Lemma COS : + forall a:R, - PI / 2 <= a -> a <= PI / 2 -> cos_lb a <= cos a <= cos_ub a. +intros; unfold cos_lb, cos_ub in |- *; apply (cos_bound a 1 H H0). Qed. (**********) -Lemma _PI2_RLT_0 : ``-(PI/2)<0``. -Rewrite <- Ropp_O; Apply Rlt_Ropp1; Apply PI2_RGT_0. +Lemma _PI2_RLT_0 : - (PI / 2) < 0. +rewrite <- Ropp_0; apply Ropp_lt_contravar; apply PI2_RGT_0. Qed. -Lemma PI4_RLT_PI2 : ``PI/4<PI/2``. -Unfold Rdiv; Apply Rlt_monotony. -Apply PI_RGT_0. -Apply Rinv_lt. -Apply Rmult_lt_pos; Sup0. -Pattern 1 ``2``; Rewrite <- Rplus_Or. -Replace ``4`` with ``2+2``; [Apply Rlt_compatibility; Sup0 | Ring]. +Lemma PI4_RLT_PI2 : PI / 4 < PI / 2. +unfold Rdiv in |- *; apply Rmult_lt_compat_l. +apply PI_RGT_0. +apply Rinv_lt_contravar. +apply Rmult_lt_0_compat; prove_sup0. +pattern 2 at 1 in |- *; rewrite <- Rplus_0_r. +replace 4 with (2 + 2); [ apply Rplus_lt_compat_l; prove_sup0 | ring ]. Qed. -Lemma PI2_Rlt_PI : ``PI/2<PI``. -Unfold Rdiv; Pattern 2 PI; Rewrite <- Rmult_1r. -Apply Rlt_monotony. -Apply PI_RGT_0. -Pattern 3 R1; Rewrite <- Rinv_R1; Apply Rinv_lt. -Rewrite Rmult_1l; Sup0. -Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1. +Lemma PI2_Rlt_PI : PI / 2 < PI. +unfold Rdiv in |- *; pattern PI at 2 in |- *; rewrite <- Rmult_1_r. +apply Rmult_lt_compat_l. +apply PI_RGT_0. +pattern 1 at 3 in |- *; rewrite <- Rinv_1; apply Rinv_lt_contravar. +rewrite Rmult_1_l; prove_sup0. +pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + apply Rlt_0_1. Qed. (********************************************) (* Increasing and decreasing of COS and SIN *) (********************************************) -Theorem sin_gt_0 : (x:R) ``0<x`` -> ``x<PI`` -> ``0<(sin x)``. -Intros; Elim (SIN x (Rlt_le R0 x H) (Rlt_le x PI H0)); Intros H1 _; Case (total_order x ``PI/2``); Intro H2. -Apply Rlt_le_trans with (sin_lb x). -Apply sin_lb_gt_0; [Assumption | Left; Assumption]. -Assumption. -Elim H2; Intro H3. -Rewrite H3; Rewrite sin_PI2; Apply Rlt_R0_R1. -Rewrite <- sin_PI_x; Generalize (Rgt_Ropp x ``PI/2`` H3); Intro H4; Generalize (Rlt_compatibility PI (Ropp x) (Ropp ``PI/2``) H4). -Replace ``PI+(-x)`` with ``PI-x``. -Replace ``PI+ -(PI/2)`` with ``PI/2``. -Intro H5; Generalize (Rlt_Ropp x PI H0); Intro H6; Change ``-PI < -x`` in H6; Generalize (Rlt_compatibility PI (Ropp PI) (Ropp x) H6). -Rewrite Rplus_Ropp_r. -Replace ``PI+ -x`` with ``PI-x``. -Intro H7; Elim (SIN ``PI-x`` (Rlt_le R0 ``PI-x`` H7) (Rlt_le ``PI-x`` PI (Rlt_trans ``PI-x`` ``PI/2`` ``PI`` H5 PI2_Rlt_PI))); Intros H8 _; Generalize (sin_lb_gt_0 ``PI-x`` H7 (Rlt_le ``PI-x`` ``PI/2`` H5)); Intro H9; Apply (Rlt_le_trans ``0`` ``(sin_lb (PI-x))`` ``(sin (PI-x))`` H9 H8). -Reflexivity. -Pattern 2 PI; Rewrite double_var; Ring. -Reflexivity. -Qed. - -Theorem cos_gt_0 : (x:R) ``-(PI/2)<x`` -> ``x<PI/2`` -> ``0<(cos x)``. -Intros; Rewrite cos_sin; Generalize (Rlt_compatibility ``PI/2`` ``-(PI/2)`` x H). -Rewrite Rplus_Ropp_r; Intro H1; Generalize (Rlt_compatibility ``PI/2`` x ``PI/2`` H0); Rewrite <- double_var; Intro H2; Apply (sin_gt_0 ``PI/2+x`` H1 H2). -Qed. - -Lemma sin_ge_0 : (x:R) ``0<=x`` -> ``x<=PI`` -> ``0<=(sin x)``. -Intros x H1 H2; Elim H1; Intro H3; [ Elim H2; Intro H4; [ Left ; Apply (sin_gt_0 x H3 H4) | Rewrite H4; Right; Symmetry; Apply sin_PI ] | Rewrite <- H3; Right; Symmetry; Apply sin_0]. -Qed. - -Lemma cos_ge_0 : (x:R) ``-(PI/2)<=x`` -> ``x<=PI/2`` -> ``0<=(cos x)``. -Intros x H1 H2; Elim H1; Intro H3; [ Elim H2; Intro H4; [ Left ; Apply (cos_gt_0 x H3 H4) | Rewrite H4; Right; Symmetry; Apply cos_PI2 ] | Rewrite <- H3; Rewrite cos_neg; Right; Symmetry; Apply cos_PI2 ]. -Qed. - -Lemma sin_le_0 : (x:R) ``PI<=x`` -> ``x<=2*PI`` -> ``(sin x)<=0``. -Intros x H1 H2; Apply Rle_sym2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (sin x)); Apply Rle_Ropp; Rewrite <- neg_sin; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``; [Rewrite -> (sin_period (Rminus x PI) (S O)); Apply sin_ge_0; [Replace ``x-PI`` with ``x+(-PI)``; [Rewrite Rplus_sym; Replace ``0`` with ``(-PI)+PI``; [Apply Rle_compatibility; Assumption | Ring] | Ring] | Replace ``x-PI`` with ``x+(-PI)``; Rewrite Rplus_sym; [Pattern 2 PI; Replace ``PI`` with ``(-PI)+2*PI``; [Apply Rle_compatibility; Assumption | Ring] | Ring]] |Unfold INR; Ring]. -Qed. - -Lemma cos_le_0 : (x:R) ``PI/2<=x``->``x<=3*(PI/2)``->``(cos x)<=0``. -Intros x H1 H2; Apply Rle_sym2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (cos x)); Apply Rle_Ropp; Rewrite <- neg_cos; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``. -Rewrite cos_period; Apply cos_ge_0. -Replace ``-(PI/2)`` with ``-PI+(PI/2)``. -Unfold Rminus; Rewrite (Rplus_sym x); Apply Rle_compatibility; Assumption. -Pattern 1 PI; Rewrite (double_var PI); Rewrite Ropp_distr1; Ring. -Unfold Rminus; Rewrite Rplus_sym; Replace ``PI/2`` with ``(-PI)+3*(PI/2)``. -Apply Rle_compatibility; Assumption. -Pattern 1 PI; Rewrite (double_var PI); Rewrite Ropp_distr1; Ring. -Unfold INR; Ring. -Qed. - -Lemma sin_lt_0 : (x:R) ``PI<x`` -> ``x<2*PI`` -> ``(sin x)<0``. -Intros x H1 H2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (sin x)); Apply Rlt_Ropp; Rewrite <- neg_sin; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``; [Rewrite -> (sin_period (Rminus x PI) (S O)); Apply sin_gt_0; [Replace ``x-PI`` with ``x+(-PI)``; [Rewrite Rplus_sym; Replace ``0`` with ``(-PI)+PI``; [Apply Rlt_compatibility; Assumption | Ring] | Ring] | Replace ``x-PI`` with ``x+(-PI)``; Rewrite Rplus_sym; [Pattern 2 PI; Replace ``PI`` with ``(-PI)+2*PI``; [Apply Rlt_compatibility; Assumption | Ring] | Ring]] |Unfold INR; Ring]. -Qed. - -Lemma sin_lt_0_var : (x:R) ``-PI<x`` -> ``x<0`` -> ``(sin x)<0``. -Intros; Generalize (Rlt_compatibility ``2*PI`` ``-PI`` x H); Replace ``2*PI+(-PI)`` with ``PI``; [Intro H1; Rewrite Rplus_sym in H1; Generalize (Rlt_compatibility ``2*PI`` x ``0`` H0); Intro H2; Rewrite (Rplus_sym ``2*PI``) in H2; Rewrite <- (Rplus_sym R0) in H2; Rewrite Rplus_Ol in H2; Rewrite <- (sin_period x (1)); Unfold INR; Replace ``2*1*PI`` with ``2*PI``; [Apply (sin_lt_0 ``x+2*PI`` H1 H2) | Ring] | Ring]. -Qed. - -Lemma cos_lt_0 : (x:R) ``PI/2<x`` -> ``x<3*(PI/2)``-> ``(cos x)<0``. -Intros x H1 H2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (cos x)); Apply Rlt_Ropp; Rewrite <- neg_cos; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``. -Rewrite cos_period; Apply cos_gt_0. -Replace ``-(PI/2)`` with ``-PI+(PI/2)``. -Unfold Rminus; Rewrite (Rplus_sym x); Apply Rlt_compatibility; Assumption. -Pattern 1 PI; Rewrite (double_var PI); Rewrite Ropp_distr1; Ring. -Unfold Rminus; Rewrite Rplus_sym; Replace ``PI/2`` with ``(-PI)+3*(PI/2)``. -Apply Rlt_compatibility; Assumption. -Pattern 1 PI; Rewrite (double_var PI); Rewrite Ropp_distr1; Ring. -Unfold INR; Ring. -Qed. - -Lemma tan_gt_0 : (x:R) ``0<x`` -> ``x<PI/2`` -> ``0<(tan x)``. -Intros x H1 H2; Unfold tan; Generalize _PI2_RLT_0; Generalize (Rlt_trans R0 x ``PI/2`` H1 H2); Intros; Generalize (Rlt_trans ``-(PI/2)`` R0 x H0 H1); Intro H5; Generalize (Rlt_trans x ``PI/2`` PI H2 PI2_Rlt_PI); Intro H7; Unfold Rdiv; Apply Rmult_lt_pos. -Apply sin_gt_0; Assumption. -Apply Rlt_Rinv; Apply cos_gt_0; Assumption. -Qed. - -Lemma tan_lt_0 : (x:R) ``-(PI/2)<x``->``x<0``->``(tan x)<0``. -Intros x H1 H2; Unfold tan; Generalize (cos_gt_0 x H1 (Rlt_trans x ``0`` ``PI/2`` H2 PI2_RGT_0)); Intro H3; Rewrite <- Ropp_O; Replace ``(sin x)/(cos x)`` with ``- ((-(sin x))/(cos x))``. -Rewrite <- sin_neg; Apply Rgt_Ropp; Change ``0<(sin (-x))/(cos x)``; Unfold Rdiv; Apply Rmult_lt_pos. -Apply sin_gt_0. -Rewrite <- Ropp_O; Apply Rgt_Ropp; Assumption. -Apply Rlt_trans with ``PI/2``. -Rewrite <- (Ropp_Ropp ``PI/2``); Apply Rgt_Ropp; Assumption. -Apply PI2_Rlt_PI. -Apply Rlt_Rinv; Assumption. -Unfold Rdiv; Ring. -Qed. - -Lemma cos_ge_0_3PI2 : (x:R) ``3*(PI/2)<=x``->``x<=2*PI``->``0<=(cos x)``. -Intros; Rewrite <- cos_neg; Rewrite <- (cos_period ``-x`` (1)); Unfold INR; Replace ``-x+2*1*PI`` with ``2*PI-x``. -Generalize (Rle_Ropp x ``2*PI`` H0); Intro H1; Generalize (Rle_sym2 ``-(2*PI)`` ``-x`` H1); Clear H1; Intro H1; Generalize (Rle_compatibility ``2*PI`` ``-(2*PI)`` ``-x`` H1). -Rewrite Rplus_Ropp_r. -Intro H2; Generalize (Rle_Ropp ``3*(PI/2)`` x H); Intro H3; Generalize (Rle_sym2 ``-x`` ``-(3*(PI/2))`` H3); Clear H3; Intro H3; Generalize (Rle_compatibility ``2*PI`` ``-x`` ``-(3*(PI/2))`` H3). -Replace ``2*PI+ -(3*PI/2)`` with ``PI/2``. -Intro H4; Apply (cos_ge_0 ``2*PI-x`` (Rlt_le ``-(PI/2)`` ``2*PI-x`` (Rlt_le_trans ``-(PI/2)`` ``0`` ``2*PI-x`` _PI2_RLT_0 H2)) H4). -Rewrite double; Pattern 2 3 PI; Rewrite double_var; Ring. -Ring. -Qed. - -Lemma form1 : (p,q:R) ``(cos p)+(cos q)==2*(cos ((p-q)/2))*(cos ((p+q)/2))``. -Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``. -Rewrite <- (cos_neg q); Replace``-q`` with ``(p-q)/2-(p+q)/2``. -Rewrite cos_plus; Rewrite cos_minus; Ring. -Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring. -Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring. -Qed. - -Lemma form2 : (p,q:R) ``(cos p)-(cos q)==-2*(sin ((p-q)/2))*(sin ((p+q)/2))``. -Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``. -Rewrite <- (cos_neg q); Replace``-q`` with ``(p-q)/2-(p+q)/2``. -Rewrite cos_plus; Rewrite cos_minus; Ring. -Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring. -Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring. -Qed. - -Lemma form3 : (p,q:R) ``(sin p)+(sin q)==2*(cos ((p-q)/2))*(sin ((p+q)/2))``. -Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``. -Pattern 3 q; Replace ``q`` with ``(p+q)/2-(p-q)/2``. -Rewrite sin_plus; Rewrite sin_minus; Ring. -Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring. -Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring. -Qed. - -Lemma form4 : (p,q:R) ``(sin p)-(sin q)==2*(cos ((p+q)/2))*(sin ((p-q)/2))``. -Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``. -Pattern 3 q; Replace ``q`` with ``(p+q)/2-(p-q)/2``. -Rewrite sin_plus; Rewrite sin_minus; Ring. -Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring. -Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring. - -Qed. - -Lemma sin_increasing_0 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``(sin x)<(sin y)``->``x<y``. -Intros; Cut ``(sin ((x-y)/2))<0``. -Intro H4; Case (total_order ``(x-y)/2`` ``0``); Intro H5. -Assert Hyp : ``0<2``. -Sup0. -Generalize (Rlt_monotony ``2`` ``(x-y)/2`` ``0`` Hyp H5). -Unfold Rdiv. -Rewrite <- Rmult_assoc. -Rewrite Rinv_r_simpl_m. -Rewrite Rmult_Or. -Clear H5; Intro H5; Apply Rminus_lt; Assumption. -DiscrR. -Elim H5; Intro H6. -Rewrite H6 in H4; Rewrite sin_0 in H4; Elim (Rlt_antirefl ``0`` H4). -Change ``0<(x-y)/2`` in H6; Generalize (Rle_Ropp ``-(PI/2)`` y H1). -Rewrite Ropp_Ropp. -Intro H7; Generalize (Rle_sym2 ``-y`` ``PI/2`` H7); Clear H7; Intro H7; Generalize (Rplus_le x ``PI/2`` ``-y`` ``PI/2`` H0 H7). -Rewrite <- double_var. -Intro H8. -Assert Hyp : ``0<2``. -Sup0. -Generalize (Rle_monotony ``(Rinv 2)`` ``x-y`` PI (Rlt_le ``0`` ``/2`` (Rlt_Rinv ``2`` Hyp)) H8). -Repeat Rewrite (Rmult_sym ``/2``). -Intro H9; Generalize (sin_gt_0 ``(x-y)/2`` H6 (Rle_lt_trans ``(x-y)/2`` ``PI/2`` PI H9 PI2_Rlt_PI)); Intro H10; Elim (Rlt_antirefl ``(sin ((x-y)/2))`` (Rlt_trans ``(sin ((x-y)/2))`` ``0`` ``(sin ((x-y)/2))`` H4 H10)). -Generalize (Rlt_minus (sin x) (sin y) H3); Clear H3; Intro H3; Rewrite form4 in H3; Generalize (Rplus_le x ``PI/2`` y ``PI/2`` H0 H2). -Rewrite <- double_var. -Assert Hyp : ``0<2``. -Sup0. -Intro H4; Generalize (Rle_monotony ``(Rinv 2)`` ``x+y`` PI (Rlt_le ``0`` ``/2`` (Rlt_Rinv ``2`` Hyp)) H4). -Repeat Rewrite (Rmult_sym ``/2``). -Clear H4; Intro H4; Generalize (Rplus_le ``-(PI/2)`` x ``-(PI/2)`` y H H1); Replace ``-(PI/2)+(-(PI/2))`` with ``-PI``. -Intro H5; Generalize (Rle_monotony ``(Rinv 2)`` ``-PI`` ``x+y`` (Rlt_le ``0`` ``/2`` (Rlt_Rinv ``2`` Hyp)) H5). -Replace ``/2*(x+y)`` with ``(x+y)/2``. -Replace ``/2*(-PI)`` with ``-(PI/2)``. -Clear H5; Intro H5; Elim H4; Intro H40. -Elim H5; Intro H50. -Generalize (cos_gt_0 ``(x+y)/2`` H50 H40); Intro H6; Generalize (Rlt_monotony ``2`` ``0`` ``(cos ((x+y)/2))`` Hyp H6). -Rewrite Rmult_Or. -Clear H6; Intro H6; Case (case_Rabsolu ``(sin ((x-y)/2))``); Intro H7. -Assumption. -Generalize (Rle_sym2 ``0`` ``(sin ((x-y)/2))`` H7); Clear H7; Intro H7; Generalize (Rmult_le_pos ``2*(cos ((x+y)/2))`` ``(sin ((x-y)/2))`` (Rlt_le ``0`` ``2*(cos ((x+y)/2))`` H6) H7); Intro H8; Generalize (Rle_lt_trans ``0`` ``2*(cos ((x+y)/2))*(sin ((x-y)/2))`` ``0`` H8 H3); Intro H9; Elim (Rlt_antirefl ``0`` H9). -Rewrite <- H50 in H3; Rewrite cos_neg in H3; Rewrite cos_PI2 in H3; Rewrite Rmult_Or in H3; Rewrite Rmult_Ol in H3; Elim (Rlt_antirefl ``0`` H3). -Unfold Rdiv in H3. -Rewrite H40 in H3; Assert H50 := cos_PI2; Unfold Rdiv in H50; Rewrite H50 in H3; Rewrite Rmult_Or in H3; Rewrite Rmult_Ol in H3; Elim (Rlt_antirefl ``0`` H3). -Unfold Rdiv. -Rewrite <- Ropp_mul1. -Apply Rmult_sym. -Unfold Rdiv; Apply Rmult_sym. -Pattern 1 PI; Rewrite double_var. -Rewrite Ropp_distr1. -Reflexivity. -Qed. - -Lemma sin_increasing_1 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``x<y``->``(sin x)<(sin y)``. -Intros; Generalize (Rlt_compatibility ``x`` ``x`` ``y`` H3); Intro H4; Generalize (Rplus_le ``-(PI/2)`` x ``-(PI/2)`` x H H); Replace ``-(PI/2)+ (-(PI/2))`` with ``-PI``. -Assert Hyp : ``0<2``. -Sup0. -Intro H5; Generalize (Rle_lt_trans ``-PI`` ``x+x`` ``x+y`` H5 H4); Intro H6; Generalize (Rlt_monotony ``(Rinv 2)`` ``-PI`` ``x+y`` (Rlt_Rinv ``2`` Hyp) H6); Replace ``/2*(-PI)`` with ``-(PI/2)``. -Replace ``/2*(x+y)`` with ``(x+y)/2``. -Clear H4 H5 H6; Intro H4; Generalize (Rlt_compatibility ``y`` ``x`` ``y`` H3); Intro H5; Rewrite Rplus_sym in H5; Generalize (Rplus_le y ``PI/2`` y ``PI/2`` H2 H2). -Rewrite <- double_var. -Intro H6; Generalize (Rlt_le_trans ``x+y`` ``y+y`` PI H5 H6); Intro H7; Generalize (Rlt_monotony ``(Rinv 2)`` ``x+y`` PI (Rlt_Rinv ``2`` Hyp) H7); Replace ``/2*PI`` with ``PI/2``. -Replace ``/2*(x+y)`` with ``(x+y)/2``. -Clear H5 H6 H7; Intro H5; Generalize (Rle_Ropp ``-(PI/2)`` y H1); Rewrite Ropp_Ropp; Clear H1; Intro H1; Generalize (Rle_sym2 ``-y`` ``PI/2`` H1); Clear H1; Intro H1; Generalize (Rle_Ropp y ``PI/2`` H2); Clear H2; Intro H2; Generalize (Rle_sym2 ``-(PI/2)`` ``-y`` H2); Clear H2; Intro H2; Generalize (Rlt_compatibility ``-y`` x y H3); Replace ``-y+x`` with ``x-y``. -Rewrite Rplus_Ropp_l. -Intro H6; Generalize (Rlt_monotony ``(Rinv 2)`` ``x-y`` ``0`` (Rlt_Rinv ``2`` Hyp) H6); Rewrite Rmult_Or; Replace ``/2*(x-y)`` with ``(x-y)/2``. -Clear H6; Intro H6; Generalize (Rplus_le ``-(PI/2)`` x ``-(PI/2)`` ``-y`` H H2); Replace ``-(PI/2)+ (-(PI/2))`` with ``-PI``. -Replace `` x+ -y`` with ``x-y``. -Intro H7; Generalize (Rle_monotony ``(Rinv 2)`` ``-PI`` ``x-y`` (Rlt_le ``0`` ``/2`` (Rlt_Rinv ``2`` Hyp)) H7); Replace ``/2*(-PI)`` with ``-(PI/2)``. -Replace ``/2*(x-y)`` with ``(x-y)/2``. -Clear H7; Intro H7; Clear H H0 H1 H2; Apply Rminus_lt; Rewrite form4; Generalize (cos_gt_0 ``(x+y)/2`` H4 H5); Intro H8; Generalize (Rmult_lt_pos ``2`` ``(cos ((x+y)/2))`` Hyp H8); Clear H8; Intro H8; Cut ``-PI< -(PI/2)``. -Intro H9; Generalize (sin_lt_0_var ``(x-y)/2`` (Rlt_le_trans ``-PI`` ``-(PI/2)`` ``(x-y)/2`` H9 H7) H6); Intro H10; Generalize (Rlt_anti_monotony ``(sin ((x-y)/2))`` ``0`` ``2*(cos ((x+y)/2))`` H10 H8); Intro H11; Rewrite Rmult_Or in H11; Rewrite Rmult_sym; Assumption. -Apply Rlt_Ropp; Apply PI2_Rlt_PI. -Unfold Rdiv; Apply Rmult_sym. -Unfold Rdiv; Rewrite <- Ropp_mul1; Apply Rmult_sym. -Reflexivity. -Pattern 1 PI; Rewrite double_var. -Rewrite Ropp_distr1. -Reflexivity. -Unfold Rdiv; Apply Rmult_sym. -Unfold Rminus; Apply Rplus_sym. -Unfold Rdiv; Apply Rmult_sym. -Unfold Rdiv; Apply Rmult_sym. -Unfold Rdiv; Apply Rmult_sym. -Unfold Rdiv. -Rewrite <- Ropp_mul1. -Apply Rmult_sym. -Pattern 1 PI; Rewrite double_var. -Rewrite Ropp_distr1. -Reflexivity. -Qed. - -Lemma sin_decreasing_0 : (x,y:R) ``x<=3*(PI/2)``-> ``PI/2<=x`` -> ``y<=3*(PI/2)``-> ``PI/2<=y`` -> ``(sin x)<(sin y)`` -> ``y<x``. -Intros; Rewrite <- (sin_PI_x x) in H3; Rewrite <- (sin_PI_x y) in H3; Generalize (Rlt_Ropp ``(sin (PI-x))`` ``(sin (PI-y))`` H3); Repeat Rewrite <- sin_neg; Generalize (Rle_compatibility ``-PI`` x ``3*(PI/2)`` H); Generalize (Rle_compatibility ``-PI`` ``PI/2`` x H0); Generalize (Rle_compatibility ``-PI`` y ``3*(PI/2)`` H1); Generalize (Rle_compatibility ``-PI`` ``PI/2`` y H2); Replace ``-PI+x`` with ``x-PI``. -Replace ``-PI+PI/2`` with ``-(PI/2)``. -Replace ``-PI+y`` with ``y-PI``. -Replace ``-PI+3*(PI/2)`` with ``PI/2``. -Replace ``-(PI-x)`` with ``x-PI``. -Replace ``-(PI-y)`` with ``y-PI``. -Intros; Change ``(sin (y-PI))<(sin (x-PI))`` in H8; Apply Rlt_anti_compatibility with ``-PI``; Rewrite Rplus_sym; Replace ``y+ (-PI)`` with ``y-PI``. -Rewrite Rplus_sym; Replace ``x+ (-PI)`` with ``x-PI``. -Apply (sin_increasing_0 ``y-PI`` ``x-PI`` H4 H5 H6 H7 H8). -Reflexivity. -Reflexivity. -Unfold Rminus; Rewrite Ropp_distr1. -Rewrite Ropp_Ropp. -Apply Rplus_sym. -Unfold Rminus; Rewrite Ropp_distr1. -Rewrite Ropp_Ropp. -Apply Rplus_sym. -Pattern 2 PI; Rewrite double_var. -Rewrite Ropp_distr1. -Ring. -Unfold Rminus; Apply Rplus_sym. -Pattern 2 PI; Rewrite double_var. -Rewrite Ropp_distr1. -Ring. -Unfold Rminus; Apply Rplus_sym. -Qed. - -Lemma sin_decreasing_1 : (x,y:R) ``x<=3*(PI/2)``-> ``PI/2<=x`` -> ``y<=3*(PI/2)``-> ``PI/2<=y`` -> ``x<y`` -> ``(sin y)<(sin x)``. -Intros; Rewrite <- (sin_PI_x x); Rewrite <- (sin_PI_x y); Generalize (Rle_compatibility ``-PI`` x ``3*(PI/2)`` H); Generalize (Rle_compatibility ``-PI`` ``PI/2`` x H0); Generalize (Rle_compatibility ``-PI`` y ``3*(PI/2)`` H1); Generalize (Rle_compatibility ``-PI`` ``PI/2`` y H2); Generalize (Rlt_compatibility ``-PI`` x y H3); Replace ``-PI+PI/2`` with ``-(PI/2)``. -Replace ``-PI+y`` with ``y-PI``. -Replace ``-PI+3*(PI/2)`` with ``PI/2``. -Replace ``-PI+x`` with ``x-PI``. -Intros; Apply Ropp_Rlt; Repeat Rewrite <- sin_neg; Replace ``-(PI-x)`` with ``x-PI``. -Replace ``-(PI-y)`` with ``y-PI``. -Apply (sin_increasing_1 ``x-PI`` ``y-PI`` H7 H8 H5 H6 H4). -Unfold Rminus; Rewrite Ropp_distr1. -Rewrite Ropp_Ropp. -Apply Rplus_sym. -Unfold Rminus; Rewrite Ropp_distr1. -Rewrite Ropp_Ropp. -Apply Rplus_sym. -Unfold Rminus; Apply Rplus_sym. -Pattern 2 PI; Rewrite double_var; Ring. -Unfold Rminus; Apply Rplus_sym. -Pattern 2 PI; Rewrite double_var; Ring. -Qed. - -Lemma cos_increasing_0 : (x,y:R) ``PI<=x`` -> ``x<=2*PI`` ->``PI<=y`` -> ``y<=2*PI`` -> ``(cos x)<(cos y)`` -> ``x<y``. -Intros x y H1 H2 H3 H4; Rewrite <- (cos_neg x); Rewrite <- (cos_neg y); Rewrite <- (cos_period ``-x`` (1)); Rewrite <- (cos_period ``-y`` (1)); Unfold INR; Replace ``-x+2*1*PI`` with ``PI/2-(x-3*(PI/2))``. -Replace ``-y+2*1*PI`` with ``PI/2-(y-3*(PI/2))``. -Repeat Rewrite cos_shift; Intro H5; Generalize (Rle_compatibility ``-3*(PI/2)`` PI x H1); Generalize (Rle_compatibility ``-3*(PI/2)`` x ``2*PI`` H2); Generalize (Rle_compatibility ``-3*(PI/2)`` PI y H3); Generalize (Rle_compatibility ``-3*(PI/2)`` y ``2*PI`` H4). -Replace ``-3*(PI/2)+y`` with ``y-3*(PI/2)``. -Replace ``-3*(PI/2)+x`` with ``x-3*(PI/2)``. -Replace ``-3*(PI/2)+2*PI`` with ``PI/2``. -Replace ``-3*PI/2+PI`` with ``-(PI/2)``. -Clear H1 H2 H3 H4; Intros H1 H2 H3 H4; Apply Rlt_anti_compatibility with ``-3*(PI/2)``; Replace ``-3*PI/2+x`` with ``x-3*(PI/2)``. -Replace ``-3*PI/2+y`` with ``y-3*(PI/2)``. -Apply (sin_increasing_0 ``x-3*(PI/2)`` ``y-3*(PI/2)`` H4 H3 H2 H1 H5). -Unfold Rminus. -Rewrite Ropp_mul1. -Apply Rplus_sym. -Unfold Rminus. -Rewrite Ropp_mul1. -Apply Rplus_sym. -Pattern 3 PI; Rewrite double_var. -Ring. -Rewrite double; Pattern 3 4 PI; Rewrite double_var. -Ring. -Unfold Rminus. -Rewrite Ropp_mul1. -Apply Rplus_sym. -Unfold Rminus. -Rewrite Ropp_mul1. -Apply Rplus_sym. -Rewrite Rmult_1r. -Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var. -Ring. -Rewrite Rmult_1r. -Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var. -Ring. -Qed. - -Lemma cos_increasing_1 : (x,y:R) ``PI<=x`` -> ``x<=2*PI`` ->``PI<=y`` -> ``y<=2*PI`` -> ``x<y`` -> ``(cos x)<(cos y)``. -Intros x y H1 H2 H3 H4 H5; Generalize (Rle_compatibility ``-3*(PI/2)`` PI x H1); Generalize (Rle_compatibility ``-3*(PI/2)`` x ``2*PI`` H2); Generalize (Rle_compatibility ``-3*(PI/2)`` PI y H3); Generalize (Rle_compatibility ``-3*(PI/2)`` y ``2*PI`` H4); Generalize (Rlt_compatibility ``-3*(PI/2)`` x y H5); Rewrite <- (cos_neg x); Rewrite <- (cos_neg y); Rewrite <- (cos_period ``-x`` (1)); Rewrite <- (cos_period ``-y`` (1)); Unfold INR; Replace ``-3*(PI/2)+x`` with ``x-3*(PI/2)``. -Replace ``-3*(PI/2)+y`` with ``y-3*(PI/2)``. -Replace ``-3*(PI/2)+PI`` with ``-(PI/2)``. -Replace ``-3*(PI/2)+2*PI`` with ``PI/2``. -Clear H1 H2 H3 H4 H5; Intros H1 H2 H3 H4 H5; Replace ``-x+2*1*PI`` with ``(PI/2)-(x-3*(PI/2))``. -Replace ``-y+2*1*PI`` with ``(PI/2)-(y-3*(PI/2))``. -Repeat Rewrite cos_shift; Apply (sin_increasing_1 ``x-3*(PI/2)`` ``y-3*(PI/2)`` H5 H4 H3 H2 H1). -Rewrite Rmult_1r. -Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var. -Ring. -Rewrite Rmult_1r. -Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var. -Ring. -Rewrite (double PI); Pattern 3 4 PI; Rewrite double_var. -Ring. -Pattern 3 PI; Rewrite double_var; Ring. -Unfold Rminus. -Rewrite <- Ropp_mul1. -Apply Rplus_sym. -Unfold Rminus. -Rewrite <- Ropp_mul1. -Apply Rplus_sym. -Qed. - -Lemma cos_decreasing_0 : (x,y:R) ``0<=x``->``x<=PI``->``0<=y``->``y<=PI``->``(cos x)<(cos y)``->``y<x``. -Intros; Generalize (Rlt_Ropp (cos x) (cos y) H3); Repeat Rewrite <- neg_cos; Intro H4; Change ``(cos (y+PI))<(cos (x+PI))`` in H4; Rewrite (Rplus_sym x) in H4; Rewrite (Rplus_sym y) in H4; Generalize (Rle_compatibility PI ``0`` x H); Generalize (Rle_compatibility PI x PI H0); Generalize (Rle_compatibility PI ``0`` y H1); Generalize (Rle_compatibility PI y PI H2); Rewrite Rplus_Or. -Rewrite <- double. -Clear H H0 H1 H2 H3; Intros; Apply Rlt_anti_compatibility with ``PI``; Apply (cos_increasing_0 ``PI+y`` ``PI+x`` H0 H H2 H1 H4). -Qed. - -Lemma cos_decreasing_1 : (x,y:R) ``0<=x``->``x<=PI``->``0<=y``->``y<=PI``->``x<y``->``(cos y)<(cos x)``. -Intros; Apply Ropp_Rlt; Repeat Rewrite <- neg_cos; Rewrite (Rplus_sym x); Rewrite (Rplus_sym y); Generalize (Rle_compatibility PI ``0`` x H); Generalize (Rle_compatibility PI x PI H0); Generalize (Rle_compatibility PI ``0`` y H1); Generalize (Rle_compatibility PI y PI H2); Rewrite Rplus_Or. -Rewrite <- double. -Generalize (Rlt_compatibility PI x y H3); Clear H H0 H1 H2 H3; Intros; Apply (cos_increasing_1 ``PI+x`` ``PI+y`` H3 H2 H1 H0 H). -Qed. - -Lemma tan_diff : (x,y:R) ~``(cos x)==0``->~``(cos y)==0``->``(tan x)-(tan y)==(sin (x-y))/((cos x)*(cos y))``. -Intros; Unfold tan;Rewrite sin_minus. -Unfold Rdiv. -Unfold Rminus. -Rewrite Rmult_Rplus_distrl. -Rewrite Rinv_Rmult. -Repeat Rewrite (Rmult_sym (sin x)). -Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym (cos y)). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Rewrite (Rmult_sym (sin x)). -Apply Rplus_plus_r. -Rewrite <- Ropp_mul1. -Rewrite <- Ropp_mul3. -Rewrite (Rmult_sym ``/(cos x)``). -Repeat Rewrite Rmult_assoc. -Rewrite (Rmult_sym (cos x)). -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Reflexivity. -Assumption. -Assumption. -Assumption. -Assumption. -Qed. - -Lemma tan_increasing_0 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI/4``->``(tan x)<(tan y)``->``x<y``. -Intros; Generalize PI4_RLT_PI2; Intro H4; Generalize (Rlt_Ropp ``PI/4`` ``PI/2`` H4); Intro H5; Change ``-(PI/2)< -(PI/4)`` in H5; Generalize (cos_gt_0 x (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` x H5 H) (Rle_lt_trans x ``PI/4`` ``PI/2`` H0 H4)); Intro HP1; Generalize (cos_gt_0 y (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` y H5 H1) (Rle_lt_trans y ``PI/4`` ``PI/2`` H2 H4)); Intro HP2; Generalize (not_sym ``0`` (cos x) (Rlt_not_eq ``0`` (cos x) (cos_gt_0 x (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` x H5 H) (Rle_lt_trans x ``PI/4`` ``PI/2`` H0 H4)))); Intro H6; Generalize (not_sym ``0`` (cos y) (Rlt_not_eq ``0`` (cos y) (cos_gt_0 y (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` y H5 H1) (Rle_lt_trans y ``PI/4`` ``PI/2`` H2 H4)))); Intro H7; Generalize (tan_diff x y H6 H7); Intro H8; Generalize (Rlt_minus (tan x) (tan y) H3); Clear H3; Intro H3; Rewrite H8 in H3; Cut ``(sin (x-y))<0``. -Intro H9; Generalize (Rle_Ropp ``-(PI/4)`` y H1); Rewrite Ropp_Ropp; Intro H10; Generalize (Rle_sym2 ``-y`` ``PI/4`` H10); Clear H10; Intro H10; Generalize (Rle_Ropp y ``PI/4`` H2); Intro H11; Generalize (Rle_sym2 ``-(PI/4)`` ``-y`` H11); Clear H11; Intro H11; Generalize (Rplus_le ``-(PI/4)`` x ``-(PI/4)`` ``-y`` H H11); Generalize (Rplus_le x ``PI/4`` ``-y`` ``PI/4`` H0 H10); Replace ``x+ -y`` with ``x-y``. -Replace ``PI/4+PI/4`` with ``PI/2``. -Replace ``-(PI/4)+ -(PI/4)`` with ``-(PI/2)``. -Intros; Case (total_order ``0`` ``x-y``); Intro H14. -Generalize (sin_gt_0 ``x-y`` H14 (Rle_lt_trans ``x-y`` ``PI/2`` PI H12 PI2_Rlt_PI)); Intro H15; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``(sin (x-y))`` ``0`` H15 H9)). -Elim H14; Intro H15. -Rewrite <- H15 in H9; Rewrite -> sin_0 in H9; Elim (Rlt_antirefl ``0`` H9). -Apply Rminus_lt; Assumption. -Pattern 1 PI; Rewrite double_var. -Unfold Rdiv. -Rewrite Rmult_Rplus_distrl. -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_Rmult. -Rewrite Ropp_distr1. -Replace ``2*2`` with ``4``. -Reflexivity. -Ring. -DiscrR. -DiscrR. -Pattern 1 PI; Rewrite double_var. -Unfold Rdiv. -Rewrite Rmult_Rplus_distrl. -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_Rmult. -Replace ``2*2`` with ``4``. -Reflexivity. -Ring. -DiscrR. -DiscrR. -Reflexivity. -Case (case_Rabsolu ``(sin (x-y))``); Intro H9. -Assumption. -Generalize (Rle_sym2 ``0`` ``(sin (x-y))`` H9); Clear H9; Intro H9; Generalize (Rlt_Rinv (cos x) HP1); Intro H10; Generalize (Rlt_Rinv (cos y) HP2); Intro H11; Generalize (Rmult_lt_pos (Rinv (cos x)) (Rinv (cos y)) H10 H11); Replace ``/(cos x)*/(cos y)`` with ``/((cos x)*(cos y))``. -Intro H12; Generalize (Rmult_le_pos ``(sin (x-y))`` ``/((cos x)*(cos y))`` H9 (Rlt_le ``0`` ``/((cos x)*(cos y))`` H12)); Intro H13; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` ``(sin (x-y))*/((cos x)*(cos y))`` ``0`` H13 H3)). -Rewrite Rinv_Rmult. -Reflexivity. -Assumption. -Assumption. -Qed. - -Lemma tan_increasing_1 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI/4``->``x<y``->``(tan x)<(tan y)``. -Intros; Apply Rminus_lt; Generalize PI4_RLT_PI2; Intro H4; Generalize (Rlt_Ropp ``PI/4`` ``PI/2`` H4); Intro H5; Change ``-(PI/2)< -(PI/4)`` in H5; Generalize (cos_gt_0 x (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` x H5 H) (Rle_lt_trans x ``PI/4`` ``PI/2`` H0 H4)); Intro HP1; Generalize (cos_gt_0 y (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` y H5 H1) (Rle_lt_trans y ``PI/4`` ``PI/2`` H2 H4)); Intro HP2; Generalize (not_sym ``0`` (cos x) (Rlt_not_eq ``0`` (cos x) (cos_gt_0 x (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` x H5 H) (Rle_lt_trans x ``PI/4`` ``PI/2`` H0 H4)))); Intro H6; Generalize (not_sym ``0`` (cos y) (Rlt_not_eq ``0`` (cos y) (cos_gt_0 y (Rlt_le_trans ``-(PI/2)`` ``-(PI/4)`` y H5 H1) (Rle_lt_trans y ``PI/4`` ``PI/2`` H2 H4)))); Intro H7; Rewrite (tan_diff x y H6 H7); Generalize (Rlt_Rinv (cos x) HP1); Intro H10; Generalize (Rlt_Rinv (cos y) HP2); Intro H11; Generalize (Rmult_lt_pos (Rinv (cos x)) (Rinv (cos y)) H10 H11); Replace ``/(cos x)*/(cos y)`` with ``/((cos x)*(cos y))``. -Clear H10 H11; Intro H8; Generalize (Rle_Ropp y ``PI/4`` H2); Intro H11; Generalize (Rle_sym2 ``-(PI/4)`` ``-y`` H11); Clear H11; Intro H11; Generalize (Rplus_le ``-(PI/4)`` x ``-(PI/4)`` ``-y`` H H11); Replace ``x+ -y`` with ``x-y``. -Replace ``-(PI/4)+ -(PI/4)`` with ``-(PI/2)``. -Clear H11; Intro H9; Generalize (Rlt_minus x y H3); Clear H3; Intro H3; Clear H H0 H1 H2 H4 H5 HP1 HP2; Generalize PI2_Rlt_PI; Intro H1; Generalize (Rlt_Ropp ``PI/2`` PI H1); Clear H1; Intro H1; Generalize (sin_lt_0_var ``x-y`` (Rlt_le_trans ``-PI`` ``-(PI/2)`` ``x-y`` H1 H9) H3); Intro H2; Generalize (Rlt_anti_monotony ``(sin (x-y))`` ``0`` ``/((cos x)*(cos y))`` H2 H8); Rewrite Rmult_Or; Intro H4; Assumption. -Pattern 1 PI; Rewrite double_var. -Unfold Rdiv. -Rewrite Rmult_Rplus_distrl. -Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_Rmult. -Replace ``2*2`` with ``4``. -Rewrite Ropp_distr1. -Reflexivity. -Ring. -DiscrR. -DiscrR. -Reflexivity. -Apply Rinv_Rmult; Assumption. -Qed. - -Lemma sin_incr_0 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``(sin x)<=(sin y)``->``x<=y``. -Intros; Case (total_order (sin x) (sin y)); Intro H4; [Left; Apply (sin_increasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (sin_increasing_1 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (sin y) H8)]] | Elim (Rlt_antirefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5))]]. -Qed. - -Lemma sin_incr_1 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``x<=y``->``(sin x)<=(sin y)``. -Intros; Case (total_order x y); Intro H4; [Left; Apply (sin_increasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (sin x) (sin y)); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (sin_increasing_0 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8)]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]]. -Qed. - -Lemma sin_decr_0 : (x,y:R) ``x<=3*(PI/2)``->``PI/2<=x``->``y<=3*(PI/2)``->``PI/2<=y``-> ``(sin x)<=(sin y)`` -> ``y<=x``. -Intros; Case (total_order (sin x) (sin y)); Intro H4; [Left; Apply (sin_decreasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Generalize (sin_decreasing_1 x y H H0 H1 H2 H6); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (sin y) H8) | Elim H6; Intro H7; [Right; Symmetry; Assumption | Left; Assumption]] | Elim (Rlt_antirefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5))]]. -Qed. - -Lemma sin_decr_1 : (x,y:R) ``x<=3*(PI/2)``-> ``PI/2<=x`` -> ``y<=3*(PI/2)``-> ``PI/2<=y`` -> ``x<=y`` -> ``(sin y)<=(sin x)``. -Intros; Case (total_order x y); Intro H4; [Left; Apply (sin_decreasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (sin x) (sin y)); Intro H6; [Generalize (sin_decreasing_0 x y H H0 H1 H2 H6); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8) | Elim H6; Intro H7; [Right; Symmetry; Assumption | Left; Assumption]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]]. -Qed. - -Lemma cos_incr_0 : (x,y:R) ``PI<=x`` -> ``x<=2*PI`` ->``PI<=y`` -> ``y<=2*PI`` -> ``(cos x)<=(cos y)`` -> ``x<=y``. -Intros; Case (total_order (cos x) (cos y)); Intro H4; [Left; Apply (cos_increasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (cos_increasing_1 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (cos y) H8)]] | Elim (Rlt_antirefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5))]]. -Qed. - -Lemma cos_incr_1 : (x,y:R) ``PI<=x`` -> ``x<=2*PI`` ->``PI<=y`` -> ``y<=2*PI`` -> ``x<=y`` -> ``(cos x)<=(cos y)``. -Intros; Case (total_order x y); Intro H4; [Left; Apply (cos_increasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (cos x) (cos y)); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (cos_increasing_0 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8)]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]]. -Qed. - -Lemma cos_decr_0 : (x,y:R) ``0<=x``->``x<=PI``->``0<=y``->``y<=PI``->``(cos x)<=(cos y)`` -> ``y<=x``. -Intros; Case (total_order (cos x) (cos y)); Intro H4; [Left; Apply (cos_decreasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Generalize (cos_decreasing_1 x y H H0 H1 H2 H6); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (cos y) H8) | Elim H6; Intro H7; [Right; Symmetry; Assumption | Left; Assumption]] | Elim (Rlt_antirefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5))]]. -Qed. - -Lemma cos_decr_1 : (x,y:R) ``0<=x``->``x<=PI``->``0<=y``->``y<=PI``->``x<=y``->``(cos y)<=(cos x)``. -Intros; Case (total_order x y); Intro H4; [Left; Apply (cos_decreasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (cos x) (cos y)); Intro H6; [Generalize (cos_decreasing_0 x y H H0 H1 H2 H6); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8) | Elim H6; Intro H7; [Right; Symmetry; Assumption | Left; Assumption]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]]. -Qed. - -Lemma tan_incr_0 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI/4``->``(tan x)<=(tan y)``->``x<=y``. -Intros; Case (total_order (tan x) (tan y)); Intro H4; [Left; Apply (tan_increasing_0 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order x y); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (tan_increasing_1 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl (tan y) H8)]] | Elim (Rlt_antirefl (tan x) (Rle_lt_trans (tan x) (tan y) (tan x) H3 H5))]]. -Qed. - -Lemma tan_incr_1 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI/4``->``x<=y``->``(tan x)<=(tan y)``. -Intros; Case (total_order x y); Intro H4; [Left; Apply (tan_increasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (tan x) (tan y)); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (tan_increasing_0 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8)]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]]. +Theorem sin_gt_0 : forall x:R, 0 < x -> x < PI -> 0 < sin x. +intros; elim (SIN x (Rlt_le 0 x H) (Rlt_le x PI H0)); intros H1 _; + case (Rtotal_order x (PI / 2)); intro H2. +apply Rlt_le_trans with (sin_lb x). +apply sin_lb_gt_0; [ assumption | left; assumption ]. +assumption. +elim H2; intro H3. +rewrite H3; rewrite sin_PI2; apply Rlt_0_1. +rewrite <- sin_PI_x; generalize (Ropp_gt_lt_contravar x (PI / 2) H3); + intro H4; generalize (Rplus_lt_compat_l PI (- x) (- (PI / 2)) H4). +replace (PI + - x) with (PI - x). +replace (PI + - (PI / 2)) with (PI / 2). +intro H5; generalize (Ropp_lt_gt_contravar x PI H0); intro H6; + change (- PI < - x) in H6; generalize (Rplus_lt_compat_l PI (- PI) (- x) H6). +rewrite Rplus_opp_r. +replace (PI + - x) with (PI - x). +intro H7; + elim + (SIN (PI - x) (Rlt_le 0 (PI - x) H7) + (Rlt_le (PI - x) PI (Rlt_trans (PI - x) (PI / 2) PI H5 PI2_Rlt_PI))); + intros H8 _; + generalize (sin_lb_gt_0 (PI - x) H7 (Rlt_le (PI - x) (PI / 2) H5)); + intro H9; apply (Rlt_le_trans 0 (sin_lb (PI - x)) (sin (PI - x)) H9 H8). +reflexivity. +pattern PI at 2 in |- *; rewrite double_var; ring. +reflexivity. +Qed. + +Theorem cos_gt_0 : forall x:R, - (PI / 2) < x -> x < PI / 2 -> 0 < cos x. +intros; rewrite cos_sin; + generalize (Rplus_lt_compat_l (PI / 2) (- (PI / 2)) x H). +rewrite Rplus_opp_r; intro H1; + generalize (Rplus_lt_compat_l (PI / 2) x (PI / 2) H0); + rewrite <- double_var; intro H2; apply (sin_gt_0 (PI / 2 + x) H1 H2). +Qed. + +Lemma sin_ge_0 : forall x:R, 0 <= x -> x <= PI -> 0 <= sin x. +intros x H1 H2; elim H1; intro H3; + [ elim H2; intro H4; + [ left; apply (sin_gt_0 x H3 H4) + | rewrite H4; right; symmetry in |- *; apply sin_PI ] + | rewrite <- H3; right; symmetry in |- *; apply sin_0 ]. +Qed. + +Lemma cos_ge_0 : forall x:R, - (PI / 2) <= x -> x <= PI / 2 -> 0 <= cos x. +intros x H1 H2; elim H1; intro H3; + [ elim H2; intro H4; + [ left; apply (cos_gt_0 x H3 H4) + | rewrite H4; right; symmetry in |- *; apply cos_PI2 ] + | rewrite <- H3; rewrite cos_neg; right; symmetry in |- *; apply cos_PI2 ]. +Qed. + +Lemma sin_le_0 : forall x:R, PI <= x -> x <= 2 * PI -> sin x <= 0. +intros x H1 H2; apply Rge_le; rewrite <- Ropp_0; + rewrite <- (Ropp_involutive (sin x)); apply Ropp_le_ge_contravar; + rewrite <- neg_sin; replace (x + PI) with (x - PI + 2 * INR 1 * PI); + [ rewrite (sin_period (x - PI) 1); apply sin_ge_0; + [ replace (x - PI) with (x + - PI); + [ rewrite Rplus_comm; replace 0 with (- PI + PI); + [ apply Rplus_le_compat_l; assumption | ring ] + | ring ] + | replace (x - PI) with (x + - PI); rewrite Rplus_comm; + [ pattern PI at 2 in |- *; replace PI with (- PI + 2 * PI); + [ apply Rplus_le_compat_l; assumption | ring ] + | ring ] ] + | unfold INR in |- *; ring ]. +Qed. + +Lemma cos_le_0 : forall x:R, PI / 2 <= x -> x <= 3 * (PI / 2) -> cos x <= 0. +intros x H1 H2; apply Rge_le; rewrite <- Ropp_0; + rewrite <- (Ropp_involutive (cos x)); apply Ropp_le_ge_contravar; + rewrite <- neg_cos; replace (x + PI) with (x - PI + 2 * INR 1 * PI). +rewrite cos_period; apply cos_ge_0. +replace (- (PI / 2)) with (- PI + PI / 2). +unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_le_compat_l; + assumption. +pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr; + ring. +unfold Rminus in |- *; rewrite Rplus_comm; + replace (PI / 2) with (- PI + 3 * (PI / 2)). +apply Rplus_le_compat_l; assumption. +pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr; + ring. +unfold INR in |- *; ring. +Qed. + +Lemma sin_lt_0 : forall x:R, PI < x -> x < 2 * PI -> sin x < 0. +intros x H1 H2; rewrite <- Ropp_0; rewrite <- (Ropp_involutive (sin x)); + apply Ropp_lt_gt_contravar; rewrite <- neg_sin; + replace (x + PI) with (x - PI + 2 * INR 1 * PI); + [ rewrite (sin_period (x - PI) 1); apply sin_gt_0; + [ replace (x - PI) with (x + - PI); + [ rewrite Rplus_comm; replace 0 with (- PI + PI); + [ apply Rplus_lt_compat_l; assumption | ring ] + | ring ] + | replace (x - PI) with (x + - PI); rewrite Rplus_comm; + [ pattern PI at 2 in |- *; replace PI with (- PI + 2 * PI); + [ apply Rplus_lt_compat_l; assumption | ring ] + | ring ] ] + | unfold INR in |- *; ring ]. +Qed. + +Lemma sin_lt_0_var : forall x:R, - PI < x -> x < 0 -> sin x < 0. +intros; generalize (Rplus_lt_compat_l (2 * PI) (- PI) x H); + replace (2 * PI + - PI) with PI; + [ intro H1; rewrite Rplus_comm in H1; + generalize (Rplus_lt_compat_l (2 * PI) x 0 H0); + intro H2; rewrite (Rplus_comm (2 * PI)) in H2; + rewrite <- (Rplus_comm 0) in H2; rewrite Rplus_0_l in H2; + rewrite <- (sin_period x 1); unfold INR in |- *; + replace (2 * 1 * PI) with (2 * PI); + [ apply (sin_lt_0 (x + 2 * PI) H1 H2) | ring ] + | ring ]. +Qed. + +Lemma cos_lt_0 : forall x:R, PI / 2 < x -> x < 3 * (PI / 2) -> cos x < 0. +intros x H1 H2; rewrite <- Ropp_0; rewrite <- (Ropp_involutive (cos x)); + apply Ropp_lt_gt_contravar; rewrite <- neg_cos; + replace (x + PI) with (x - PI + 2 * INR 1 * PI). +rewrite cos_period; apply cos_gt_0. +replace (- (PI / 2)) with (- PI + PI / 2). +unfold Rminus in |- *; rewrite (Rplus_comm x); apply Rplus_lt_compat_l; + assumption. +pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr; + ring. +unfold Rminus in |- *; rewrite Rplus_comm; + replace (PI / 2) with (- PI + 3 * (PI / 2)). +apply Rplus_lt_compat_l; assumption. +pattern PI at 1 in |- *; rewrite (double_var PI); rewrite Ropp_plus_distr; + ring. +unfold INR in |- *; ring. +Qed. + +Lemma tan_gt_0 : forall x:R, 0 < x -> x < PI / 2 -> 0 < tan x. +intros x H1 H2; unfold tan in |- *; generalize _PI2_RLT_0; + generalize (Rlt_trans 0 x (PI / 2) H1 H2); intros; + generalize (Rlt_trans (- (PI / 2)) 0 x H0 H1); intro H5; + generalize (Rlt_trans x (PI / 2) PI H2 PI2_Rlt_PI); + intro H7; unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply sin_gt_0; assumption. +apply Rinv_0_lt_compat; apply cos_gt_0; assumption. +Qed. + +Lemma tan_lt_0 : forall x:R, - (PI / 2) < x -> x < 0 -> tan x < 0. +intros x H1 H2; unfold tan in |- *; + generalize (cos_gt_0 x H1 (Rlt_trans x 0 (PI / 2) H2 PI2_RGT_0)); + intro H3; rewrite <- Ropp_0; + replace (sin x / cos x) with (- (- sin x / cos x)). +rewrite <- sin_neg; apply Ropp_gt_lt_contravar; + change (0 < sin (- x) / cos x) in |- *; unfold Rdiv in |- *; + apply Rmult_lt_0_compat. +apply sin_gt_0. +rewrite <- Ropp_0; apply Ropp_gt_lt_contravar; assumption. +apply Rlt_trans with (PI / 2). +rewrite <- (Ropp_involutive (PI / 2)); apply Ropp_gt_lt_contravar; assumption. +apply PI2_Rlt_PI. +apply Rinv_0_lt_compat; assumption. +unfold Rdiv in |- *; ring. +Qed. + +Lemma cos_ge_0_3PI2 : + forall x:R, 3 * (PI / 2) <= x -> x <= 2 * PI -> 0 <= cos x. +intros; rewrite <- cos_neg; rewrite <- (cos_period (- x) 1); + unfold INR in |- *; replace (- x + 2 * 1 * PI) with (2 * PI - x). +generalize (Ropp_le_ge_contravar x (2 * PI) H0); intro H1; + generalize (Rge_le (- x) (- (2 * PI)) H1); clear H1; + intro H1; generalize (Rplus_le_compat_l (2 * PI) (- (2 * PI)) (- x) H1). +rewrite Rplus_opp_r. +intro H2; generalize (Ropp_le_ge_contravar (3 * (PI / 2)) x H); intro H3; + generalize (Rge_le (- (3 * (PI / 2))) (- x) H3); clear H3; + intro H3; + generalize (Rplus_le_compat_l (2 * PI) (- x) (- (3 * (PI / 2))) H3). +replace (2 * PI + - (3 * (PI / 2))) with (PI / 2). +intro H4; + apply + (cos_ge_0 (2 * PI - x) + (Rlt_le (- (PI / 2)) (2 * PI - x) + (Rlt_le_trans (- (PI / 2)) 0 (2 * PI - x) _PI2_RLT_0 H2)) H4). +rewrite double; pattern PI at 2 3 in |- *; rewrite double_var; ring. +ring. +Qed. + +Lemma form1 : + forall p q:R, cos p + cos q = 2 * cos ((p - q) / 2) * cos ((p + q) / 2). +intros p q; pattern p at 1 in |- *; + replace p with ((p - q) / 2 + (p + q) / 2). +rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2). +rewrite cos_plus; rewrite cos_minus; ring. +pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +Qed. + +Lemma form2 : + forall p q:R, cos p - cos q = -2 * sin ((p - q) / 2) * sin ((p + q) / 2). +intros p q; pattern p at 1 in |- *; + replace p with ((p - q) / 2 + (p + q) / 2). +rewrite <- (cos_neg q); replace (- q) with ((p - q) / 2 - (p + q) / 2). +rewrite cos_plus; rewrite cos_minus; ring. +pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +Qed. + +Lemma form3 : + forall p q:R, sin p + sin q = 2 * cos ((p - q) / 2) * sin ((p + q) / 2). +intros p q; pattern p at 1 in |- *; + replace p with ((p - q) / 2 + (p + q) / 2). +pattern q at 3 in |- *; replace q with ((p + q) / 2 - (p - q) / 2). +rewrite sin_plus; rewrite sin_minus; ring. +pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +Qed. + +Lemma form4 : + forall p q:R, sin p - sin q = 2 * cos ((p + q) / 2) * sin ((p - q) / 2). +intros p q; pattern p at 1 in |- *; + replace p with ((p - q) / 2 + (p + q) / 2). +pattern q at 3 in |- *; replace q with ((p + q) / 2 - (p - q) / 2). +rewrite sin_plus; rewrite sin_minus; ring. +pattern q at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. +pattern p at 3 in |- *; rewrite double_var; unfold Rdiv in |- *; ring. + +Qed. + +Lemma sin_increasing_0 : + forall x y:R, + - (PI / 2) <= x -> + x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> sin x < sin y -> x < y. +intros; cut (sin ((x - y) / 2) < 0). +intro H4; case (Rtotal_order ((x - y) / 2) 0); intro H5. +assert (Hyp : 0 < 2). +prove_sup0. +generalize (Rmult_lt_compat_l 2 ((x - y) / 2) 0 Hyp H5). +unfold Rdiv in |- *. +rewrite <- Rmult_assoc. +rewrite Rinv_r_simpl_m. +rewrite Rmult_0_r. +clear H5; intro H5; apply Rminus_lt; assumption. +discrR. +elim H5; intro H6. +rewrite H6 in H4; rewrite sin_0 in H4; elim (Rlt_irrefl 0 H4). +change (0 < (x - y) / 2) in H6; + generalize (Ropp_le_ge_contravar (- (PI / 2)) y H1). +rewrite Ropp_involutive. +intro H7; generalize (Rge_le (PI / 2) (- y) H7); clear H7; intro H7; + generalize (Rplus_le_compat x (PI / 2) (- y) (PI / 2) H0 H7). +rewrite <- double_var. +intro H8. +assert (Hyp : 0 < 2). +prove_sup0. +generalize + (Rmult_le_compat_l (/ 2) (x - y) PI + (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H8). +repeat rewrite (Rmult_comm (/ 2)). +intro H9; + generalize + (sin_gt_0 ((x - y) / 2) H6 + (Rle_lt_trans ((x - y) / 2) (PI / 2) PI H9 PI2_Rlt_PI)); + intro H10; + elim + (Rlt_irrefl (sin ((x - y) / 2)) + (Rlt_trans (sin ((x - y) / 2)) 0 (sin ((x - y) / 2)) H4 H10)). +generalize (Rlt_minus (sin x) (sin y) H3); clear H3; intro H3; + rewrite form4 in H3; + generalize (Rplus_le_compat x (PI / 2) y (PI / 2) H0 H2). +rewrite <- double_var. +assert (Hyp : 0 < 2). +prove_sup0. +intro H4; + generalize + (Rmult_le_compat_l (/ 2) (x + y) PI + (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H4). +repeat rewrite (Rmult_comm (/ 2)). +clear H4; intro H4; + generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) y H H1); + replace (- (PI / 2) + - (PI / 2)) with (- PI). +intro H5; + generalize + (Rmult_le_compat_l (/ 2) (- PI) (x + y) + (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H5). +replace (/ 2 * (x + y)) with ((x + y) / 2). +replace (/ 2 * - PI) with (- (PI / 2)). +clear H5; intro H5; elim H4; intro H40. +elim H5; intro H50. +generalize (cos_gt_0 ((x + y) / 2) H50 H40); intro H6; + generalize (Rmult_lt_compat_l 2 0 (cos ((x + y) / 2)) Hyp H6). +rewrite Rmult_0_r. +clear H6; intro H6; case (Rcase_abs (sin ((x - y) / 2))); intro H7. +assumption. +generalize (Rge_le (sin ((x - y) / 2)) 0 H7); clear H7; intro H7; + generalize + (Rmult_le_pos (2 * cos ((x + y) / 2)) (sin ((x - y) / 2)) + (Rlt_le 0 (2 * cos ((x + y) / 2)) H6) H7); intro H8; + generalize + (Rle_lt_trans 0 (2 * cos ((x + y) / 2) * sin ((x - y) / 2)) 0 H8 H3); + intro H9; elim (Rlt_irrefl 0 H9). +rewrite <- H50 in H3; rewrite cos_neg in H3; rewrite cos_PI2 in H3; + rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3; + elim (Rlt_irrefl 0 H3). +unfold Rdiv in H3. +rewrite H40 in H3; assert (H50 := cos_PI2); unfold Rdiv in H50; + rewrite H50 in H3; rewrite Rmult_0_r in H3; rewrite Rmult_0_l in H3; + elim (Rlt_irrefl 0 H3). +unfold Rdiv in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +apply Rmult_comm. +unfold Rdiv in |- *; apply Rmult_comm. +pattern PI at 1 in |- *; rewrite double_var. +rewrite Ropp_plus_distr. +reflexivity. +Qed. + +Lemma sin_increasing_1 : + forall x y:R, + - (PI / 2) <= x -> + x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> x < y -> sin x < sin y. +intros; generalize (Rplus_lt_compat_l x x y H3); intro H4; + generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) x H H); + replace (- (PI / 2) + - (PI / 2)) with (- PI). +assert (Hyp : 0 < 2). +prove_sup0. +intro H5; generalize (Rle_lt_trans (- PI) (x + x) (x + y) H5 H4); intro H6; + generalize + (Rmult_lt_compat_l (/ 2) (- PI) (x + y) (Rinv_0_lt_compat 2 Hyp) H6); + replace (/ 2 * - PI) with (- (PI / 2)). +replace (/ 2 * (x + y)) with ((x + y) / 2). +clear H4 H5 H6; intro H4; generalize (Rplus_lt_compat_l y x y H3); intro H5; + rewrite Rplus_comm in H5; + generalize (Rplus_le_compat y (PI / 2) y (PI / 2) H2 H2). +rewrite <- double_var. +intro H6; generalize (Rlt_le_trans (x + y) (y + y) PI H5 H6); intro H7; + generalize (Rmult_lt_compat_l (/ 2) (x + y) PI (Rinv_0_lt_compat 2 Hyp) H7); + replace (/ 2 * PI) with (PI / 2). +replace (/ 2 * (x + y)) with ((x + y) / 2). +clear H5 H6 H7; intro H5; generalize (Ropp_le_ge_contravar (- (PI / 2)) y H1); + rewrite Ropp_involutive; clear H1; intro H1; + generalize (Rge_le (PI / 2) (- y) H1); clear H1; intro H1; + generalize (Ropp_le_ge_contravar y (PI / 2) H2); clear H2; + intro H2; generalize (Rge_le (- y) (- (PI / 2)) H2); + clear H2; intro H2; generalize (Rplus_lt_compat_l (- y) x y H3); + replace (- y + x) with (x - y). +rewrite Rplus_opp_l. +intro H6; + generalize (Rmult_lt_compat_l (/ 2) (x - y) 0 (Rinv_0_lt_compat 2 Hyp) H6); + rewrite Rmult_0_r; replace (/ 2 * (x - y)) with ((x - y) / 2). +clear H6; intro H6; + generalize (Rplus_le_compat (- (PI / 2)) x (- (PI / 2)) (- y) H H2); + replace (- (PI / 2) + - (PI / 2)) with (- PI). +replace (x + - y) with (x - y). +intro H7; + generalize + (Rmult_le_compat_l (/ 2) (- PI) (x - y) + (Rlt_le 0 (/ 2) (Rinv_0_lt_compat 2 Hyp)) H7); + replace (/ 2 * - PI) with (- (PI / 2)). +replace (/ 2 * (x - y)) with ((x - y) / 2). +clear H7; intro H7; clear H H0 H1 H2; apply Rminus_lt; rewrite form4; + generalize (cos_gt_0 ((x + y) / 2) H4 H5); intro H8; + generalize (Rmult_lt_0_compat 2 (cos ((x + y) / 2)) Hyp H8); + clear H8; intro H8; cut (- PI < - (PI / 2)). +intro H9; + generalize + (sin_lt_0_var ((x - y) / 2) + (Rlt_le_trans (- PI) (- (PI / 2)) ((x - y) / 2) H9 H7) H6); + intro H10; + generalize + (Rmult_lt_gt_compat_neg_l (sin ((x - y) / 2)) 0 ( + 2 * cos ((x + y) / 2)) H10 H8); intro H11; rewrite Rmult_0_r in H11; + rewrite Rmult_comm; assumption. +apply Ropp_lt_gt_contravar; apply PI2_Rlt_PI. +unfold Rdiv in |- *; apply Rmult_comm. +unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse; apply Rmult_comm. +reflexivity. +pattern PI at 1 in |- *; rewrite double_var. +rewrite Ropp_plus_distr. +reflexivity. +unfold Rdiv in |- *; apply Rmult_comm. +unfold Rminus in |- *; apply Rplus_comm. +unfold Rdiv in |- *; apply Rmult_comm. +unfold Rdiv in |- *; apply Rmult_comm. +unfold Rdiv in |- *; apply Rmult_comm. +unfold Rdiv in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +apply Rmult_comm. +pattern PI at 1 in |- *; rewrite double_var. +rewrite Ropp_plus_distr. +reflexivity. +Qed. + +Lemma sin_decreasing_0 : + forall x y:R, + x <= 3 * (PI / 2) -> + PI / 2 <= x -> y <= 3 * (PI / 2) -> PI / 2 <= y -> sin x < sin y -> y < x. +intros; rewrite <- (sin_PI_x x) in H3; rewrite <- (sin_PI_x y) in H3; + generalize (Ropp_lt_gt_contravar (sin (PI - x)) (sin (PI - y)) H3); + repeat rewrite <- sin_neg; + generalize (Rplus_le_compat_l (- PI) x (3 * (PI / 2)) H); + generalize (Rplus_le_compat_l (- PI) (PI / 2) x H0); + generalize (Rplus_le_compat_l (- PI) y (3 * (PI / 2)) H1); + generalize (Rplus_le_compat_l (- PI) (PI / 2) y H2); + replace (- PI + x) with (x - PI). +replace (- PI + PI / 2) with (- (PI / 2)). +replace (- PI + y) with (y - PI). +replace (- PI + 3 * (PI / 2)) with (PI / 2). +replace (- (PI - x)) with (x - PI). +replace (- (PI - y)) with (y - PI). +intros; change (sin (y - PI) < sin (x - PI)) in H8; + apply Rplus_lt_reg_r with (- PI); rewrite Rplus_comm; + replace (y + - PI) with (y - PI). +rewrite Rplus_comm; replace (x + - PI) with (x - PI). +apply (sin_increasing_0 (y - PI) (x - PI) H4 H5 H6 H7 H8). +reflexivity. +reflexivity. +unfold Rminus in |- *; rewrite Ropp_plus_distr. +rewrite Ropp_involutive. +apply Rplus_comm. +unfold Rminus in |- *; rewrite Ropp_plus_distr. +rewrite Ropp_involutive. +apply Rplus_comm. +pattern PI at 2 in |- *; rewrite double_var. +rewrite Ropp_plus_distr. +ring. +unfold Rminus in |- *; apply Rplus_comm. +pattern PI at 2 in |- *; rewrite double_var. +rewrite Ropp_plus_distr. +ring. +unfold Rminus in |- *; apply Rplus_comm. +Qed. + +Lemma sin_decreasing_1 : + forall x y:R, + x <= 3 * (PI / 2) -> + PI / 2 <= x -> y <= 3 * (PI / 2) -> PI / 2 <= y -> x < y -> sin y < sin x. +intros; rewrite <- (sin_PI_x x); rewrite <- (sin_PI_x y); + generalize (Rplus_le_compat_l (- PI) x (3 * (PI / 2)) H); + generalize (Rplus_le_compat_l (- PI) (PI / 2) x H0); + generalize (Rplus_le_compat_l (- PI) y (3 * (PI / 2)) H1); + generalize (Rplus_le_compat_l (- PI) (PI / 2) y H2); + generalize (Rplus_lt_compat_l (- PI) x y H3); + replace (- PI + PI / 2) with (- (PI / 2)). +replace (- PI + y) with (y - PI). +replace (- PI + 3 * (PI / 2)) with (PI / 2). +replace (- PI + x) with (x - PI). +intros; apply Ropp_lt_cancel; repeat rewrite <- sin_neg; + replace (- (PI - x)) with (x - PI). +replace (- (PI - y)) with (y - PI). +apply (sin_increasing_1 (x - PI) (y - PI) H7 H8 H5 H6 H4). +unfold Rminus in |- *; rewrite Ropp_plus_distr. +rewrite Ropp_involutive. +apply Rplus_comm. +unfold Rminus in |- *; rewrite Ropp_plus_distr. +rewrite Ropp_involutive. +apply Rplus_comm. +unfold Rminus in |- *; apply Rplus_comm. +pattern PI at 2 in |- *; rewrite double_var; ring. +unfold Rminus in |- *; apply Rplus_comm. +pattern PI at 2 in |- *; rewrite double_var; ring. +Qed. + +Lemma cos_increasing_0 : + forall x y:R, + PI <= x -> x <= 2 * PI -> PI <= y -> y <= 2 * PI -> cos x < cos y -> x < y. +intros x y H1 H2 H3 H4; rewrite <- (cos_neg x); rewrite <- (cos_neg y); + rewrite <- (cos_period (- x) 1); rewrite <- (cos_period (- y) 1); + unfold INR in |- *; + replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))). +replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))). +repeat rewrite cos_shift; intro H5; + generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI x H1); + generalize (Rplus_le_compat_l (-3 * (PI / 2)) x (2 * PI) H2); + generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI y H3); + generalize (Rplus_le_compat_l (-3 * (PI / 2)) y (2 * PI) H4). +replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)). +replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)). +replace (-3 * (PI / 2) + 2 * PI) with (PI / 2). +replace (-3 * (PI / 2) + PI) with (- (PI / 2)). +clear H1 H2 H3 H4; intros H1 H2 H3 H4; + apply Rplus_lt_reg_r with (-3 * (PI / 2)); + replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)). +replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)). +apply (sin_increasing_0 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H4 H3 H2 H1 H5). +unfold Rminus in |- *. +rewrite Ropp_mult_distr_l_reverse. +apply Rplus_comm. +unfold Rminus in |- *. +rewrite Ropp_mult_distr_l_reverse. +apply Rplus_comm. +pattern PI at 3 in |- *; rewrite double_var. +ring. +rewrite double; pattern PI at 3 4 in |- *; rewrite double_var. +ring. +unfold Rminus in |- *. +rewrite Ropp_mult_distr_l_reverse. +apply Rplus_comm. +unfold Rminus in |- *. +rewrite Ropp_mult_distr_l_reverse. +apply Rplus_comm. +rewrite Rmult_1_r. +rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. +ring. +rewrite Rmult_1_r. +rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. +ring. +Qed. + +Lemma cos_increasing_1 : + forall x y:R, + PI <= x -> x <= 2 * PI -> PI <= y -> y <= 2 * PI -> x < y -> cos x < cos y. +intros x y H1 H2 H3 H4 H5; + generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI x H1); + generalize (Rplus_le_compat_l (-3 * (PI / 2)) x (2 * PI) H2); + generalize (Rplus_le_compat_l (-3 * (PI / 2)) PI y H3); + generalize (Rplus_le_compat_l (-3 * (PI / 2)) y (2 * PI) H4); + generalize (Rplus_lt_compat_l (-3 * (PI / 2)) x y H5); + rewrite <- (cos_neg x); rewrite <- (cos_neg y); + rewrite <- (cos_period (- x) 1); rewrite <- (cos_period (- y) 1); + unfold INR in |- *; replace (-3 * (PI / 2) + x) with (x - 3 * (PI / 2)). +replace (-3 * (PI / 2) + y) with (y - 3 * (PI / 2)). +replace (-3 * (PI / 2) + PI) with (- (PI / 2)). +replace (-3 * (PI / 2) + 2 * PI) with (PI / 2). +clear H1 H2 H3 H4 H5; intros H1 H2 H3 H4 H5; + replace (- x + 2 * 1 * PI) with (PI / 2 - (x - 3 * (PI / 2))). +replace (- y + 2 * 1 * PI) with (PI / 2 - (y - 3 * (PI / 2))). +repeat rewrite cos_shift; + apply + (sin_increasing_1 (x - 3 * (PI / 2)) (y - 3 * (PI / 2)) H5 H4 H3 H2 H1). +rewrite Rmult_1_r. +rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. +ring. +rewrite Rmult_1_r. +rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. +ring. +rewrite (double PI); pattern PI at 3 4 in |- *; rewrite double_var. +ring. +pattern PI at 3 in |- *; rewrite double_var; ring. +unfold Rminus in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +apply Rplus_comm. +unfold Rminus in |- *. +rewrite <- Ropp_mult_distr_l_reverse. +apply Rplus_comm. +Qed. + +Lemma cos_decreasing_0 : + forall x y:R, + 0 <= x -> x <= PI -> 0 <= y -> y <= PI -> cos x < cos y -> y < x. +intros; generalize (Ropp_lt_gt_contravar (cos x) (cos y) H3); + repeat rewrite <- neg_cos; intro H4; + change (cos (y + PI) < cos (x + PI)) in H4; rewrite (Rplus_comm x) in H4; + rewrite (Rplus_comm y) in H4; generalize (Rplus_le_compat_l PI 0 x H); + generalize (Rplus_le_compat_l PI x PI H0); + generalize (Rplus_le_compat_l PI 0 y H1); + generalize (Rplus_le_compat_l PI y PI H2); rewrite Rplus_0_r. +rewrite <- double. +clear H H0 H1 H2 H3; intros; apply Rplus_lt_reg_r with PI; + apply (cos_increasing_0 (PI + y) (PI + x) H0 H H2 H1 H4). +Qed. + +Lemma cos_decreasing_1 : + forall x y:R, + 0 <= x -> x <= PI -> 0 <= y -> y <= PI -> x < y -> cos y < cos x. +intros; apply Ropp_lt_cancel; repeat rewrite <- neg_cos; + rewrite (Rplus_comm x); rewrite (Rplus_comm y); + generalize (Rplus_le_compat_l PI 0 x H); + generalize (Rplus_le_compat_l PI x PI H0); + generalize (Rplus_le_compat_l PI 0 y H1); + generalize (Rplus_le_compat_l PI y PI H2); rewrite Rplus_0_r. +rewrite <- double. +generalize (Rplus_lt_compat_l PI x y H3); clear H H0 H1 H2 H3; intros; + apply (cos_increasing_1 (PI + x) (PI + y) H3 H2 H1 H0 H). +Qed. + +Lemma tan_diff : + forall x y:R, + cos x <> 0 -> cos y <> 0 -> tan x - tan y = sin (x - y) / (cos x * cos y). +intros; unfold tan in |- *; rewrite sin_minus. +unfold Rdiv in |- *. +unfold Rminus in |- *. +rewrite Rmult_plus_distr_r. +rewrite Rinv_mult_distr. +repeat rewrite (Rmult_comm (sin x)). +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (cos y)). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +rewrite (Rmult_comm (sin x)). +apply Rplus_eq_compat_l. +rewrite <- Ropp_mult_distr_l_reverse. +rewrite <- Ropp_mult_distr_r_reverse. +rewrite (Rmult_comm (/ cos x)). +repeat rewrite Rmult_assoc. +rewrite (Rmult_comm (cos x)). +repeat rewrite Rmult_assoc. +rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +reflexivity. +assumption. +assumption. +assumption. +assumption. +Qed. + +Lemma tan_increasing_0 : + forall x y:R, + - (PI / 4) <= x -> + x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> tan x < tan y -> x < y. +intros; generalize PI4_RLT_PI2; intro H4; + generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4); + intro H5; change (- (PI / 2) < - (PI / 4)) in H5; + generalize + (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H) + (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)); intro HP1; + generalize + (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1) + (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)); intro HP2; + generalize + (sym_not_eq + (Rlt_not_eq 0 (cos x) + (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H) + (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)))); + intro H6; + generalize + (sym_not_eq + (Rlt_not_eq 0 (cos y) + (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1) + (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)))); + intro H7; generalize (tan_diff x y H6 H7); intro H8; + generalize (Rlt_minus (tan x) (tan y) H3); clear H3; + intro H3; rewrite H8 in H3; cut (sin (x - y) < 0). +intro H9; generalize (Ropp_le_ge_contravar (- (PI / 4)) y H1); + rewrite Ropp_involutive; intro H10; generalize (Rge_le (PI / 4) (- y) H10); + clear H10; intro H10; generalize (Ropp_le_ge_contravar y (PI / 4) H2); + intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11); + clear H11; intro H11; + generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11); + generalize (Rplus_le_compat x (PI / 4) (- y) (PI / 4) H0 H10); + replace (x + - y) with (x - y). +replace (PI / 4 + PI / 4) with (PI / 2). +replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)). +intros; case (Rtotal_order 0 (x - y)); intro H14. +generalize + (sin_gt_0 (x - y) H14 (Rle_lt_trans (x - y) (PI / 2) PI H12 PI2_Rlt_PI)); + intro H15; elim (Rlt_irrefl 0 (Rlt_trans 0 (sin (x - y)) 0 H15 H9)). +elim H14; intro H15. +rewrite <- H15 in H9; rewrite sin_0 in H9; elim (Rlt_irrefl 0 H9). +apply Rminus_lt; assumption. +pattern PI at 1 in |- *; rewrite double_var. +unfold Rdiv in |- *. +rewrite Rmult_plus_distr_r. +repeat rewrite Rmult_assoc. +rewrite <- Rinv_mult_distr. +rewrite Ropp_plus_distr. +replace 4 with 4. +reflexivity. +ring. +discrR. +discrR. +pattern PI at 1 in |- *; rewrite double_var. +unfold Rdiv in |- *. +rewrite Rmult_plus_distr_r. +repeat rewrite Rmult_assoc. +rewrite <- Rinv_mult_distr. +replace 4 with 4. +reflexivity. +ring. +discrR. +discrR. +reflexivity. +case (Rcase_abs (sin (x - y))); intro H9. +assumption. +generalize (Rge_le (sin (x - y)) 0 H9); clear H9; intro H9; + generalize (Rinv_0_lt_compat (cos x) HP1); intro H10; + generalize (Rinv_0_lt_compat (cos y) HP2); intro H11; + generalize (Rmult_lt_0_compat (/ cos x) (/ cos y) H10 H11); + replace (/ cos x * / cos y) with (/ (cos x * cos y)). +intro H12; + generalize + (Rmult_le_pos (sin (x - y)) (/ (cos x * cos y)) H9 + (Rlt_le 0 (/ (cos x * cos y)) H12)); intro H13; + elim + (Rlt_irrefl 0 (Rle_lt_trans 0 (sin (x - y) * / (cos x * cos y)) 0 H13 H3)). +rewrite Rinv_mult_distr. +reflexivity. +assumption. +assumption. +Qed. + +Lemma tan_increasing_1 : + forall x y:R, + - (PI / 4) <= x -> + x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> x < y -> tan x < tan y. +intros; apply Rminus_lt; generalize PI4_RLT_PI2; intro H4; + generalize (Ropp_lt_gt_contravar (PI / 4) (PI / 2) H4); + intro H5; change (- (PI / 2) < - (PI / 4)) in H5; + generalize + (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H) + (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)); intro HP1; + generalize + (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1) + (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)); intro HP2; + generalize + (sym_not_eq + (Rlt_not_eq 0 (cos x) + (cos_gt_0 x (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) x H5 H) + (Rle_lt_trans x (PI / 4) (PI / 2) H0 H4)))); + intro H6; + generalize + (sym_not_eq + (Rlt_not_eq 0 (cos y) + (cos_gt_0 y (Rlt_le_trans (- (PI / 2)) (- (PI / 4)) y H5 H1) + (Rle_lt_trans y (PI / 4) (PI / 2) H2 H4)))); + intro H7; rewrite (tan_diff x y H6 H7); + generalize (Rinv_0_lt_compat (cos x) HP1); intro H10; + generalize (Rinv_0_lt_compat (cos y) HP2); intro H11; + generalize (Rmult_lt_0_compat (/ cos x) (/ cos y) H10 H11); + replace (/ cos x * / cos y) with (/ (cos x * cos y)). +clear H10 H11; intro H8; generalize (Ropp_le_ge_contravar y (PI / 4) H2); + intro H11; generalize (Rge_le (- y) (- (PI / 4)) H11); + clear H11; intro H11; + generalize (Rplus_le_compat (- (PI / 4)) x (- (PI / 4)) (- y) H H11); + replace (x + - y) with (x - y). +replace (- (PI / 4) + - (PI / 4)) with (- (PI / 2)). +clear H11; intro H9; generalize (Rlt_minus x y H3); clear H3; intro H3; + clear H H0 H1 H2 H4 H5 HP1 HP2; generalize PI2_Rlt_PI; + intro H1; generalize (Ropp_lt_gt_contravar (PI / 2) PI H1); + clear H1; intro H1; + generalize + (sin_lt_0_var (x - y) (Rlt_le_trans (- PI) (- (PI / 2)) (x - y) H1 H9) H3); + intro H2; + generalize + (Rmult_lt_gt_compat_neg_l (sin (x - y)) 0 (/ (cos x * cos y)) H2 H8); + rewrite Rmult_0_r; intro H4; assumption. +pattern PI at 1 in |- *; rewrite double_var. +unfold Rdiv in |- *. +rewrite Rmult_plus_distr_r. +repeat rewrite Rmult_assoc. +rewrite <- Rinv_mult_distr. +replace 4 with 4. +rewrite Ropp_plus_distr. +reflexivity. +ring. +discrR. +discrR. +reflexivity. +apply Rinv_mult_distr; assumption. +Qed. + +Lemma sin_incr_0 : + forall x y:R, + - (PI / 2) <= x -> + x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> sin x <= sin y -> x <= y. +intros; case (Rtotal_order (sin x) (sin y)); intro H4; + [ left; apply (sin_increasing_0 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order x y); intro H6; + [ left; assumption + | elim H6; intro H7; + [ right; assumption + | generalize (sin_increasing_1 y x H1 H2 H H0 H7); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl (sin y) H8) ] ] + | elim (Rlt_irrefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5)) ] ]. +Qed. + +Lemma sin_incr_1 : + forall x y:R, + - (PI / 2) <= x -> + x <= PI / 2 -> - (PI / 2) <= y -> y <= PI / 2 -> x <= y -> sin x <= sin y. +intros; case (Rtotal_order x y); intro H4; + [ left; apply (sin_increasing_1 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order (sin x) (sin y)); intro H6; + [ left; assumption + | elim H6; intro H7; + [ right; assumption + | generalize (sin_increasing_0 y x H1 H2 H H0 H7); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ] + | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ]. +Qed. + +Lemma sin_decr_0 : + forall x y:R, + x <= 3 * (PI / 2) -> + PI / 2 <= x -> + y <= 3 * (PI / 2) -> PI / 2 <= y -> sin x <= sin y -> y <= x. +intros; case (Rtotal_order (sin x) (sin y)); intro H4; + [ left; apply (sin_decreasing_0 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order x y); intro H6; + [ generalize (sin_decreasing_1 x y H H0 H1 H2 H6); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl (sin y) H8) + | elim H6; intro H7; + [ right; symmetry in |- *; assumption | left; assumption ] ] + | elim (Rlt_irrefl (sin x) (Rle_lt_trans (sin x) (sin y) (sin x) H3 H5)) ] ]. +Qed. + +Lemma sin_decr_1 : + forall x y:R, + x <= 3 * (PI / 2) -> + PI / 2 <= x -> + y <= 3 * (PI / 2) -> PI / 2 <= y -> x <= y -> sin y <= sin x. +intros; case (Rtotal_order x y); intro H4; + [ left; apply (sin_decreasing_1 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order (sin x) (sin y)); intro H6; + [ generalize (sin_decreasing_0 x y H H0 H1 H2 H6); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl y H8) + | elim H6; intro H7; + [ right; symmetry in |- *; assumption | left; assumption ] ] + | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ]. +Qed. + +Lemma cos_incr_0 : + forall x y:R, + PI <= x -> + x <= 2 * PI -> PI <= y -> y <= 2 * PI -> cos x <= cos y -> x <= y. +intros; case (Rtotal_order (cos x) (cos y)); intro H4; + [ left; apply (cos_increasing_0 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order x y); intro H6; + [ left; assumption + | elim H6; intro H7; + [ right; assumption + | generalize (cos_increasing_1 y x H1 H2 H H0 H7); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl (cos y) H8) ] ] + | elim (Rlt_irrefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5)) ] ]. +Qed. + +Lemma cos_incr_1 : + forall x y:R, + PI <= x -> + x <= 2 * PI -> PI <= y -> y <= 2 * PI -> x <= y -> cos x <= cos y. +intros; case (Rtotal_order x y); intro H4; + [ left; apply (cos_increasing_1 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order (cos x) (cos y)); intro H6; + [ left; assumption + | elim H6; intro H7; + [ right; assumption + | generalize (cos_increasing_0 y x H1 H2 H H0 H7); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ] + | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ]. +Qed. + +Lemma cos_decr_0 : + forall x y:R, + 0 <= x -> x <= PI -> 0 <= y -> y <= PI -> cos x <= cos y -> y <= x. +intros; case (Rtotal_order (cos x) (cos y)); intro H4; + [ left; apply (cos_decreasing_0 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order x y); intro H6; + [ generalize (cos_decreasing_1 x y H H0 H1 H2 H6); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl (cos y) H8) + | elim H6; intro H7; + [ right; symmetry in |- *; assumption | left; assumption ] ] + | elim (Rlt_irrefl (cos x) (Rle_lt_trans (cos x) (cos y) (cos x) H3 H5)) ] ]. +Qed. + +Lemma cos_decr_1 : + forall x y:R, + 0 <= x -> x <= PI -> 0 <= y -> y <= PI -> x <= y -> cos y <= cos x. +intros; case (Rtotal_order x y); intro H4; + [ left; apply (cos_decreasing_1 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order (cos x) (cos y)); intro H6; + [ generalize (cos_decreasing_0 x y H H0 H1 H2 H6); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl y H8) + | elim H6; intro H7; + [ right; symmetry in |- *; assumption | left; assumption ] ] + | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ]. +Qed. + +Lemma tan_incr_0 : + forall x y:R, + - (PI / 4) <= x -> + x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> tan x <= tan y -> x <= y. +intros; case (Rtotal_order (tan x) (tan y)); intro H4; + [ left; apply (tan_increasing_0 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order x y); intro H6; + [ left; assumption + | elim H6; intro H7; + [ right; assumption + | generalize (tan_increasing_1 y x H1 H2 H H0 H7); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl (tan y) H8) ] ] + | elim (Rlt_irrefl (tan x) (Rle_lt_trans (tan x) (tan y) (tan x) H3 H5)) ] ]. +Qed. + +Lemma tan_incr_1 : + forall x y:R, + - (PI / 4) <= x -> + x <= PI / 4 -> - (PI / 4) <= y -> y <= PI / 4 -> x <= y -> tan x <= tan y. +intros; case (Rtotal_order x y); intro H4; + [ left; apply (tan_increasing_1 x y H H0 H1 H2 H4) + | elim H4; intro H5; + [ case (Rtotal_order (tan x) (tan y)); intro H6; + [ left; assumption + | elim H6; intro H7; + [ right; assumption + | generalize (tan_increasing_0 y x H1 H2 H H0 H7); intro H8; + rewrite H5 in H8; elim (Rlt_irrefl y H8) ] ] + | elim (Rlt_irrefl x (Rle_lt_trans x y x H3 H5)) ] ]. Qed. (**********) -Lemma sin_eq_0_1 : (x:R) (EXT k:Z | x==(Rmult (IZR k) PI)) -> (sin x)==R0. -Intros. -Elim H; Intros. -Apply (Zcase_sign x0). -Intro. -Rewrite H1 in H0. -Simpl in H0. -Rewrite H0; Rewrite Rmult_Ol; Apply sin_0. -Intro. -Cut `0<=x0`. -Intro. -Elim (IZN x0 H2); Intros. -Rewrite H3 in H0. -Rewrite <- INR_IZR_INZ in H0. -Rewrite H0. -Elim (even_odd_cor x1); Intros. -Elim H4; Intro. -Rewrite H5. -Rewrite mult_INR. -Simpl. -Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``). -Rewrite sin_period. -Apply sin_0. -Rewrite H5. -Rewrite S_INR; Rewrite mult_INR. -Simpl. -Rewrite Rmult_Rplus_distrl. -Rewrite Rmult_1l; Rewrite sin_plus. -Rewrite sin_PI. -Rewrite Rmult_Or. -Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``). -Rewrite sin_period. -Rewrite sin_0; Ring. -Apply le_IZR. -Left; Apply IZR_lt. -Assert H2 := Zgt_iff_lt. -Elim (H2 x0 `0`); Intros. -Apply H3; Assumption. -Intro. -Rewrite H0. -Replace ``(sin ((IZR x0)*PI))`` with ``-(sin (-(IZR x0)*PI))``. -Cut `0<=-x0`. -Intro. -Rewrite <- Ropp_Ropp_IZR. -Elim (IZN `-x0` H2); Intros. -Rewrite H3. -Rewrite <- INR_IZR_INZ. -Elim (even_odd_cor x1); Intros. -Elim H4; Intro. -Rewrite H5. -Rewrite mult_INR. -Simpl. -Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``). -Rewrite sin_period. -Rewrite sin_0; Ring. -Rewrite H5. -Rewrite S_INR; Rewrite mult_INR. -Simpl. -Rewrite Rmult_Rplus_distrl. -Rewrite Rmult_1l; Rewrite sin_plus. -Rewrite sin_PI. -Rewrite Rmult_Or. -Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``). -Rewrite sin_period. -Rewrite sin_0; Ring. -Apply le_IZR. -Apply Rle_anti_compatibility with ``(IZR x0)``. -Rewrite Rplus_Or. -Rewrite Ropp_Ropp_IZR. -Rewrite Rplus_Ropp_r. -Left; Replace R0 with (IZR `0`); [Apply IZR_lt | Reflexivity]. -Assumption. -Rewrite <- sin_neg. -Rewrite Ropp_mul1. -Rewrite Ropp_Ropp. -Reflexivity. -Qed. - -Lemma sin_eq_0_0 : (x:R) (sin x)==R0 -> (EXT k:Z | x==(Rmult (IZR k) PI)). -Intros. -Assert H0 := (euclidian_division x PI PI_neq0). -Elim H0; Intros q H1. -Elim H1; Intros r H2. -Exists q. -Cut r==R0. -Intro. -Elim H2; Intros H4 _; Rewrite H4; Rewrite H3. -Apply Rplus_Or. -Elim H2; Intros. -Rewrite H3 in H. -Rewrite sin_plus in H. -Cut ``(sin ((IZR q)*PI))==0``. -Intro. -Rewrite H5 in H. -Rewrite Rmult_Ol in H. -Rewrite Rplus_Ol in H. -Assert H6 := (without_div_Od ? ? H). -Elim H6; Intro. -Assert H8 := (sin2_cos2 ``(IZR q)*PI``). -Rewrite H5 in H8; Rewrite H7 in H8. -Rewrite Rsqr_O in H8. -Rewrite Rplus_Or in H8. -Elim R1_neq_R0; Symmetry; Assumption. -Cut r==R0\/``0<r<PI``. -Intro; Elim H8; Intro. -Assumption. -Elim H9; Intros. -Assert H12 := (sin_gt_0 ? H10 H11). -Rewrite H7 in H12; Elim (Rlt_antirefl ? H12). -Rewrite Rabsolu_right in H4. -Elim H4; Intros. -Case (total_order R0 r); Intro. -Right; Split; Assumption. -Elim H10; Intro. -Left; Symmetry; Assumption. -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H8 H11)). -Apply Rle_sym1. -Left; Apply PI_RGT_0. -Apply sin_eq_0_1. -Exists q; Reflexivity. -Qed. - -Lemma cos_eq_0_0 : (x:R) (cos x)==R0 -> (EXT k : Z | ``x==(IZR k)*PI+PI/2``). -Intros x H; Rewrite -> cos_sin in H; Generalize (sin_eq_0_0 (Rplus (Rdiv PI (INR (2))) x) H); Intro H2; Elim H2; Intros x0 H3; Exists (Zminus x0 (inject_nat (S O))); Rewrite <- Z_R_minus; Ring; Rewrite Rmult_sym; Rewrite <- H3; Unfold INR. -Rewrite (double_var ``-PI``); Unfold Rdiv; Ring. -Qed. - -Lemma cos_eq_0_1 : (x:R) (EXT k : Z | ``x==(IZR k)*PI+PI/2``) -> ``(cos x)==0``. -Intros x H1; Rewrite cos_sin; Elim H1; Intros x0 H2; Rewrite H2; Replace ``PI/2+((IZR x0)*PI+PI/2)`` with ``(IZR x0)*PI+PI``. -Rewrite neg_sin; Rewrite <- Ropp_O. -Apply eq_Ropp; Apply sin_eq_0_1; Exists x0; Reflexivity. -Pattern 2 PI; Rewrite (double_var PI); Ring. -Qed. - -Lemma sin_eq_O_2PI_0 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``(sin x)==0`` -> ``x==0``\/``x==PI``\/``x==2*PI``. -Intros; Generalize (sin_eq_0_0 x H1); Intro. -Elim H2; Intros k0 H3. -Case (total_order PI x); Intro. -Rewrite H3 in H4; Rewrite H3 in H0. -Right; Right. -Generalize (Rlt_monotony_r ``/PI`` ``PI`` ``(IZR k0)*PI`` (Rlt_Rinv ``PI`` PI_RGT_0) H4); Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Intro; Generalize (Rle_monotony_r ``/PI`` ``(IZR k0)*PI`` ``2*PI`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv ``PI`` PI_RGT_0)) H0); Repeat Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. -Repeat Rewrite Rmult_1r; Intro; Generalize (Rlt_compatibility (IZR `-2`) ``1`` (IZR k0) H5); Rewrite <- plus_IZR. -Replace ``(IZR (NEG (xO xH)))+1`` with ``-1``. -Intro; Generalize (Rle_compatibility (IZR `-2`) (IZR k0) ``2`` H6); Rewrite <- plus_IZR. -Replace ``(IZR (NEG (xO xH)))+2`` with ``0``. -Intro; Cut ``-1 < (IZR (Zplus (NEG (xO xH)) k0)) < 1``. -Intro; Generalize (one_IZR_lt1 (Zplus (NEG (xO xH)) k0) H9); Intro. -Cut k0=`2`. -Intro; Rewrite H11 in H3; Rewrite H3; Simpl. -Reflexivity. -Rewrite <- (Zplus_inverse_l `2`) in H10; Generalize (Zsimpl_plus_l `-2` k0 `2` H10); Intro; Assumption. -Split. -Assumption. -Apply Rle_lt_trans with ``0``. -Assumption. -Apply Rlt_R0_R1. -Simpl; Ring. -Simpl; Ring. -Apply PI_neq0. -Apply PI_neq0. -Elim H4; Intro. -Right; Left. -Symmetry; Assumption. -Left. -Rewrite H3 in H5; Rewrite H3 in H; Generalize (Rlt_monotony_r ``/PI`` ``(IZR k0)*PI`` PI (Rlt_Rinv ``PI`` PI_RGT_0) H5); Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Intro; Generalize (Rle_monotony_r ``/PI`` ``0`` ``(IZR k0)*PI`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv ``PI`` PI_RGT_0)) H); Repeat Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Rewrite Rmult_Ol; Intro. -Cut ``-1 < (IZR (k0)) < 1``. -Intro; Generalize (one_IZR_lt1 k0 H8); Intro; Rewrite H9 in H3; Rewrite H3; Simpl; Apply Rmult_Ol. -Split. -Apply Rlt_le_trans with ``0``. -Rewrite <- Ropp_O; Apply Rgt_Ropp; Apply Rlt_R0_R1. -Assumption. -Assumption. -Apply PI_neq0. -Apply PI_neq0. -Qed. - -Lemma sin_eq_O_2PI_1 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``x==0``\/``x==PI``\/``x==2*PI`` -> ``(sin x)==0``. -Intros x H1 H2 H3; Elim H3; Intro H4; [ Rewrite H4; Rewrite -> sin_0; Reflexivity | Elim H4; Intro H5; [Rewrite H5; Rewrite -> sin_PI; Reflexivity | Rewrite H5; Rewrite -> sin_2PI; Reflexivity]]. -Qed. - -Lemma cos_eq_0_2PI_0 : (x:R) ``R0<=x`` -> ``x<=2*PI`` -> ``(cos x)==0`` -> ``x==(PI/2)``\/``x==3*(PI/2)``. -Intros; Case (total_order x ``3*(PI/2)``); Intro. -Rewrite cos_sin in H1. -Cut ``0<=PI/2+x``. -Cut ``PI/2+x<=2*PI``. -Intros; Generalize (sin_eq_O_2PI_0 ``PI/2+x`` H4 H3 H1); Intros. -Decompose [or] H5. -Generalize (Rle_compatibility ``PI/2`` ``0`` x H); Rewrite Rplus_Or; Rewrite H6; Intro. -Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``PI/2`` ``0`` PI2_RGT_0 H7)). -Left. -Generalize (Rplus_plus_r ``-(PI/2)`` ``PI/2+x`` PI H7). -Replace ``-(PI/2)+(PI/2+x)`` with x. -Replace ``-(PI/2)+PI`` with ``PI/2``. -Intro; Assumption. -Pattern 3 PI; Rewrite (double_var PI); Ring. -Ring. -Right. -Generalize (Rplus_plus_r ``-(PI/2)`` ``PI/2+x`` ``2*PI`` H7). -Replace ``-(PI/2)+(PI/2+x)`` with x. -Replace ``-(PI/2)+2*PI`` with ``3*(PI/2)``. -Intro; Assumption. -Rewrite double; Pattern 3 4 PI; Rewrite (double_var PI); Ring. -Ring. -Left; Replace ``2*PI`` with ``PI/2+3*(PI/2)``. -Apply Rlt_compatibility; Assumption. -Rewrite (double PI); Pattern 3 4 PI; Rewrite (double_var PI); Ring. -Apply ge0_plus_ge0_is_ge0. -Left; Unfold Rdiv; Apply Rmult_lt_pos. -Apply PI_RGT_0. -Apply Rlt_Rinv; Sup0. -Assumption. -Elim H2; Intro. -Right; Assumption. -Generalize (cos_eq_0_0 x H1); Intro; Elim H4; Intros k0 H5. -Rewrite H5 in H3; Rewrite H5 in H0; Generalize (Rlt_compatibility ``-(PI/2)`` ``3*PI/2`` ``(IZR k0)*PI+PI/2`` H3); Generalize (Rle_compatibility ``-(PI/2)`` ``(IZR k0)*PI+PI/2`` ``2*PI`` H0). -Replace ``-(PI/2)+3*PI/2`` with PI. -Replace ``-(PI/2)+((IZR k0)*PI+PI/2)`` with ``(IZR k0)*PI``. -Replace ``-(PI/2)+2*PI`` with ``3*(PI/2)``. -Intros; Generalize (Rlt_monotony ``/PI`` ``PI`` ``(IZR k0)*PI`` (Rlt_Rinv PI PI_RGT_0) H7); Generalize (Rle_monotony ``/PI`` ``(IZR k0)*PI`` ``3*(PI/2)`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv PI PI_RGT_0)) H6). -Replace ``/PI*((IZR k0)*PI)`` with (IZR k0). -Replace ``/PI*(3*PI/2)`` with ``3*/2``. -Rewrite <- Rinv_l_sym. -Intros; Generalize (Rlt_compatibility (IZR `-2`) ``1`` (IZR k0) H9); Rewrite <- plus_IZR. -Replace ``(IZR (NEG (xO xH)))+1`` with ``-1``. -Intro; Generalize (Rle_compatibility (IZR `-2`) (IZR k0) ``3*/2`` H8); Rewrite <- plus_IZR. -Replace ``(IZR (NEG (xO xH)))+2`` with ``0``. -Intro; Cut `` -1 < (IZR (Zplus (NEG (xO xH)) k0)) < 1``. -Intro; Generalize (one_IZR_lt1 (Zplus (NEG (xO xH)) k0) H12); Intro. -Cut k0=`2`. -Intro; Rewrite H14 in H8. -Assert Hyp : ``0<2``. -Sup0. -Generalize (Rle_monotony ``2`` ``(IZR (POS (xO xH)))`` ``3*/2`` (Rlt_le ``0`` ``2`` Hyp) H8); Simpl. -Replace ``2*2`` with ``4``. -Replace ``2*(3*/2)`` with ``3``. -Intro; Cut ``3<4``. -Intro; Elim (Rlt_antirefl ``3`` (Rlt_le_trans ``3`` ``4`` ``3`` H16 H15)). -Generalize (Rlt_compatibility ``3`` ``0`` ``1`` Rlt_R0_R1); Rewrite Rplus_Or. -Replace ``3+1`` with ``4``. -Intro; Assumption. -Ring. -Symmetry; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m. -DiscrR. -Ring. -Rewrite <- (Zplus_inverse_l `2`) in H13; Generalize (Zsimpl_plus_l `-2` k0 `2` H13); Intro; Assumption. -Split. -Assumption. -Apply Rle_lt_trans with ``(IZR (NEG (xO xH)))+3*/2``. -Assumption. -Simpl; Replace ``-2+3*/2`` with ``-(1*/2)``. -Apply Rlt_trans with ``0``. -Rewrite <- Ropp_O; Apply Rlt_Ropp. -Apply Rmult_lt_pos; [Apply Rlt_R0_R1 | Apply Rlt_Rinv; Sup0]. -Apply Rlt_R0_R1. -Rewrite Rmult_1l; Apply r_Rmult_mult with ``2``. -Rewrite Ropp_mul3; Rewrite <- Rinv_r_sym. -Rewrite Rmult_Rplus_distr; Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m. -Ring. -DiscrR. -DiscrR. -DiscrR. -Simpl; Ring. -Simpl; Ring. -Apply PI_neq0. -Unfold Rdiv; Pattern 1 ``3``; Rewrite (Rmult_sym ``3``); Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Apply Rmult_sym. -Apply PI_neq0. -Symmetry; Rewrite (Rmult_sym ``/PI``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Apply Rmult_1r. -Apply PI_neq0. -Rewrite double; Pattern 3 4 PI; Rewrite double_var; Ring. -Ring. -Pattern 1 PI; Rewrite double_var; Ring. -Qed. - -Lemma cos_eq_0_2PI_1 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``x==PI/2``\/``x==3*(PI/2)`` -> ``(cos x)==0``. -Intros x H1 H2 H3; Elim H3; Intro H4; [ Rewrite H4; Rewrite -> cos_PI2; Reflexivity | Rewrite H4; Rewrite -> cos_3PI2; Reflexivity ]. -Qed. +Lemma sin_eq_0_1 : forall x:R, ( exists k : Z | x = IZR k * PI) -> sin x = 0. +intros. +elim H; intros. +apply (Zcase_sign x0). +intro. +rewrite H1 in H0. +simpl in H0. +rewrite H0; rewrite Rmult_0_l; apply sin_0. +intro. +cut (0 <= x0)%Z. +intro. +elim (IZN x0 H2); intros. +rewrite H3 in H0. +rewrite <- INR_IZR_INZ in H0. +rewrite H0. +elim (even_odd_cor x1); intros. +elim H4; intro. +rewrite H5. +rewrite mult_INR. +simpl in |- *. +rewrite <- (Rplus_0_l (2 * INR x2 * PI)). +rewrite sin_period. +apply sin_0. +rewrite H5. +rewrite S_INR; rewrite mult_INR. +simpl in |- *. +rewrite Rmult_plus_distr_r. +rewrite Rmult_1_l; rewrite sin_plus. +rewrite sin_PI. +rewrite Rmult_0_r. +rewrite <- (Rplus_0_l (2 * INR x2 * PI)). +rewrite sin_period. +rewrite sin_0; ring. +apply le_IZR. +left; apply IZR_lt. +assert (H2 := Zorder.Zgt_iff_lt). +elim (H2 x0 0%Z); intros. +apply H3; assumption. +intro. +rewrite H0. +replace (sin (IZR x0 * PI)) with (- sin (- IZR x0 * PI)). +cut (0 <= - x0)%Z. +intro. +rewrite <- Ropp_Ropp_IZR. +elim (IZN (- x0) H2); intros. +rewrite H3. +rewrite <- INR_IZR_INZ. +elim (even_odd_cor x1); intros. +elim H4; intro. +rewrite H5. +rewrite mult_INR. +simpl in |- *. +rewrite <- (Rplus_0_l (2 * INR x2 * PI)). +rewrite sin_period. +rewrite sin_0; ring. +rewrite H5. +rewrite S_INR; rewrite mult_INR. +simpl in |- *. +rewrite Rmult_plus_distr_r. +rewrite Rmult_1_l; rewrite sin_plus. +rewrite sin_PI. +rewrite Rmult_0_r. +rewrite <- (Rplus_0_l (2 * INR x2 * PI)). +rewrite sin_period. +rewrite sin_0; ring. +apply le_IZR. +apply Rplus_le_reg_l with (IZR x0). +rewrite Rplus_0_r. +rewrite Ropp_Ropp_IZR. +rewrite Rplus_opp_r. +left; replace 0 with (IZR 0); [ apply IZR_lt | reflexivity ]. +assumption. +rewrite <- sin_neg. +rewrite Ropp_mult_distr_l_reverse. +rewrite Ropp_involutive. +reflexivity. +Qed. + +Lemma sin_eq_0_0 : forall x:R, sin x = 0 -> exists k : Z | x = IZR k * PI. +intros. +assert (H0 := euclidian_division x PI PI_neq0). +elim H0; intros q H1. +elim H1; intros r H2. +exists q. +cut (r = 0). +intro. +elim H2; intros H4 _; rewrite H4; rewrite H3. +apply Rplus_0_r. +elim H2; intros. +rewrite H3 in H. +rewrite sin_plus in H. +cut (sin (IZR q * PI) = 0). +intro. +rewrite H5 in H. +rewrite Rmult_0_l in H. +rewrite Rplus_0_l in H. +assert (H6 := Rmult_integral _ _ H). +elim H6; intro. +assert (H8 := sin2_cos2 (IZR q * PI)). +rewrite H5 in H8; rewrite H7 in H8. +rewrite Rsqr_0 in H8. +rewrite Rplus_0_r in H8. +elim R1_neq_R0; symmetry in |- *; assumption. +cut (r = 0 \/ 0 < r < PI). +intro; elim H8; intro. +assumption. +elim H9; intros. +assert (H12 := sin_gt_0 _ H10 H11). +rewrite H7 in H12; elim (Rlt_irrefl _ H12). +rewrite Rabs_right in H4. +elim H4; intros. +case (Rtotal_order 0 r); intro. +right; split; assumption. +elim H10; intro. +left; symmetry in |- *; assumption. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H8 H11)). +apply Rle_ge. +left; apply PI_RGT_0. +apply sin_eq_0_1. +exists q; reflexivity. +Qed. + +Lemma cos_eq_0_0 : + forall x:R, cos x = 0 -> exists k : Z | x = IZR k * PI + PI / 2. +intros x H; rewrite cos_sin in H; generalize (sin_eq_0_0 (PI / INR 2 + x) H); + intro H2; elim H2; intros x0 H3; exists (x0 - Z_of_nat 1)%Z; + rewrite <- Z_R_minus; ring; rewrite Rmult_comm; rewrite <- H3; + unfold INR in |- *. +rewrite (double_var (- PI)); unfold Rdiv in |- *; ring. +Qed. + +Lemma cos_eq_0_1 : + forall x:R, ( exists k : Z | x = IZR k * PI + PI / 2) -> cos x = 0. +intros x H1; rewrite cos_sin; elim H1; intros x0 H2; rewrite H2; + replace (PI / 2 + (IZR x0 * PI + PI / 2)) with (IZR x0 * PI + PI). +rewrite neg_sin; rewrite <- Ropp_0. +apply Ropp_eq_compat; apply sin_eq_0_1; exists x0; reflexivity. +pattern PI at 2 in |- *; rewrite (double_var PI); ring. +Qed. + +Lemma sin_eq_O_2PI_0 : + forall x:R, + 0 <= x -> x <= 2 * PI -> sin x = 0 -> x = 0 \/ x = PI \/ x = 2 * PI. +intros; generalize (sin_eq_0_0 x H1); intro. +elim H2; intros k0 H3. +case (Rtotal_order PI x); intro. +rewrite H3 in H4; rewrite H3 in H0. +right; right. +generalize + (Rmult_lt_compat_r (/ PI) PI (IZR k0 * PI) (Rinv_0_lt_compat PI PI_RGT_0) H4); + rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; intro; + generalize + (Rmult_le_compat_r (/ PI) (IZR k0 * PI) (2 * PI) + (Rlt_le 0 (/ PI) (Rinv_0_lt_compat PI PI_RGT_0)) H0); + repeat rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym. +repeat rewrite Rmult_1_r; intro; + generalize (Rplus_lt_compat_l (IZR (-2)) 1 (IZR k0) H5); + rewrite <- plus_IZR. +replace (IZR (-2) + 1) with (-1). +intro; generalize (Rplus_le_compat_l (IZR (-2)) (IZR k0) 2 H6); + rewrite <- plus_IZR. +replace (IZR (-2) + 2) with 0. +intro; cut (-1 < IZR (-2 + k0) < 1). +intro; generalize (one_IZR_lt1 (-2 + k0) H9); intro. +cut (k0 = 2%Z). +intro; rewrite H11 in H3; rewrite H3; simpl in |- *. +reflexivity. +rewrite <- (Zplus_opp_l 2) in H10; generalize (Zplus_reg_l (-2) k0 2 H10); + intro; assumption. +split. +assumption. +apply Rle_lt_trans with 0. +assumption. +apply Rlt_0_1. +simpl in |- *; ring. +simpl in |- *; ring. +apply PI_neq0. +apply PI_neq0. +elim H4; intro. +right; left. +symmetry in |- *; assumption. +left. +rewrite H3 in H5; rewrite H3 in H; + generalize + (Rmult_lt_compat_r (/ PI) (IZR k0 * PI) PI (Rinv_0_lt_compat PI PI_RGT_0) + H5); rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; intro; + generalize + (Rmult_le_compat_r (/ PI) 0 (IZR k0 * PI) + (Rlt_le 0 (/ PI) (Rinv_0_lt_compat PI PI_RGT_0)) H); + repeat rewrite Rmult_assoc; repeat rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; rewrite Rmult_0_l; intro. +cut (-1 < IZR k0 < 1). +intro; generalize (one_IZR_lt1 k0 H8); intro; rewrite H9 in H3; rewrite H3; + simpl in |- *; apply Rmult_0_l. +split. +apply Rlt_le_trans with 0. +rewrite <- Ropp_0; apply Ropp_gt_lt_contravar; apply Rlt_0_1. +assumption. +assumption. +apply PI_neq0. +apply PI_neq0. +Qed. + +Lemma sin_eq_O_2PI_1 : + forall x:R, + 0 <= x -> x <= 2 * PI -> x = 0 \/ x = PI \/ x = 2 * PI -> sin x = 0. +intros x H1 H2 H3; elim H3; intro H4; + [ rewrite H4; rewrite sin_0; reflexivity + | elim H4; intro H5; + [ rewrite H5; rewrite sin_PI; reflexivity + | rewrite H5; rewrite sin_2PI; reflexivity ] ]. +Qed. + +Lemma cos_eq_0_2PI_0 : + forall x:R, + 0 <= x -> x <= 2 * PI -> cos x = 0 -> x = PI / 2 \/ x = 3 * (PI / 2). +intros; case (Rtotal_order x (3 * (PI / 2))); intro. +rewrite cos_sin in H1. +cut (0 <= PI / 2 + x). +cut (PI / 2 + x <= 2 * PI). +intros; generalize (sin_eq_O_2PI_0 (PI / 2 + x) H4 H3 H1); intros. +decompose [or] H5. +generalize (Rplus_le_compat_l (PI / 2) 0 x H); rewrite Rplus_0_r; rewrite H6; + intro. +elim (Rlt_irrefl 0 (Rlt_le_trans 0 (PI / 2) 0 PI2_RGT_0 H7)). +left. +generalize (Rplus_eq_compat_l (- (PI / 2)) (PI / 2 + x) PI H7). +replace (- (PI / 2) + (PI / 2 + x)) with x. +replace (- (PI / 2) + PI) with (PI / 2). +intro; assumption. +pattern PI at 3 in |- *; rewrite (double_var PI); ring. +ring. +right. +generalize (Rplus_eq_compat_l (- (PI / 2)) (PI / 2 + x) (2 * PI) H7). +replace (- (PI / 2) + (PI / 2 + x)) with x. +replace (- (PI / 2) + 2 * PI) with (3 * (PI / 2)). +intro; assumption. +rewrite double; pattern PI at 3 4 in |- *; rewrite (double_var PI); ring. +ring. +left; replace (2 * PI) with (PI / 2 + 3 * (PI / 2)). +apply Rplus_lt_compat_l; assumption. +rewrite (double PI); pattern PI at 3 4 in |- *; rewrite (double_var PI); ring. +apply Rplus_le_le_0_compat. +left; unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply PI_RGT_0. +apply Rinv_0_lt_compat; prove_sup0. +assumption. +elim H2; intro. +right; assumption. +generalize (cos_eq_0_0 x H1); intro; elim H4; intros k0 H5. +rewrite H5 in H3; rewrite H5 in H0; + generalize + (Rplus_lt_compat_l (- (PI / 2)) (3 * (PI / 2)) (IZR k0 * PI + PI / 2) H3); + generalize + (Rplus_le_compat_l (- (PI / 2)) (IZR k0 * PI + PI / 2) (2 * PI) H0). +replace (- (PI / 2) + 3 * (PI / 2)) with PI. +replace (- (PI / 2) + (IZR k0 * PI + PI / 2)) with (IZR k0 * PI). +replace (- (PI / 2) + 2 * PI) with (3 * (PI / 2)). +intros; + generalize + (Rmult_lt_compat_l (/ PI) PI (IZR k0 * PI) (Rinv_0_lt_compat PI PI_RGT_0) + H7); + generalize + (Rmult_le_compat_l (/ PI) (IZR k0 * PI) (3 * (PI / 2)) + (Rlt_le 0 (/ PI) (Rinv_0_lt_compat PI PI_RGT_0)) H6). +replace (/ PI * (IZR k0 * PI)) with (IZR k0). +replace (/ PI * (3 * (PI / 2))) with (3 * / 2). +rewrite <- Rinv_l_sym. +intros; generalize (Rplus_lt_compat_l (IZR (-2)) 1 (IZR k0) H9); + rewrite <- plus_IZR. +replace (IZR (-2) + 1) with (-1). +intro; generalize (Rplus_le_compat_l (IZR (-2)) (IZR k0) (3 * / 2) H8); + rewrite <- plus_IZR. +replace (IZR (-2) + 2) with 0. +intro; cut (-1 < IZR (-2 + k0) < 1). +intro; generalize (one_IZR_lt1 (-2 + k0) H12); intro. +cut (k0 = 2%Z). +intro; rewrite H14 in H8. +assert (Hyp : 0 < 2). +prove_sup0. +generalize (Rmult_le_compat_l 2 (IZR 2) (3 * / 2) (Rlt_le 0 2 Hyp) H8); + simpl in |- *. +replace 4 with 4. +replace (2 * (3 * / 2)) with 3. +intro; cut (3 < 4). +intro; elim (Rlt_irrefl 3 (Rlt_le_trans 3 4 3 H16 H15)). +generalize (Rplus_lt_compat_l 3 0 1 Rlt_0_1); rewrite Rplus_0_r. +replace (3 + 1) with 4. +intro; assumption. +ring. +symmetry in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. +discrR. +ring. +rewrite <- (Zplus_opp_l 2) in H13; generalize (Zplus_reg_l (-2) k0 2 H13); + intro; assumption. +split. +assumption. +apply Rle_lt_trans with (IZR (-2) + 3 * / 2). +assumption. +simpl in |- *; replace (-2 + 3 * / 2) with (- (1 * / 2)). +apply Rlt_trans with 0. +rewrite <- Ropp_0; apply Ropp_lt_gt_contravar. +apply Rmult_lt_0_compat; + [ apply Rlt_0_1 | apply Rinv_0_lt_compat; prove_sup0 ]. +apply Rlt_0_1. +rewrite Rmult_1_l; apply Rmult_eq_reg_l with 2. +rewrite Ropp_mult_distr_r_reverse; rewrite <- Rinv_r_sym. +rewrite Rmult_plus_distr_l; rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m. +ring. +discrR. +discrR. +discrR. +simpl in |- *; ring. +simpl in |- *; ring. +apply PI_neq0. +unfold Rdiv in |- *; pattern 3 at 1 in |- *; rewrite (Rmult_comm 3); + repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; apply Rmult_comm. +apply PI_neq0. +symmetry in |- *; rewrite (Rmult_comm (/ PI)); rewrite Rmult_assoc; + rewrite <- Rinv_r_sym. +apply Rmult_1_r. +apply PI_neq0. +rewrite double; pattern PI at 3 4 in |- *; rewrite double_var; ring. +ring. +pattern PI at 1 in |- *; rewrite double_var; ring. +Qed. + +Lemma cos_eq_0_2PI_1 : + forall x:R, + 0 <= x -> x <= 2 * PI -> x = PI / 2 \/ x = 3 * (PI / 2) -> cos x = 0. +intros x H1 H2 H3; elim H3; intro H4; + [ rewrite H4; rewrite cos_PI2; reflexivity + | rewrite H4; rewrite cos_3PI2; reflexivity ]. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Rtrigo_alt.v b/theories/Reals/Rtrigo_alt.v index 4fdc39106..c1ffc68ea 100644 --- a/theories/Reals/Rtrigo_alt.v +++ b/theories/Reals/Rtrigo_alt.v @@ -8,287 +8,419 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo_def. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo_def. Open Local Scope R_scope. (*****************************************************************) (* Using series definitions of cos and sin *) (*****************************************************************) -Definition sin_term [a:R] : nat->R := [i:nat] ``(pow (-1) i)*(pow a (plus (mult (S (S O)) i) (S O)))/(INR (fact (plus (mult (S (S O)) i) (S O))))``. +Definition sin_term (a:R) (i:nat) : R := + (-1) ^ i * (a ^ (2 * i + 1) / INR (fact (2 * i + 1))). -Definition cos_term [a:R] : nat->R := [i:nat] ``(pow (-1) i)*(pow a (mult (S (S O)) i))/(INR (fact (mult (S (S O)) i)))``. +Definition cos_term (a:R) (i:nat) : R := + (-1) ^ i * (a ^ (2 * i) / INR (fact (2 * i))). -Definition sin_approx [a:R;n:nat] : R := (sum_f_R0 (sin_term a) n). +Definition sin_approx (a:R) (n:nat) : R := sum_f_R0 (sin_term a) n. -Definition cos_approx [a:R;n:nat] : R := (sum_f_R0 (cos_term a) n). +Definition cos_approx (a:R) (n:nat) : R := sum_f_R0 (cos_term a) n. (**********) -Lemma PI_4 : ``PI<=4``. -Assert H0 := (PI_ineq O). -Elim H0; Clear H0; Intros _ H0. -Unfold tg_alt PI_tg in H0; Simpl in H0. -Rewrite Rinv_R1 in H0; Rewrite Rmult_1r in H0; Unfold Rdiv in H0. -Apply Rle_monotony_contra with ``/4``. -Apply Rlt_Rinv; Sup0. -Rewrite <- Rinv_l_sym; [Rewrite Rmult_sym; Assumption | DiscrR]. +Lemma PI_4 : PI <= 4. +assert (H0 := PI_ineq 0). +elim H0; clear H0; intros _ H0. +unfold tg_alt, PI_tg in H0; simpl in H0. +rewrite Rinv_1 in H0; rewrite Rmult_1_r in H0; unfold Rdiv in H0. +apply Rmult_le_reg_l with (/ 4). +apply Rinv_0_lt_compat; prove_sup0. +rewrite <- Rinv_l_sym; [ rewrite Rmult_comm; assumption | discrR ]. Qed. (**********) -Theorem sin_bound : (a:R; n:nat) ``0 <= a``->``a <= PI``->``(sin_approx a (plus (mult (S (S O)) n) (S O))) <= (sin a)<= (sin_approx a (mult (S (S O)) (plus n (S O))))``. -Intros; Case (Req_EM a R0); Intro Hyp_a. -Rewrite Hyp_a; Rewrite sin_0; Split; Right; Unfold sin_approx; Apply sum_eq_R0 Orelse (Symmetry; Apply sum_eq_R0); Intros; Unfold sin_term; Rewrite pow_add; Simpl; Unfold Rdiv; Rewrite Rmult_Ol; Ring. -Unfold sin_approx; Cut ``0<a``. -Intro Hyp_a_pos. -Rewrite (decomp_sum (sin_term a) (plus (mult (S (S O)) n) (S O))). -Rewrite (decomp_sum (sin_term a) (mult (S (S O)) (plus n (S O)))). -Replace (sin_term a O) with a. -Cut (Rle (sum_f_R0 [i:nat](sin_term a (S i)) (pred (plus (mult (S (S O)) n) (S O)))) ``(sin a)-a``)/\(Rle ``(sin a)-a`` (sum_f_R0 [i:nat](sin_term a (S i)) (pred (mult (S (S O)) (plus n (S O)))))) -> (Rle (Rplus a (sum_f_R0 [i:nat](sin_term a (S i)) (pred (plus (mult (S (S O)) n) (S O))))) (sin a))/\(Rle (sin a) (Rplus a (sum_f_R0 [i:nat](sin_term a (S i)) (pred (mult (S (S O)) (plus n (S O))))))). -Intro; Apply H1. -Pose Un := [n:nat]``(pow a (plus (mult (S (S O)) (S n)) (S O)))/(INR (fact (plus (mult (S (S O)) (S n)) (S O))))``. -Replace (pred (plus (mult (S (S O)) n) (S O))) with (mult (S (S O)) n). -Replace (pred (mult (S (S O)) (plus n (S O)))) with (S (mult (S (S O)) n)). -Replace (sum_f_R0 [i:nat](sin_term a (S i)) (mult (S (S O)) n)) with ``-(sum_f_R0 (tg_alt Un) (mult (S (S O)) n))``. -Replace (sum_f_R0 [i:nat](sin_term a (S i)) (S (mult (S (S O)) n))) with ``-(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))``. -Cut ``(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))<=a-(sin a)<=(sum_f_R0 (tg_alt Un) (mult (S (S O)) n))``->`` -(sum_f_R0 (tg_alt Un) (mult (S (S O)) n)) <= (sin a)-a <= -(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))``. -Intro; Apply H2. -Apply alternated_series_ineq. -Unfold Un_decreasing Un; Intro; Cut (plus (mult (S (S O)) (S (S n0))) (S O))=(S (S (plus (mult (S (S O)) (S n0)) (S O)))). -Intro; Rewrite H3. -Replace ``(pow a (S (S (plus (mult (S (S O)) (S n0)) (S O)))))`` with ``(pow a (plus (mult (S (S O)) (S n0)) (S O)))*(a*a)``. -Unfold Rdiv; Rewrite Rmult_assoc; Apply Rle_monotony. -Left; Apply pow_lt; Assumption. -Apply Rle_monotony_contra with ``(INR (fact (S (S (plus (mult (S (S O)) (S n0)) (S O))))))``. -Rewrite <- H3; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H5 := (sym_eq ? ? ? H4); Elim (fact_neq_0 ? H5). -Rewrite <- H3; Rewrite (Rmult_sym ``(INR (fact (plus (mult (S (S O)) (S (S n0))) (S O))))``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite H3; Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r. -Do 2 Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Simpl; Replace ``((0+1+1)*((INR n0)+1)+(0+1)+1+1)*((0+1+1)*((INR n0)+1)+(0+1)+1)`` with ``4*(INR n0)*(INR n0)+18*(INR n0)+20``; [Idtac | Ring]. -Apply Rle_trans with ``20``. -Apply Rle_trans with ``16``. -Replace ``16`` with ``(Rsqr 4)``; [Idtac | SqRing]. -Replace ``a*a`` with (Rsqr a); [Idtac | Reflexivity]. -Apply Rsqr_incr_1. -Apply Rle_trans with PI; [Assumption | Apply PI_4]. -Assumption. -Left; Sup0. -Rewrite <- (Rplus_Or ``16``); Replace ``20`` with ``16+4``; [Apply Rle_compatibility; Left; Sup0 | Ring]. -Rewrite <- (Rplus_sym ``20``); Pattern 1 ``20``; Rewrite <- Rplus_Or; Apply Rle_compatibility. -Apply ge0_plus_ge0_is_ge0. -Repeat Apply Rmult_le_pos. -Left; Sup0. -Left; Sup0. -Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. -Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. -Apply Rmult_le_pos. -Left; Sup0. -Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Simpl; Ring. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite plus_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Assert H3 := (cv_speed_pow_fact a); Unfold Un; Unfold Un_cv in H3; Unfold R_dist in H3; Unfold Un_cv; Unfold R_dist; Intros; Elim (H3 eps H4); Intros N H5. -Exists N; Intros; Apply H5. -Replace (plus (mult (2) (S n0)) (1)) with (S (mult (2) (S n0))). -Unfold ge; Apply le_trans with (mult (2) (S n0)). -Apply le_trans with (mult (2) (S N)). -Apply le_trans with (mult (2) N). -Apply le_n_2n. -Apply mult_le; Apply le_n_Sn. -Apply mult_le; Apply le_n_S; Assumption. -Apply le_n_Sn. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Reflexivity. -Assert X := (exist_sin (Rsqr a)); Elim X; Intros. -Cut ``x==(sin a)/a``. -Intro; Rewrite H3 in p; Unfold sin_in in p; Unfold infinit_sum in p; Unfold R_dist in p; Unfold Un_cv; Unfold R_dist; Intros. -Cut ``0<eps/(Rabsolu a)``. -Intro; Elim (p ? H5); Intros N H6. -Exists N; Intros. -Replace (sum_f_R0 (tg_alt Un) n0) with (Rmult a (Rminus R1 (sum_f_R0 [i:nat]``(sin_n i)*(pow (Rsqr a) i)`` (S n0)))). -Unfold Rminus; Rewrite Rmult_Rplus_distr; Rewrite Rmult_1r; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Repeat Rewrite Rplus_assoc; Rewrite (Rplus_sym a); Rewrite (Rplus_sym ``-a``); Repeat Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply Rlt_monotony_contra with ``/(Rabsolu a)``. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. -Pattern 1 ``/(Rabsolu a)``; Rewrite <- (Rabsolu_Rinv a Hyp_a). -Rewrite <- Rabsolu_mult; Rewrite Rmult_Rplus_distr; Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym; [Rewrite Rmult_1l | Assumption]; Rewrite (Rmult_sym ``/a``); Rewrite (Rmult_sym ``/(Rabsolu a)``); Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Unfold Rminus Rdiv in H6; Apply H6; Unfold ge; Apply le_trans with n0; [Exact H7 | Apply le_n_Sn]. -Rewrite (decomp_sum [i:nat]``(sin_n i)*(pow (Rsqr a) i)`` (S n0)). -Replace (sin_n O) with R1. -Simpl; Rewrite Rmult_1r; Unfold Rminus; Rewrite Ropp_distr1; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Ol; Rewrite Ropp_mul3; Rewrite <- Ropp_mul1; Rewrite scal_sum; Apply sum_eq. -Intros; Unfold sin_n Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-(pow (-1) i)``. -Replace ``(pow a (plus (mult (S (S O)) (S i)) (S O)))`` with ``(Rsqr a)*(pow (Rsqr a) i)*a``. -Unfold Rdiv; Ring. -Rewrite pow_add; Rewrite pow_Rsqr; Simpl; Ring. -Simpl; Ring. -Unfold sin_n; Unfold Rdiv; Simpl; Rewrite Rinv_R1; Rewrite Rmult_1r; Reflexivity. -Apply lt_O_Sn. -Unfold Rdiv; Apply Rmult_lt_pos. -Assumption. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. -Unfold sin; Case (exist_sin (Rsqr a)). -Intros; Cut x==x0. -Intro; Rewrite H3; Unfold Rdiv. -Symmetry; Apply Rinv_r_simpl_m; Assumption. -Unfold sin_in in p; Unfold sin_in in s; EApply unicity_sum. -Apply p. -Apply s. -Intros; Elim H2; Intros. -Replace ``(sin a)-a`` with ``-(a-(sin a))``; [Idtac | Ring]. -Split; Apply Rle_Ropp1; Assumption. -Replace ``-(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))`` with ``-1*(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n)))``; [Rewrite scal_sum | Ring]. -Apply sum_eq; Intros; Unfold sin_term Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i)``. -Unfold Rdiv; Ring. -Reflexivity. -Replace ``-(sum_f_R0 (tg_alt Un) (mult (S (S O)) n))`` with ``-1*(sum_f_R0 (tg_alt Un) (mult (S (S O)) n))``; [Rewrite scal_sum | Ring]. -Apply sum_eq; Intros. -Unfold sin_term Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i)``. -Unfold Rdiv; Ring. -Reflexivity. -Replace (mult (2) (plus n (1))) with (S (S (mult (2) n))). -Reflexivity. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Rewrite plus_INR; Repeat Rewrite S_INR; Ring. -Replace (plus (mult (2) n) (1)) with (S (mult (2) n)). -Reflexivity. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Intro; Elim H1; Intros. -Split. -Apply Rle_anti_compatibility with ``-a``. -Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite (Rplus_sym ``-a``); Apply H2. -Apply Rle_anti_compatibility with ``-a``. -Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite (Rplus_sym ``-a``); Apply H3. -Unfold sin_term; Simpl; Unfold Rdiv; Rewrite Rinv_R1; Ring. -Replace (mult (2) (plus n (1))) with (S (S (mult (2) n))). -Apply lt_O_Sn. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Rewrite plus_INR; Repeat Rewrite S_INR; Ring. -Replace (plus (mult (2) n) (1)) with (S (mult (2) n)). -Apply lt_O_Sn. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Inversion H; [Assumption | Elim Hyp_a; Symmetry; Assumption]. +Theorem sin_bound : + forall (a:R) (n:nat), + 0 <= a -> + a <= PI -> sin_approx a (2 * n + 1) <= sin a <= sin_approx a (2 * (n + 1)). +intros; case (Req_dec a 0); intro Hyp_a. +rewrite Hyp_a; rewrite sin_0; split; right; unfold sin_approx in |- *; + apply sum_eq_R0 || (symmetry in |- *; apply sum_eq_R0); + intros; unfold sin_term in |- *; rewrite pow_add; + simpl in |- *; unfold Rdiv in |- *; rewrite Rmult_0_l; + ring. +unfold sin_approx in |- *; cut (0 < a). +intro Hyp_a_pos. +rewrite (decomp_sum (sin_term a) (2 * n + 1)). +rewrite (decomp_sum (sin_term a) (2 * (n + 1))). +replace (sin_term a 0) with a. +cut + (sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * n + 1)) <= sin a - a /\ + sin a - a <= sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * (n + 1))) -> + a + sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * n + 1)) <= sin a /\ + sin a <= a + sum_f_R0 (fun i:nat => sin_term a (S i)) (pred (2 * (n + 1)))). +intro; apply H1. +pose (Un := fun n:nat => a ^ (2 * S n + 1) / INR (fact (2 * S n + 1))). +replace (pred (2 * n + 1)) with (2 * n)%nat. +replace (pred (2 * (n + 1))) with (S (2 * n)). +replace (sum_f_R0 (fun i:nat => sin_term a (S i)) (2 * n)) with + (- sum_f_R0 (tg_alt Un) (2 * n)). +replace (sum_f_R0 (fun i:nat => sin_term a (S i)) (S (2 * n))) with + (- sum_f_R0 (tg_alt Un) (S (2 * n))). +cut + (sum_f_R0 (tg_alt Un) (S (2 * n)) <= a - sin a <= + sum_f_R0 (tg_alt Un) (2 * n) -> + - sum_f_R0 (tg_alt Un) (2 * n) <= sin a - a <= + - sum_f_R0 (tg_alt Un) (S (2 * n))). +intro; apply H2. +apply alternated_series_ineq. +unfold Un_decreasing, Un in |- *; intro; + cut ((2 * S (S n0) + 1)%nat = S (S (2 * S n0 + 1))). +intro; rewrite H3. +replace (a ^ S (S (2 * S n0 + 1))) with (a ^ (2 * S n0 + 1) * (a * a)). +unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l. +left; apply pow_lt; assumption. +apply Rmult_le_reg_l with (INR (fact (S (S (2 * S n0 + 1))))). +rewrite <- H3; apply lt_INR_0; apply neq_O_lt; red in |- *; intro; + assert (H5 := sym_eq H4); elim (fact_neq_0 _ H5). +rewrite <- H3; rewrite (Rmult_comm (INR (fact (2 * S (S n0) + 1)))); + rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite H3; do 2 rewrite fact_simpl; do 2 rewrite mult_INR; + repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_r. +do 2 rewrite S_INR; rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR; + simpl in |- *; + replace + (((0 + 1 + 1) * (INR n0 + 1) + (0 + 1) + 1 + 1) * + ((0 + 1 + 1) * (INR n0 + 1) + (0 + 1) + 1)) with + (4 * INR n0 * INR n0 + 18 * INR n0 + 20); [ idtac | ring ]. +apply Rle_trans with 20. +apply Rle_trans with 16. +replace 16 with (Rsqr 4); [ idtac | ring_Rsqr ]. +replace (a * a) with (Rsqr a); [ idtac | reflexivity ]. +apply Rsqr_incr_1. +apply Rle_trans with PI; [ assumption | apply PI_4 ]. +assumption. +left; prove_sup0. +rewrite <- (Rplus_0_r 16); replace 20 with (16 + 4); + [ apply Rplus_le_compat_l; left; prove_sup0 | ring ]. +rewrite <- (Rplus_comm 20); pattern 20 at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l. +apply Rplus_le_le_0_compat. +repeat apply Rmult_le_pos. +left; prove_sup0. +left; prove_sup0. +replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. +replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. +apply Rmult_le_pos. +left; prove_sup0. +replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +simpl in |- *; ring. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite plus_INR; + do 2 rewrite mult_INR; repeat rewrite S_INR; ring. +assert (H3 := cv_speed_pow_fact a); unfold Un in |- *; unfold Un_cv in H3; + unfold R_dist in H3; unfold Un_cv in |- *; unfold R_dist in |- *; + intros; elim (H3 eps H4); intros N H5. +exists N; intros; apply H5. +replace (2 * S n0 + 1)%nat with (S (2 * S n0)). +unfold ge in |- *; apply le_trans with (2 * S n0)%nat. +apply le_trans with (2 * S N)%nat. +apply le_trans with (2 * N)%nat. +apply le_n_2n. +apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_Sn. +apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_S; assumption. +apply le_n_Sn. +apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR; reflexivity. +assert (X := exist_sin (Rsqr a)); elim X; intros. +cut (x = sin a / a). +intro; rewrite H3 in p; unfold sin_in in p; unfold infinit_sum in p; + unfold R_dist in p; unfold Un_cv in |- *; unfold R_dist in |- *; + intros. +cut (0 < eps / Rabs a). +intro; elim (p _ H5); intros N H6. +exists N; intros. +replace (sum_f_R0 (tg_alt Un) n0) with + (a * (1 - sum_f_R0 (fun i:nat => sin_n i * Rsqr a ^ i) (S n0))). +unfold Rminus in |- *; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r; + rewrite Ropp_plus_distr; rewrite Ropp_involutive; + repeat rewrite Rplus_assoc; rewrite (Rplus_comm a); + rewrite (Rplus_comm (- a)); repeat rewrite Rplus_assoc; + rewrite Rplus_opp_l; rewrite Rplus_0_r; apply Rmult_lt_reg_l with (/ Rabs a). +apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. +pattern (/ Rabs a) at 1 in |- *; rewrite <- (Rabs_Rinv a Hyp_a). +rewrite <- Rabs_mult; rewrite Rmult_plus_distr_l; rewrite <- Rmult_assoc; + rewrite <- Rinv_l_sym; [ rewrite Rmult_1_l | assumption ]; + rewrite (Rmult_comm (/ a)); rewrite (Rmult_comm (/ Rabs a)); + rewrite <- Rabs_Ropp; rewrite Ropp_plus_distr; rewrite Ropp_involutive; + unfold Rminus, Rdiv in H6; apply H6; unfold ge in |- *; + apply le_trans with n0; [ exact H7 | apply le_n_Sn ]. +rewrite (decomp_sum (fun i:nat => sin_n i * Rsqr a ^ i) (S n0)). +replace (sin_n 0) with 1. +simpl in |- *; rewrite Rmult_1_r; unfold Rminus in |- *; + rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r; + rewrite Rplus_0_l; rewrite Ropp_mult_distr_r_reverse; + rewrite <- Ropp_mult_distr_l_reverse; rewrite scal_sum; + apply sum_eq. +intros; unfold sin_n, Un, tg_alt in |- *; + replace ((-1) ^ S i) with (- (-1) ^ i). +replace (a ^ (2 * S i + 1)) with (Rsqr a * Rsqr a ^ i * a). +unfold Rdiv in |- *; ring. +rewrite pow_add; rewrite pow_Rsqr; simpl in |- *; ring. +simpl in |- *; ring. +unfold sin_n in |- *; unfold Rdiv in |- *; simpl in |- *; rewrite Rinv_1; + rewrite Rmult_1_r; reflexivity. +apply lt_O_Sn. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +assumption. +apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. +unfold sin in |- *; case (exist_sin (Rsqr a)). +intros; cut (x = x0). +intro; rewrite H3; unfold Rdiv in |- *. +symmetry in |- *; apply Rinv_r_simpl_m; assumption. +unfold sin_in in p; unfold sin_in in s; eapply uniqueness_sum. +apply p. +apply s. +intros; elim H2; intros. +replace (sin a - a) with (- (a - sin a)); [ idtac | ring ]. +split; apply Ropp_le_contravar; assumption. +replace (- sum_f_R0 (tg_alt Un) (S (2 * n))) with + (-1 * sum_f_R0 (tg_alt Un) (S (2 * n))); [ rewrite scal_sum | ring ]. +apply sum_eq; intros; unfold sin_term, Un, tg_alt in |- *; + replace ((-1) ^ S i) with (-1 * (-1) ^ i). +unfold Rdiv in |- *; ring. +reflexivity. +replace (- sum_f_R0 (tg_alt Un) (2 * n)) with + (-1 * sum_f_R0 (tg_alt Un) (2 * n)); [ rewrite scal_sum | ring ]. +apply sum_eq; intros. +unfold sin_term, Un, tg_alt in |- *; + replace ((-1) ^ S i) with (-1 * (-1) ^ i). +unfold Rdiv in |- *; ring. +reflexivity. +replace (2 * (n + 1))%nat with (S (S (2 * n))). +reflexivity. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; rewrite plus_INR; + repeat rewrite S_INR; ring. +replace (2 * n + 1)%nat with (S (2 * n)). +reflexivity. +apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR; + repeat rewrite S_INR; ring. +intro; elim H1; intros. +split. +apply Rplus_le_reg_l with (- a). +rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; + rewrite (Rplus_comm (- a)); apply H2. +apply Rplus_le_reg_l with (- a). +rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; + rewrite (Rplus_comm (- a)); apply H3. +unfold sin_term in |- *; simpl in |- *; unfold Rdiv in |- *; rewrite Rinv_1; + ring. +replace (2 * (n + 1))%nat with (S (S (2 * n))). +apply lt_O_Sn. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; rewrite plus_INR; + repeat rewrite S_INR; ring. +replace (2 * n + 1)%nat with (S (2 * n)). +apply lt_O_Sn. +apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR; + repeat rewrite S_INR; ring. +inversion H; [ assumption | elim Hyp_a; symmetry in |- *; assumption ]. Qed. (**********) -Lemma cos_bound : (a:R; n:nat) `` -PI/2 <= a``->``a <= PI/2``->``(cos_approx a (plus (mult (S (S O)) n) (S O))) <= (cos a) <= (cos_approx a (mult (S (S O)) (plus n (S O))))``. -Cut ((a:R; n:nat) ``0 <= a``->``a <= PI/2``->``(cos_approx a (plus (mult (S (S O)) n) (S O))) <= (cos a) <= (cos_approx a (mult (S (S O)) (plus n (S O))))``) -> ((a:R; n:nat) `` -PI/2 <= a``->``a <= PI/2``->``(cos_approx a (plus (mult (S (S O)) n) (S O))) <= (cos a) <= (cos_approx a (mult (S (S O)) (plus n (S O))))``). -Intros H a n; Apply H. -Intros; Unfold cos_approx. -Rewrite (decomp_sum (cos_term a0) (plus (mult (S (S O)) n0) (S O))). -Rewrite (decomp_sum (cos_term a0) (mult (S (S O)) (plus n0 (S O)))). -Replace (cos_term a0 O) with R1. -Cut (Rle (sum_f_R0 [i:nat](cos_term a0 (S i)) (pred (plus (mult (S (S O)) n0) (S O)))) ``(cos a0)-1``)/\(Rle ``(cos a0)-1`` (sum_f_R0 [i:nat](cos_term a0 (S i)) (pred (mult (S (S O)) (plus n0 (S O)))))) -> (Rle (Rplus R1 (sum_f_R0 [i:nat](cos_term a0 (S i)) (pred (plus (mult (S (S O)) n0) (S O))))) (cos a0))/\(Rle (cos a0) (Rplus R1 (sum_f_R0 [i:nat](cos_term a0 (S i)) (pred (mult (S (S O)) (plus n0 (S O))))))). -Intro; Apply H2. -Pose Un := [n:nat]``(pow a0 (mult (S (S O)) (S n)))/(INR (fact (mult (S (S O)) (S n))))``. -Replace (pred (plus (mult (S (S O)) n0) (S O))) with (mult (S (S O)) n0). -Replace (pred (mult (S (S O)) (plus n0 (S O)))) with (S (mult (S (S O)) n0)). -Replace (sum_f_R0 [i:nat](cos_term a0 (S i)) (mult (S (S O)) n0)) with ``-(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0))``. -Replace (sum_f_R0 [i:nat](cos_term a0 (S i)) (S (mult (S (S O)) n0))) with ``-(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))``. -Cut ``(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))<=1-(cos a0)<=(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0))``->`` -(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0)) <= (cos a0)-1 <= -(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))``. -Intro; Apply H3. -Apply alternated_series_ineq. -Unfold Un_decreasing; Intro; Unfold Un. -Cut (mult (S (S O)) (S (S n1)))=(S (S (mult (S (S O)) (S n1)))). -Intro; Rewrite H4; Replace ``(pow a0 (S (S (mult (S (S O)) (S n1)))))`` with ``(pow a0 (mult (S (S O)) (S n1)))*(a0*a0)``. -Unfold Rdiv; Rewrite Rmult_assoc; Apply Rle_monotony. -Apply pow_le; Assumption. -Apply Rle_monotony_contra with ``(INR (fact (S (S (mult (S (S O)) (S n1))))))``. -Rewrite <- H4; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H6 := (sym_eq ? ? ? H5); Elim (fact_neq_0 ? H6). -Rewrite <- H4; Rewrite (Rmult_sym ``(INR (fact (mult (S (S O)) (S (S n1)))))``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite H4; Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Do 2 Rewrite S_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Simpl; Replace ``((0+1+1)*((INR n1)+1)+1+1)*((0+1+1)*((INR n1)+1)+1)`` with ``4*(INR n1)*(INR n1)+14*(INR n1)+12``; [Idtac | Ring]. -Apply Rle_trans with ``12``. -Apply Rle_trans with ``4``. -Replace ``4`` with ``(Rsqr 2)``; [Idtac | SqRing]. -Replace ``a0*a0`` with (Rsqr a0); [Idtac | Reflexivity]. -Apply Rsqr_incr_1. -Apply Rle_trans with ``PI/2``. -Assumption. -Unfold Rdiv; Apply Rle_monotony_contra with ``2``. -Sup0. -Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m. -Replace ``2*2`` with ``4``; [Apply PI_4 | Ring]. -DiscrR. -Assumption. -Left; Sup0. -Pattern 1 ``4``; Rewrite <- Rplus_Or; Replace ``12`` with ``4+8``; [Apply Rle_compatibility; Left; Sup0 | Ring]. -Rewrite <- (Rplus_sym ``12``); Pattern 1 ``12``; Rewrite <- Rplus_Or; Apply Rle_compatibility. -Apply ge0_plus_ge0_is_ge0. -Repeat Apply Rmult_le_pos. -Left; Sup0. -Left; Sup0. -Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. -Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. -Apply Rmult_le_pos. -Left; Sup0. -Replace R0 with (INR O); [Apply le_INR; Apply le_O_n | Reflexivity]. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Simpl; Ring. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Assert H4 := (cv_speed_pow_fact a0); Unfold Un; Unfold Un_cv in H4; Unfold R_dist in H4; Unfold Un_cv; Unfold R_dist; Intros; Elim (H4 eps H5); Intros N H6; Exists N; Intros. -Apply H6; Unfold ge; Apply le_trans with (mult (2) (S N)). -Apply le_trans with (mult (2) N). -Apply le_n_2n. -Apply mult_le; Apply le_n_Sn. -Apply mult_le; Apply le_n_S; Assumption. -Assert X := (exist_cos (Rsqr a0)); Elim X; Intros. -Cut ``x==(cos a0)``. -Intro; Rewrite H4 in p; Unfold cos_in in p; Unfold infinit_sum in p; Unfold R_dist in p; Unfold Un_cv; Unfold R_dist; Intros. -Elim (p ? H5); Intros N H6. -Exists N; Intros. -Replace (sum_f_R0 (tg_alt Un) n1) with (Rminus R1 (sum_f_R0 [i:nat]``(cos_n i)*(pow (Rsqr a0) i)`` (S n1))). -Unfold Rminus; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Repeat Rewrite Rplus_assoc; Rewrite (Rplus_sym R1); Rewrite (Rplus_sym ``-1``); Repeat Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr1; Rewrite Ropp_Ropp; Unfold Rminus in H6; Apply H6. -Unfold ge; Apply le_trans with n1. -Exact H7. -Apply le_n_Sn. -Rewrite (decomp_sum [i:nat]``(cos_n i)*(pow (Rsqr a0) i)`` (S n1)). -Replace (cos_n O) with R1. -Simpl; Rewrite Rmult_1r; Unfold Rminus; Rewrite Ropp_distr1; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Ol; Replace (Ropp (sum_f_R0 [i:nat]``(cos_n (S i))*((Rsqr a0)*(pow (Rsqr a0) i))`` n1)) with (Rmult ``-1`` (sum_f_R0 [i:nat]``(cos_n (S i))*((Rsqr a0)*(pow (Rsqr a0) i))`` n1)); [Idtac | Ring]; Rewrite scal_sum; Apply sum_eq; Intros; Unfold cos_n Un tg_alt. -Replace ``(pow (-1) (S i))`` with ``-(pow (-1) i)``. -Replace ``(pow a0 (mult (S (S O)) (S i)))`` with ``(Rsqr a0)*(pow (Rsqr a0) i)``. -Unfold Rdiv; Ring. -Rewrite pow_Rsqr; Reflexivity. -Simpl; Ring. -Unfold cos_n; Unfold Rdiv; Simpl; Rewrite Rinv_R1; Rewrite Rmult_1r; Reflexivity. -Apply lt_O_Sn. -Unfold cos; Case (exist_cos (Rsqr a0)); Intros; Unfold cos_in in p; Unfold cos_in in c; EApply unicity_sum. -Apply p. -Apply c. -Intros; Elim H3; Intros; Replace ``(cos a0)-1`` with ``-(1-(cos a0))``; [Idtac | Ring]. -Split; Apply Rle_Ropp1; Assumption. -Replace ``-(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))`` with ``-1*(sum_f_R0 (tg_alt Un) (S (mult (S (S O)) n0)))``; [Rewrite scal_sum | Ring]. -Apply sum_eq; Intros; Unfold cos_term Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i)``. -Unfold Rdiv; Ring. -Reflexivity. -Replace ``-(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0))`` with ``-1*(sum_f_R0 (tg_alt Un) (mult (S (S O)) n0))``; [Rewrite scal_sum | Ring]; Apply sum_eq; Intros; Unfold cos_term Un tg_alt; Replace ``(pow (-1) (S i))`` with ``-1*(pow (-1) i)``. -Unfold Rdiv; Ring. -Reflexivity. -Replace (mult (2) (plus n0 (1))) with (S (S (mult (2) n0))). -Reflexivity. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Rewrite plus_INR; Repeat Rewrite S_INR; Ring. -Replace (plus (mult (2) n0) (1)) with (S (mult (2) n0)). -Reflexivity. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Intro; Elim H2; Intros; Split. -Apply Rle_anti_compatibility with ``-1``. -Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite (Rplus_sym ``-1``); Apply H3. -Apply Rle_anti_compatibility with ``-1``. -Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Rewrite (Rplus_sym ``-1``); Apply H4. -Unfold cos_term; Simpl; Unfold Rdiv; Rewrite Rinv_R1; Ring. -Replace (mult (2) (plus n0 (1))) with (S (S (mult (2) n0))). -Apply lt_O_Sn. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Rewrite plus_INR; Repeat Rewrite S_INR; Ring. -Replace (plus (mult (2) n0) (1)) with (S (mult (2) n0)). -Apply lt_O_Sn. -Apply INR_eq; Rewrite S_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Intros; Case (total_order_T R0 a); Intro. -Elim s; Intro. -Apply H; [Left; Assumption | Assumption]. -Apply H; [Right; Assumption | Assumption]. -Cut ``0< -a``. -Intro; Cut (x:R;n:nat) (cos_approx x n)==(cos_approx ``-x`` n). -Intro; Rewrite H3; Rewrite (H3 a (mult (S (S O)) (plus n (S O)))); Rewrite cos_sym; Apply H. -Left; Assumption. -Rewrite <- (Ropp_Ropp ``PI/2``); Apply Rle_Ropp1; Unfold Rdiv; Unfold Rdiv in H0; Rewrite <- Ropp_mul1; Exact H0. -Intros; Unfold cos_approx; Apply sum_eq; Intros; Unfold cos_term; Do 2 Rewrite pow_Rsqr; Rewrite Rsqr_neg; Unfold Rdiv; Reflexivity. -Apply Rgt_RO_Ropp; Assumption. -Qed. +Lemma cos_bound : + forall (a:R) (n:nat), + - PI / 2 <= a -> + a <= PI / 2 -> + cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1)). +cut + ((forall (a:R) (n:nat), + 0 <= a -> + a <= PI / 2 -> + cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1))) -> + forall (a:R) (n:nat), + - PI / 2 <= a -> + a <= PI / 2 -> + cos_approx a (2 * n + 1) <= cos a <= cos_approx a (2 * (n + 1))). +intros H a n; apply H. +intros; unfold cos_approx in |- *. +rewrite (decomp_sum (cos_term a0) (2 * n0 + 1)). +rewrite (decomp_sum (cos_term a0) (2 * (n0 + 1))). +replace (cos_term a0 0) with 1. +cut + (sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * n0 + 1)) <= cos a0 - 1 /\ + cos a0 - 1 <= + sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * (n0 + 1))) -> + 1 + sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * n0 + 1)) <= cos a0 /\ + cos a0 <= + 1 + sum_f_R0 (fun i:nat => cos_term a0 (S i)) (pred (2 * (n0 + 1)))). +intro; apply H2. +pose (Un := fun n:nat => a0 ^ (2 * S n) / INR (fact (2 * S n))). +replace (pred (2 * n0 + 1)) with (2 * n0)%nat. +replace (pred (2 * (n0 + 1))) with (S (2 * n0)). +replace (sum_f_R0 (fun i:nat => cos_term a0 (S i)) (2 * n0)) with + (- sum_f_R0 (tg_alt Un) (2 * n0)). +replace (sum_f_R0 (fun i:nat => cos_term a0 (S i)) (S (2 * n0))) with + (- sum_f_R0 (tg_alt Un) (S (2 * n0))). +cut + (sum_f_R0 (tg_alt Un) (S (2 * n0)) <= 1 - cos a0 <= + sum_f_R0 (tg_alt Un) (2 * n0) -> + - sum_f_R0 (tg_alt Un) (2 * n0) <= cos a0 - 1 <= + - sum_f_R0 (tg_alt Un) (S (2 * n0))). +intro; apply H3. +apply alternated_series_ineq. +unfold Un_decreasing in |- *; intro; unfold Un in |- *. +cut ((2 * S (S n1))%nat = S (S (2 * S n1))). +intro; rewrite H4; + replace (a0 ^ S (S (2 * S n1))) with (a0 ^ (2 * S n1) * (a0 * a0)). +unfold Rdiv in |- *; rewrite Rmult_assoc; apply Rmult_le_compat_l. +apply pow_le; assumption. +apply Rmult_le_reg_l with (INR (fact (S (S (2 * S n1))))). +rewrite <- H4; apply lt_INR_0; apply neq_O_lt; red in |- *; intro; + assert (H6 := sym_eq H5); elim (fact_neq_0 _ H6). +rewrite <- H4; rewrite (Rmult_comm (INR (fact (2 * S (S n1))))); + rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; rewrite H4; do 2 rewrite fact_simpl; do 2 rewrite mult_INR; + repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; do 2 rewrite S_INR; rewrite mult_INR; repeat rewrite S_INR; + simpl in |- *; + replace + (((0 + 1 + 1) * (INR n1 + 1) + 1 + 1) * ((0 + 1 + 1) * (INR n1 + 1) + 1)) + with (4 * INR n1 * INR n1 + 14 * INR n1 + 12); [ idtac | ring ]. +apply Rle_trans with 12. +apply Rle_trans with 4. +replace 4 with (Rsqr 2); [ idtac | ring_Rsqr ]. +replace (a0 * a0) with (Rsqr a0); [ idtac | reflexivity ]. +apply Rsqr_incr_1. +apply Rle_trans with (PI / 2). +assumption. +unfold Rdiv in |- *; apply Rmult_le_reg_l with 2. +prove_sup0. +rewrite <- Rmult_assoc; rewrite Rinv_r_simpl_m. +replace 4 with 4; [ apply PI_4 | ring ]. +discrR. +assumption. +left; prove_sup0. +pattern 4 at 1 in |- *; rewrite <- Rplus_0_r; replace 12 with (4 + 8); + [ apply Rplus_le_compat_l; left; prove_sup0 | ring ]. +rewrite <- (Rplus_comm 12); pattern 12 at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_le_compat_l. +apply Rplus_le_le_0_compat. +repeat apply Rmult_le_pos. +left; prove_sup0. +left; prove_sup0. +replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. +replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. +apply Rmult_le_pos. +left; prove_sup0. +replace 0 with (INR 0); [ apply le_INR; apply le_O_n | reflexivity ]. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +simpl in |- *; ring. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +assert (H4 := cv_speed_pow_fact a0); unfold Un in |- *; unfold Un_cv in H4; + unfold R_dist in H4; unfold Un_cv in |- *; unfold R_dist in |- *; + intros; elim (H4 eps H5); intros N H6; exists N; intros. +apply H6; unfold ge in |- *; apply le_trans with (2 * S N)%nat. +apply le_trans with (2 * N)%nat. +apply le_n_2n. +apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_Sn. +apply (fun m n p:nat => mult_le_compat_l p n m); apply le_n_S; assumption. +assert (X := exist_cos (Rsqr a0)); elim X; intros. +cut (x = cos a0). +intro; rewrite H4 in p; unfold cos_in in p; unfold infinit_sum in p; + unfold R_dist in p; unfold Un_cv in |- *; unfold R_dist in |- *; + intros. +elim (p _ H5); intros N H6. +exists N; intros. +replace (sum_f_R0 (tg_alt Un) n1) with + (1 - sum_f_R0 (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)). +unfold Rminus in |- *; rewrite Ropp_plus_distr; rewrite Ropp_involutive; + repeat rewrite Rplus_assoc; rewrite (Rplus_comm 1); + rewrite (Rplus_comm (-1)); repeat rewrite Rplus_assoc; + rewrite Rplus_opp_l; rewrite Rplus_0_r; rewrite <- Rabs_Ropp; + rewrite Ropp_plus_distr; rewrite Ropp_involutive; + unfold Rminus in H6; apply H6. +unfold ge in |- *; apply le_trans with n1. +exact H7. +apply le_n_Sn. +rewrite (decomp_sum (fun i:nat => cos_n i * Rsqr a0 ^ i) (S n1)). +replace (cos_n 0) with 1. +simpl in |- *; rewrite Rmult_1_r; unfold Rminus in |- *; + rewrite Ropp_plus_distr; rewrite <- Rplus_assoc; rewrite Rplus_opp_r; + rewrite Rplus_0_l; + replace (- sum_f_R0 (fun i:nat => cos_n (S i) * (Rsqr a0 * Rsqr a0 ^ i)) n1) + with + (-1 * sum_f_R0 (fun i:nat => cos_n (S i) * (Rsqr a0 * Rsqr a0 ^ i)) n1); + [ idtac | ring ]; rewrite scal_sum; apply sum_eq; + intros; unfold cos_n, Un, tg_alt in |- *. +replace ((-1) ^ S i) with (- (-1) ^ i). +replace (a0 ^ (2 * S i)) with (Rsqr a0 * Rsqr a0 ^ i). +unfold Rdiv in |- *; ring. +rewrite pow_Rsqr; reflexivity. +simpl in |- *; ring. +unfold cos_n in |- *; unfold Rdiv in |- *; simpl in |- *; rewrite Rinv_1; + rewrite Rmult_1_r; reflexivity. +apply lt_O_Sn. +unfold cos in |- *; case (exist_cos (Rsqr a0)); intros; unfold cos_in in p; + unfold cos_in in c; eapply uniqueness_sum. +apply p. +apply c. +intros; elim H3; intros; replace (cos a0 - 1) with (- (1 - cos a0)); + [ idtac | ring ]. +split; apply Ropp_le_contravar; assumption. +replace (- sum_f_R0 (tg_alt Un) (S (2 * n0))) with + (-1 * sum_f_R0 (tg_alt Un) (S (2 * n0))); [ rewrite scal_sum | ring ]. +apply sum_eq; intros; unfold cos_term, Un, tg_alt in |- *; + replace ((-1) ^ S i) with (-1 * (-1) ^ i). +unfold Rdiv in |- *; ring. +reflexivity. +replace (- sum_f_R0 (tg_alt Un) (2 * n0)) with + (-1 * sum_f_R0 (tg_alt Un) (2 * n0)); [ rewrite scal_sum | ring ]; + apply sum_eq; intros; unfold cos_term, Un, tg_alt in |- *; + replace ((-1) ^ S i) with (-1 * (-1) ^ i). +unfold Rdiv in |- *; ring. +reflexivity. +replace (2 * (n0 + 1))%nat with (S (S (2 * n0))). +reflexivity. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; rewrite plus_INR; + repeat rewrite S_INR; ring. +replace (2 * n0 + 1)%nat with (S (2 * n0)). +reflexivity. +apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR; + repeat rewrite S_INR; ring. +intro; elim H2; intros; split. +apply Rplus_le_reg_l with (-1). +rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; + rewrite (Rplus_comm (-1)); apply H3. +apply Rplus_le_reg_l with (-1). +rewrite <- Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_l; + rewrite (Rplus_comm (-1)); apply H4. +unfold cos_term in |- *; simpl in |- *; unfold Rdiv in |- *; rewrite Rinv_1; + ring. +replace (2 * (n0 + 1))%nat with (S (S (2 * n0))). +apply lt_O_Sn. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; rewrite plus_INR; + repeat rewrite S_INR; ring. +replace (2 * n0 + 1)%nat with (S (2 * n0)). +apply lt_O_Sn. +apply INR_eq; rewrite S_INR; rewrite plus_INR; rewrite mult_INR; + repeat rewrite S_INR; ring. +intros; case (total_order_T 0 a); intro. +elim s; intro. +apply H; [ left; assumption | assumption ]. +apply H; [ right; assumption | assumption ]. +cut (0 < - a). +intro; cut (forall (x:R) (n:nat), cos_approx x n = cos_approx (- x) n). +intro; rewrite H3; rewrite (H3 a (2 * (n + 1))%nat); rewrite cos_sym; apply H. +left; assumption. +rewrite <- (Ropp_involutive (PI / 2)); apply Ropp_le_contravar; + unfold Rdiv in |- *; unfold Rdiv in H0; rewrite <- Ropp_mult_distr_l_reverse; + exact H0. +intros; unfold cos_approx in |- *; apply sum_eq; intros; + unfold cos_term in |- *; do 2 rewrite pow_Rsqr; rewrite Rsqr_neg; + unfold Rdiv in |- *; reflexivity. +apply Ropp_0_gt_lt_contravar; assumption. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Rtrigo_calc.v b/theories/Reals/Rtrigo_calc.v index 8ede9fc1c..28cb27a58 100644 --- a/theories/Reals/Rtrigo_calc.v +++ b/theories/Reals/Rtrigo_calc.v @@ -8,343 +8,427 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo. -Require R_sqrt. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import R_sqrt. Open Local Scope R_scope. -Lemma tan_PI : ``(tan PI)==0``. -Unfold tan; Rewrite sin_PI; Rewrite cos_PI; Unfold Rdiv; Apply Rmult_Ol. -Qed. - -Lemma sin_3PI2 : ``(sin (3*(PI/2)))==(-1)``. -Replace ``3*(PI/2)`` with ``PI+(PI/2)``. -Rewrite sin_plus; Rewrite sin_PI; Rewrite cos_PI; Rewrite sin_PI2; Ring. -Pattern 1 PI; Rewrite (double_var PI); Ring. -Qed. - -Lemma tan_2PI : ``(tan (2*PI))==0``. -Unfold tan; Rewrite sin_2PI; Unfold Rdiv; Apply Rmult_Ol. -Qed. - -Lemma sin_cos_PI4 : ``(sin (PI/4)) == (cos (PI/4))``. -Proof with Trivial. -Rewrite cos_sin. -Replace ``PI/2+PI/4`` with ``-(PI/4)+PI``. -Rewrite neg_sin; Rewrite sin_neg; Ring. -Cut ``PI==PI/2+PI/2``; [Intro | Apply double_var]. -Pattern 2 3 PI; Rewrite H; Pattern 2 3 PI; Rewrite H. -Assert H0 : ``2<>0``; [DiscrR | Unfold Rdiv; Rewrite Rinv_Rmult; Try Ring]. -Qed. - -Lemma sin_PI3_cos_PI6 : ``(sin (PI/3))==(cos (PI/6))``. -Proof with Trivial. -Replace ``PI/6`` with ``(PI/2)-(PI/3)``. -Rewrite cos_shift. -Assert H0 : ``6<>0``; [DiscrR | Idtac]. -Assert H1 : ``3<>0``; [DiscrR | Idtac]. -Assert H2 : ``2<>0``; [DiscrR | Idtac]. -Apply r_Rmult_mult with ``6``. -Rewrite Rminus_distr; Repeat Rewrite (Rmult_sym ``6``). -Unfold Rdiv; Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite (Rmult_sym ``/3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Pattern 2 PI; Rewrite (Rmult_sym PI); Repeat Rewrite Rmult_1r; Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Ring. -Qed. - -Lemma sin_PI6_cos_PI3 : ``(cos (PI/3))==(sin (PI/6))``. -Proof with Trivial. -Replace ``PI/6`` with ``(PI/2)-(PI/3)``. -Rewrite sin_shift. -Assert H0 : ``6<>0``; [DiscrR | Idtac]. -Assert H1 : ``3<>0``; [DiscrR | Idtac]. -Assert H2 : ``2<>0``; [DiscrR | Idtac]. -Apply r_Rmult_mult with ``6``. -Rewrite Rminus_distr; Repeat Rewrite (Rmult_sym ``6``). -Unfold Rdiv; Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite (Rmult_sym ``/3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Pattern 2 PI; Rewrite (Rmult_sym PI); Repeat Rewrite Rmult_1r; Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Ring. -Qed. - -Lemma PI6_RGT_0 : ``0<PI/6``. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]. -Qed. - -Lemma PI6_RLT_PI2 : ``PI/6<PI/2``. -Unfold Rdiv; Apply Rlt_monotony. -Apply PI_RGT_0. -Apply Rinv_lt; Sup. -Qed. - -Lemma sin_PI6 : ``(sin (PI/6))==1/2``. -Proof with Trivial. -Assert H : ``2<>0``; [DiscrR | Idtac]. -Apply r_Rmult_mult with ``2*(cos (PI/6))``. -Replace ``2*(cos (PI/6))*(sin (PI/6))`` with ``2*(sin (PI/6))*(cos (PI/6))``. -Rewrite <- sin_2a; Replace ``2*(PI/6)`` with ``PI/3``. -Rewrite sin_PI3_cos_PI6. -Unfold Rdiv; Rewrite Rmult_1l; Rewrite Rmult_assoc; Pattern 2 ``2``; Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Unfold Rdiv; Rewrite Rinv_Rmult. -Rewrite (Rmult_sym ``/2``); Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -DiscrR. -Ring. -Apply prod_neq_R0. -Cut ``0<(cos (PI/6))``; [Intro H1; Auto with real | Apply cos_gt_0; [Apply (Rlt_trans ``-(PI/2)`` ``0`` ``PI/6`` _PI2_RLT_0 PI6_RGT_0) | Apply PI6_RLT_PI2]]. -Qed. - -Lemma sqrt2_neq_0 : ~``(sqrt 2)==0``. -Assert Hyp:``0<2``; [Sup0 | Generalize (Rlt_le ``0`` ``2`` Hyp); Intro H1; Red; Intro H2; Generalize (sqrt_eq_0 ``2`` H1 H2); Intro H; Absurd ``2==0``; [ DiscrR | Assumption]]. -Qed. - -Lemma R1_sqrt2_neq_0 : ~``1/(sqrt 2)==0``. -Generalize (Rinv_neq_R0 ``(sqrt 2)`` sqrt2_neq_0); Intro H; Generalize (prod_neq_R0 ``1`` ``(Rinv (sqrt 2))`` R1_neq_R0 H); Intro H0; Assumption. -Qed. - -Lemma sqrt3_2_neq_0 : ~``2*(sqrt 3)==0``. -Apply prod_neq_R0; [DiscrR | Assert Hyp:``0<3``; [Sup0 | Generalize (Rlt_le ``0`` ``3`` Hyp); Intro H1; Red; Intro H2; Generalize (sqrt_eq_0 ``3`` H1 H2); Intro H; Absurd ``3==0``; [ DiscrR | Assumption]]]. -Qed. - -Lemma Rlt_sqrt2_0 : ``0<(sqrt 2)``. -Assert Hyp:``0<2``; [Sup0 | Generalize (sqrt_positivity ``2`` (Rlt_le ``0`` ``2`` Hyp)); Intro H1; Elim H1; Intro H2; [Assumption | Absurd ``0 == (sqrt 2)``; [Apply not_sym; Apply sqrt2_neq_0 | Assumption]]]. -Qed. - -Lemma Rlt_sqrt3_0 : ``0<(sqrt 3)``. -Cut ~(O=(1)); [Intro H0; Assert Hyp:``0<2``; [Sup0 | Generalize (Rlt_le ``0`` ``2`` Hyp); Intro H1; Assert Hyp2:``0<3``; [Sup0 | Generalize (Rlt_le ``0`` ``3`` Hyp2); Intro H2; Generalize (lt_INR_0 (1) (neq_O_lt (1) H0)); Unfold INR; Intro H3; Generalize (Rlt_compatibility ``2`` ``0`` ``1`` H3); Rewrite Rplus_sym; Rewrite Rplus_Ol; Replace ``2+1`` with ``3``; [Intro H4; Generalize (sqrt_lt_1 ``2`` ``3`` H1 H2 H4); Clear H3; Intro H3; Apply (Rlt_trans ``0`` ``(sqrt 2)`` ``(sqrt 3)`` Rlt_sqrt2_0 H3) | Ring]]] | Discriminate]. -Qed. - -Lemma PI4_RGT_0 : ``0<PI/4``. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]. -Qed. - -Lemma cos_PI4 : ``(cos (PI/4))==1/(sqrt 2)``. -Proof with Trivial. -Apply Rsqr_inj. -Apply cos_ge_0. -Left; Apply (Rlt_trans ``-(PI/2)`` R0 ``PI/4`` _PI2_RLT_0 PI4_RGT_0). -Left; Apply PI4_RLT_PI2. -Left; Apply (Rmult_lt_pos R1 ``(Rinv (sqrt 2))``). -Sup. -Apply Rlt_Rinv; Apply Rlt_sqrt2_0. -Rewrite Rsqr_div. -Rewrite Rsqr_1; Rewrite Rsqr_sqrt. -Assert H : ``2<>0``; [DiscrR | Idtac]. -Unfold Rsqr; Pattern 1 ``(cos (PI/4))``; Rewrite <- sin_cos_PI4; Replace ``(sin (PI/4))*(cos (PI/4))`` with ``(1/2)*(2*(sin (PI/4))*(cos (PI/4)))``. -Rewrite <- sin_2a; Replace ``2*(PI/4)`` with ``PI/2``. -Rewrite sin_PI2. -Apply Rmult_1r. -Unfold Rdiv; Rewrite (Rmult_sym ``2``); Rewrite Rinv_Rmult. -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Unfold Rdiv; Rewrite Rmult_1l; Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Left; Sup. -Apply sqrt2_neq_0. -Qed. - -Lemma sin_PI4 : ``(sin (PI/4))==1/(sqrt 2)``. -Rewrite sin_cos_PI4; Apply cos_PI4. -Qed. - -Lemma tan_PI4 : ``(tan (PI/4))==1``. -Unfold tan; Rewrite sin_cos_PI4. -Unfold Rdiv; Apply Rinv_r. -Change ``(cos (PI/4))<>0``; Rewrite cos_PI4; Apply R1_sqrt2_neq_0. -Qed. - -Lemma cos3PI4 : ``(cos (3*(PI/4)))==-1/(sqrt 2)``. -Proof with Trivial. -Replace ``3*(PI/4)`` with ``(PI/2)-(-(PI/4))``. -Rewrite cos_shift; Rewrite sin_neg; Rewrite sin_PI4. -Unfold Rdiv; Rewrite Ropp_mul1. -Unfold Rminus; Rewrite Ropp_Ropp; Pattern 1 PI; Rewrite double_var; Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_Rmult; [Ring | DiscrR | DiscrR]. -Qed. - -Lemma sin3PI4 : ``(sin (3*(PI/4)))==1/(sqrt 2)``. -Proof with Trivial. -Replace ``3*(PI/4)`` with ``(PI/2)-(-(PI/4))``. -Rewrite sin_shift; Rewrite cos_neg; Rewrite cos_PI4. -Unfold Rminus; Rewrite Ropp_Ropp; Pattern 1 PI; Rewrite double_var; Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_Rmult; [Ring | DiscrR | DiscrR]. -Qed. - -Lemma cos_PI6 : ``(cos (PI/6))==(sqrt 3)/2``. -Proof with Trivial. -Apply Rsqr_inj. -Apply cos_ge_0. -Left; Apply (Rlt_trans ``-(PI/2)`` R0 ``PI/6`` _PI2_RLT_0 PI6_RGT_0). -Left; Apply PI6_RLT_PI2. -Left; Apply (Rmult_lt_pos ``(sqrt 3)`` ``(Rinv 2)``). -Apply Rlt_sqrt3_0. -Apply Rlt_Rinv; Sup0. -Assert H : ``2<>0``; [DiscrR | Idtac]. -Assert H1 : ``4<>0``; [Apply prod_neq_R0 | Idtac]. -Rewrite Rsqr_div. -Rewrite cos2; Unfold Rsqr; Rewrite sin_PI6; Rewrite sqrt_def. -Unfold Rdiv; Rewrite Rmult_1l; Apply r_Rmult_mult with ``4``. -Rewrite Rminus_distr; Rewrite (Rmult_sym ``3``); Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite Rmult_1r. -Rewrite <- (Rmult_sym ``/2``); Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Rewrite <- Rinv_r_sym. -Ring. -Left; Sup0. -Qed. - -Lemma tan_PI6 : ``(tan (PI/6))==1/(sqrt 3)``. -Unfold tan; Rewrite sin_PI6; Rewrite cos_PI6; Unfold Rdiv; Repeat Rewrite Rmult_1l; Rewrite Rinv_Rmult. -Rewrite Rinv_Rinv. -Rewrite (Rmult_sym ``/2``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Apply Rmult_1r. -DiscrR. -DiscrR. -Red; Intro; Assert H1 := Rlt_sqrt3_0; Rewrite H in H1; Elim (Rlt_antirefl ``0`` H1). -Apply Rinv_neq_R0; DiscrR. -Qed. - -Lemma sin_PI3 : ``(sin (PI/3))==(sqrt 3)/2``. -Rewrite sin_PI3_cos_PI6; Apply cos_PI6. -Qed. - -Lemma cos_PI3 : ``(cos (PI/3))==1/2``. -Rewrite sin_PI6_cos_PI3; Apply sin_PI6. -Qed. - -Lemma tan_PI3 : ``(tan (PI/3))==(sqrt 3)``. -Unfold tan; Rewrite sin_PI3; Rewrite cos_PI3; Unfold Rdiv; Rewrite Rmult_1l; Rewrite Rinv_Rinv. -Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Apply Rmult_1r. -DiscrR. -DiscrR. -Qed. - -Lemma sin_2PI3 : ``(sin (2*(PI/3)))==(sqrt 3)/2``. -Rewrite double; Rewrite sin_plus; Rewrite sin_PI3; Rewrite cos_PI3; Unfold Rdiv; Repeat Rewrite Rmult_1l; Rewrite (Rmult_sym ``/2``); Repeat Rewrite <- Rmult_assoc; Rewrite double_var; Reflexivity. -Qed. - -Lemma cos_2PI3 : ``(cos (2*(PI/3)))==-1/2``. -Proof with Trivial. -Assert H : ``2<>0``; [DiscrR | Idtac]. -Assert H0 : ``4<>0``; [Apply prod_neq_R0 | Idtac]. -Rewrite double; Rewrite cos_plus; Rewrite sin_PI3; Rewrite cos_PI3; Unfold Rdiv; Rewrite Rmult_1l; Apply r_Rmult_mult with ``4``. -Rewrite Rminus_distr; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym ``2``). -Repeat Rewrite Rmult_assoc; Rewrite <- (Rinv_l_sym). -Rewrite Rmult_1r; Rewrite <- Rinv_r_sym. -Pattern 4 ``2``; Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite Ropp_mul3; Rewrite Rmult_1r. -Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite (Rmult_sym ``2``); Rewrite (Rmult_sym ``/2``). -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite sqrt_def. -Ring. -Left; Sup. -Qed. - -Lemma tan_2PI3 : ``(tan (2*(PI/3)))==-(sqrt 3)``. -Proof with Trivial. -Assert H : ``2<>0``; [DiscrR | Idtac]. -Unfold tan; Rewrite sin_2PI3; Rewrite cos_2PI3; Unfold Rdiv; Rewrite Ropp_mul1; Rewrite Rmult_1l; Rewrite <- Ropp_Rinv. -Rewrite Rinv_Rinv. -Rewrite Rmult_assoc; Rewrite Ropp_mul3; Rewrite <- Rinv_l_sym. -Ring. -Apply Rinv_neq_R0. -Qed. - -Lemma cos_5PI4 : ``(cos (5*(PI/4)))==-1/(sqrt 2)``. -Proof with Trivial. -Replace ``5*(PI/4)`` with ``(PI/4)+(PI)``. -Rewrite neg_cos; Rewrite cos_PI4; Unfold Rdiv; Rewrite Ropp_mul1. -Pattern 2 PI; Rewrite double_var; Pattern 2 3 PI; Rewrite double_var; Assert H : ``2<>0``; [DiscrR | Unfold Rdiv; Repeat Rewrite Rinv_Rmult; Try Ring]. -Qed. - -Lemma sin_5PI4 : ``(sin (5*(PI/4)))==-1/(sqrt 2)``. -Proof with Trivial. -Replace ``5*(PI/4)`` with ``(PI/4)+(PI)``. -Rewrite neg_sin; Rewrite sin_PI4; Unfold Rdiv; Rewrite Ropp_mul1. -Pattern 2 PI; Rewrite double_var; Pattern 2 3 PI; Rewrite double_var; Assert H : ``2<>0``; [DiscrR | Unfold Rdiv; Repeat Rewrite Rinv_Rmult; Try Ring]. -Qed. - -Lemma sin_cos5PI4 : ``(cos (5*(PI/4)))==(sin (5*(PI/4)))``. -Rewrite cos_5PI4; Rewrite sin_5PI4; Reflexivity. -Qed. - -Lemma Rgt_3PI2_0 : ``0<3*(PI/2)``. -Apply Rmult_lt_pos; [Sup0 | Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]]. -Qed. - -Lemma Rgt_2PI_0 : ``0<2*PI``. -Apply Rmult_lt_pos; [Sup0 | Apply PI_RGT_0]. -Qed. - -Lemma Rlt_PI_3PI2 : ``PI<3*(PI/2)``. -Generalize PI2_RGT_0; Intro H1; Generalize (Rlt_compatibility PI ``0`` ``PI/2`` H1); Replace ``PI+(PI/2)`` with ``3*(PI/2)``. -Rewrite Rplus_Or; Intro H2; Assumption. -Pattern 2 PI; Rewrite double_var; Ring. +Lemma tan_PI : tan PI = 0. +unfold tan in |- *; rewrite sin_PI; rewrite cos_PI; unfold Rdiv in |- *; + apply Rmult_0_l. +Qed. + +Lemma sin_3PI2 : sin (3 * (PI / 2)) = -1. +replace (3 * (PI / 2)) with (PI + PI / 2). +rewrite sin_plus; rewrite sin_PI; rewrite cos_PI; rewrite sin_PI2; ring. +pattern PI at 1 in |- *; rewrite (double_var PI); ring. +Qed. + +Lemma tan_2PI : tan (2 * PI) = 0. +unfold tan in |- *; rewrite sin_2PI; unfold Rdiv in |- *; apply Rmult_0_l. +Qed. + +Lemma sin_cos_PI4 : sin (PI / 4) = cos (PI / 4). +Proof with trivial. +rewrite cos_sin... +replace (PI / 2 + PI / 4) with (- (PI / 4) + PI)... +rewrite neg_sin; rewrite sin_neg; ring... +cut (PI = PI / 2 + PI / 2); [ intro | apply double_var ]... +pattern PI at 2 3 in |- *; rewrite H; pattern PI at 2 3 in |- *; rewrite H... +assert (H0 : 2 <> 0); + [ discrR | unfold Rdiv in |- *; rewrite Rinv_mult_distr; try ring ]... +Qed. + +Lemma sin_PI3_cos_PI6 : sin (PI / 3) = cos (PI / 6). +Proof with trivial. +replace (PI / 6) with (PI / 2 - PI / 3)... +rewrite cos_shift... +assert (H0 : 6 <> 0); [ discrR | idtac ]... +assert (H1 : 3 <> 0); [ discrR | idtac ]... +assert (H2 : 2 <> 0); [ discrR | idtac ]... +apply Rmult_eq_reg_l with 6... +rewrite Rmult_minus_distr_l; repeat rewrite (Rmult_comm 6)... +unfold Rdiv in |- *; repeat rewrite Rmult_assoc... +rewrite <- Rinv_l_sym... +rewrite (Rmult_comm (/ 3)); repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym... +pattern PI at 2 in |- *; rewrite (Rmult_comm PI); repeat rewrite Rmult_1_r; + repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym... +ring... +Qed. + +Lemma sin_PI6_cos_PI3 : cos (PI / 3) = sin (PI / 6). +Proof with trivial. +replace (PI / 6) with (PI / 2 - PI / 3)... +rewrite sin_shift... +assert (H0 : 6 <> 0); [ discrR | idtac ]... +assert (H1 : 3 <> 0); [ discrR | idtac ]... +assert (H2 : 2 <> 0); [ discrR | idtac ]... +apply Rmult_eq_reg_l with 6... +rewrite Rmult_minus_distr_l; repeat rewrite (Rmult_comm 6)... +unfold Rdiv in |- *; repeat rewrite Rmult_assoc... +rewrite <- Rinv_l_sym... +rewrite (Rmult_comm (/ 3)); repeat rewrite Rmult_assoc; rewrite <- Rinv_r_sym... +pattern PI at 2 in |- *; rewrite (Rmult_comm PI); repeat rewrite Rmult_1_r; + repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym... +ring... +Qed. + +Lemma PI6_RGT_0 : 0 < PI / 6. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup0 ]. +Qed. + +Lemma PI6_RLT_PI2 : PI / 6 < PI / 2. +unfold Rdiv in |- *; apply Rmult_lt_compat_l. +apply PI_RGT_0. +apply Rinv_lt_contravar; prove_sup. +Qed. + +Lemma sin_PI6 : sin (PI / 6) = 1 / 2. +Proof with trivial. +assert (H : 2 <> 0); [ discrR | idtac ]... +apply Rmult_eq_reg_l with (2 * cos (PI / 6))... +replace (2 * cos (PI / 6) * sin (PI / 6)) with + (2 * sin (PI / 6) * cos (PI / 6))... +rewrite <- sin_2a; replace (2 * (PI / 6)) with (PI / 3)... +rewrite sin_PI3_cos_PI6... +unfold Rdiv in |- *; rewrite Rmult_1_l; rewrite Rmult_assoc; + pattern 2 at 2 in |- *; rewrite (Rmult_comm 2); rewrite Rmult_assoc; + rewrite <- Rinv_l_sym... +rewrite Rmult_1_r... +unfold Rdiv in |- *; rewrite Rinv_mult_distr... +rewrite (Rmult_comm (/ 2)); rewrite (Rmult_comm 2); + repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... +rewrite Rmult_1_r... +discrR... +ring... +apply prod_neq_R0... +cut (0 < cos (PI / 6)); + [ intro H1; auto with real + | apply cos_gt_0; + [ apply (Rlt_trans (- (PI / 2)) 0 (PI / 6) _PI2_RLT_0 PI6_RGT_0) + | apply PI6_RLT_PI2 ] ]... +Qed. + +Lemma sqrt2_neq_0 : sqrt 2 <> 0. +assert (Hyp : 0 < 2); + [ prove_sup0 + | generalize (Rlt_le 0 2 Hyp); intro H1; red in |- *; intro H2; + generalize (sqrt_eq_0 2 H1 H2); intro H; absurd (2 = 0); + [ discrR | assumption ] ]. +Qed. + +Lemma R1_sqrt2_neq_0 : 1 / sqrt 2 <> 0. +generalize (Rinv_neq_0_compat (sqrt 2) sqrt2_neq_0); intro H; + generalize (prod_neq_R0 1 (/ sqrt 2) R1_neq_R0 H); + intro H0; assumption. +Qed. + +Lemma sqrt3_2_neq_0 : 2 * sqrt 3 <> 0. +apply prod_neq_R0; + [ discrR + | assert (Hyp : 0 < 3); + [ prove_sup0 + | generalize (Rlt_le 0 3 Hyp); intro H1; red in |- *; intro H2; + generalize (sqrt_eq_0 3 H1 H2); intro H; absurd (3 = 0); + [ discrR | assumption ] ] ]. +Qed. + +Lemma Rlt_sqrt2_0 : 0 < sqrt 2. +assert (Hyp : 0 < 2); + [ prove_sup0 + | generalize (sqrt_positivity 2 (Rlt_le 0 2 Hyp)); intro H1; elim H1; + intro H2; + [ assumption + | absurd (0 = sqrt 2); + [ apply (sym_not_eq (A:=R)); apply sqrt2_neq_0 | assumption ] ] ]. +Qed. + +Lemma Rlt_sqrt3_0 : 0 < sqrt 3. +cut (0%nat <> 1%nat); + [ intro H0; assert (Hyp : 0 < 2); + [ prove_sup0 + | generalize (Rlt_le 0 2 Hyp); intro H1; assert (Hyp2 : 0 < 3); + [ prove_sup0 + | generalize (Rlt_le 0 3 Hyp2); intro H2; + generalize (lt_INR_0 1 (neq_O_lt 1 H0)); + unfold INR in |- *; intro H3; + generalize (Rplus_lt_compat_l 2 0 1 H3); + rewrite Rplus_comm; rewrite Rplus_0_l; replace (2 + 1) with 3; + [ intro H4; generalize (sqrt_lt_1 2 3 H1 H2 H4); clear H3; intro H3; + apply (Rlt_trans 0 (sqrt 2) (sqrt 3) Rlt_sqrt2_0 H3) + | ring ] ] ] + | discriminate ]. +Qed. + +Lemma PI4_RGT_0 : 0 < PI / 4. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup0 ]. +Qed. + +Lemma cos_PI4 : cos (PI / 4) = 1 / sqrt 2. +Proof with trivial. +apply Rsqr_inj... +apply cos_ge_0... +left; apply (Rlt_trans (- (PI / 2)) 0 (PI / 4) _PI2_RLT_0 PI4_RGT_0)... +left; apply PI4_RLT_PI2... +left; apply (Rmult_lt_0_compat 1 (/ sqrt 2))... +prove_sup... +apply Rinv_0_lt_compat; apply Rlt_sqrt2_0... +rewrite Rsqr_div... +rewrite Rsqr_1; rewrite Rsqr_sqrt... +assert (H : 2 <> 0); [ discrR | idtac ]... +unfold Rsqr in |- *; pattern (cos (PI / 4)) at 1 in |- *; + rewrite <- sin_cos_PI4; + replace (sin (PI / 4) * cos (PI / 4)) with + (1 / 2 * (2 * sin (PI / 4) * cos (PI / 4)))... +rewrite <- sin_2a; replace (2 * (PI / 4)) with (PI / 2)... +rewrite sin_PI2... +apply Rmult_1_r... +unfold Rdiv in |- *; rewrite (Rmult_comm 2); rewrite Rinv_mult_distr... +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... +rewrite Rmult_1_r... +unfold Rdiv in |- *; rewrite Rmult_1_l; repeat rewrite <- Rmult_assoc... +rewrite <- Rinv_l_sym... +rewrite Rmult_1_l... +left; prove_sup... +apply sqrt2_neq_0... +Qed. + +Lemma sin_PI4 : sin (PI / 4) = 1 / sqrt 2. +rewrite sin_cos_PI4; apply cos_PI4. +Qed. + +Lemma tan_PI4 : tan (PI / 4) = 1. +unfold tan in |- *; rewrite sin_cos_PI4. +unfold Rdiv in |- *; apply Rinv_r. +change (cos (PI / 4) <> 0) in |- *; rewrite cos_PI4; apply R1_sqrt2_neq_0. +Qed. + +Lemma cos3PI4 : cos (3 * (PI / 4)) = -1 / sqrt 2. +Proof with trivial. +replace (3 * (PI / 4)) with (PI / 2 - - (PI / 4))... +rewrite cos_shift; rewrite sin_neg; rewrite sin_PI4... +unfold Rdiv in |- *; rewrite Ropp_mult_distr_l_reverse... +unfold Rminus in |- *; rewrite Ropp_involutive; pattern PI at 1 in |- *; + rewrite double_var; unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; + repeat rewrite Rmult_assoc; rewrite <- Rinv_mult_distr; + [ ring | discrR | discrR ]... +Qed. + +Lemma sin3PI4 : sin (3 * (PI / 4)) = 1 / sqrt 2. +Proof with trivial. +replace (3 * (PI / 4)) with (PI / 2 - - (PI / 4))... +rewrite sin_shift; rewrite cos_neg; rewrite cos_PI4... +unfold Rminus in |- *; rewrite Ropp_involutive; pattern PI at 1 in |- *; + rewrite double_var; unfold Rdiv in |- *; rewrite Rmult_plus_distr_r; + repeat rewrite Rmult_assoc; rewrite <- Rinv_mult_distr; + [ ring | discrR | discrR ]... +Qed. + +Lemma cos_PI6 : cos (PI / 6) = sqrt 3 / 2. +Proof with trivial. +apply Rsqr_inj... +apply cos_ge_0... +left; apply (Rlt_trans (- (PI / 2)) 0 (PI / 6) _PI2_RLT_0 PI6_RGT_0)... +left; apply PI6_RLT_PI2... +left; apply (Rmult_lt_0_compat (sqrt 3) (/ 2))... +apply Rlt_sqrt3_0... +apply Rinv_0_lt_compat; prove_sup0... +assert (H : 2 <> 0); [ discrR | idtac ]... +assert (H1 : 4 <> 0); [ apply prod_neq_R0 | idtac ]... +rewrite Rsqr_div... +rewrite cos2; unfold Rsqr in |- *; rewrite sin_PI6; rewrite sqrt_def... +unfold Rdiv in |- *; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4... +rewrite Rmult_minus_distr_l; rewrite (Rmult_comm 3); + repeat rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym... +rewrite Rmult_1_l; rewrite Rmult_1_r... +rewrite <- (Rmult_comm (/ 2)); repeat rewrite <- Rmult_assoc... +rewrite <- Rinv_l_sym... +rewrite Rmult_1_l; rewrite <- Rinv_r_sym... +ring... +left; prove_sup0... +Qed. + +Lemma tan_PI6 : tan (PI / 6) = 1 / sqrt 3. +unfold tan in |- *; rewrite sin_PI6; rewrite cos_PI6; unfold Rdiv in |- *; + repeat rewrite Rmult_1_l; rewrite Rinv_mult_distr. +rewrite Rinv_involutive. +rewrite (Rmult_comm (/ 2)); rewrite Rmult_assoc; rewrite <- Rinv_r_sym. +apply Rmult_1_r. +discrR. +discrR. +red in |- *; intro; assert (H1 := Rlt_sqrt3_0); rewrite H in H1; + elim (Rlt_irrefl 0 H1). +apply Rinv_neq_0_compat; discrR. +Qed. + +Lemma sin_PI3 : sin (PI / 3) = sqrt 3 / 2. +rewrite sin_PI3_cos_PI6; apply cos_PI6. +Qed. + +Lemma cos_PI3 : cos (PI / 3) = 1 / 2. +rewrite sin_PI6_cos_PI3; apply sin_PI6. +Qed. + +Lemma tan_PI3 : tan (PI / 3) = sqrt 3. +unfold tan in |- *; rewrite sin_PI3; rewrite cos_PI3; unfold Rdiv in |- *; + rewrite Rmult_1_l; rewrite Rinv_involutive. +rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +apply Rmult_1_r. +discrR. +discrR. +Qed. + +Lemma sin_2PI3 : sin (2 * (PI / 3)) = sqrt 3 / 2. +rewrite double; rewrite sin_plus; rewrite sin_PI3; rewrite cos_PI3; + unfold Rdiv in |- *; repeat rewrite Rmult_1_l; rewrite (Rmult_comm (/ 2)); + repeat rewrite <- Rmult_assoc; rewrite double_var; + reflexivity. +Qed. + +Lemma cos_2PI3 : cos (2 * (PI / 3)) = -1 / 2. +Proof with trivial. +assert (H : 2 <> 0); [ discrR | idtac ]... +assert (H0 : 4 <> 0); [ apply prod_neq_R0 | idtac ]... +rewrite double; rewrite cos_plus; rewrite sin_PI3; rewrite cos_PI3; + unfold Rdiv in |- *; rewrite Rmult_1_l; apply Rmult_eq_reg_l with 4... +rewrite Rmult_minus_distr_l; repeat rewrite Rmult_assoc; + rewrite (Rmult_comm 2)... +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... +rewrite Rmult_1_r; rewrite <- Rinv_r_sym... +pattern 2 at 4 in |- *; rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym... +rewrite Rmult_1_r; rewrite Ropp_mult_distr_r_reverse; rewrite Rmult_1_r... +rewrite (Rmult_comm 2); repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... +rewrite Rmult_1_r; rewrite (Rmult_comm 2); rewrite (Rmult_comm (/ 2))... +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym... +rewrite Rmult_1_r; rewrite sqrt_def... +ring... +left; prove_sup... +Qed. + +Lemma tan_2PI3 : tan (2 * (PI / 3)) = - sqrt 3. +Proof with trivial. +assert (H : 2 <> 0); [ discrR | idtac ]... +unfold tan in |- *; rewrite sin_2PI3; rewrite cos_2PI3; unfold Rdiv in |- *; + rewrite Ropp_mult_distr_l_reverse; rewrite Rmult_1_l; + rewrite <- Ropp_inv_permute... +rewrite Rinv_involutive... +rewrite Rmult_assoc; rewrite Ropp_mult_distr_r_reverse; rewrite <- Rinv_l_sym... +ring... +apply Rinv_neq_0_compat... +Qed. + +Lemma cos_5PI4 : cos (5 * (PI / 4)) = -1 / sqrt 2. +Proof with trivial. +replace (5 * (PI / 4)) with (PI / 4 + PI)... +rewrite neg_cos; rewrite cos_PI4; unfold Rdiv in |- *; + rewrite Ropp_mult_distr_l_reverse... +pattern PI at 2 in |- *; rewrite double_var; pattern PI at 2 3 in |- *; + rewrite double_var; assert (H : 2 <> 0); + [ discrR | unfold Rdiv in |- *; repeat rewrite Rinv_mult_distr; try ring ]... +Qed. + +Lemma sin_5PI4 : sin (5 * (PI / 4)) = -1 / sqrt 2. +Proof with trivial. +replace (5 * (PI / 4)) with (PI / 4 + PI)... +rewrite neg_sin; rewrite sin_PI4; unfold Rdiv in |- *; + rewrite Ropp_mult_distr_l_reverse... +pattern PI at 2 in |- *; rewrite double_var; pattern PI at 2 3 in |- *; + rewrite double_var; assert (H : 2 <> 0); + [ discrR | unfold Rdiv in |- *; repeat rewrite Rinv_mult_distr; try ring ]... +Qed. + +Lemma sin_cos5PI4 : cos (5 * (PI / 4)) = sin (5 * (PI / 4)). +rewrite cos_5PI4; rewrite sin_5PI4; reflexivity. +Qed. + +Lemma Rgt_3PI2_0 : 0 < 3 * (PI / 2). +apply Rmult_lt_0_compat; + [ prove_sup0 + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply PI_RGT_0 | apply Rinv_0_lt_compat; prove_sup0 ] ]. +Qed. + +Lemma Rgt_2PI_0 : 0 < 2 * PI. +apply Rmult_lt_0_compat; [ prove_sup0 | apply PI_RGT_0 ]. +Qed. + +Lemma Rlt_PI_3PI2 : PI < 3 * (PI / 2). +generalize PI2_RGT_0; intro H1; + generalize (Rplus_lt_compat_l PI 0 (PI / 2) H1); + replace (PI + PI / 2) with (3 * (PI / 2)). +rewrite Rplus_0_r; intro H2; assumption. +pattern PI at 2 in |- *; rewrite double_var; ring. Qed. -Lemma Rlt_3PI2_2PI : ``3*(PI/2)<2*PI``. -Generalize PI2_RGT_0; Intro H1; Generalize (Rlt_compatibility ``3*(PI/2)`` ``0`` ``PI/2`` H1); Replace ``3*(PI/2)+(PI/2)`` with ``2*PI``. -Rewrite Rplus_Or; Intro H2; Assumption. -Rewrite double; Pattern 1 2 PI; Rewrite double_var; Ring. +Lemma Rlt_3PI2_2PI : 3 * (PI / 2) < 2 * PI. +generalize PI2_RGT_0; intro H1; + generalize (Rplus_lt_compat_l (3 * (PI / 2)) 0 (PI / 2) H1); + replace (3 * (PI / 2) + PI / 2) with (2 * PI). +rewrite Rplus_0_r; intro H2; assumption. +rewrite double; pattern PI at 1 2 in |- *; rewrite double_var; ring. Qed. (***************************************************************) (* Radian -> Degree | Degree -> Radian *) (***************************************************************) -Definition plat : R := ``180``. -Definition toRad [x:R] : R := ``x*PI*/plat``. -Definition toDeg [x:R] : R := ``x*plat*/PI``. +Definition plat : R := 180. +Definition toRad (x:R) : R := x * PI * / plat. +Definition toDeg (x:R) : R := x * plat * / PI. -Lemma rad_deg : (x:R) (toRad (toDeg x))==x. -Intro; Unfold toRad toDeg; Replace ``x*plat*/PI*PI*/plat`` with ``x*(plat*/plat)*(PI*/PI)``; [Idtac | Ring]. -Repeat Rewrite <- Rinv_r_sym. -Ring. -Apply PI_neq0. -Unfold plat; DiscrR. +Lemma rad_deg : forall x:R, toRad (toDeg x) = x. +intro; unfold toRad, toDeg in |- *; + replace (x * plat * / PI * PI * / plat) with + (x * (plat * / plat) * (PI * / PI)); [ idtac | ring ]. +repeat rewrite <- Rinv_r_sym. +ring. +apply PI_neq0. +unfold plat in |- *; discrR. Qed. -Lemma toRad_inj : (x,y:R) (toRad x)==(toRad y) -> x==y. -Intros; Unfold toRad in H; Apply r_Rmult_mult with PI. -Rewrite <- (Rmult_sym x); Rewrite <- (Rmult_sym y). -Apply r_Rmult_mult with ``/plat``. -Rewrite <- (Rmult_sym ``x*PI``); Rewrite <- (Rmult_sym ``y*PI``); Assumption. -Apply Rinv_neq_R0; Unfold plat; DiscrR. -Apply PI_neq0. +Lemma toRad_inj : forall x y:R, toRad x = toRad y -> x = y. +intros; unfold toRad in H; apply Rmult_eq_reg_l with PI. +rewrite <- (Rmult_comm x); rewrite <- (Rmult_comm y). +apply Rmult_eq_reg_l with (/ plat). +rewrite <- (Rmult_comm (x * PI)); rewrite <- (Rmult_comm (y * PI)); + assumption. +apply Rinv_neq_0_compat; unfold plat in |- *; discrR. +apply PI_neq0. Qed. -Lemma deg_rad : (x:R) (toDeg (toRad x))==x. -Intro x; Apply toRad_inj; Rewrite -> (rad_deg (toRad x)); Reflexivity. +Lemma deg_rad : forall x:R, toDeg (toRad x) = x. +intro x; apply toRad_inj; rewrite (rad_deg (toRad x)); reflexivity. Qed. -Definition sind [x:R] : R := (sin (toRad x)). -Definition cosd [x:R] : R := (cos (toRad x)). -Definition tand [x:R] : R := (tan (toRad x)). +Definition sind (x:R) : R := sin (toRad x). +Definition cosd (x:R) : R := cos (toRad x). +Definition tand (x:R) : R := tan (toRad x). -Lemma Rsqr_sin_cos_d_one : (x:R) ``(Rsqr (sind x))+(Rsqr (cosd x))==1``. -Intro x; Unfold sind; Unfold cosd; Apply sin2_cos2. +Lemma Rsqr_sin_cos_d_one : forall x:R, Rsqr (sind x) + Rsqr (cosd x) = 1. +intro x; unfold sind in |- *; unfold cosd in |- *; apply sin2_cos2. Qed. (***************************************************) (* Other properties *) (***************************************************) -Lemma sin_lb_ge_0 : (a:R) ``0<=a``->``a<=PI/2``->``0<=(sin_lb a)``. -Intros; Case (total_order R0 a); Intro. -Left; Apply sin_lb_gt_0; Assumption. -Elim H1; Intro. -Rewrite <- H2; Unfold sin_lb; Unfold sin_approx; Unfold sum_f_R0; Unfold sin_term; Repeat Rewrite pow_ne_zero. -Unfold Rdiv; Repeat Rewrite Rmult_Ol; Repeat Rewrite Rmult_Or; Repeat Rewrite Rplus_Or; Right; Reflexivity. -Discriminate. -Discriminate. -Discriminate. -Discriminate. -Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` a ``0`` H H2)). -Qed. +Lemma sin_lb_ge_0 : forall a:R, 0 <= a -> a <= PI / 2 -> 0 <= sin_lb a. +intros; case (Rtotal_order 0 a); intro. +left; apply sin_lb_gt_0; assumption. +elim H1; intro. +rewrite <- H2; unfold sin_lb in |- *; unfold sin_approx in |- *; + unfold sum_f_R0 in |- *; unfold sin_term in |- *; + repeat rewrite pow_ne_zero. +unfold Rdiv in |- *; repeat rewrite Rmult_0_l; repeat rewrite Rmult_0_r; + repeat rewrite Rplus_0_r; right; reflexivity. +discriminate. +discriminate. +discriminate. +discriminate. +elim (Rlt_irrefl 0 (Rle_lt_trans 0 a 0 H H2)). +Qed.
\ No newline at end of file diff --git a/theories/Reals/Rtrigo_def.v b/theories/Reals/Rtrigo_def.v index 82c63a7b2..f18e9188e 100644 --- a/theories/Reals/Rtrigo_def.v +++ b/theories/Reals/Rtrigo_def.v @@ -8,350 +8,405 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo_fun. -Require Max. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo_fun. +Require Import Max. Open Local Scope R_scope. (*****************************) (* Definition of exponential *) (*****************************) -Definition exp_in:R->R->Prop := [x,l:R](infinit_sum [i:nat]``/(INR (fact i))*(pow x i)`` l). +Definition exp_in (x l:R) : Prop := + infinit_sum (fun i:nat => / INR (fact i) * x ^ i) l. -Lemma exp_cof_no_R0 : (n:nat) ``/(INR (fact n))<>0``. -Intro. -Apply Rinv_neq_R0. -Apply INR_fact_neq_0. +Lemma exp_cof_no_R0 : forall n:nat, / INR (fact n) <> 0. +intro. +apply Rinv_neq_0_compat. +apply INR_fact_neq_0. Qed. -Lemma exist_exp : (x:R)(SigT R [l:R](exp_in x l)). -Intro; Generalize (Alembert_C3 [n:nat](Rinv (INR (fact n))) x exp_cof_no_R0 Alembert_exp). -Unfold Pser exp_in. -Trivial. +Lemma exist_exp : forall x:R, sigT (fun l:R => exp_in x l). +intro; + generalize + (Alembert_C3 (fun n:nat => / INR (fact n)) x exp_cof_no_R0 Alembert_exp). +unfold Pser, exp_in in |- *. +trivial. Defined. -Definition exp : R -> R := [x:R](projT1 ? ? (exist_exp x)). +Definition exp (x:R) : R := projT1 (exist_exp x). -Lemma pow_i : (i:nat) (lt O i) -> (pow R0 i)==R0. -Intros; Apply pow_ne_zero. -Red; Intro; Rewrite H0 in H; Elim (lt_n_n ? H). +Lemma pow_i : forall i:nat, (0 < i)%nat -> 0 ^ i = 0. +intros; apply pow_ne_zero. +red in |- *; intro; rewrite H0 in H; elim (lt_irrefl _ H). Qed. (*i Calculus of $e^0$ *) -Lemma exist_exp0 : (SigT R [l:R](exp_in R0 l)). -Apply Specif.existT with R1. -Unfold exp_in; Unfold infinit_sum; Intros. -Exists O. -Intros; Replace (sum_f_R0 ([i:nat]``/(INR (fact i))*(pow R0 i)``) n) with R1. -Unfold R_dist; Replace ``1-1`` with R0; [Rewrite Rabsolu_R0; Assumption | Ring]. -Induction n. -Simpl; Rewrite Rinv_R1; Ring. -Rewrite tech5. -Rewrite <- Hrecn. -Simpl. -Ring. -Unfold ge; Apply le_O_n. +Lemma exist_exp0 : sigT (fun l:R => exp_in 0 l). +apply existT with 1. +unfold exp_in in |- *; unfold infinit_sum in |- *; intros. +exists 0%nat. +intros; replace (sum_f_R0 (fun i:nat => / INR (fact i) * 0 ^ i) n) with 1. +unfold R_dist in |- *; replace (1 - 1) with 0; + [ rewrite Rabs_R0; assumption | ring ]. +induction n as [| n Hrecn]. +simpl in |- *; rewrite Rinv_1; ring. +rewrite tech5. +rewrite <- Hrecn. +simpl in |- *. +ring. +unfold ge in |- *; apply le_O_n. Defined. -Lemma exp_0 : ``(exp 0)==1``. -Cut (exp_in R0 (exp R0)). -Cut (exp_in R0 R1). -Unfold exp_in; Intros; EApply unicity_sum. -Apply H0. -Apply H. -Exact (projT2 ? ? exist_exp0). -Exact (projT2 ? ? (exist_exp R0)). +Lemma exp_0 : exp 0 = 1. +cut (exp_in 0 (exp 0)). +cut (exp_in 0 1). +unfold exp_in in |- *; intros; eapply uniqueness_sum. +apply H0. +apply H. +exact (projT2 exist_exp0). +exact (projT2 (exist_exp 0)). Qed. (**************************************) (* Definition of hyperbolic functions *) (**************************************) -Definition cosh : R->R := [x:R]``((exp x)+(exp (-x)))/2``. -Definition sinh : R->R := [x:R]``((exp x)-(exp (-x)))/2``. -Definition tanh : R->R := [x:R]``(sinh x)/(cosh x)``. +Definition cosh (x:R) : R := (exp x + exp (- x)) / 2. +Definition sinh (x:R) : R := (exp x - exp (- x)) / 2. +Definition tanh (x:R) : R := sinh x / cosh x. -Lemma cosh_0 : ``(cosh 0)==1``. -Unfold cosh; Rewrite Ropp_O; Rewrite exp_0. -Unfold Rdiv; Rewrite <- Rinv_r_sym; [Reflexivity | DiscrR]. +Lemma cosh_0 : cosh 0 = 1. +unfold cosh in |- *; rewrite Ropp_0; rewrite exp_0. +unfold Rdiv in |- *; rewrite <- Rinv_r_sym; [ reflexivity | discrR ]. Qed. -Lemma sinh_0 : ``(sinh 0)==0``. -Unfold sinh; Rewrite Ropp_O; Rewrite exp_0. -Unfold Rminus Rdiv; Rewrite Rplus_Ropp_r; Apply Rmult_Ol. +Lemma sinh_0 : sinh 0 = 0. +unfold sinh in |- *; rewrite Ropp_0; rewrite exp_0. +unfold Rminus, Rdiv in |- *; rewrite Rplus_opp_r; apply Rmult_0_l. Qed. -Definition cos_n [n:nat] : R := ``(pow (-1) n)/(INR (fact (mult (S (S O)) n)))``. - -Lemma simpl_cos_n : (n:nat) (Rdiv (cos_n (S n)) (cos_n n))==(Ropp (Rinv (INR (mult (mult (2) (S n)) (plus (mult (2) n) (1)))))). -Intro; Unfold cos_n; Replace (S n) with (plus n (1)); [Idtac | Ring]. -Rewrite pow_add; Unfold Rdiv; Rewrite Rinv_Rmult. -Rewrite Rinv_Rinv. -Replace ``(pow ( -1) n)*(pow ( -1) (S O))*/(INR (fact (mult (S (S O)) (plus n (S O)))))*(/(pow ( -1) n)*(INR (fact (mult (S (S O)) n))))`` with ``((pow ( -1) n)*/(pow ( -1) n))*/(INR (fact (mult (S (S O)) (plus n (S O)))))*(INR (fact (mult (S (S O)) n)))*(pow (-1) (S O))``; [Idtac | Ring]. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Unfold pow; Rewrite Rmult_1r. -Replace (mult (S (S O)) (plus n (S O))) with (S (S (mult (S (S O)) n))); [Idtac | Ring]. -Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR; Repeat Rewrite Rinv_Rmult; Try (Apply not_O_INR; Discriminate). -Rewrite <- (Rmult_sym ``-1``). -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Replace (S (mult (S (S O)) n)) with (plus (mult (S (S O)) n) (S O)); [Idtac | Ring]. -Rewrite mult_INR; Rewrite Rinv_Rmult. -Ring. -Apply not_O_INR; Discriminate. -Replace (plus (mult (S (S O)) n) (S O)) with (S (mult (S (S O)) n)); [Apply not_O_INR; Discriminate | Ring]. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply prod_neq_R0; [Apply not_O_INR; Discriminate | Apply INR_fact_neq_0]. -Apply pow_nonzero; DiscrR. -Apply INR_fact_neq_0. -Apply pow_nonzero; DiscrR. -Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Definition cos_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n)). + +Lemma simpl_cos_n : + forall n:nat, cos_n (S n) / cos_n n = - / INR (2 * S n * (2 * n + 1)). +intro; unfold cos_n in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. +rewrite pow_add; unfold Rdiv in |- *; rewrite Rinv_mult_distr. +rewrite Rinv_involutive. +replace + ((-1) ^ n * (-1) ^ 1 * / INR (fact (2 * (n + 1))) * + (/ (-1) ^ n * INR (fact (2 * n)))) with + ((-1) ^ n * / (-1) ^ n * / INR (fact (2 * (n + 1))) * INR (fact (2 * n)) * + (-1) ^ 1); [ idtac | ring ]. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; unfold pow in |- *; rewrite Rmult_1_r. +replace (2 * (n + 1))%nat with (S (S (2 * n))); [ idtac | ring ]. +do 2 rewrite fact_simpl; do 2 rewrite mult_INR; + repeat rewrite Rinv_mult_distr; try (apply not_O_INR; discriminate). +rewrite <- (Rmult_comm (-1)). +repeat rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r. +replace (S (2 * n)) with (2 * n + 1)%nat; [ idtac | ring ]. +rewrite mult_INR; rewrite Rinv_mult_distr. +ring. +apply not_O_INR; discriminate. +replace (2 * n + 1)%nat with (S (2 * n)); + [ apply not_O_INR; discriminate | ring ]. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ]. +apply pow_nonzero; discrR. +apply INR_fact_neq_0. +apply pow_nonzero; discrR. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. Qed. -Lemma archimed_cor1 : (eps:R) ``0<eps`` -> (EX N : nat | ``/(INR N) < eps``/\(lt O N)). -Intros; Cut ``/eps < (IZR (up (/eps)))``. -Intro; Cut `0<=(up (Rinv eps))`. -Intro; Assert H2 := (IZN ? H1); Elim H2; Intros; Exists (max x (1)). -Split. -Cut ``0<(IZR (INZ x))``. -Intro; Rewrite INR_IZR_INZ; Apply Rle_lt_trans with ``/(IZR (INZ x))``. -Apply Rle_monotony_contra with (IZR (INZ x)). -Assumption. -Rewrite <- Rinv_r_sym; [Idtac | Red; Intro; Rewrite H5 in H4; Elim (Rlt_antirefl ? H4)]. -Apply Rle_monotony_contra with (IZR (INZ (max x (1)))). -Apply Rlt_le_trans with (IZR (INZ x)). -Assumption. -Repeat Rewrite <- INR_IZR_INZ; Apply le_INR; Apply le_max_l. -Rewrite Rmult_1r; Rewrite (Rmult_sym (IZR (INZ (max x (S O))))); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Repeat Rewrite <- INR_IZR_INZ; Apply le_INR; Apply le_max_l. -Rewrite <- INR_IZR_INZ; Apply not_O_INR. -Red; Intro;Assert H6 := (le_max_r x (1)); Cut (lt O (1)); [Intro | Apply lt_O_Sn]; Assert H8 := (lt_le_trans ? ? ? H7 H6); Rewrite H5 in H8; Elim (lt_n_n ? H8). -Pattern 1 eps; Rewrite <- Rinv_Rinv. -Apply Rinv_lt. -Apply Rmult_lt_pos; [Apply Rlt_Rinv; Assumption | Assumption]. -Rewrite H3 in H0; Assumption. -Red; Intro; Rewrite H5 in H; Elim (Rlt_antirefl ? H). -Apply Rlt_trans with ``/eps``. -Apply Rlt_Rinv; Assumption. -Rewrite H3 in H0; Assumption. -Apply lt_le_trans with (1); [Apply lt_O_Sn | Apply le_max_r]. -Apply le_IZR; Replace (IZR `0`) with R0; [Idtac | Reflexivity]; Left; Apply Rlt_trans with ``/eps``; [Apply Rlt_Rinv; Assumption | Assumption]. -Assert H0 := (archimed ``/eps``). -Elim H0; Intros; Assumption. +Lemma archimed_cor1 : + forall eps:R, 0 < eps -> exists N : nat | / INR N < eps /\ (0 < N)%nat. +intros; cut (/ eps < IZR (up (/ eps))). +intro; cut (0 <= up (/ eps))%Z. +intro; assert (H2 := IZN _ H1); elim H2; intros; exists (max x 1). +split. +cut (0 < IZR (Z_of_nat x)). +intro; rewrite INR_IZR_INZ; apply Rle_lt_trans with (/ IZR (Z_of_nat x)). +apply Rmult_le_reg_l with (IZR (Z_of_nat x)). +assumption. +rewrite <- Rinv_r_sym; + [ idtac | red in |- *; intro; rewrite H5 in H4; elim (Rlt_irrefl _ H4) ]. +apply Rmult_le_reg_l with (IZR (Z_of_nat (max x 1))). +apply Rlt_le_trans with (IZR (Z_of_nat x)). +assumption. +repeat rewrite <- INR_IZR_INZ; apply le_INR; apply le_max_l. +rewrite Rmult_1_r; rewrite (Rmult_comm (IZR (Z_of_nat (max x 1)))); + rewrite Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; repeat rewrite <- INR_IZR_INZ; apply le_INR; + apply le_max_l. +rewrite <- INR_IZR_INZ; apply not_O_INR. +red in |- *; intro; assert (H6 := le_max_r x 1); cut (0 < 1)%nat; + [ intro | apply lt_O_Sn ]; assert (H8 := lt_le_trans _ _ _ H7 H6); + rewrite H5 in H8; elim (lt_irrefl _ H8). +pattern eps at 1 in |- *; rewrite <- Rinv_involutive. +apply Rinv_lt_contravar. +apply Rmult_lt_0_compat; [ apply Rinv_0_lt_compat; assumption | assumption ]. +rewrite H3 in H0; assumption. +red in |- *; intro; rewrite H5 in H; elim (Rlt_irrefl _ H). +apply Rlt_trans with (/ eps). +apply Rinv_0_lt_compat; assumption. +rewrite H3 in H0; assumption. +apply lt_le_trans with 1%nat; [ apply lt_O_Sn | apply le_max_r ]. +apply le_IZR; replace (IZR 0) with 0; [ idtac | reflexivity ]; left; + apply Rlt_trans with (/ eps); + [ apply Rinv_0_lt_compat; assumption | assumption ]. +assert (H0 := archimed (/ eps)). +elim H0; intros; assumption. Qed. -Lemma Alembert_cos : (Un_cv [n:nat]``(Rabsolu (cos_n (S n))/(cos_n n))`` R0). -Unfold Un_cv; Intros. -Assert H0 := (archimed_cor1 eps H). -Elim H0; Intros; Exists x. -Intros; Rewrite simpl_cos_n; Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Rewrite Rabsolu_Ropp; Rewrite Rabsolu_right. -Rewrite mult_INR; Rewrite Rinv_Rmult. -Cut ``/(INR (mult (S (S O)) (S n)))<1``. -Intro; Cut ``/(INR (plus (mult (S (S O)) n) (S O)))<eps``. -Intro; Rewrite <- (Rmult_1l eps). -Apply Rmult_lt; Try Assumption. -Change ``0</(INR (plus (mult (S (S O)) n) (S O)))``; Apply Rlt_Rinv; Apply lt_INR_0. -Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_O_Sn | Ring]. -Apply Rlt_R0_R1. -Cut (lt x (plus (mult (2) n) (1))). -Intro; Assert H5 := (lt_INR ? ? H4). -Apply Rlt_trans with ``/(INR x)``. -Apply Rinv_lt. -Apply Rmult_lt_pos. -Apply lt_INR_0. -Elim H1; Intros; Assumption. -Apply lt_INR_0; Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Apply lt_O_Sn | Ring]. -Assumption. -Elim H1; Intros; Assumption. -Apply lt_le_trans with (S n). -Unfold ge in H2; Apply le_lt_n_Sm; Assumption. -Replace (plus (mult (2) n) (1)) with (S (mult (2) n)); [Idtac | Ring]. -Apply le_n_S; Apply le_n_2n. -Apply Rlt_monotony_contra with (INR (mult (S (S O)) (S n))). -Apply lt_INR_0; Replace (mult (2) (S n)) with (S (S (mult (2) n))). -Apply lt_O_Sn. -Replace (S n) with (plus n (1)); [Idtac | Ring]. -Ring. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Replace R1 with (INR (1)); [Apply lt_INR | Reflexivity]. -Replace (mult (2) (S n)) with (S (S (mult (2) n))). -Apply lt_n_S; Apply lt_O_Sn. -Replace (S n) with (plus n (1)); [Ring | Ring]. -Apply not_O_INR; Discriminate. -Apply not_O_INR; Discriminate. -Replace (plus (mult (S (S O)) n) (S O)) with (S (mult (2) n)); [Apply not_O_INR; Discriminate | Ring]. -Apply Rle_sym1; Left; Apply Rlt_Rinv. -Apply lt_INR_0. -Replace (mult (mult (2) (S n)) (plus (mult (2) n) (1))) with (S (S (plus (mult (4) (mult n n)) (mult (6) n)))). -Apply lt_O_Sn. -Apply INR_eq. -Repeat Rewrite S_INR; Rewrite plus_INR; Repeat Rewrite mult_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Replace (INR O) with R0; [Ring | Reflexivity]. +Lemma Alembert_cos : Un_cv (fun n:nat => Rabs (cos_n (S n) / cos_n n)) 0. +unfold Un_cv in |- *; intros. +assert (H0 := archimed_cor1 eps H). +elim H0; intros; exists x. +intros; rewrite simpl_cos_n; unfold R_dist in |- *; unfold Rminus in |- *; + rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; + rewrite Rabs_Ropp; rewrite Rabs_right. +rewrite mult_INR; rewrite Rinv_mult_distr. +cut (/ INR (2 * S n) < 1). +intro; cut (/ INR (2 * n + 1) < eps). +intro; rewrite <- (Rmult_1_l eps). +apply Rmult_gt_0_lt_compat; try assumption. +change (0 < / INR (2 * n + 1)) in |- *; apply Rinv_0_lt_compat; + apply lt_INR_0. +replace (2 * n + 1)%nat with (S (2 * n)); [ apply lt_O_Sn | ring ]. +apply Rlt_0_1. +cut (x < 2 * n + 1)%nat. +intro; assert (H5 := lt_INR _ _ H4). +apply Rlt_trans with (/ INR x). +apply Rinv_lt_contravar. +apply Rmult_lt_0_compat. +apply lt_INR_0. +elim H1; intros; assumption. +apply lt_INR_0; replace (2 * n + 1)%nat with (S (2 * n)); + [ apply lt_O_Sn | ring ]. +assumption. +elim H1; intros; assumption. +apply lt_le_trans with (S n). +unfold ge in H2; apply le_lt_n_Sm; assumption. +replace (2 * n + 1)%nat with (S (2 * n)); [ idtac | ring ]. +apply le_n_S; apply le_n_2n. +apply Rmult_lt_reg_l with (INR (2 * S n)). +apply lt_INR_0; replace (2 * S n)%nat with (S (S (2 * n))). +apply lt_O_Sn. +replace (S n) with (n + 1)%nat; [ idtac | ring ]. +ring. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; replace 1 with (INR 1); [ apply lt_INR | reflexivity ]. +replace (2 * S n)%nat with (S (S (2 * n))). +apply lt_n_S; apply lt_O_Sn. +replace (S n) with (n + 1)%nat; [ ring | ring ]. +apply not_O_INR; discriminate. +apply not_O_INR; discriminate. +replace (2 * n + 1)%nat with (S (2 * n)); + [ apply not_O_INR; discriminate | ring ]. +apply Rle_ge; left; apply Rinv_0_lt_compat. +apply lt_INR_0. +replace (2 * S n * (2 * n + 1))%nat with (S (S (4 * (n * n) + 6 * n))). +apply lt_O_Sn. +apply INR_eq. +repeat rewrite S_INR; rewrite plus_INR; repeat rewrite mult_INR; + rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR; + replace (INR 0) with 0; [ ring | reflexivity ]. Qed. -Lemma cosn_no_R0 : (n:nat)``(cos_n n)<>0``. -Intro; Unfold cos_n; Unfold Rdiv; Apply prod_neq_R0. -Apply pow_nonzero; DiscrR. -Apply Rinv_neq_R0. -Apply INR_fact_neq_0. +Lemma cosn_no_R0 : forall n:nat, cos_n n <> 0. +intro; unfold cos_n in |- *; unfold Rdiv in |- *; apply prod_neq_R0. +apply pow_nonzero; discrR. +apply Rinv_neq_0_compat. +apply INR_fact_neq_0. Qed. (**********) -Definition cos_in:R->R->Prop := [x,l:R](infinit_sum [i:nat]``(cos_n i)*(pow x i)`` l). +Definition cos_in (x l:R) : Prop := + infinit_sum (fun i:nat => cos_n i * x ^ i) l. (**********) -Lemma exist_cos : (x:R)(SigT R [l:R](cos_in x l)). -Intro; Generalize (Alembert_C3 cos_n x cosn_no_R0 Alembert_cos). -Unfold Pser cos_in; Trivial. +Lemma exist_cos : forall x:R, sigT (fun l:R => cos_in x l). +intro; generalize (Alembert_C3 cos_n x cosn_no_R0 Alembert_cos). +unfold Pser, cos_in in |- *; trivial. Qed. (* Definition of cosinus *) (*************************) -Definition cos : R -> R := [x:R](Cases (exist_cos (Rsqr x)) of (Specif.existT a b) => a end). - - -Definition sin_n [n:nat] : R := ``(pow (-1) n)/(INR (fact (plus (mult (S (S O)) n) (S O))))``. - -Lemma simpl_sin_n : (n:nat) (Rdiv (sin_n (S n)) (sin_n n))==(Ropp (Rinv (INR (mult (plus (mult (2) (S n)) (1)) (mult (2) (S n)))))). -Intro; Unfold sin_n; Replace (S n) with (plus n (1)); [Idtac | Ring]. -Rewrite pow_add; Unfold Rdiv; Rewrite Rinv_Rmult. -Rewrite Rinv_Rinv. -Replace ``(pow ( -1) n)*(pow ( -1) (S O))*/(INR (fact (plus (mult (S (S O)) (plus n (S O))) (S O))))*(/(pow ( -1) n)*(INR (fact (plus (mult (S (S O)) n) (S O)))))`` with ``((pow ( -1) n)*/(pow ( -1) n))*/(INR (fact (plus (mult (S (S O)) (plus n (S O))) (S O))))*(INR (fact (plus (mult (S (S O)) n) (S O))))*(pow (-1) (S O))``; [Idtac | Ring]. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Unfold pow; Rewrite Rmult_1r; Replace (plus (mult (S (S O)) (plus n (S O))) (S O)) with (S (S (plus (mult (S (S O)) n) (S O)))). -Do 2 Rewrite fact_simpl; Do 2 Rewrite mult_INR; Repeat Rewrite Rinv_Rmult. -Rewrite <- (Rmult_sym ``-1``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Replace (S (plus (mult (S (S O)) n) (S O))) with (mult (S (S O)) (plus n (S O))). -Repeat Rewrite mult_INR; Repeat Rewrite Rinv_Rmult. -Ring. -Apply not_O_INR; Discriminate. -Replace (plus n (S O)) with (S n); [Apply not_O_INR; Discriminate | Ring]. -Apply not_O_INR; Discriminate. -Apply prod_neq_R0. -Apply not_O_INR; Discriminate. -Replace (plus n (S O)) with (S n); [Apply not_O_INR; Discriminate | Ring]. -Apply not_O_INR; Discriminate. -Replace (plus n (S O)) with (S n); [Apply not_O_INR; Discriminate | Ring]. -Rewrite mult_plus_distr_r; Cut (n:nat) (S n)=(plus n (1)). -Intros; Rewrite (H (plus (mult (2) n) (1))). -Ring. -Intros; Ring. -Apply INR_fact_neq_0. -Apply not_O_INR; Discriminate. -Apply INR_fact_neq_0. -Apply not_O_INR; Discriminate. -Apply prod_neq_R0; [Apply not_O_INR; Discriminate | Apply INR_fact_neq_0]. -Cut (n:nat) (S (S n))=(plus n (2)); [Intros; Rewrite (H (plus (mult (2) n) (1))); Ring | Intros; Ring]. -Apply pow_nonzero; DiscrR. -Apply INR_fact_neq_0. -Apply pow_nonzero; DiscrR. -Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Definition cos (x:R) : R := + match exist_cos (Rsqr x) with + | existT a b => a + end. + + +Definition sin_n (n:nat) : R := (-1) ^ n / INR (fact (2 * n + 1)). + +Lemma simpl_sin_n : + forall n:nat, sin_n (S n) / sin_n n = - / INR ((2 * S n + 1) * (2 * S n)). +intro; unfold sin_n in |- *; replace (S n) with (n + 1)%nat; [ idtac | ring ]. +rewrite pow_add; unfold Rdiv in |- *; rewrite Rinv_mult_distr. +rewrite Rinv_involutive. +replace + ((-1) ^ n * (-1) ^ 1 * / INR (fact (2 * (n + 1) + 1)) * + (/ (-1) ^ n * INR (fact (2 * n + 1)))) with + ((-1) ^ n * / (-1) ^ n * / INR (fact (2 * (n + 1) + 1)) * + INR (fact (2 * n + 1)) * (-1) ^ 1); [ idtac | ring ]. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; unfold pow in |- *; rewrite Rmult_1_r; + replace (2 * (n + 1) + 1)%nat with (S (S (2 * n + 1))). +do 2 rewrite fact_simpl; do 2 rewrite mult_INR; + repeat rewrite Rinv_mult_distr. +rewrite <- (Rmult_comm (-1)); repeat rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; replace (S (2 * n + 1)) with (2 * (n + 1))%nat. +repeat rewrite mult_INR; repeat rewrite Rinv_mult_distr. +ring. +apply not_O_INR; discriminate. +replace (n + 1)%nat with (S n); [ apply not_O_INR; discriminate | ring ]. +apply not_O_INR; discriminate. +apply prod_neq_R0. +apply not_O_INR; discriminate. +replace (n + 1)%nat with (S n); [ apply not_O_INR; discriminate | ring ]. +apply not_O_INR; discriminate. +replace (n + 1)%nat with (S n); [ apply not_O_INR; discriminate | ring ]. +rewrite mult_plus_distr_l; cut (forall n:nat, S n = (n + 1)%nat). +intros; rewrite (H (2 * n + 1)%nat). +ring. +intros; ring. +apply INR_fact_neq_0. +apply not_O_INR; discriminate. +apply INR_fact_neq_0. +apply not_O_INR; discriminate. +apply prod_neq_R0; [ apply not_O_INR; discriminate | apply INR_fact_neq_0 ]. +cut (forall n:nat, S (S n) = (n + 2)%nat); + [ intros; rewrite (H (2 * n + 1)%nat); ring | intros; ring ]. +apply pow_nonzero; discrR. +apply INR_fact_neq_0. +apply pow_nonzero; discrR. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. Qed. -Lemma Alembert_sin : (Un_cv [n:nat]``(Rabsolu (sin_n (S n))/(sin_n n))`` R0). -Unfold Un_cv; Intros; Assert H0 := (archimed_cor1 eps H). -Elim H0; Intros; Exists x. -Intros; Rewrite simpl_sin_n; Unfold R_dist; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_Rabsolu; Rewrite Rabsolu_Ropp; Rewrite Rabsolu_right. -Rewrite mult_INR; Rewrite Rinv_Rmult. -Cut ``/(INR (mult (S (S O)) (S n)))<1``. -Intro; Cut ``/(INR (plus (mult (S (S O)) (S n)) (S O)))<eps``. -Intro; Rewrite <- (Rmult_1l eps); Rewrite (Rmult_sym ``/(INR (plus (mult (S (S O)) (S n)) (S O)))``); Apply Rmult_lt; Try Assumption. -Change ``0</(INR (plus (mult (S (S O)) (S n)) (S O)))``; Apply Rlt_Rinv; Apply lt_INR_0; Replace (plus (mult (2) (S n)) (1)) with (S (mult (2) (S n))); [Apply lt_O_Sn | Ring]. -Apply Rlt_R0_R1. -Cut (lt x (plus (mult (2) (S n)) (1))). -Intro; Assert H5 := (lt_INR ? ? H4); Apply Rlt_trans with ``/(INR x)``. -Apply Rinv_lt. -Apply Rmult_lt_pos. -Apply lt_INR_0; Elim H1; Intros; Assumption. -Apply lt_INR_0; Replace (plus (mult (2) (S n)) (1)) with (S (mult (2) (S n))); [Apply lt_O_Sn | Ring]. -Assumption. -Elim H1; Intros; Assumption. -Apply lt_le_trans with (S n). -Unfold ge in H2; Apply le_lt_n_Sm; Assumption. -Replace (plus (mult (2) (S n)) (1)) with (S (mult (2) (S n))); [Idtac | Ring]. -Apply le_S; Apply le_n_2n. -Apply Rlt_monotony_contra with (INR (mult (S (S O)) (S n))). -Apply lt_INR_0; Replace (mult (2) (S n)) with (S (S (mult (2) n))); [Apply lt_O_Sn | Replace (S n) with (plus n (1)); [Idtac | Ring]; Ring]. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Replace R1 with (INR (1)); [Apply lt_INR | Reflexivity]. -Replace (mult (2) (S n)) with (S (S (mult (2) n))). -Apply lt_n_S; Apply lt_O_Sn. -Replace (S n) with (plus n (1)); [Ring | Ring]. -Apply not_O_INR; Discriminate. -Apply not_O_INR; Discriminate. -Apply not_O_INR; Discriminate. -Left; Change ``0</(INR (mult (plus (mult (S (S O)) (S n)) (S O)) (mult (S (S O)) (S n))))``; Apply Rlt_Rinv. -Apply lt_INR_0. -Replace (mult (plus (mult (2) (S n)) (1)) (mult (2) (S n))) with (S (S (S (S (S (S (plus (mult (4) (mult n n)) (mult (10) n)))))))). -Apply lt_O_Sn. -Apply INR_eq; Repeat Rewrite S_INR; Rewrite plus_INR; Repeat Rewrite mult_INR; Rewrite plus_INR; Rewrite mult_INR; Repeat Rewrite S_INR; Replace (INR O) with R0; [Ring | Reflexivity]. +Lemma Alembert_sin : Un_cv (fun n:nat => Rabs (sin_n (S n) / sin_n n)) 0. +unfold Un_cv in |- *; intros; assert (H0 := archimed_cor1 eps H). +elim H0; intros; exists x. +intros; rewrite simpl_sin_n; unfold R_dist in |- *; unfold Rminus in |- *; + rewrite Ropp_0; rewrite Rplus_0_r; rewrite Rabs_Rabsolu; + rewrite Rabs_Ropp; rewrite Rabs_right. +rewrite mult_INR; rewrite Rinv_mult_distr. +cut (/ INR (2 * S n) < 1). +intro; cut (/ INR (2 * S n + 1) < eps). +intro; rewrite <- (Rmult_1_l eps); rewrite (Rmult_comm (/ INR (2 * S n + 1))); + apply Rmult_gt_0_lt_compat; try assumption. +change (0 < / INR (2 * S n + 1)) in |- *; apply Rinv_0_lt_compat; + apply lt_INR_0; replace (2 * S n + 1)%nat with (S (2 * S n)); + [ apply lt_O_Sn | ring ]. +apply Rlt_0_1. +cut (x < 2 * S n + 1)%nat. +intro; assert (H5 := lt_INR _ _ H4); apply Rlt_trans with (/ INR x). +apply Rinv_lt_contravar. +apply Rmult_lt_0_compat. +apply lt_INR_0; elim H1; intros; assumption. +apply lt_INR_0; replace (2 * S n + 1)%nat with (S (2 * S n)); + [ apply lt_O_Sn | ring ]. +assumption. +elim H1; intros; assumption. +apply lt_le_trans with (S n). +unfold ge in H2; apply le_lt_n_Sm; assumption. +replace (2 * S n + 1)%nat with (S (2 * S n)); [ idtac | ring ]. +apply le_S; apply le_n_2n. +apply Rmult_lt_reg_l with (INR (2 * S n)). +apply lt_INR_0; replace (2 * S n)%nat with (S (S (2 * n))); + [ apply lt_O_Sn | replace (S n) with (n + 1)%nat; [ idtac | ring ]; ring ]. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; replace 1 with (INR 1); [ apply lt_INR | reflexivity ]. +replace (2 * S n)%nat with (S (S (2 * n))). +apply lt_n_S; apply lt_O_Sn. +replace (S n) with (n + 1)%nat; [ ring | ring ]. +apply not_O_INR; discriminate. +apply not_O_INR; discriminate. +apply not_O_INR; discriminate. +left; change (0 < / INR ((2 * S n + 1) * (2 * S n))) in |- *; + apply Rinv_0_lt_compat. +apply lt_INR_0. +replace ((2 * S n + 1) * (2 * S n))%nat with + (S (S (S (S (S (S (4 * (n * n) + 10 * n))))))). +apply lt_O_Sn. +apply INR_eq; repeat rewrite S_INR; rewrite plus_INR; repeat rewrite mult_INR; + rewrite plus_INR; rewrite mult_INR; repeat rewrite S_INR; + replace (INR 0) with 0; [ ring | reflexivity ]. Qed. -Lemma sin_no_R0 : (n:nat)``(sin_n n)<>0``. -Intro; Unfold sin_n; Unfold Rdiv; Apply prod_neq_R0. -Apply pow_nonzero; DiscrR. -Apply Rinv_neq_R0; Apply INR_fact_neq_0. +Lemma sin_no_R0 : forall n:nat, sin_n n <> 0. +intro; unfold sin_n in |- *; unfold Rdiv in |- *; apply prod_neq_R0. +apply pow_nonzero; discrR. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. Qed. (**********) -Definition sin_in:R->R->Prop := [x,l:R](infinit_sum [i:nat]``(sin_n i)*(pow x i)`` l). +Definition sin_in (x l:R) : Prop := + infinit_sum (fun i:nat => sin_n i * x ^ i) l. (**********) -Lemma exist_sin : (x:R)(SigT R [l:R](sin_in x l)). -Intro; Generalize (Alembert_C3 sin_n x sin_no_R0 Alembert_sin). -Unfold Pser sin_n; Trivial. +Lemma exist_sin : forall x:R, sigT (fun l:R => sin_in x l). +intro; generalize (Alembert_C3 sin_n x sin_no_R0 Alembert_sin). +unfold Pser, sin_n in |- *; trivial. Qed. (***********************) (* Definition of sinus *) -Definition sin : R -> R := [x:R](Cases (exist_sin (Rsqr x)) of (Specif.existT a b) => ``x*a`` end). +Definition sin (x:R) : R := + match exist_sin (Rsqr x) with + | existT a b => x * a + end. (*********************************************) (* PROPERTIES *) (*********************************************) -Lemma cos_sym : (x:R) ``(cos x)==(cos (-x))``. -Intros; Unfold cos; Replace ``(Rsqr (-x))`` with (Rsqr x). -Reflexivity. -Apply Rsqr_neg. +Lemma cos_sym : forall x:R, cos x = cos (- x). +intros; unfold cos in |- *; replace (Rsqr (- x)) with (Rsqr x). +reflexivity. +apply Rsqr_neg. Qed. -Lemma sin_antisym : (x:R)``(sin (-x))==-(sin x)``. -Intro; Unfold sin; Replace ``(Rsqr (-x))`` with (Rsqr x); [Idtac | Apply Rsqr_neg]. -Case (exist_sin (Rsqr x)); Intros; Ring. +Lemma sin_antisym : forall x:R, sin (- x) = - sin x. +intro; unfold sin in |- *; replace (Rsqr (- x)) with (Rsqr x); + [ idtac | apply Rsqr_neg ]. +case (exist_sin (Rsqr x)); intros; ring. Qed. -Lemma sin_0 : ``(sin 0)==0``. -Unfold sin; Case (exist_sin (Rsqr R0)). -Intros; Ring. +Lemma sin_0 : sin 0 = 0. +unfold sin in |- *; case (exist_sin (Rsqr 0)). +intros; ring. Qed. -Lemma exist_cos0 : (SigT R [l:R](cos_in R0 l)). -Apply Specif.existT with R1. -Unfold cos_in; Unfold infinit_sum; Intros; Exists O. -Intros. -Unfold R_dist. -Induction n. -Unfold cos_n; Simpl. -Unfold Rdiv; Rewrite Rinv_R1. -Do 2 Rewrite Rmult_1r. -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Rewrite tech5. -Replace ``(cos_n (S n))*(pow 0 (S n))`` with R0. -Rewrite Rplus_Or. -Apply Hrecn; Unfold ge; Apply le_O_n. -Simpl; Ring. +Lemma exist_cos0 : sigT (fun l:R => cos_in 0 l). +apply existT with 1. +unfold cos_in in |- *; unfold infinit_sum in |- *; intros; exists 0%nat. +intros. +unfold R_dist in |- *. +induction n as [| n Hrecn]. +unfold cos_n in |- *; simpl in |- *. +unfold Rdiv in |- *; rewrite Rinv_1. +do 2 rewrite Rmult_1_r. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. +rewrite tech5. +replace (cos_n (S n) * 0 ^ S n) with 0. +rewrite Rplus_0_r. +apply Hrecn; unfold ge in |- *; apply le_O_n. +simpl in |- *; ring. Defined. (* Calculus of (cos 0) *) -Lemma cos_0 : ``(cos 0)==1``. -Cut (cos_in R0 (cos R0)). -Cut (cos_in R0 R1). -Unfold cos_in; Intros; EApply unicity_sum. -Apply H0. -Apply H. -Exact (projT2 ? ? exist_cos0). -Assert H := (projT2 ? ? (exist_cos (Rsqr R0))); Unfold cos; Pattern 1 R0; Replace R0 with (Rsqr R0); [Exact H | Apply Rsqr_O]. -Qed. +Lemma cos_0 : cos 0 = 1. +cut (cos_in 0 (cos 0)). +cut (cos_in 0 1). +unfold cos_in in |- *; intros; eapply uniqueness_sum. +apply H0. +apply H. +exact (projT2 exist_cos0). +assert (H := projT2 (exist_cos (Rsqr 0))); unfold cos in |- *; + pattern 0 at 1 in |- *; replace 0 with (Rsqr 0); [ exact H | apply Rsqr_0 ]. +Qed.
\ No newline at end of file diff --git a/theories/Reals/Rtrigo_fun.v b/theories/Reals/Rtrigo_fun.v index 33c3f6a5f..6470dd581 100644 --- a/theories/Reals/Rtrigo_fun.v +++ b/theories/Reals/Rtrigo_fun.v @@ -8,10 +8,9 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. Open Local Scope R_scope. (*****************************************************************) @@ -24,95 +23,87 @@ Open Local Scope R_scope. (*****************************************************************) (*********) -Lemma Alembert_exp:(Un_cv - [n:nat](Rabsolu (Rmult (Rinv (INR (fact (S n)))) - (Rinv (Rinv (INR (fact n)))))) R0). -Unfold Un_cv;Intros;Elim (total_order_Rgt eps R1);Intro. -Split with O;Intros;Rewrite (simpl_fact n);Unfold R_dist; - Rewrite (minus_R0 (Rabsolu (Rinv (INR (S n))))); - Rewrite (Rabsolu_Rabsolu (Rinv (INR (S n)))); - Cut (Rgt (Rinv (INR (S n))) R0). -Intro; Rewrite (Rabsolu_pos_eq (Rinv (INR (S n)))). -Cut (Rlt (Rminus (Rinv eps) R1) R0). -Intro;Generalize (Rlt_le_trans (Rminus (Rinv eps) R1) R0 (INR n) H2 - (pos_INR n));Clear H2;Intro; - Unfold Rminus in H2;Generalize (Rlt_compatibility R1 - (Rplus (Rinv eps) (Ropp R1)) (INR n) H2); - Replace (Rplus R1 (Rplus (Rinv eps) (Ropp R1))) with (Rinv eps); - [Clear H2;Intro|Ring]. -Rewrite (Rplus_sym R1 (INR n)) in H2;Rewrite <-(S_INR n) in H2; - Generalize (Rmult_gt (Rinv (INR (S n))) eps H1 H);Intro; - Unfold Rgt in H3; - Generalize (Rlt_monotony (Rmult (Rinv (INR (S n))) eps) (Rinv eps) - (INR (S n)) H3 H2);Intro; - Rewrite (Rmult_assoc (Rinv (INR (S n))) eps (Rinv eps)) in H4; - Rewrite (Rinv_r eps (imp_not_Req eps R0 - (or_intror (Rlt eps R0) (Rgt eps R0) H))) - in H4;Rewrite (let (H1,H2)=(Rmult_ne (Rinv (INR (S n)))) in H1) - in H4;Rewrite (Rmult_sym (Rinv (INR (S n)))) in H4; - Rewrite (Rmult_assoc eps (Rinv (INR (S n))) (INR (S n))) in H4; - Rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) - (sym_not_equal nat O (S n) (O_S n)))) in H4; - Rewrite (let (H1,H2)=(Rmult_ne eps) in H1) in H4;Assumption. -Apply Rlt_minus;Unfold Rgt in a;Rewrite <- Rinv_R1; - Apply (Rinv_lt R1 eps);Auto; - Rewrite (let (H1,H2)=(Rmult_ne eps) in H2);Unfold Rgt in H;Assumption. -Unfold Rgt in H1;Apply Rlt_le;Assumption. -Unfold Rgt;Apply Rlt_Rinv; Apply lt_INR_0;Apply lt_O_Sn. +Lemma Alembert_exp : + Un_cv (fun n:nat => Rabs (/ INR (fact (S n)) * / / INR (fact n))) 0. +unfold Un_cv in |- *; intros; elim (Rgt_dec eps 1); intro. +split with 0%nat; intros; rewrite (simpl_fact n); unfold R_dist in |- *; + rewrite (Rminus_0_r (Rabs (/ INR (S n)))); + rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0). +intro; rewrite (Rabs_pos_eq (/ INR (S n))). +cut (/ eps - 1 < 0). +intro; generalize (Rlt_le_trans (/ eps - 1) 0 (INR n) H2 (pos_INR n)); + clear H2; intro; unfold Rminus in H2; + generalize (Rplus_lt_compat_l 1 (/ eps + -1) (INR n) H2); + replace (1 + (/ eps + -1)) with (/ eps); [ clear H2; intro | ring ]. +rewrite (Rplus_comm 1 (INR n)) in H2; rewrite <- (S_INR n) in H2; + generalize (Rmult_gt_0_compat (/ INR (S n)) eps H1 H); + intro; unfold Rgt in H3; + generalize (Rmult_lt_compat_l (/ INR (S n) * eps) (/ eps) (INR (S n)) H3 H2); + intro; rewrite (Rmult_assoc (/ INR (S n)) eps (/ eps)) in H4; + rewrite (Rinv_r eps (Rlt_dichotomy_converse eps 0 (or_intror (eps < 0) H))) + in H4; rewrite (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1) in H4; + rewrite (Rmult_comm (/ INR (S n))) in H4; + rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H4; + rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (sym_not_equal (O_S n)))) in H4; + rewrite (let (H1, H2) := Rmult_ne eps in H1) in H4; + assumption. +apply Rlt_minus; unfold Rgt in a; rewrite <- Rinv_1; + apply (Rinv_lt_contravar 1 eps); auto; + rewrite (let (H1, H2) := Rmult_ne eps in H2); unfold Rgt in H; + assumption. +unfold Rgt in H1; apply Rlt_le; assumption. +unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. (**) -Cut `0<=(up (Rminus (Rinv eps) R1))`. -Intro;Elim (IZN (up (Rminus (Rinv eps) R1)) H0);Intros; - Split with x;Intros;Rewrite (simpl_fact n);Unfold R_dist; - Rewrite (minus_R0 (Rabsolu (Rinv (INR (S n))))); - Rewrite (Rabsolu_Rabsolu (Rinv (INR (S n)))); - Cut (Rgt (Rinv (INR (S n))) R0). -Intro; Rewrite (Rabsolu_pos_eq (Rinv (INR (S n)))). -Cut (Rlt (Rminus (Rinv eps) R1) (INR x)). -Intro;Generalize (Rlt_le_trans (Rminus (Rinv eps) R1) (INR x) (INR n) - H4 (le_INR x n ([n,m:nat; H:(ge m n)]H x n H2))); - Clear H4;Intro;Unfold Rminus in H4;Generalize (Rlt_compatibility R1 - (Rplus (Rinv eps) (Ropp R1)) (INR n) H4); - Replace (Rplus R1 (Rplus (Rinv eps) (Ropp R1))) with (Rinv eps); - [Clear H4;Intro|Ring]. -Rewrite (Rplus_sym R1 (INR n)) in H4;Rewrite <-(S_INR n) in H4; - Generalize (Rmult_gt (Rinv (INR (S n))) eps H3 H);Intro; - Unfold Rgt in H5; - Generalize (Rlt_monotony (Rmult (Rinv (INR (S n))) eps) (Rinv eps) - (INR (S n)) H5 H4);Intro; - Rewrite (Rmult_assoc (Rinv (INR (S n))) eps (Rinv eps)) in H6; - Rewrite (Rinv_r eps (imp_not_Req eps R0 - (or_intror (Rlt eps R0) (Rgt eps R0) H))) - in H6;Rewrite (let (H1,H2)=(Rmult_ne (Rinv (INR (S n)))) in H1) - in H6;Rewrite (Rmult_sym (Rinv (INR (S n)))) in H6; - Rewrite (Rmult_assoc eps (Rinv (INR (S n))) (INR (S n))) in H6; - Rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) - (sym_not_equal nat O (S n) (O_S n)))) in H6; - Rewrite (let (H1,H2)=(Rmult_ne eps) in H1) in H6;Assumption. -Cut (IZR (up (Rminus (Rinv eps) R1)))==(IZR (INZ x)); - [Intro|Rewrite H1;Trivial]. -Elim (archimed (Rminus (Rinv eps) R1));Intros;Clear H6; - Unfold Rgt in H5;Rewrite H4 in H5;Rewrite INR_IZR_INZ;Assumption. -Unfold Rgt in H1;Apply Rlt_le;Assumption. -Unfold Rgt;Apply Rlt_Rinv; Apply lt_INR_0;Apply lt_O_Sn. -Apply (le_O_IZR (up (Rminus (Rinv eps) R1))); - Apply (Rle_trans R0 (Rminus (Rinv eps) R1) - (IZR (up (Rminus (Rinv eps) R1)))). -Generalize (Rgt_not_le eps R1 b);Clear b;Unfold Rle;Intro;Elim H0; - Clear H0;Intro. -Left;Unfold Rgt in H; - Generalize (Rlt_monotony (Rinv eps) eps R1 (Rlt_Rinv eps H) H0); - Rewrite (Rinv_l eps (sym_not_eqT R R0 eps - (imp_not_Req R0 eps (or_introl (Rlt R0 eps) (Rgt R0 eps) H)))); - Rewrite (let (H1,H2)=(Rmult_ne (Rinv eps)) in H1);Intro; - Fold (Rgt (Rminus (Rinv eps) R1) R0);Apply Rgt_minus;Unfold Rgt; - Assumption. -Right;Rewrite H0;Rewrite Rinv_R1;Apply sym_eqT;Apply eq_Rminus;Auto. -Elim (archimed (Rminus (Rinv eps) R1));Intros;Clear H1; - Unfold Rgt in H0;Apply Rlt_le;Assumption. +cut (0 <= up (/ eps - 1))%Z. +intro; elim (IZN (up (/ eps - 1)) H0); intros; split with x; intros; + rewrite (simpl_fact n); unfold R_dist in |- *; + rewrite (Rminus_0_r (Rabs (/ INR (S n)))); + rewrite (Rabs_Rabsolu (/ INR (S n))); cut (/ INR (S n) > 0). +intro; rewrite (Rabs_pos_eq (/ INR (S n))). +cut (/ eps - 1 < INR x). +intro; + generalize + (Rlt_le_trans (/ eps - 1) (INR x) (INR n) H4 + (le_INR x n ((fun (n m:nat) (H:(m >= n)%nat) => H) x n H2))); + clear H4; intro; unfold Rminus in H4; + generalize (Rplus_lt_compat_l 1 (/ eps + -1) (INR n) H4); + replace (1 + (/ eps + -1)) with (/ eps); [ clear H4; intro | ring ]. +rewrite (Rplus_comm 1 (INR n)) in H4; rewrite <- (S_INR n) in H4; + generalize (Rmult_gt_0_compat (/ INR (S n)) eps H3 H); + intro; unfold Rgt in H5; + generalize (Rmult_lt_compat_l (/ INR (S n) * eps) (/ eps) (INR (S n)) H5 H4); + intro; rewrite (Rmult_assoc (/ INR (S n)) eps (/ eps)) in H6; + rewrite (Rinv_r eps (Rlt_dichotomy_converse eps 0 (or_intror (eps < 0) H))) + in H6; rewrite (let (H1, H2) := Rmult_ne (/ INR (S n)) in H1) in H6; + rewrite (Rmult_comm (/ INR (S n))) in H6; + rewrite (Rmult_assoc eps (/ INR (S n)) (INR (S n))) in H6; + rewrite (Rinv_l (INR (S n)) (not_O_INR (S n) (sym_not_equal (O_S n)))) in H6; + rewrite (let (H1, H2) := Rmult_ne eps in H1) in H6; + assumption. +cut (IZR (up (/ eps - 1)) = IZR (Z_of_nat x)); + [ intro | rewrite H1; trivial ]. +elim (archimed (/ eps - 1)); intros; clear H6; unfold Rgt in H5; + rewrite H4 in H5; rewrite INR_IZR_INZ; assumption. +unfold Rgt in H1; apply Rlt_le; assumption. +unfold Rgt in |- *; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. +apply (le_O_IZR (up (/ eps - 1))); + apply (Rle_trans 0 (/ eps - 1) (IZR (up (/ eps - 1)))). +generalize (Rnot_gt_le eps 1 b); clear b; unfold Rle in |- *; intro; elim H0; + clear H0; intro. +left; unfold Rgt in H; + generalize (Rmult_lt_compat_l (/ eps) eps 1 (Rinv_0_lt_compat eps H) H0); + rewrite + (Rinv_l eps + (sym_not_eq (Rlt_dichotomy_converse 0 eps (or_introl (0 > eps) H)))) + ; rewrite (let (H1, H2) := Rmult_ne (/ eps) in H1); + intro; fold (/ eps - 1 > 0) in |- *; apply Rgt_minus; + unfold Rgt in |- *; assumption. +right; rewrite H0; rewrite Rinv_1; apply sym_eq; apply Rminus_diag_eq; auto. +elim (archimed (/ eps - 1)); intros; clear H1; unfold Rgt in H0; apply Rlt_le; + assumption. Qed. - diff --git a/theories/Reals/Rtrigo_reg.v b/theories/Reals/Rtrigo_reg.v index 1155a05a0..ca0eb33dc 100644 --- a/theories/Reals/Rtrigo_reg.v +++ b/theories/Reals/Rtrigo_reg.v @@ -8,490 +8,601 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo. -Require Ranalysis1. -Require PSeries_reg. -V7only [Import nat_scope. Import Z_scope. Import R_scope.]. +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import Ranalysis1. +Require Import PSeries_reg. Open Local Scope nat_scope. Open Local Scope R_scope. -Lemma CVN_R_cos : (fn:nat->R->R) (fn == [N:nat][x:R]``(pow (-1) N)/(INR (fact (mult (S (S O)) N)))*(pow x (mult (S (S O)) N))``) -> (CVN_R fn). -Unfold CVN_R; Intros. -Cut (r::R)<>``0``. -Intro hyp_r; Unfold CVN_r. -Apply Specif.existT with [n:nat]``/(INR (fact (mult (S (S O)) n)))*(pow r (mult (S (S O)) n))``. -Cut (SigT ? [l:R](Un_cv [n:nat](sum_f_R0 [k:nat](Rabsolu ``/(INR (fact (mult (S (S O)) k)))*(pow r (mult (S (S O)) k))``) n) l)). -Intro; Elim X; Intros. -Apply existTT with x. -Split. -Apply p. -Intros; Rewrite H; Unfold Rdiv; Do 2 Rewrite Rabsolu_mult. -Rewrite pow_1_abs; Rewrite Rmult_1l. -Cut ``0</(INR (fact (mult (S (S O)) n)))``. -Intro; Rewrite (Rabsolu_right ? (Rle_sym1 ? ? (Rlt_le ? ? H1))). -Apply Rle_monotony. -Left; Apply H1. -Rewrite <- Pow_Rabsolu; Apply pow_maj_Rabs. -Rewrite Rabsolu_Rabsolu. -Unfold Boule in H0; Rewrite minus_R0 in H0. -Left; Apply H0. -Apply Rlt_Rinv; Apply INR_fact_lt_0. -Apply Alembert_C2. -Intro; Apply Rabsolu_no_R0. -Apply prod_neq_R0. -Apply Rinv_neq_R0. -Apply INR_fact_neq_0. -Apply pow_nonzero; Assumption. -Assert H0 := Alembert_cos. -Unfold cos_n in H0; Unfold Un_cv in H0; Unfold Un_cv; Intros. -Cut ``0<eps/(Rsqr r)``. -Intro; Elim (H0 ? H2); Intros N0 H3. -Exists N0; Intros. -Unfold R_dist; Assert H5 := (H3 ? H4). -Unfold R_dist in H5; Replace ``(Rabsolu ((Rabsolu (/(INR (fact (mult (S (S O)) (S n))))*(pow r (mult (S (S O)) (S n)))))/(Rabsolu (/(INR (fact (mult (S (S O)) n)))*(pow r (mult (S (S O)) n))))))`` with ``(Rsqr r)*(Rabsolu ((pow ( -1) (S n))/(INR (fact (mult (S (S O)) (S n))))/((pow ( -1) n)/(INR (fact (mult (S (S O)) n))))))``. -Apply Rlt_monotony_contra with ``/(Rsqr r)``. -Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption. -Pattern 1 ``/(Rsqr r)``; Replace ``/(Rsqr r)`` with ``(Rabsolu (/(Rsqr r)))``. -Rewrite <- Rabsolu_mult; Rewrite Rminus_distr; Rewrite Rmult_Or; Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps); Apply H5. -Unfold Rsqr; Apply prod_neq_R0; Assumption. -Rewrite Rabsolu_Rinv. -Rewrite Rabsolu_right. -Reflexivity. -Apply Rle_sym1; Apply pos_Rsqr. -Unfold Rsqr; Apply prod_neq_R0; Assumption. -Rewrite (Rmult_sym (Rsqr r)); Unfold Rdiv; Repeat Rewrite Rabsolu_mult; Rewrite Rabsolu_Rabsolu; Rewrite pow_1_abs; Rewrite Rmult_1l; Repeat Rewrite Rmult_assoc; Apply Rmult_mult_r. -Rewrite Rabsolu_Rinv. -Rewrite Rabsolu_mult; Rewrite (pow_1_abs n); Rewrite Rmult_1l; Rewrite <- Rabsolu_Rinv. -Rewrite Rinv_Rinv. -Rewrite Rinv_Rmult. -Rewrite Rabsolu_Rinv. -Rewrite Rinv_Rinv. -Rewrite (Rmult_sym ``(Rabsolu (Rabsolu (pow r (mult (S (S O)) (S n)))))``); Rewrite Rabsolu_mult; Rewrite Rabsolu_Rabsolu; Rewrite Rmult_assoc; Apply Rmult_mult_r. -Rewrite Rabsolu_Rinv. -Do 2 Rewrite Rabsolu_Rabsolu; Repeat Rewrite Rabsolu_right. -Replace ``(pow r (mult (S (S O)) (S n)))`` with ``(pow r (mult (S (S O)) n))*r*r``. -Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Unfold Rsqr; Ring. -Apply pow_nonzero; Assumption. -Replace (mult (2) (S n)) with (S (S (mult (2) n))). -Simpl; Ring. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Apply Rle_sym1; Apply pow_le; Left; Apply (cond_pos r). -Apply Rle_sym1; Apply pow_le; Left; Apply (cond_pos r). -Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. -Apply Rabsolu_no_R0; Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply Rabsolu_no_R0; Apply Rinv_neq_R0; Apply INR_fact_neq_0. -Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. -Apply INR_fact_neq_0. -Apply Rinv_neq_R0; Apply INR_fact_neq_0. -Apply prod_neq_R0. -Apply pow_nonzero; DiscrR. -Apply Rinv_neq_R0; Apply INR_fact_neq_0. -Unfold Rdiv; Apply Rmult_lt_pos. -Apply H1. -Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption. -Assert H0 := (cond_pos r); Red; Intro; Rewrite H1 in H0; Elim (Rlt_antirefl ? H0). +Lemma CVN_R_cos : + forall fn:nat -> R -> R, + fn = (fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)) -> + CVN_R fn. +unfold CVN_R in |- *; intros. +cut ((r:R) <> 0). +intro hyp_r; unfold CVN_r in |- *. +apply existT with (fun n:nat => / INR (fact (2 * n)) * r ^ (2 * n)). +cut + (sigT + (fun l:R => + Un_cv + (fun n:nat => + sum_f_R0 (fun k:nat => Rabs (/ INR (fact (2 * k)) * r ^ (2 * k))) + n) l)). +intro; elim X; intros. +apply existT with x. +split. +apply p. +intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult. +rewrite pow_1_abs; rewrite Rmult_1_l. +cut (0 < / INR (fact (2 * n))). +intro; rewrite (Rabs_right _ (Rle_ge _ _ (Rlt_le _ _ H1))). +apply Rmult_le_compat_l. +left; apply H1. +rewrite <- RPow_abs; apply pow_maj_Rabs. +rewrite Rabs_Rabsolu. +unfold Boule in H0; rewrite Rminus_0_r in H0. +left; apply H0. +apply Rinv_0_lt_compat; apply INR_fact_lt_0. +apply Alembert_C2. +intro; apply Rabs_no_R0. +apply prod_neq_R0. +apply Rinv_neq_0_compat. +apply INR_fact_neq_0. +apply pow_nonzero; assumption. +assert (H0 := Alembert_cos). +unfold cos_n in H0; unfold Un_cv in H0; unfold Un_cv in |- *; intros. +cut (0 < eps / Rsqr r). +intro; elim (H0 _ H2); intros N0 H3. +exists N0; intros. +unfold R_dist in |- *; assert (H5 := H3 _ H4). +unfold R_dist in H5; + replace + (Rabs + (Rabs (/ INR (fact (2 * S n)) * r ^ (2 * S n)) / + Rabs (/ INR (fact (2 * n)) * r ^ (2 * n)))) with + (Rsqr r * + Rabs ((-1) ^ S n / INR (fact (2 * S n)) / ((-1) ^ n / INR (fact (2 * n))))). +apply Rmult_lt_reg_l with (/ Rsqr r). +apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption. +pattern (/ Rsqr r) at 1 in |- *; replace (/ Rsqr r) with (Rabs (/ Rsqr r)). +rewrite <- Rabs_mult; rewrite Rmult_minus_distr_l; rewrite Rmult_0_r; + rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite <- (Rmult_comm eps); apply H5. +unfold Rsqr in |- *; apply prod_neq_R0; assumption. +rewrite Rabs_Rinv. +rewrite Rabs_right. +reflexivity. +apply Rle_ge; apply Rle_0_sqr. +unfold Rsqr in |- *; apply prod_neq_R0; assumption. +rewrite (Rmult_comm (Rsqr r)); unfold Rdiv in |- *; repeat rewrite Rabs_mult; + rewrite Rabs_Rabsolu; rewrite pow_1_abs; rewrite Rmult_1_l; + repeat rewrite Rmult_assoc; apply Rmult_eq_compat_l. +rewrite Rabs_Rinv. +rewrite Rabs_mult; rewrite (pow_1_abs n); rewrite Rmult_1_l; + rewrite <- Rabs_Rinv. +rewrite Rinv_involutive. +rewrite Rinv_mult_distr. +rewrite Rabs_Rinv. +rewrite Rinv_involutive. +rewrite (Rmult_comm (Rabs (Rabs (r ^ (2 * S n))))); rewrite Rabs_mult; + rewrite Rabs_Rabsolu; rewrite Rmult_assoc; apply Rmult_eq_compat_l. +rewrite Rabs_Rinv. +do 2 rewrite Rabs_Rabsolu; repeat rewrite Rabs_right. +replace (r ^ (2 * S n)) with (r ^ (2 * n) * r * r). +repeat rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +unfold Rsqr in |- *; ring. +apply pow_nonzero; assumption. +replace (2 * S n)%nat with (S (S (2 * n))). +simpl in |- *; ring. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +apply Rle_ge; apply pow_le; left; apply (cond_pos r). +apply Rle_ge; apply pow_le; left; apply (cond_pos r). +apply Rabs_no_R0; apply pow_nonzero; assumption. +apply Rabs_no_R0; apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply Rabs_no_R0; apply Rinv_neq_0_compat; apply INR_fact_neq_0. +apply Rabs_no_R0; apply pow_nonzero; assumption. +apply INR_fact_neq_0. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. +apply prod_neq_R0. +apply pow_nonzero; discrR. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply H1. +apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption. +assert (H0 := cond_pos r); red in |- *; intro; rewrite H1 in H0; + elim (Rlt_irrefl _ H0). Qed. (**********) -Lemma continuity_cos : (continuity cos). -Pose fn := [N:nat][x:R]``(pow (-1) N)/(INR (fact (mult (S (S O)) N)))*(pow x (mult (S (S O)) N))``. -Cut (CVN_R fn). -Intro; Cut (x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l)). -Intro cv; Cut ((n:nat)(continuity (fn n))). -Intro; Cut (x:R)(cos x)==(SFL fn cv x). -Intro; Cut (continuity (SFL fn cv))->(continuity cos). -Intro; Apply H1. -Apply SFL_continuity; Assumption. -Unfold continuity; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. -Elim (H1 x ? H2); Intros. -Exists x0; Intros. -Elim H3; Intros. -Split. -Apply H4. -Intros; Rewrite (H0 x); Rewrite (H0 x1); Apply H5; Apply H6. -Intro; Unfold cos SFL. -Case (cv x); Case (exist_cos (Rsqr x)); Intros. -Symmetry; EApply UL_sequence. -Apply u. -Unfold cos_in in c; Unfold infinit_sum in c; Unfold Un_cv; Intros. -Elim (c ? H0); Intros N0 H1. -Exists N0; Intros. -Unfold R_dist in H1; Unfold R_dist SP. -Replace (sum_f_R0 [k:nat](fn k x) n) with (sum_f_R0 [i:nat]``(cos_n i)*(pow (Rsqr x) i)`` n). -Apply H1; Assumption. -Apply sum_eq; Intros. -Unfold cos_n fn; Apply Rmult_mult_r. -Unfold Rsqr; Rewrite pow_sqr; Reflexivity. -Intro; Unfold fn; Replace [x:R]``(pow ( -1) n)/(INR (fact (mult (S (S O)) n)))*(pow x (mult (S (S O)) n))`` with (mult_fct (fct_cte ``(pow ( -1) n)/(INR (fact (mult (S (S O)) n)))``) (pow_fct (mult (S (S O)) n))); [Idtac | Reflexivity]. -Apply continuity_mult. -Apply derivable_continuous; Apply derivable_const. -Apply derivable_continuous; Apply (derivable_pow (mult (2) n)). -Apply CVN_R_CVS; Apply X. -Apply CVN_R_cos; Unfold fn; Reflexivity. +Lemma continuity_cos : continuity cos. +pose (fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N)) * x ^ (2 * N)). +cut (CVN_R fn). +intro; cut (forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)). +intro cv; cut (forall n:nat, continuity (fn n)). +intro; cut (forall x:R, cos x = SFL fn cv x). +intro; cut (continuity (SFL fn cv) -> continuity cos). +intro; apply H1. +apply SFL_continuity; assumption. +unfold continuity in |- *; unfold continuity_pt in |- *; + unfold continue_in in |- *; unfold limit1_in in |- *; + unfold limit_in in |- *; simpl in |- *; unfold R_dist in |- *; + intros. +elim (H1 x _ H2); intros. +exists x0; intros. +elim H3; intros. +split. +apply H4. +intros; rewrite (H0 x); rewrite (H0 x1); apply H5; apply H6. +intro; unfold cos, SFL in |- *. +case (cv x); case (exist_cos (Rsqr x)); intros. +symmetry in |- *; eapply UL_sequence. +apply u. +unfold cos_in in c; unfold infinit_sum in c; unfold Un_cv in |- *; intros. +elim (c _ H0); intros N0 H1. +exists N0; intros. +unfold R_dist in H1; unfold R_dist, SP in |- *. +replace (sum_f_R0 (fun k:nat => fn k x) n) with + (sum_f_R0 (fun i:nat => cos_n i * Rsqr x ^ i) n). +apply H1; assumption. +apply sum_eq; intros. +unfold cos_n, fn in |- *; apply Rmult_eq_compat_l. +unfold Rsqr in |- *; rewrite pow_sqr; reflexivity. +intro; unfold fn in |- *; + replace (fun x:R => (-1) ^ n / INR (fact (2 * n)) * x ^ (2 * n)) with + (fct_cte ((-1) ^ n / INR (fact (2 * n))) * pow_fct (2 * n))%F; + [ idtac | reflexivity ]. +apply continuity_mult. +apply derivable_continuous; apply derivable_const. +apply derivable_continuous; apply (derivable_pow (2 * n)). +apply CVN_R_CVS; apply X. +apply CVN_R_cos; unfold fn in |- *; reflexivity. Qed. (**********) -Lemma continuity_sin : (continuity sin). -Unfold continuity; Intro. -Assert H0 := (continuity_cos ``PI/2-x``). -Unfold continuity_pt in H0; Unfold continue_in in H0; Unfold limit1_in in H0; Unfold limit_in in H0; Simpl in H0; Unfold R_dist in H0; Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. -Elim (H0 ? H); Intros. -Exists x0; Intros. -Elim H1; Intros. -Split. -Assumption. -Intros; Rewrite <- (cos_shift x); Rewrite <- (cos_shift x1); Apply H3. -Elim H4; Intros. -Split. -Unfold D_x no_cond; Split. -Trivial. -Red; Intro; Unfold D_x no_cond in H5; Elim H5; Intros _ H8; Elim H8; Rewrite <- (Ropp_Ropp x); Rewrite <- (Ropp_Ropp x1); Apply eq_Ropp; Apply r_Rplus_plus with ``PI/2``; Apply H7. -Replace ``PI/2-x1-(PI/2-x)`` with ``x-x1``; [Idtac | Ring]; Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr3; Apply H6. +Lemma continuity_sin : continuity sin. +unfold continuity in |- *; intro. +assert (H0 := continuity_cos (PI / 2 - x)). +unfold continuity_pt in H0; unfold continue_in in H0; unfold limit1_in in H0; + unfold limit_in in H0; simpl in H0; unfold R_dist in H0; + unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +elim (H0 _ H); intros. +exists x0; intros. +elim H1; intros. +split. +assumption. +intros; rewrite <- (cos_shift x); rewrite <- (cos_shift x1); apply H3. +elim H4; intros. +split. +unfold D_x, no_cond in |- *; split. +trivial. +red in |- *; intro; unfold D_x, no_cond in H5; elim H5; intros _ H8; elim H8; + rewrite <- (Ropp_involutive x); rewrite <- (Ropp_involutive x1); + apply Ropp_eq_compat; apply Rplus_eq_reg_l with (PI / 2); + apply H7. +replace (PI / 2 - x1 - (PI / 2 - x)) with (x - x1); [ idtac | ring ]; + rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr'; apply H6. Qed. -Lemma CVN_R_sin : (fn:nat->R->R) (fn == [N:nat][x:R]``(pow ( -1) N)/(INR (fact (plus (mult (S (S O)) N) (S O))))*(pow x (mult (S (S O)) N))``) -> (CVN_R fn). -Unfold CVN_R; Unfold CVN_r; Intros fn H r. -Apply Specif.existT with [n:nat]``/(INR (fact (plus (mult (S (S O)) n) (S O))))*(pow r (mult (S (S O)) n))``. -Cut (SigT ? [l:R](Un_cv [n:nat](sum_f_R0 [k:nat](Rabsolu ``/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow r (mult (S (S O)) k))``) n) l)). -Intro; Elim X; Intros. -Apply existTT with x. -Split. -Apply p. -Intros; Rewrite H; Unfold Rdiv; Do 2 Rewrite Rabsolu_mult; Rewrite pow_1_abs; Rewrite Rmult_1l. -Cut ``0</(INR (fact (plus (mult (S (S O)) n) (S O))))``. -Intro; Rewrite (Rabsolu_right ? (Rle_sym1 ? ? (Rlt_le ? ? H1))). -Apply Rle_monotony. -Left; Apply H1. -Rewrite <- Pow_Rabsolu; Apply pow_maj_Rabs. -Rewrite Rabsolu_Rabsolu; Unfold Boule in H0; Rewrite minus_R0 in H0; Left; Apply H0. -Apply Rlt_Rinv; Apply INR_fact_lt_0. -Cut (r::R)<>``0``. -Intro; Apply Alembert_C2. -Intro; Apply Rabsolu_no_R0. -Apply prod_neq_R0. -Apply Rinv_neq_R0; Apply INR_fact_neq_0. -Apply pow_nonzero; Assumption. -Assert H1 := Alembert_sin. -Unfold sin_n in H1; Unfold Un_cv in H1; Unfold Un_cv; Intros. -Cut ``0<eps/(Rsqr r)``. -Intro; Elim (H1 ? H3); Intros N0 H4. -Exists N0; Intros. -Unfold R_dist; Assert H6 := (H4 ? H5). -Unfold R_dist in H5; Replace ``(Rabsolu ((Rabsolu (/(INR (fact (plus (mult (S (S O)) (S n)) (S O))))*(pow r (mult (S (S O)) (S n)))))/(Rabsolu (/(INR (fact (plus (mult (S (S O)) n) (S O))))*(pow r (mult (S (S O)) n))))))`` with ``(Rsqr r)*(Rabsolu ((pow ( -1) (S n))/(INR (fact (plus (mult (S (S O)) (S n)) (S O))))/((pow ( -1) n)/(INR (fact (plus (mult (S (S O)) n) (S O)))))))``. -Apply Rlt_monotony_contra with ``/(Rsqr r)``. -Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption. -Pattern 1 ``/(Rsqr r)``; Rewrite <- (Rabsolu_right ``/(Rsqr r)``). -Rewrite <- Rabsolu_mult. -Rewrite Rminus_distr. -Rewrite Rmult_Or; Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Rewrite <- (Rmult_sym eps). -Apply H6. -Unfold Rsqr; Apply prod_neq_R0; Assumption. -Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption. -Unfold Rdiv; Rewrite (Rmult_sym (Rsqr r)); Repeat Rewrite Rabsolu_mult; Rewrite Rabsolu_Rabsolu; Rewrite pow_1_abs. -Rewrite Rmult_1l. -Repeat Rewrite Rmult_assoc; Apply Rmult_mult_r. -Rewrite Rinv_Rmult. -Rewrite Rinv_Rinv. -Rewrite Rabsolu_mult. -Rewrite Rabsolu_Rinv. -Rewrite pow_1_abs; Rewrite Rinv_R1; Rewrite Rmult_1l. -Rewrite Rinv_Rmult. -Rewrite <- Rabsolu_Rinv. -Rewrite Rinv_Rinv. -Rewrite Rabsolu_mult. -Do 2 Rewrite Rabsolu_Rabsolu. -Rewrite (Rmult_sym ``(Rabsolu (pow r (mult (S (S O)) (S n))))``). -Rewrite Rmult_assoc; Apply Rmult_mult_r. -Rewrite Rabsolu_Rinv. -Rewrite Rabsolu_Rabsolu. -Repeat Rewrite Rabsolu_right. -Replace ``(pow r (mult (S (S O)) (S n)))`` with ``(pow r (mult (S (S O)) n))*r*r``. -Do 2 Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Unfold Rsqr; Ring. -Apply pow_nonzero; Assumption. -Replace (mult (2) (S n)) with (S (S (mult (2) n))). -Simpl; Ring. -Apply INR_eq; Do 2 Rewrite S_INR; Do 2 Rewrite mult_INR; Repeat Rewrite S_INR; Ring. -Apply Rle_sym1; Apply pow_le; Left; Apply (cond_pos r). -Apply Rle_sym1; Apply pow_le; Left; Apply (cond_pos r). -Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. -Apply INR_fact_neq_0. -Apply Rinv_neq_R0; Apply INR_fact_neq_0. -Apply Rabsolu_no_R0; Apply Rinv_neq_R0; Apply INR_fact_neq_0. -Apply Rabsolu_no_R0; Apply pow_nonzero; Assumption. -Apply pow_nonzero; DiscrR. -Apply INR_fact_neq_0. -Apply pow_nonzero; DiscrR. -Apply Rinv_neq_R0; Apply INR_fact_neq_0. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rsqr_pos_lt; Assumption]. -Assert H0 := (cond_pos r); Red; Intro; Rewrite H1 in H0; Elim (Rlt_antirefl ? H0). +Lemma CVN_R_sin : + forall fn:nat -> R -> R, + fn = + (fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)) -> + CVN_R fn. +unfold CVN_R in |- *; unfold CVN_r in |- *; intros fn H r. +apply existT with (fun n:nat => / INR (fact (2 * n + 1)) * r ^ (2 * n)). +cut + (sigT + (fun l:R => + Un_cv + (fun n:nat => + sum_f_R0 + (fun k:nat => Rabs (/ INR (fact (2 * k + 1)) * r ^ (2 * k))) n) + l)). +intro; elim X; intros. +apply existT with x. +split. +apply p. +intros; rewrite H; unfold Rdiv in |- *; do 2 rewrite Rabs_mult; + rewrite pow_1_abs; rewrite Rmult_1_l. +cut (0 < / INR (fact (2 * n + 1))). +intro; rewrite (Rabs_right _ (Rle_ge _ _ (Rlt_le _ _ H1))). +apply Rmult_le_compat_l. +left; apply H1. +rewrite <- RPow_abs; apply pow_maj_Rabs. +rewrite Rabs_Rabsolu; unfold Boule in H0; rewrite Rminus_0_r in H0; left; + apply H0. +apply Rinv_0_lt_compat; apply INR_fact_lt_0. +cut ((r:R) <> 0). +intro; apply Alembert_C2. +intro; apply Rabs_no_R0. +apply prod_neq_R0. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. +apply pow_nonzero; assumption. +assert (H1 := Alembert_sin). +unfold sin_n in H1; unfold Un_cv in H1; unfold Un_cv in |- *; intros. +cut (0 < eps / Rsqr r). +intro; elim (H1 _ H3); intros N0 H4. +exists N0; intros. +unfold R_dist in |- *; assert (H6 := H4 _ H5). +unfold R_dist in H5; + replace + (Rabs + (Rabs (/ INR (fact (2 * S n + 1)) * r ^ (2 * S n)) / + Rabs (/ INR (fact (2 * n + 1)) * r ^ (2 * n)))) with + (Rsqr r * + Rabs + ((-1) ^ S n / INR (fact (2 * S n + 1)) / + ((-1) ^ n / INR (fact (2 * n + 1))))). +apply Rmult_lt_reg_l with (/ Rsqr r). +apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption. +pattern (/ Rsqr r) at 1 in |- *; rewrite <- (Rabs_right (/ Rsqr r)). +rewrite <- Rabs_mult. +rewrite Rmult_minus_distr_l. +rewrite Rmult_0_r; rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite <- (Rmult_comm eps). +apply H6. +unfold Rsqr in |- *; apply prod_neq_R0; assumption. +apply Rle_ge; left; apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption. +unfold Rdiv in |- *; rewrite (Rmult_comm (Rsqr r)); repeat rewrite Rabs_mult; + rewrite Rabs_Rabsolu; rewrite pow_1_abs. +rewrite Rmult_1_l. +repeat rewrite Rmult_assoc; apply Rmult_eq_compat_l. +rewrite Rinv_mult_distr. +rewrite Rinv_involutive. +rewrite Rabs_mult. +rewrite Rabs_Rinv. +rewrite pow_1_abs; rewrite Rinv_1; rewrite Rmult_1_l. +rewrite Rinv_mult_distr. +rewrite <- Rabs_Rinv. +rewrite Rinv_involutive. +rewrite Rabs_mult. +do 2 rewrite Rabs_Rabsolu. +rewrite (Rmult_comm (Rabs (r ^ (2 * S n)))). +rewrite Rmult_assoc; apply Rmult_eq_compat_l. +rewrite Rabs_Rinv. +rewrite Rabs_Rabsolu. +repeat rewrite Rabs_right. +replace (r ^ (2 * S n)) with (r ^ (2 * n) * r * r). +do 2 rewrite <- Rmult_assoc. +rewrite <- Rinv_l_sym. +unfold Rsqr in |- *; ring. +apply pow_nonzero; assumption. +replace (2 * S n)%nat with (S (S (2 * n))). +simpl in |- *; ring. +apply INR_eq; do 2 rewrite S_INR; do 2 rewrite mult_INR; repeat rewrite S_INR; + ring. +apply Rle_ge; apply pow_le; left; apply (cond_pos r). +apply Rle_ge; apply pow_le; left; apply (cond_pos r). +apply Rabs_no_R0; apply pow_nonzero; assumption. +apply INR_fact_neq_0. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. +apply Rabs_no_R0; apply Rinv_neq_0_compat; apply INR_fact_neq_0. +apply Rabs_no_R0; apply pow_nonzero; assumption. +apply pow_nonzero; discrR. +apply INR_fact_neq_0. +apply pow_nonzero; discrR. +apply Rinv_neq_0_compat; apply INR_fact_neq_0. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; apply Rsqr_pos_lt; assumption ]. +assert (H0 := cond_pos r); red in |- *; intro; rewrite H1 in H0; + elim (Rlt_irrefl _ H0). Qed. (* (sin h)/h -> 1 when h -> 0 *) -Lemma derivable_pt_lim_sin_0 : (derivable_pt_lim sin R0 R1). -Unfold derivable_pt_lim; Intros. -Pose fn := [N:nat][x:R]``(pow ( -1) N)/(INR (fact (plus (mult (S (S O)) N) (S O))))*(pow x (mult (S (S O)) N))``. -Cut (CVN_R fn). -Intro; Cut (x:R)(sigTT ? [l:R](Un_cv [N:nat](SP fn N x) l)). -Intro cv. -Pose r := (mkposreal ? Rlt_R0_R1). -Cut (CVN_r fn r). -Intro; Cut ((n:nat; y:R)(Boule ``0`` r y)->(continuity_pt (fn n) y)). -Intro; Cut (Boule R0 r R0). -Intro; Assert H2 := (SFL_continuity_pt ? cv ? X0 H0 ? H1). -Unfold continuity_pt in H2; Unfold continue_in in H2; Unfold limit1_in in H2; Unfold limit_in in H2; Simpl in H2; Unfold R_dist in H2. -Elim (H2 ? H); Intros alp H3. -Elim H3; Intros. -Exists (mkposreal ? H4). -Simpl; Intros. -Rewrite sin_0; Rewrite Rplus_Ol; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or. -Cut ``(Rabsolu ((SFL fn cv h)-(SFL fn cv 0))) < eps``. -Intro; Cut (SFL fn cv R0)==R1. -Intro; Cut (SFL fn cv h)==``(sin h)/h``. -Intro; Rewrite H9 in H8; Rewrite H10 in H8. -Apply H8. -Unfold SFL sin. -Case (cv h); Intros. -Case (exist_sin (Rsqr h)); Intros. -Unfold Rdiv; Rewrite (Rinv_r_simpl_m h x0 H6). -EApply UL_sequence. -Apply u. -Unfold sin_in in s; Unfold sin_n infinit_sum in s; Unfold SP fn Un_cv; Intros. -Elim (s ? H10); Intros N0 H11. -Exists N0; Intros. -Unfold R_dist; Unfold R_dist in H11. -Replace (sum_f_R0 [k:nat]``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow h (mult (S (S O)) k))`` n) with (sum_f_R0 [i:nat]``(pow ( -1) i)/(INR (fact (plus (mult (S (S O)) i) (S O))))*(pow (Rsqr h) i)`` n). -Apply H11; Assumption. -Apply sum_eq; Intros; Apply Rmult_mult_r; Unfold Rsqr; Rewrite pow_sqr; Reflexivity. -Unfold SFL sin. -Case (cv R0); Intros. -EApply UL_sequence. -Apply u. -Unfold SP fn; Unfold Un_cv; Intros; Exists (S O); Intros. -Unfold R_dist; Replace (sum_f_R0 [k:nat]``(pow ( -1) k)/(INR (fact (plus (mult (S (S O)) k) (S O))))*(pow 0 (mult (S (S O)) k))`` n) with R1. -Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Assumption. -Rewrite decomp_sum. -Simpl; Rewrite Rmult_1r; Unfold Rdiv; Rewrite Rinv_R1; Rewrite Rmult_1r; Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rplus_plus_r. -Symmetry; Apply sum_eq_R0; Intros. -Rewrite Rmult_Ol; Rewrite Rmult_Or; Reflexivity. -Unfold ge in H10; Apply lt_le_trans with (1); [Apply lt_n_Sn | Apply H10]. -Apply H5. -Split. -Unfold D_x no_cond; Split. -Trivial. -Apply not_sym; Apply H6. -Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply H7. -Unfold Boule; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_R0; Apply (cond_pos r). -Intros; Unfold fn; Replace [x:R]``(pow ( -1) n)/(INR (fact (plus (mult (S (S O)) n) (S O))))*(pow x (mult (S (S O)) n))`` with (mult_fct (fct_cte ``(pow ( -1) n)/(INR (fact (plus (mult (S (S O)) n) (S O))))``) (pow_fct (mult (S (S O)) n))); [Idtac | Reflexivity]. -Apply continuity_pt_mult. -Apply derivable_continuous_pt. -Apply derivable_pt_const. -Apply derivable_continuous_pt. -Apply (derivable_pt_pow (mult (2) n) y). -Apply (X r). -Apply (CVN_R_CVS ? X). -Apply CVN_R_sin; Unfold fn; Reflexivity. +Lemma derivable_pt_lim_sin_0 : derivable_pt_lim sin 0 1. +unfold derivable_pt_lim in |- *; intros. +pose + (fn := fun (N:nat) (x:R) => (-1) ^ N / INR (fact (2 * N + 1)) * x ^ (2 * N)). +cut (CVN_R fn). +intro; cut (forall x:R, sigT (fun l:R => Un_cv (fun N:nat => SP fn N x) l)). +intro cv. +pose (r := mkposreal _ Rlt_0_1). +cut (CVN_r fn r). +intro; cut (forall (n:nat) (y:R), Boule 0 r y -> continuity_pt (fn n) y). +intro; cut (Boule 0 r 0). +intro; assert (H2 := SFL_continuity_pt _ cv _ X0 H0 _ H1). +unfold continuity_pt in H2; unfold continue_in in H2; unfold limit1_in in H2; + unfold limit_in in H2; simpl in H2; unfold R_dist in H2. +elim (H2 _ H); intros alp H3. +elim H3; intros. +exists (mkposreal _ H4). +simpl in |- *; intros. +rewrite sin_0; rewrite Rplus_0_l; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r. +cut (Rabs (SFL fn cv h - SFL fn cv 0) < eps). +intro; cut (SFL fn cv 0 = 1). +intro; cut (SFL fn cv h = sin h / h). +intro; rewrite H9 in H8; rewrite H10 in H8. +apply H8. +unfold SFL, sin in |- *. +case (cv h); intros. +case (exist_sin (Rsqr h)); intros. +unfold Rdiv in |- *; rewrite (Rinv_r_simpl_m h x0 H6). +eapply UL_sequence. +apply u. +unfold sin_in in s; unfold sin_n, infinit_sum in s; + unfold SP, fn, Un_cv in |- *; intros. +elim (s _ H10); intros N0 H11. +exists N0; intros. +unfold R_dist in |- *; unfold R_dist in H11. +replace + (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * h ^ (2 * k)) n) + with + (sum_f_R0 (fun i:nat => (-1) ^ i / INR (fact (2 * i + 1)) * Rsqr h ^ i) n). +apply H11; assumption. +apply sum_eq; intros; apply Rmult_eq_compat_l; unfold Rsqr in |- *; + rewrite pow_sqr; reflexivity. +unfold SFL, sin in |- *. +case (cv 0); intros. +eapply UL_sequence. +apply u. +unfold SP, fn in |- *; unfold Un_cv in |- *; intros; exists 1%nat; intros. +unfold R_dist in |- *; + replace + (sum_f_R0 (fun k:nat => (-1) ^ k / INR (fact (2 * k + 1)) * 0 ^ (2 * k)) n) + with 1. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rabs_R0; assumption. +rewrite decomp_sum. +simpl in |- *; rewrite Rmult_1_r; unfold Rdiv in |- *; rewrite Rinv_1; + rewrite Rmult_1_r; pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; + apply Rplus_eq_compat_l. +symmetry in |- *; apply sum_eq_R0; intros. +rewrite Rmult_0_l; rewrite Rmult_0_r; reflexivity. +unfold ge in H10; apply lt_le_trans with 1%nat; [ apply lt_n_Sn | apply H10 ]. +apply H5. +split. +unfold D_x, no_cond in |- *; split. +trivial. +apply (sym_not_eq (A:=R)); apply H6. +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; apply H7. +unfold Boule in |- *; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; rewrite Rabs_R0; apply (cond_pos r). +intros; unfold fn in |- *; + replace (fun x:R => (-1) ^ n / INR (fact (2 * n + 1)) * x ^ (2 * n)) with + (fct_cte ((-1) ^ n / INR (fact (2 * n + 1))) * pow_fct (2 * n))%F; + [ idtac | reflexivity ]. +apply continuity_pt_mult. +apply derivable_continuous_pt. +apply derivable_pt_const. +apply derivable_continuous_pt. +apply (derivable_pt_pow (2 * n) y). +apply (X r). +apply (CVN_R_CVS _ X). +apply CVN_R_sin; unfold fn in |- *; reflexivity. Qed. (* ((cos h)-1)/h -> 0 when h -> 0 *) -Lemma derivable_pt_lim_cos_0 : (derivable_pt_lim cos ``0`` ``0``). -Unfold derivable_pt_lim; Intros. -Assert H0 := derivable_pt_lim_sin_0. -Unfold derivable_pt_lim in H0. -Cut ``0<eps/2``. -Intro; Elim (H0 ? H1); Intros del H2. -Cut (continuity_pt sin ``0``). -Intro; Unfold continuity_pt in H3; Unfold continue_in in H3; Unfold limit1_in in H3; Unfold limit_in in H3; Simpl in H3; Unfold R_dist in H3. -Cut ``0<eps/2``; [Intro | Assumption]. -Elim (H3 ? H4); Intros del_c H5. -Cut ``0<(Rmin del del_c)``. -Intro; Pose delta := (mkposreal ? H6). -Exists delta; Intros. -Rewrite Rplus_Ol; Replace ``((cos h)-(cos 0))`` with ``-2*(Rsqr (sin (h/2)))``. -Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or. -Unfold Rdiv; Do 2 Rewrite Ropp_mul1. -Rewrite Rabsolu_Ropp. -Replace ``2*(Rsqr (sin (h*/2)))*/h`` with ``(sin (h/2))*((sin (h/2))/(h/2)-1)+(sin (h/2))``. -Apply Rle_lt_trans with ``(Rabsolu ((sin (h/2))*((sin (h/2))/(h/2)-1)))+(Rabsolu ((sin (h/2))))``. -Apply Rabsolu_triang. -Rewrite (double_var eps); Apply Rplus_lt. -Apply Rle_lt_trans with ``(Rabsolu ((sin (h/2))/(h/2)-1))``. -Rewrite Rabsolu_mult; Rewrite Rmult_sym; Pattern 2 ``(Rabsolu ((sin (h/2))/(h/2)-1))``; Rewrite <- Rmult_1r; Apply Rle_monotony. -Apply Rabsolu_pos. -Assert H9 := (SIN_bound ``h/2``). -Unfold Rabsolu; Case (case_Rabsolu ``(sin (h/2))``); Intro. -Pattern 3 R1; Rewrite <- (Ropp_Ropp ``1``). -Apply Rle_Ropp1. -Elim H9; Intros; Assumption. -Elim H9; Intros; Assumption. -Cut ``(Rabsolu (h/2))<del``. -Intro; Cut ``h/2<>0``. -Intro; Assert H11 := (H2 ? H10 H9). -Rewrite Rplus_Ol in H11; Rewrite sin_0 in H11. -Rewrite minus_R0 in H11; Apply H11. -Unfold Rdiv; Apply prod_neq_R0. -Apply H7. -Apply Rinv_neq_R0; DiscrR. -Apply Rlt_trans with ``del/2``. -Unfold Rdiv; Rewrite Rabsolu_mult. -Rewrite (Rabsolu_right ``/2``). -Do 2 Rewrite <- (Rmult_sym ``/2``); Apply Rlt_monotony. -Apply Rlt_Rinv; Sup0. -Apply Rlt_le_trans with (pos delta). -Apply H8. -Unfold delta; Simpl; Apply Rmin_l. -Apply Rle_sym1; Left; Apply Rlt_Rinv; Sup0. -Rewrite <- (Rplus_Or ``del/2``); Pattern 1 del; Rewrite (double_var del); Apply Rlt_compatibility; Unfold Rdiv; Apply Rmult_lt_pos. -Apply (cond_pos del). -Apply Rlt_Rinv; Sup0. -Elim H5; Intros; Assert H11 := (H10 ``h/2``). -Rewrite sin_0 in H11; Do 2 Rewrite minus_R0 in H11. -Apply H11. -Split. -Unfold D_x no_cond; Split. -Trivial. -Apply not_sym; Unfold Rdiv; Apply prod_neq_R0. -Apply H7. -Apply Rinv_neq_R0; DiscrR. -Apply Rlt_trans with ``del_c/2``. -Unfold Rdiv; Rewrite Rabsolu_mult. -Rewrite (Rabsolu_right ``/2``). -Do 2 Rewrite <- (Rmult_sym ``/2``). -Apply Rlt_monotony. -Apply Rlt_Rinv; Sup0. -Apply Rlt_le_trans with (pos delta). -Apply H8. -Unfold delta; Simpl; Apply Rmin_r. -Apply Rle_sym1; Left; Apply Rlt_Rinv; Sup0. -Rewrite <- (Rplus_Or ``del_c/2``); Pattern 2 del_c; Rewrite (double_var del_c); Apply Rlt_compatibility. -Unfold Rdiv; Apply Rmult_lt_pos. -Apply H9. -Apply Rlt_Rinv; Sup0. -Rewrite Rminus_distr; Rewrite Rmult_1r; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Rewrite (Rmult_sym ``2``); Unfold Rdiv Rsqr. -Repeat Rewrite Rmult_assoc. -Repeat Apply Rmult_mult_r. -Rewrite Rinv_Rmult. -Rewrite Rinv_Rinv. -Apply Rmult_sym. -DiscrR. -Apply H7. -Apply Rinv_neq_R0; DiscrR. -Pattern 2 h; Replace h with ``2*(h/2)``. -Rewrite (cos_2a_sin ``h/2``). -Rewrite cos_0; Unfold Rsqr; Ring. -Unfold Rdiv; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m. -DiscrR. -Unfold Rmin; Case (total_order_Rle del del_c); Intro. -Apply (cond_pos del). -Elim H5; Intros; Assumption. -Apply continuity_sin. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]. +Lemma derivable_pt_lim_cos_0 : derivable_pt_lim cos 0 0. +unfold derivable_pt_lim in |- *; intros. +assert (H0 := derivable_pt_lim_sin_0). +unfold derivable_pt_lim in H0. +cut (0 < eps / 2). +intro; elim (H0 _ H1); intros del H2. +cut (continuity_pt sin 0). +intro; unfold continuity_pt in H3; unfold continue_in in H3; + unfold limit1_in in H3; unfold limit_in in H3; simpl in H3; + unfold R_dist in H3. +cut (0 < eps / 2); [ intro | assumption ]. +elim (H3 _ H4); intros del_c H5. +cut (0 < Rmin del del_c). +intro; pose (delta := mkposreal _ H6). +exists delta; intros. +rewrite Rplus_0_l; replace (cos h - cos 0) with (-2 * Rsqr (sin (h / 2))). +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r. +unfold Rdiv in |- *; do 2 rewrite Ropp_mult_distr_l_reverse. +rewrite Rabs_Ropp. +replace (2 * Rsqr (sin (h * / 2)) * / h) with + (sin (h / 2) * (sin (h / 2) / (h / 2) - 1) + sin (h / 2)). +apply Rle_lt_trans with + (Rabs (sin (h / 2) * (sin (h / 2) / (h / 2) - 1)) + Rabs (sin (h / 2))). +apply Rabs_triang. +rewrite (double_var eps); apply Rplus_lt_compat. +apply Rle_lt_trans with (Rabs (sin (h / 2) / (h / 2) - 1)). +rewrite Rabs_mult; rewrite Rmult_comm; + pattern (Rabs (sin (h / 2) / (h / 2) - 1)) at 2 in |- *; + rewrite <- Rmult_1_r; apply Rmult_le_compat_l. +apply Rabs_pos. +assert (H9 := SIN_bound (h / 2)). +unfold Rabs in |- *; case (Rcase_abs (sin (h / 2))); intro. +pattern 1 at 3 in |- *; rewrite <- (Ropp_involutive 1). +apply Ropp_le_contravar. +elim H9; intros; assumption. +elim H9; intros; assumption. +cut (Rabs (h / 2) < del). +intro; cut (h / 2 <> 0). +intro; assert (H11 := H2 _ H10 H9). +rewrite Rplus_0_l in H11; rewrite sin_0 in H11. +rewrite Rminus_0_r in H11; apply H11. +unfold Rdiv in |- *; apply prod_neq_R0. +apply H7. +apply Rinv_neq_0_compat; discrR. +apply Rlt_trans with (del / 2). +unfold Rdiv in |- *; rewrite Rabs_mult. +rewrite (Rabs_right (/ 2)). +do 2 rewrite <- (Rmult_comm (/ 2)); apply Rmult_lt_compat_l. +apply Rinv_0_lt_compat; prove_sup0. +apply Rlt_le_trans with (pos delta). +apply H8. +unfold delta in |- *; simpl in |- *; apply Rmin_l. +apply Rle_ge; left; apply Rinv_0_lt_compat; prove_sup0. +rewrite <- (Rplus_0_r (del / 2)); pattern del at 1 in |- *; + rewrite (double_var del); apply Rplus_lt_compat_l; + unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply (cond_pos del). +apply Rinv_0_lt_compat; prove_sup0. +elim H5; intros; assert (H11 := H10 (h / 2)). +rewrite sin_0 in H11; do 2 rewrite Rminus_0_r in H11. +apply H11. +split. +unfold D_x, no_cond in |- *; split. +trivial. +apply (sym_not_eq (A:=R)); unfold Rdiv in |- *; apply prod_neq_R0. +apply H7. +apply Rinv_neq_0_compat; discrR. +apply Rlt_trans with (del_c / 2). +unfold Rdiv in |- *; rewrite Rabs_mult. +rewrite (Rabs_right (/ 2)). +do 2 rewrite <- (Rmult_comm (/ 2)). +apply Rmult_lt_compat_l. +apply Rinv_0_lt_compat; prove_sup0. +apply Rlt_le_trans with (pos delta). +apply H8. +unfold delta in |- *; simpl in |- *; apply Rmin_r. +apply Rle_ge; left; apply Rinv_0_lt_compat; prove_sup0. +rewrite <- (Rplus_0_r (del_c / 2)); pattern del_c at 2 in |- *; + rewrite (double_var del_c); apply Rplus_lt_compat_l. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply H9. +apply Rinv_0_lt_compat; prove_sup0. +rewrite Rmult_minus_distr_l; rewrite Rmult_1_r; unfold Rminus in |- *; + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; + rewrite (Rmult_comm 2); unfold Rdiv, Rsqr in |- *. +repeat rewrite Rmult_assoc. +repeat apply Rmult_eq_compat_l. +rewrite Rinv_mult_distr. +rewrite Rinv_involutive. +apply Rmult_comm. +discrR. +apply H7. +apply Rinv_neq_0_compat; discrR. +pattern h at 2 in |- *; replace h with (2 * (h / 2)). +rewrite (cos_2a_sin (h / 2)). +rewrite cos_0; unfold Rsqr in |- *; ring. +unfold Rdiv in |- *; rewrite <- Rmult_assoc; apply Rinv_r_simpl_m. +discrR. +unfold Rmin in |- *; case (Rle_dec del del_c); intro. +apply (cond_pos del). +elim H5; intros; assumption. +apply continuity_sin. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ]. Qed. (**********) -Theorem derivable_pt_lim_sin : (x:R)(derivable_pt_lim sin x (cos x)). -Intro; Assert H0 := derivable_pt_lim_sin_0. -Assert H := derivable_pt_lim_cos_0. -Unfold derivable_pt_lim in H0 H. -Unfold derivable_pt_lim; Intros. -Cut ``0<eps/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Apply H1 | Apply Rlt_Rinv; Sup0]]. -Elim (H0 ? H2); Intros alp1 H3. -Elim (H ? H2); Intros alp2 H4. -Pose alp := (Rmin alp1 alp2). -Cut ``0<alp``. -Intro; Exists (mkposreal ? H5); Intros. -Replace ``((sin (x+h))-(sin x))/h-(cos x)`` with ``(sin x)*((cos h)-1)/h+(cos x)*((sin h)/h-1)``. -Apply Rle_lt_trans with ``(Rabsolu ((sin x)*((cos h)-1)/h))+(Rabsolu ((cos x)*((sin h)/h-1)))``. -Apply Rabsolu_triang. -Rewrite (double_var eps); Apply Rplus_lt. -Apply Rle_lt_trans with ``(Rabsolu ((cos h)-1)/h)``. -Rewrite Rabsolu_mult; Rewrite Rmult_sym; Pattern 2 ``(Rabsolu (((cos h)-1)/h))``; Rewrite <- Rmult_1r; Apply Rle_monotony. -Apply Rabsolu_pos. -Assert H8 := (SIN_bound x); Elim H8; Intros. -Unfold Rabsolu; Case (case_Rabsolu (sin x)); Intro. -Rewrite <- (Ropp_Ropp R1). -Apply Rle_Ropp1; Assumption. -Assumption. -Cut ``(Rabsolu h)<alp2``. -Intro; Assert H9 := (H4 ? H6 H8). -Rewrite cos_0 in H9; Rewrite Rplus_Ol in H9; Rewrite minus_R0 in H9; Apply H9. -Apply Rlt_le_trans with alp. -Apply H7. -Unfold alp; Apply Rmin_r. -Apply Rle_lt_trans with ``(Rabsolu ((sin h)/h-1))``. -Rewrite Rabsolu_mult; Rewrite Rmult_sym; Pattern 2 ``(Rabsolu ((sin h)/h-1))``; Rewrite <- Rmult_1r; Apply Rle_monotony. -Apply Rabsolu_pos. -Assert H8 := (COS_bound x); Elim H8; Intros. -Unfold Rabsolu; Case (case_Rabsolu (cos x)); Intro. -Rewrite <- (Ropp_Ropp R1); Apply Rle_Ropp1; Assumption. -Assumption. -Cut ``(Rabsolu h)<alp1``. -Intro; Assert H9 := (H3 ? H6 H8). -Rewrite sin_0 in H9; Rewrite Rplus_Ol in H9; Rewrite minus_R0 in H9; Apply H9. -Apply Rlt_le_trans with alp. -Apply H7. -Unfold alp; Apply Rmin_l. -Rewrite sin_plus; Unfold Rminus Rdiv; Repeat Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_Rplus_distr; Repeat Rewrite Rmult_assoc; Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. -Rewrite (Rplus_sym ``(sin x)*( -1*/h)``); Repeat Rewrite Rplus_assoc; Apply Rplus_plus_r. -Rewrite Ropp_mul3; Rewrite Ropp_mul1; Rewrite Rmult_1r; Rewrite Rmult_1l; Rewrite Ropp_mul3; Rewrite <- Ropp_mul1; Apply Rplus_sym. -Unfold alp; Unfold Rmin; Case (total_order_Rle alp1 alp2); Intro. -Apply (cond_pos alp1). -Apply (cond_pos alp2). +Theorem derivable_pt_lim_sin : forall x:R, derivable_pt_lim sin x (cos x). +intro; assert (H0 := derivable_pt_lim_sin_0). +assert (H := derivable_pt_lim_cos_0). +unfold derivable_pt_lim in H0, H. +unfold derivable_pt_lim in |- *; intros. +cut (0 < eps / 2); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ apply H1 | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H0 _ H2); intros alp1 H3. +elim (H _ H2); intros alp2 H4. +pose (alp := Rmin alp1 alp2). +cut (0 < alp). +intro; exists (mkposreal _ H5); intros. +replace ((sin (x + h) - sin x) / h - cos x) with + (sin x * ((cos h - 1) / h) + cos x * (sin h / h - 1)). +apply Rle_lt_trans with + (Rabs (sin x * ((cos h - 1) / h)) + Rabs (cos x * (sin h / h - 1))). +apply Rabs_triang. +rewrite (double_var eps); apply Rplus_lt_compat. +apply Rle_lt_trans with (Rabs ((cos h - 1) / h)). +rewrite Rabs_mult; rewrite Rmult_comm; + pattern (Rabs ((cos h - 1) / h)) at 2 in |- *; rewrite <- Rmult_1_r; + apply Rmult_le_compat_l. +apply Rabs_pos. +assert (H8 := SIN_bound x); elim H8; intros. +unfold Rabs in |- *; case (Rcase_abs (sin x)); intro. +rewrite <- (Ropp_involutive 1). +apply Ropp_le_contravar; assumption. +assumption. +cut (Rabs h < alp2). +intro; assert (H9 := H4 _ H6 H8). +rewrite cos_0 in H9; rewrite Rplus_0_l in H9; rewrite Rminus_0_r in H9; + apply H9. +apply Rlt_le_trans with alp. +apply H7. +unfold alp in |- *; apply Rmin_r. +apply Rle_lt_trans with (Rabs (sin h / h - 1)). +rewrite Rabs_mult; rewrite Rmult_comm; + pattern (Rabs (sin h / h - 1)) at 2 in |- *; rewrite <- Rmult_1_r; + apply Rmult_le_compat_l. +apply Rabs_pos. +assert (H8 := COS_bound x); elim H8; intros. +unfold Rabs in |- *; case (Rcase_abs (cos x)); intro. +rewrite <- (Ropp_involutive 1); apply Ropp_le_contravar; assumption. +assumption. +cut (Rabs h < alp1). +intro; assert (H9 := H3 _ H6 H8). +rewrite sin_0 in H9; rewrite Rplus_0_l in H9; rewrite Rminus_0_r in H9; + apply H9. +apply Rlt_le_trans with alp. +apply H7. +unfold alp in |- *; apply Rmin_l. +rewrite sin_plus; unfold Rminus, Rdiv in |- *; + repeat rewrite Rmult_plus_distr_r; repeat rewrite Rmult_plus_distr_l; + repeat rewrite Rmult_assoc; repeat rewrite Rplus_assoc; + apply Rplus_eq_compat_l. +rewrite (Rplus_comm (sin x * (-1 * / h))); repeat rewrite Rplus_assoc; + apply Rplus_eq_compat_l. +rewrite Ropp_mult_distr_r_reverse; rewrite Ropp_mult_distr_l_reverse; + rewrite Rmult_1_r; rewrite Rmult_1_l; rewrite Ropp_mult_distr_r_reverse; + rewrite <- Ropp_mult_distr_l_reverse; apply Rplus_comm. +unfold alp in |- *; unfold Rmin in |- *; case (Rle_dec alp1 alp2); intro. +apply (cond_pos alp1). +apply (cond_pos alp2). Qed. -Lemma derivable_pt_lim_cos : (x:R) (derivable_pt_lim cos x ``-(sin x)``). -Intro; Cut (h:R)``(sin (h+PI/2))``==(cos h). -Intro; Replace ``-(sin x)`` with (Rmult (cos ``x+PI/2``) (Rplus R1 R0)). -Generalize (derivable_pt_lim_comp (plus_fct id (fct_cte ``PI/2``)) sin); Intros. -Cut (derivable_pt_lim (plus_fct id (fct_cte ``PI/2``)) x ``1+0``). -Cut (derivable_pt_lim sin (plus_fct id (fct_cte ``PI/2``) x) ``(cos (x+PI/2))``). -Intros; Generalize (H0 ? ? ? H2 H1); Replace (comp sin (plus_fct id (fct_cte ``PI/2``))) with [x:R]``(sin (x+PI/2))``; [Idtac | Reflexivity]. -Unfold derivable_pt_lim; Intros. -Elim (H3 eps H4); Intros. -Exists x0. -Intros; Rewrite <- (H ``x+h``); Rewrite <- (H x); Apply H5; Assumption. -Apply derivable_pt_lim_sin. -Apply derivable_pt_lim_plus. -Apply derivable_pt_lim_id. -Apply derivable_pt_lim_const. -Rewrite sin_cos; Rewrite <- (Rplus_sym x); Ring. -Intro; Rewrite cos_sin; Rewrite Rplus_sym; Reflexivity. +Lemma derivable_pt_lim_cos : forall x:R, derivable_pt_lim cos x (- sin x). +intro; cut (forall h:R, sin (h + PI / 2) = cos h). +intro; replace (- sin x) with (cos (x + PI / 2) * (1 + 0)). +generalize (derivable_pt_lim_comp (id + fct_cte (PI / 2))%F sin); intros. +cut (derivable_pt_lim (id + fct_cte (PI / 2)) x (1 + 0)). +cut (derivable_pt_lim sin ((id + fct_cte (PI / 2))%F x) (cos (x + PI / 2))). +intros; generalize (H0 _ _ _ H2 H1); + replace (comp sin (id + fct_cte (PI / 2))%F) with + (fun x:R => sin (x + PI / 2)); [ idtac | reflexivity ]. +unfold derivable_pt_lim in |- *; intros. +elim (H3 eps H4); intros. +exists x0. +intros; rewrite <- (H (x + h)); rewrite <- (H x); apply H5; assumption. +apply derivable_pt_lim_sin. +apply derivable_pt_lim_plus. +apply derivable_pt_lim_id. +apply derivable_pt_lim_const. +rewrite sin_cos; rewrite <- (Rplus_comm x); ring. +intro; rewrite cos_sin; rewrite Rplus_comm; reflexivity. Qed. -Lemma derivable_pt_sin : (x:R) (derivable_pt sin x). -Unfold derivable_pt; Intro. -Apply Specif.existT with (cos x). -Apply derivable_pt_lim_sin. +Lemma derivable_pt_sin : forall x:R, derivable_pt sin x. +unfold derivable_pt in |- *; intro. +apply existT with (cos x). +apply derivable_pt_lim_sin. Qed. -Lemma derivable_pt_cos : (x:R) (derivable_pt cos x). -Unfold derivable_pt; Intro. -Apply Specif.existT with ``-(sin x)``. -Apply derivable_pt_lim_cos. +Lemma derivable_pt_cos : forall x:R, derivable_pt cos x. +unfold derivable_pt in |- *; intro. +apply existT with (- sin x). +apply derivable_pt_lim_cos. Qed. -Lemma derivable_sin : (derivable sin). -Unfold derivable; Intro; Apply derivable_pt_sin. +Lemma derivable_sin : derivable sin. +unfold derivable in |- *; intro; apply derivable_pt_sin. Qed. -Lemma derivable_cos : (derivable cos). -Unfold derivable; Intro; Apply derivable_pt_cos. +Lemma derivable_cos : derivable cos. +unfold derivable in |- *; intro; apply derivable_pt_cos. Qed. -Lemma derive_pt_sin : (x:R) ``(derive_pt sin x (derivable_pt_sin ?))==(cos x)``. -Intros; Apply derive_pt_eq_0. -Apply derivable_pt_lim_sin. +Lemma derive_pt_sin : + forall x:R, derive_pt sin x (derivable_pt_sin _) = cos x. +intros; apply derive_pt_eq_0. +apply derivable_pt_lim_sin. Qed. -Lemma derive_pt_cos : (x:R) ``(derive_pt cos x (derivable_pt_cos ?))==-(sin x)``. -Intros; Apply derive_pt_eq_0. -Apply derivable_pt_lim_cos. -Qed. +Lemma derive_pt_cos : + forall x:R, derive_pt cos x (derivable_pt_cos _) = - sin x. +intros; apply derive_pt_eq_0. +apply derivable_pt_lim_cos. +Qed.
\ No newline at end of file diff --git a/theories/Reals/SeqProp.v b/theories/Reals/SeqProp.v index 7bd6b8a47..1175543b6 100644 --- a/theories/Reals/SeqProp.v +++ b/theories/Reals/SeqProp.v @@ -8,1082 +8,1288 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Rseries. -Require Classical. -Require Max. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Require Import Rbase. +Require Import Rfunctions. +Require Import Rseries. +Require Import Classical. +Require Import Max. Open Local Scope R_scope. -Definition Un_decreasing [Un:nat->R] : Prop := (n:nat) (Rle (Un (S n)) (Un n)). -Definition opp_seq [Un:nat->R] : nat->R := [n:nat]``-(Un n)``. -Definition has_ub [Un:nat->R] : Prop := (bound (EUn Un)). -Definition has_lb [Un:nat->R] : Prop := (bound (EUn (opp_seq Un))). +Definition Un_decreasing (Un:nat -> R) : Prop := + forall n:nat, Un (S n) <= Un n. +Definition opp_seq (Un:nat -> R) (n:nat) : R := - Un n. +Definition has_ub (Un:nat -> R) : Prop := bound (EUn Un). +Definition has_lb (Un:nat -> R) : Prop := bound (EUn (opp_seq Un)). (**********) -Lemma growing_cv : (Un:nat->R) (Un_growing Un) -> (has_ub Un) -> (sigTT R [l:R](Un_cv Un l)). -Unfold Un_growing Un_cv;Intros; - NewDestruct (complet (EUn Un) H0 (EUn_noempty Un)) as [x [H2 H3]]. - Exists x;Intros eps H1. - Unfold is_upper_bound in H2 H3. -Assert H5:(n:nat)(Rle (Un n) x). - Intro n; Apply (H2 (Un n) (Un_in_EUn Un n)). -Cut (Ex [N:nat] (Rlt (Rminus x eps) (Un N))). -Intro H6;NewDestruct H6 as [N H6];Exists N. -Intros n H7;Unfold R_dist;Apply (Rabsolu_def1 (Rminus (Un n) x) eps). -Unfold Rgt in H1. - Apply (Rle_lt_trans (Rminus (Un n) x) R0 eps - (Rle_minus (Un n) x (H5 n)) H1). -Fold Un_growing in H;Generalize (growing_prop Un n N H H7);Intro H8. - Generalize (Rlt_le_trans (Rminus x eps) (Un N) (Un n) H6 - (Rle_sym2 (Un N) (Un n) H8));Intro H9; - Generalize (Rlt_compatibility (Ropp x) (Rminus x eps) (Un n) H9); - Unfold Rminus;Rewrite <-(Rplus_assoc (Ropp x) x (Ropp eps)); - Rewrite (Rplus_sym (Ropp x) (Un n));Fold (Rminus (Un n) x); - Rewrite Rplus_Ropp_l;Rewrite (let (H1,H2)=(Rplus_ne (Ropp eps)) in H2); - Trivial. -Cut ~((N:nat)(Rle (Un N) (Rminus x eps))). -Intro H6;Apply (not_all_not_ex nat ([N:nat](Rlt (Rminus x eps) (Un N)))). - Intro H7; Apply H6; Intro N; Apply Rnot_lt_le; Apply H7. -Intro H7;Generalize (Un_bound_imp Un (Rminus x eps) H7);Intro H8; - Unfold is_upper_bound in H8;Generalize (H3 (Rminus x eps) H8); - Apply Rlt_le_not; Apply tech_Rgt_minus; Exact H1. +Lemma growing_cv : + forall Un:nat -> R, + Un_growing Un -> has_ub Un -> sigT (fun l:R => Un_cv Un l). +unfold Un_growing, Un_cv in |- *; intros; + destruct (completeness (EUn Un) H0 (EUn_noempty Un)) as [x [H2 H3]]. + exists x; intros eps H1. + unfold is_upper_bound in H2, H3. +assert (H5 : forall n:nat, Un n <= x). + intro n; apply (H2 (Un n) (Un_in_EUn Un n)). +cut ( exists N : nat | x - eps < Un N). +intro H6; destruct H6 as [N H6]; exists N. +intros n H7; unfold R_dist in |- *; apply (Rabs_def1 (Un n - x) eps). +unfold Rgt in H1. + apply (Rle_lt_trans (Un n - x) 0 eps (Rle_minus (Un n) x (H5 n)) H1). +fold Un_growing in H; generalize (growing_prop Un n N H H7); intro H8. + generalize + (Rlt_le_trans (x - eps) (Un N) (Un n) H6 (Rge_le (Un n) (Un N) H8)); + intro H9; generalize (Rplus_lt_compat_l (- x) (x - eps) (Un n) H9); + unfold Rminus in |- *; rewrite <- (Rplus_assoc (- x) x (- eps)); + rewrite (Rplus_comm (- x) (Un n)); fold (Un n - x) in |- *; + rewrite Rplus_opp_l; rewrite (let (H1, H2) := Rplus_ne (- eps) in H2); + trivial. +cut (~ (forall N:nat, Un N <= x - eps)). +intro H6; apply (not_all_not_ex nat (fun N:nat => x - eps < Un N)). + intro H7; apply H6; intro N; apply Rnot_lt_le; apply H7. +intro H7; generalize (Un_bound_imp Un (x - eps) H7); intro H8; + unfold is_upper_bound in H8; generalize (H3 (x - eps) H8); + apply Rlt_not_le; apply tech_Rgt_minus; exact H1. Qed. -Lemma decreasing_growing : (Un:nat->R) (Un_decreasing Un) -> (Un_growing (opp_seq Un)). -Intro. -Unfold Un_growing opp_seq Un_decreasing. -Intros. -Apply Rle_Ropp1. -Apply H. +Lemma decreasing_growing : + forall Un:nat -> R, Un_decreasing Un -> Un_growing (opp_seq Un). +intro. +unfold Un_growing, opp_seq, Un_decreasing in |- *. +intros. +apply Ropp_le_contravar. +apply H. Qed. -Lemma decreasing_cv : (Un:nat->R) (Un_decreasing Un) -> (has_lb Un) -> (sigTT R [l:R](Un_cv Un l)). -Intros. -Cut (sigTT R [l:R](Un_cv (opp_seq Un) l)) -> (sigTT R [l:R](Un_cv Un l)). -Intro. -Apply X. -Apply growing_cv. -Apply decreasing_growing; Assumption. -Exact H0. -Intro. -Elim X; Intros. -Apply existTT with ``-x``. -Unfold Un_cv in p. -Unfold R_dist in p. -Unfold opp_seq in p. -Unfold Un_cv. -Unfold R_dist. -Intros. -Elim (p eps H1); Intros. -Exists x0; Intros. -Assert H4 := (H2 n H3). -Rewrite <- Rabsolu_Ropp. -Replace ``-((Un n)- -x)`` with ``-(Un n)-x``; [Assumption | Ring]. +Lemma decreasing_cv : + forall Un:nat -> R, + Un_decreasing Un -> has_lb Un -> sigT (fun l:R => Un_cv Un l). +intros. +cut (sigT (fun l:R => Un_cv (opp_seq Un) l) -> sigT (fun l:R => Un_cv Un l)). +intro. +apply X. +apply growing_cv. +apply decreasing_growing; assumption. +exact H0. +intro. +elim X; intros. +apply existT with (- x). +unfold Un_cv in p. +unfold R_dist in p. +unfold opp_seq in p. +unfold Un_cv in |- *. +unfold R_dist in |- *. +intros. +elim (p eps H1); intros. +exists x0; intros. +assert (H4 := H2 n H3). +rewrite <- Rabs_Ropp. +replace (- (Un n - - x)) with (- Un n - x); [ assumption | ring ]. Qed. (***********) -Lemma maj_sup : (Un:nat->R) (has_ub Un) -> (sigTT R [l:R](is_lub (EUn Un) l)). -Intros. -Unfold has_ub in H. -Apply complet. -Assumption. -Exists (Un O). -Unfold EUn. -Exists O; Reflexivity. +Lemma maj_sup : + forall Un:nat -> R, has_ub Un -> sigT (fun l:R => is_lub (EUn Un) l). +intros. +unfold has_ub in H. +apply completeness. +assumption. +exists (Un 0%nat). +unfold EUn in |- *. +exists 0%nat; reflexivity. Qed. (**********) -Lemma min_inf : (Un:nat->R) (has_lb Un) -> (sigTT R [l:R](is_lub (EUn (opp_seq Un)) l)). -Intros; Unfold has_lb in H. -Apply complet. -Assumption. -Exists ``-(Un O)``. -Exists O. -Reflexivity. +Lemma min_inf : + forall Un:nat -> R, + has_lb Un -> sigT (fun l:R => is_lub (EUn (opp_seq Un)) l). +intros; unfold has_lb in H. +apply completeness. +assumption. +exists (- Un 0%nat). +exists 0%nat. +reflexivity. Qed. -Definition majorant [Un:nat->R;pr:(has_ub Un)] : R := Cases (maj_sup Un pr) of (existTT a b) => a end. +Definition majorant (Un:nat -> R) (pr:has_ub Un) : R := + match maj_sup Un pr with + | existT a b => a + end. -Definition minorant [Un:nat->R;pr:(has_lb Un)] : R := Cases (min_inf Un pr) of (existTT a b) => ``-a`` end. +Definition minorant (Un:nat -> R) (pr:has_lb Un) : R := + match min_inf Un pr with + | existT a b => - a + end. -Lemma maj_ss : (Un:nat->R;k:nat) (has_ub Un) -> (has_ub [i:nat](Un (plus k i))). -Intros. -Unfold has_ub in H. -Unfold bound in H. -Elim H; Intros. -Unfold is_upper_bound in H0. -Unfold has_ub. -Exists x. -Unfold is_upper_bound. -Intros. -Apply H0. -Elim H1; Intros. -Exists (plus k x1); Assumption. +Lemma maj_ss : + forall (Un:nat -> R) (k:nat), + has_ub Un -> has_ub (fun i:nat => Un (k + i)%nat). +intros. +unfold has_ub in H. +unfold bound in H. +elim H; intros. +unfold is_upper_bound in H0. +unfold has_ub in |- *. +exists x. +unfold is_upper_bound in |- *. +intros. +apply H0. +elim H1; intros. +exists (k + x1)%nat; assumption. Qed. -Lemma min_ss : (Un:nat->R;k:nat) (has_lb Un) -> (has_lb [i:nat](Un (plus k i))). -Intros. -Unfold has_lb in H. -Unfold bound in H. -Elim H; Intros. -Unfold is_upper_bound in H0. -Unfold has_lb. -Exists x. -Unfold is_upper_bound. -Intros. -Apply H0. -Elim H1; Intros. -Exists (plus k x1); Assumption. +Lemma min_ss : + forall (Un:nat -> R) (k:nat), + has_lb Un -> has_lb (fun i:nat => Un (k + i)%nat). +intros. +unfold has_lb in H. +unfold bound in H. +elim H; intros. +unfold is_upper_bound in H0. +unfold has_lb in |- *. +exists x. +unfold is_upper_bound in |- *. +intros. +apply H0. +elim H1; intros. +exists (k + x1)%nat; assumption. Qed. -Definition sequence_majorant [Un:nat->R;pr:(has_ub Un)] : nat -> R := [i:nat](majorant [k:nat](Un (plus i k)) (maj_ss Un i pr)). +Definition sequence_majorant (Un:nat -> R) (pr:has_ub Un) + (i:nat) : R := majorant (fun k:nat => Un (i + k)%nat) (maj_ss Un i pr). -Definition sequence_minorant [Un:nat->R;pr:(has_lb Un)] : nat -> R := [i:nat](minorant [k:nat](Un (plus i k)) (min_ss Un i pr)). +Definition sequence_minorant (Un:nat -> R) (pr:has_lb Un) + (i:nat) : R := minorant (fun k:nat => Un (i + k)%nat) (min_ss Un i pr). -Lemma Wn_decreasing : (Un:nat->R;pr:(has_ub Un)) (Un_decreasing (sequence_majorant Un pr)). -Intros. -Unfold Un_decreasing. -Intro. -Unfold sequence_majorant. -Assert H := (maj_sup [k:nat](Un (plus (S n) k)) (maj_ss Un (S n) pr)). -Assert H0 := (maj_sup [k:nat](Un (plus n k)) (maj_ss Un n pr)). -Elim H; Intros. -Elim H0; Intros. -Cut (majorant ([k:nat](Un (plus (S n) k))) (maj_ss Un (S n) pr)) == x; [Intro Maj1; Rewrite Maj1 | Idtac]. -Cut (majorant ([k:nat](Un (plus n k))) (maj_ss Un n pr)) == x0; [Intro Maj2; Rewrite Maj2 | Idtac]. -Unfold is_lub in p. -Unfold is_lub in p0. -Elim p; Intros. -Apply H2. -Elim p0; Intros. -Unfold is_upper_bound. -Intros. -Unfold is_upper_bound in H3. -Apply H3. -Elim H5; Intros. -Exists (plus (1) x2). -Replace (plus n (plus (S O) x2)) with (plus (S n) x2). -Assumption. -Replace (S n) with (plus (1) n); [Ring | Ring]. -Cut (is_lub (EUn [k:nat](Un (plus n k))) (majorant ([k:nat](Un (plus n k))) (maj_ss Un n pr))). -Intro. -Unfold is_lub in p0; Unfold is_lub in H1. -Elim p0; Intros; Elim H1; Intros. -Assert H6 := (H5 x0 H2). -Assert H7 := (H3 (majorant ([k:nat](Un (plus n k))) (maj_ss Un n pr)) H4). -Apply Rle_antisym; Assumption. -Unfold majorant. -Case (maj_sup [k:nat](Un (plus n k)) (maj_ss Un n pr)). -Trivial. -Cut (is_lub (EUn [k:nat](Un (plus (S n) k))) (majorant ([k:nat](Un (plus (S n) k))) (maj_ss Un (S n) pr))). -Intro. -Unfold is_lub in p; Unfold is_lub in H1. -Elim p; Intros; Elim H1; Intros. -Assert H6 := (H5 x H2). -Assert H7 := (H3 (majorant ([k:nat](Un (plus (S n) k))) (maj_ss Un (S n) pr)) H4). -Apply Rle_antisym; Assumption. -Unfold majorant. -Case (maj_sup [k:nat](Un (plus (S n) k)) (maj_ss Un (S n) pr)). -Trivial. +Lemma Wn_decreasing : + forall (Un:nat -> R) (pr:has_ub Un), Un_decreasing (sequence_majorant Un pr). +intros. +unfold Un_decreasing in |- *. +intro. +unfold sequence_majorant in |- *. +assert (H := maj_sup (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)). +assert (H0 := maj_sup (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)). +elim H; intros. +elim H0; intros. +cut (majorant (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr) = x); + [ intro Maj1; rewrite Maj1 | idtac ]. +cut (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr) = x0); + [ intro Maj2; rewrite Maj2 | idtac ]. +unfold is_lub in p. +unfold is_lub in p0. +elim p; intros. +apply H2. +elim p0; intros. +unfold is_upper_bound in |- *. +intros. +unfold is_upper_bound in H3. +apply H3. +elim H5; intros. +exists (1 + x2)%nat. +replace (n + (1 + x2))%nat with (S n + x2)%nat. +assumption. +replace (S n) with (1 + n)%nat; [ ring | ring ]. +cut + (is_lub (EUn (fun k:nat => Un (n + k)%nat)) + (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr))). +intro. +unfold is_lub in p0; unfold is_lub in H1. +elim p0; intros; elim H1; intros. +assert (H6 := H5 x0 H2). +assert + (H7 := H3 (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)) H4). +apply Rle_antisym; assumption. +unfold majorant in |- *. +case (maj_sup (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr)). +trivial. +cut + (is_lub (EUn (fun k:nat => Un (S n + k)%nat)) + (majorant (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr))). +intro. +unfold is_lub in p; unfold is_lub in H1. +elim p; intros; elim H1; intros. +assert (H6 := H5 x H2). +assert + (H7 := + H3 (majorant (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)) H4). +apply Rle_antisym; assumption. +unfold majorant in |- *. +case (maj_sup (fun k:nat => Un (S n + k)%nat) (maj_ss Un (S n) pr)). +trivial. Qed. -Lemma Vn_growing : (Un:nat->R;pr:(has_lb Un)) (Un_growing (sequence_minorant Un pr)). -Intros. -Unfold Un_growing. -Intro. -Unfold sequence_minorant. -Assert H := (min_inf [k:nat](Un (plus (S n) k)) (min_ss Un (S n) pr)). -Assert H0 := (min_inf [k:nat](Un (plus n k)) (min_ss Un n pr)). -Elim H; Intros. -Elim H0; Intros. -Cut (minorant ([k:nat](Un (plus (S n) k))) (min_ss Un (S n) pr)) == ``-x``; [Intro Maj1; Rewrite Maj1 | Idtac]. -Cut (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr)) == ``-x0``; [Intro Maj2; Rewrite Maj2 | Idtac]. -Unfold is_lub in p. -Unfold is_lub in p0. -Elim p; Intros. -Apply Rle_Ropp1. -Apply H2. -Elim p0; Intros. -Unfold is_upper_bound. -Intros. -Unfold is_upper_bound in H3. -Apply H3. -Elim H5; Intros. -Exists (plus (1) x2). -Unfold opp_seq in H6. -Unfold opp_seq. -Replace (plus n (plus (S O) x2)) with (plus (S n) x2). -Assumption. -Replace (S n) with (plus (1) n); [Ring | Ring]. -Cut (is_lub (EUn (opp_seq [k:nat](Un (plus n k)))) (Ropp (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr)))). -Intro. -Unfold is_lub in p0; Unfold is_lub in H1. -Elim p0; Intros; Elim H1; Intros. -Assert H6 := (H5 x0 H2). -Assert H7 := (H3 (Ropp (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr))) H4). -Rewrite <- (Ropp_Ropp (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr))). -Apply eq_Ropp; Apply Rle_antisym; Assumption. -Unfold minorant. -Case (min_inf [k:nat](Un (plus n k)) (min_ss Un n pr)). -Intro; Rewrite Ropp_Ropp. -Trivial. -Cut (is_lub (EUn (opp_seq [k:nat](Un (plus (S n) k)))) (Ropp (minorant ([k:nat](Un (plus (S n) k))) (min_ss Un (S n) pr)))). -Intro. -Unfold is_lub in p; Unfold is_lub in H1. -Elim p; Intros; Elim H1; Intros. -Assert H6 := (H5 x H2). -Assert H7 := (H3 (Ropp (minorant ([k:nat](Un (plus (S n) k))) (min_ss Un (S n) pr))) H4). -Rewrite <- (Ropp_Ropp (minorant ([k:nat](Un (plus (S n) k))) (min_ss Un (S n) pr))). -Apply eq_Ropp; Apply Rle_antisym; Assumption. -Unfold minorant. -Case (min_inf [k:nat](Un (plus (S n) k)) (min_ss Un (S n) pr)). -Intro; Rewrite Ropp_Ropp. -Trivial. +Lemma Vn_growing : + forall (Un:nat -> R) (pr:has_lb Un), Un_growing (sequence_minorant Un pr). +intros. +unfold Un_growing in |- *. +intro. +unfold sequence_minorant in |- *. +assert (H := min_inf (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)). +assert (H0 := min_inf (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)). +elim H; intros. +elim H0; intros. +cut (minorant (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr) = - x); + [ intro Maj1; rewrite Maj1 | idtac ]. +cut (minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr) = - x0); + [ intro Maj2; rewrite Maj2 | idtac ]. +unfold is_lub in p. +unfold is_lub in p0. +elim p; intros. +apply Ropp_le_contravar. +apply H2. +elim p0; intros. +unfold is_upper_bound in |- *. +intros. +unfold is_upper_bound in H3. +apply H3. +elim H5; intros. +exists (1 + x2)%nat. +unfold opp_seq in H6. +unfold opp_seq in |- *. +replace (n + (1 + x2))%nat with (S n + x2)%nat. +assumption. +replace (S n) with (1 + n)%nat; [ ring | ring ]. +cut + (is_lub (EUn (opp_seq (fun k:nat => Un (n + k)%nat))) + (- minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr))). +intro. +unfold is_lub in p0; unfold is_lub in H1. +elim p0; intros; elim H1; intros. +assert (H6 := H5 x0 H2). +assert + (H7 := H3 (- minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)) H4). +rewrite <- + (Ropp_involutive (minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr))) + . +apply Ropp_eq_compat; apply Rle_antisym; assumption. +unfold minorant in |- *. +case (min_inf (fun k:nat => Un (n + k)%nat) (min_ss Un n pr)). +intro; rewrite Ropp_involutive. +trivial. +cut + (is_lub (EUn (opp_seq (fun k:nat => Un (S n + k)%nat))) + (- minorant (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr))). +intro. +unfold is_lub in p; unfold is_lub in H1. +elim p; intros; elim H1; intros. +assert (H6 := H5 x H2). +assert + (H7 := + H3 (- minorant (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)) H4). +rewrite <- + (Ropp_involutive + (minorant (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr))) + . +apply Ropp_eq_compat; apply Rle_antisym; assumption. +unfold minorant in |- *. +case (min_inf (fun k:nat => Un (S n + k)%nat) (min_ss Un (S n) pr)). +intro; rewrite Ropp_involutive. +trivial. Qed. (**********) -Lemma Vn_Un_Wn_order : (Un:nat->R;pr1:(has_ub Un);pr2:(has_lb Un)) (n:nat) ``((sequence_minorant Un pr2) n)<=(Un n)<=((sequence_majorant Un pr1) n)``. -Intros. -Split. -Unfold sequence_minorant. -Cut (sigTT R [l:R](is_lub (EUn (opp_seq [i:nat](Un (plus n i)))) l)). -Intro. -Elim X; Intros. -Replace (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr2)) with ``-x``. -Unfold is_lub in p. -Elim p; Intros. -Unfold is_upper_bound in H. -Rewrite <- (Ropp_Ropp (Un n)). -Apply Rle_Ropp1. -Apply H. -Exists O. -Unfold opp_seq. -Replace (plus n O) with n; [Reflexivity | Ring]. -Cut (is_lub (EUn (opp_seq [k:nat](Un (plus n k)))) (Ropp (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr2)))). -Intro. -Unfold is_lub in p; Unfold is_lub in H. -Elim p; Intros; Elim H; Intros. -Assert H4 := (H3 x H0). -Assert H5 := (H1 (Ropp (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr2))) H2). -Rewrite <- (Ropp_Ropp (minorant ([k:nat](Un (plus n k))) (min_ss Un n pr2))). -Apply eq_Ropp; Apply Rle_antisym; Assumption. -Unfold minorant. -Case (min_inf [k:nat](Un (plus n k)) (min_ss Un n pr2)). -Intro; Rewrite Ropp_Ropp. -Trivial. -Apply min_inf. -Apply min_ss; Assumption. -Unfold sequence_majorant. -Cut (sigTT R [l:R](is_lub (EUn [i:nat](Un (plus n i))) l)). -Intro. -Elim X; Intros. -Replace (majorant ([k:nat](Un (plus n k))) (maj_ss Un n pr1)) with ``x``. -Unfold is_lub in p. -Elim p; Intros. -Unfold is_upper_bound in H. -Apply H. -Exists O. -Replace (plus n O) with n; [Reflexivity | Ring]. -Cut (is_lub (EUn [k:nat](Un (plus n k))) (majorant ([k:nat](Un (plus n k))) (maj_ss Un n pr1))). -Intro. -Unfold is_lub in p; Unfold is_lub in H. -Elim p; Intros; Elim H; Intros. -Assert H4 := (H3 x H0). -Assert H5 := (H1 (majorant ([k:nat](Un (plus n k))) (maj_ss Un n pr1)) H2). -Apply Rle_antisym; Assumption. -Unfold majorant. -Case (maj_sup [k:nat](Un (plus n k)) (maj_ss Un n pr1)). -Intro; Trivial. -Apply maj_sup. -Apply maj_ss; Assumption. +Lemma Vn_Un_Wn_order : + forall (Un:nat -> R) (pr1:has_ub Un) (pr2:has_lb Un) + (n:nat), sequence_minorant Un pr2 n <= Un n <= sequence_majorant Un pr1 n. +intros. +split. +unfold sequence_minorant in |- *. +cut + (sigT (fun l:R => is_lub (EUn (opp_seq (fun i:nat => Un (n + i)%nat))) l)). +intro. +elim X; intros. +replace (minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)) with (- x). +unfold is_lub in p. +elim p; intros. +unfold is_upper_bound in H. +rewrite <- (Ropp_involutive (Un n)). +apply Ropp_le_contravar. +apply H. +exists 0%nat. +unfold opp_seq in |- *. +replace (n + 0)%nat with n; [ reflexivity | ring ]. +cut + (is_lub (EUn (opp_seq (fun k:nat => Un (n + k)%nat))) + (- minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2))). +intro. +unfold is_lub in p; unfold is_lub in H. +elim p; intros; elim H; intros. +assert (H4 := H3 x H0). +assert + (H5 := H1 (- minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)) H2). +rewrite <- + (Ropp_involutive (minorant (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2))) + . +apply Ropp_eq_compat; apply Rle_antisym; assumption. +unfold minorant in |- *. +case (min_inf (fun k:nat => Un (n + k)%nat) (min_ss Un n pr2)). +intro; rewrite Ropp_involutive. +trivial. +apply min_inf. +apply min_ss; assumption. +unfold sequence_majorant in |- *. +cut (sigT (fun l:R => is_lub (EUn (fun i:nat => Un (n + i)%nat)) l)). +intro. +elim X; intros. +replace (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)) with x. +unfold is_lub in p. +elim p; intros. +unfold is_upper_bound in H. +apply H. +exists 0%nat. +replace (n + 0)%nat with n; [ reflexivity | ring ]. +cut + (is_lub (EUn (fun k:nat => Un (n + k)%nat)) + (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1))). +intro. +unfold is_lub in p; unfold is_lub in H. +elim p; intros; elim H; intros. +assert (H4 := H3 x H0). +assert + (H5 := H1 (majorant (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)) H2). +apply Rle_antisym; assumption. +unfold majorant in |- *. +case (maj_sup (fun k:nat => Un (n + k)%nat) (maj_ss Un n pr1)). +intro; trivial. +apply maj_sup. +apply maj_ss; assumption. Qed. -Lemma min_maj : (Un:nat->R;pr1:(has_ub Un);pr2:(has_lb Un)) (has_ub (sequence_minorant Un pr2)). -Intros. -Assert H := (Vn_Un_Wn_order Un pr1 pr2). -Unfold has_ub. -Unfold bound. -Unfold has_ub in pr1. -Unfold bound in pr1. -Elim pr1; Intros. -Exists x. -Unfold is_upper_bound. -Intros. -Unfold is_upper_bound in H0. -Elim H1; Intros. -Rewrite H2. -Apply Rle_trans with (Un x1). -Assert H3 := (H x1); Elim H3; Intros; Assumption. -Apply H0. -Exists x1; Reflexivity. +Lemma min_maj : + forall (Un:nat -> R) (pr1:has_ub Un) (pr2:has_lb Un), + has_ub (sequence_minorant Un pr2). +intros. +assert (H := Vn_Un_Wn_order Un pr1 pr2). +unfold has_ub in |- *. +unfold bound in |- *. +unfold has_ub in pr1. +unfold bound in pr1. +elim pr1; intros. +exists x. +unfold is_upper_bound in |- *. +intros. +unfold is_upper_bound in H0. +elim H1; intros. +rewrite H2. +apply Rle_trans with (Un x1). +assert (H3 := H x1); elim H3; intros; assumption. +apply H0. +exists x1; reflexivity. Qed. -Lemma maj_min : (Un:nat->R;pr1:(has_ub Un);pr2:(has_lb Un)) (has_lb (sequence_majorant Un pr1)). -Intros. -Assert H := (Vn_Un_Wn_order Un pr1 pr2). -Unfold has_lb. -Unfold bound. -Unfold has_lb in pr2. -Unfold bound in pr2. -Elim pr2; Intros. -Exists x. -Unfold is_upper_bound. -Intros. -Unfold is_upper_bound in H0. -Elim H1; Intros. -Rewrite H2. -Apply Rle_trans with ((opp_seq Un) x1). -Assert H3 := (H x1); Elim H3; Intros. -Unfold opp_seq; Apply Rle_Ropp1. -Assumption. -Apply H0. -Exists x1; Reflexivity. +Lemma maj_min : + forall (Un:nat -> R) (pr1:has_ub Un) (pr2:has_lb Un), + has_lb (sequence_majorant Un pr1). +intros. +assert (H := Vn_Un_Wn_order Un pr1 pr2). +unfold has_lb in |- *. +unfold bound in |- *. +unfold has_lb in pr2. +unfold bound in pr2. +elim pr2; intros. +exists x. +unfold is_upper_bound in |- *. +intros. +unfold is_upper_bound in H0. +elim H1; intros. +rewrite H2. +apply Rle_trans with (opp_seq Un x1). +assert (H3 := H x1); elim H3; intros. +unfold opp_seq in |- *; apply Ropp_le_contravar. +assumption. +apply H0. +exists x1; reflexivity. Qed. (**********) -Lemma cauchy_maj : (Un:nat->R) (Cauchy_crit Un) -> (has_ub Un). -Intros. -Unfold has_ub. -Apply cauchy_bound. -Assumption. +Lemma cauchy_maj : forall Un:nat -> R, Cauchy_crit Un -> has_ub Un. +intros. +unfold has_ub in |- *. +apply cauchy_bound. +assumption. Qed. (**********) -Lemma cauchy_opp : (Un:nat->R) (Cauchy_crit Un) -> (Cauchy_crit (opp_seq Un)). -Intro. -Unfold Cauchy_crit. -Unfold R_dist. -Intros. -Elim (H eps H0); Intros. -Exists x; Intros. -Unfold opp_seq. -Rewrite <- Rabsolu_Ropp. -Replace ``-( -(Un n)- -(Un m))`` with ``(Un n)-(Un m)``; [Apply H1; Assumption | Ring]. +Lemma cauchy_opp : + forall Un:nat -> R, Cauchy_crit Un -> Cauchy_crit (opp_seq Un). +intro. +unfold Cauchy_crit in |- *. +unfold R_dist in |- *. +intros. +elim (H eps H0); intros. +exists x; intros. +unfold opp_seq in |- *. +rewrite <- Rabs_Ropp. +replace (- (- Un n - - Un m)) with (Un n - Un m); + [ apply H1; assumption | ring ]. Qed. (**********) -Lemma cauchy_min : (Un:nat->R) (Cauchy_crit Un) -> (has_lb Un). -Intros. -Unfold has_lb. -Assert H0 := (cauchy_opp ? H). -Apply cauchy_bound. -Assumption. +Lemma cauchy_min : forall Un:nat -> R, Cauchy_crit Un -> has_lb Un. +intros. +unfold has_lb in |- *. +assert (H0 := cauchy_opp _ H). +apply cauchy_bound. +assumption. Qed. (**********) -Lemma maj_cv : (Un:nat->R;pr:(Cauchy_crit Un)) (sigTT R [l:R](Un_cv (sequence_majorant Un (cauchy_maj Un pr)) l)). -Intros. -Apply decreasing_cv. -Apply Wn_decreasing. -Apply maj_min. -Apply cauchy_min. -Assumption. +Lemma maj_cv : + forall (Un:nat -> R) (pr:Cauchy_crit Un), + sigT (fun l:R => Un_cv (sequence_majorant Un (cauchy_maj Un pr)) l). +intros. +apply decreasing_cv. +apply Wn_decreasing. +apply maj_min. +apply cauchy_min. +assumption. Qed. (**********) -Lemma min_cv : (Un:nat->R;pr:(Cauchy_crit Un)) (sigTT R [l:R](Un_cv (sequence_minorant Un (cauchy_min Un pr)) l)). -Intros. -Apply growing_cv. -Apply Vn_growing. -Apply min_maj. -Apply cauchy_maj. -Assumption. +Lemma min_cv : + forall (Un:nat -> R) (pr:Cauchy_crit Un), + sigT (fun l:R => Un_cv (sequence_minorant Un (cauchy_min Un pr)) l). +intros. +apply growing_cv. +apply Vn_growing. +apply min_maj. +apply cauchy_maj. +assumption. Qed. -Lemma cond_eq : (x,y:R) ((eps:R)``0<eps``->``(Rabsolu (x-y))<eps``) -> x==y. -Intros. -Case (total_order_T x y); Intro. -Elim s; Intro. -Cut ``0<y-x``. -Intro. -Assert H1 := (H ``y-x`` H0). -Rewrite <- Rabsolu_Ropp in H1. -Cut ``-(x-y)==y-x``; [Intro; Rewrite H2 in H1 | Ring]. -Rewrite Rabsolu_right in H1. -Elim (Rlt_antirefl ? H1). -Left; Assumption. -Apply Rlt_anti_compatibility with x. -Rewrite Rplus_Or; Replace ``x+(y-x)`` with y; [Assumption | Ring]. -Assumption. -Cut ``0<x-y``. -Intro. -Assert H1 := (H ``x-y`` H0). -Rewrite Rabsolu_right in H1. -Elim (Rlt_antirefl ? H1). -Left; Assumption. -Apply Rlt_anti_compatibility with y. -Rewrite Rplus_Or; Replace ``y+(x-y)`` with x; [Assumption | Ring]. +Lemma cond_eq : + forall x y:R, (forall eps:R, 0 < eps -> Rabs (x - y) < eps) -> x = y. +intros. +case (total_order_T x y); intro. +elim s; intro. +cut (0 < y - x). +intro. +assert (H1 := H (y - x) H0). +rewrite <- Rabs_Ropp in H1. +cut (- (x - y) = y - x); [ intro; rewrite H2 in H1 | ring ]. +rewrite Rabs_right in H1. +elim (Rlt_irrefl _ H1). +left; assumption. +apply Rplus_lt_reg_r with x. +rewrite Rplus_0_r; replace (x + (y - x)) with y; [ assumption | ring ]. +assumption. +cut (0 < x - y). +intro. +assert (H1 := H (x - y) H0). +rewrite Rabs_right in H1. +elim (Rlt_irrefl _ H1). +left; assumption. +apply Rplus_lt_reg_r with y. +rewrite Rplus_0_r; replace (y + (x - y)) with x; [ assumption | ring ]. Qed. -Lemma not_Rlt : (r1,r2:R)~(``r1<r2``)->``r1>=r2``. -Intros r1 r2 ; Generalize (total_order r1 r2) ; Unfold Rge. -Tauto. +Lemma not_Rlt : forall r1 r2:R, ~ r1 < r2 -> r1 >= r2. +intros r1 r2; generalize (Rtotal_order r1 r2); unfold Rge in |- *. +tauto. Qed. (**********) -Lemma approx_maj : (Un:nat->R;pr:(has_ub Un)) (eps:R) ``0<eps`` -> (EX k : nat | ``(Rabsolu ((majorant Un pr)-(Un k))) < eps``). -Intros. -Pose P := [k:nat]``(Rabsolu ((majorant Un pr)-(Un k))) < eps``. -Unfold P. -Cut (EX k:nat | (P k)) -> (EX k:nat | ``(Rabsolu ((majorant Un pr)-(Un k))) < eps``). -Intros. -Apply H0. -Apply not_all_not_ex. -Red; Intro. -2:Unfold P; Trivial. -Unfold P in H1. -Cut (n:nat)``(Rabsolu ((majorant Un pr)-(Un n))) >= eps``. -Intro. -Cut (is_lub (EUn Un) (majorant Un pr)). -Intro. -Unfold is_lub in H3. -Unfold is_upper_bound in H3. -Elim H3; Intros. -Cut (n:nat)``eps<=(majorant Un pr)-(Un n)``. -Intro. -Cut (n:nat)``(Un n)<=(majorant Un pr)-eps``. -Intro. -Cut ((x:R)(EUn Un x)->``x <= (majorant Un pr)-eps``). -Intro. -Assert H9 := (H5 ``(majorant Un pr)-eps`` H8). -Cut ``eps<=0``. -Intro. -Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H H10)). -Apply Rle_anti_compatibility with ``(majorant Un pr)-eps``. -Rewrite Rplus_Or. -Replace ``(majorant Un pr)-eps+eps`` with (majorant Un pr); [Assumption | Ring]. -Intros. -Unfold EUn in H8. -Elim H8; Intros. -Rewrite H9; Apply H7. -Intro. -Assert H7 := (H6 n). -Apply Rle_anti_compatibility with ``eps-(Un n)``. -Replace ``eps-(Un n)+(Un n)`` with ``eps``. -Replace ``eps-(Un n)+((majorant Un pr)-eps)`` with ``(majorant Un pr)-(Un n)``. -Assumption. -Ring. -Ring. -Intro. -Assert H6 := (H2 n). -Rewrite Rabsolu_right in H6. -Apply Rle_sym2. -Assumption. -Apply Rle_sym1. -Apply Rle_anti_compatibility with (Un n). -Rewrite Rplus_Or; Replace ``(Un n)+((majorant Un pr)-(Un n))`` with (majorant Un pr); [Apply H4 | Ring]. -Exists n; Reflexivity. -Unfold majorant. -Case (maj_sup Un pr). -Trivial. -Intro. -Assert H2 := (H1 n). -Apply not_Rlt; Assumption. +Lemma approx_maj : + forall (Un:nat -> R) (pr:has_ub Un) (eps:R), + 0 < eps -> exists k : nat | Rabs (majorant Un pr - Un k) < eps. +intros. +pose (P := fun k:nat => Rabs (majorant Un pr - Un k) < eps). +unfold P in |- *. +cut + (( exists k : nat | P k) -> + exists k : nat | Rabs (majorant Un pr - Un k) < eps). +intros. +apply H0. +apply not_all_not_ex. +red in |- *; intro. +2: unfold P in |- *; trivial. +unfold P in H1. +cut (forall n:nat, Rabs (majorant Un pr - Un n) >= eps). +intro. +cut (is_lub (EUn Un) (majorant Un pr)). +intro. +unfold is_lub in H3. +unfold is_upper_bound in H3. +elim H3; intros. +cut (forall n:nat, eps <= majorant Un pr - Un n). +intro. +cut (forall n:nat, Un n <= majorant Un pr - eps). +intro. +cut (forall x:R, EUn Un x -> x <= majorant Un pr - eps). +intro. +assert (H9 := H5 (majorant Un pr - eps) H8). +cut (eps <= 0). +intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H H10)). +apply Rplus_le_reg_l with (majorant Un pr - eps). +rewrite Rplus_0_r. +replace (majorant Un pr - eps + eps) with (majorant Un pr); + [ assumption | ring ]. +intros. +unfold EUn in H8. +elim H8; intros. +rewrite H9; apply H7. +intro. +assert (H7 := H6 n). +apply Rplus_le_reg_l with (eps - Un n). +replace (eps - Un n + Un n) with eps. +replace (eps - Un n + (majorant Un pr - eps)) with (majorant Un pr - Un n). +assumption. +ring. +ring. +intro. +assert (H6 := H2 n). +rewrite Rabs_right in H6. +apply Rge_le. +assumption. +apply Rle_ge. +apply Rplus_le_reg_l with (Un n). +rewrite Rplus_0_r; + replace (Un n + (majorant Un pr - Un n)) with (majorant Un pr); + [ apply H4 | ring ]. +exists n; reflexivity. +unfold majorant in |- *. +case (maj_sup Un pr). +trivial. +intro. +assert (H2 := H1 n). +apply not_Rlt; assumption. Qed. (**********) -Lemma approx_min : (Un:nat->R;pr:(has_lb Un)) (eps:R) ``0<eps`` -> (EX k :nat | ``(Rabsolu ((minorant Un pr)-(Un k))) < eps``). -Intros. -Pose P := [k:nat]``(Rabsolu ((minorant Un pr)-(Un k))) < eps``. -Unfold P. -Cut (EX k:nat | (P k)) -> (EX k:nat | ``(Rabsolu ((minorant Un pr)-(Un k))) < eps``). -Intros. -Apply H0. -Apply not_all_not_ex. -Red; Intro. -2:Unfold P; Trivial. -Unfold P in H1. -Cut (n:nat)``(Rabsolu ((minorant Un pr)-(Un n))) >= eps``. -Intro. -Cut (is_lub (EUn (opp_seq Un)) ``-(minorant Un pr)``). -Intro. -Unfold is_lub in H3. -Unfold is_upper_bound in H3. -Elim H3; Intros. -Cut (n:nat)``eps<=(Un n)-(minorant Un pr)``. -Intro. -Cut (n:nat)``((opp_seq Un) n)<=-(minorant Un pr)-eps``. -Intro. -Cut ((x:R)(EUn (opp_seq Un) x)->``x <= -(minorant Un pr)-eps``). -Intro. -Assert H9 := (H5 ``-(minorant Un pr)-eps`` H8). -Cut ``eps<=0``. -Intro. -Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H H10)). -Apply Rle_anti_compatibility with ``-(minorant Un pr)-eps``. -Rewrite Rplus_Or. -Replace ``-(minorant Un pr)-eps+eps`` with ``-(minorant Un pr)``; [Assumption | Ring]. -Intros. -Unfold EUn in H8. -Elim H8; Intros. -Rewrite H9; Apply H7. -Intro. -Assert H7 := (H6 n). -Unfold opp_seq. -Apply Rle_anti_compatibility with ``eps+(Un n)``. -Replace ``eps+(Un n)+ -(Un n)`` with ``eps``. -Replace ``eps+(Un n)+(-(minorant Un pr)-eps)`` with ``(Un n)-(minorant Un pr)``. -Assumption. -Ring. -Ring. -Intro. -Assert H6 := (H2 n). -Rewrite Rabsolu_left1 in H6. -Apply Rle_sym2. -Replace ``(Un n)-(minorant Un pr)`` with `` -((minorant Un pr)-(Un n))``; [Assumption | Ring]. -Apply Rle_anti_compatibility with ``-(minorant Un pr)``. -Rewrite Rplus_Or; Replace ``-(minorant Un pr)+((minorant Un pr)-(Un n))`` with ``-(Un n)``. -Apply H4. -Exists n; Reflexivity. -Ring. -Unfold minorant. -Case (min_inf Un pr). -Intro. -Rewrite Ropp_Ropp. -Trivial. -Intro. -Assert H2 := (H1 n). -Apply not_Rlt; Assumption. +Lemma approx_min : + forall (Un:nat -> R) (pr:has_lb Un) (eps:R), + 0 < eps -> exists k : nat | Rabs (minorant Un pr - Un k) < eps. +intros. +pose (P := fun k:nat => Rabs (minorant Un pr - Un k) < eps). +unfold P in |- *. +cut + (( exists k : nat | P k) -> + exists k : nat | Rabs (minorant Un pr - Un k) < eps). +intros. +apply H0. +apply not_all_not_ex. +red in |- *; intro. +2: unfold P in |- *; trivial. +unfold P in H1. +cut (forall n:nat, Rabs (minorant Un pr - Un n) >= eps). +intro. +cut (is_lub (EUn (opp_seq Un)) (- minorant Un pr)). +intro. +unfold is_lub in H3. +unfold is_upper_bound in H3. +elim H3; intros. +cut (forall n:nat, eps <= Un n - minorant Un pr). +intro. +cut (forall n:nat, opp_seq Un n <= - minorant Un pr - eps). +intro. +cut (forall x:R, EUn (opp_seq Un) x -> x <= - minorant Un pr - eps). +intro. +assert (H9 := H5 (- minorant Un pr - eps) H8). +cut (eps <= 0). +intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H H10)). +apply Rplus_le_reg_l with (- minorant Un pr - eps). +rewrite Rplus_0_r. +replace (- minorant Un pr - eps + eps) with (- minorant Un pr); + [ assumption | ring ]. +intros. +unfold EUn in H8. +elim H8; intros. +rewrite H9; apply H7. +intro. +assert (H7 := H6 n). +unfold opp_seq in |- *. +apply Rplus_le_reg_l with (eps + Un n). +replace (eps + Un n + - Un n) with eps. +replace (eps + Un n + (- minorant Un pr - eps)) with (Un n - minorant Un pr). +assumption. +ring. +ring. +intro. +assert (H6 := H2 n). +rewrite Rabs_left1 in H6. +apply Rge_le. +replace (Un n - minorant Un pr) with (- (minorant Un pr - Un n)); + [ assumption | ring ]. +apply Rplus_le_reg_l with (- minorant Un pr). +rewrite Rplus_0_r; + replace (- minorant Un pr + (minorant Un pr - Un n)) with (- Un n). +apply H4. +exists n; reflexivity. +ring. +unfold minorant in |- *. +case (min_inf Un pr). +intro. +rewrite Ropp_involutive. +trivial. +intro. +assert (H2 := H1 n). +apply not_Rlt; assumption. Qed. (* Unicity of limit for convergent sequences *) -Lemma UL_sequence : (Un:nat->R;l1,l2:R) (Un_cv Un l1) -> (Un_cv Un l2) -> l1==l2. -Intros Un l1 l2; Unfold Un_cv; Unfold R_dist; Intros. -Apply cond_eq. -Intros; Cut ``0<eps/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. -Elim (H ``eps/2`` H2); Intros. -Elim (H0 ``eps/2`` H2); Intros. -Pose N := (max x x0). -Apply Rle_lt_trans with ``(Rabsolu (l1 -(Un N)))+(Rabsolu ((Un N)-l2))``. -Replace ``l1-l2`` with ``(l1-(Un N))+((Un N)-l2)``; [Apply Rabsolu_triang | Ring]. -Rewrite (double_var eps); Apply Rplus_lt. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H3; Unfold ge N; Apply le_max_l. -Apply H4; Unfold ge N; Apply le_max_r. +Lemma UL_sequence : + forall (Un:nat -> R) (l1 l2:R), Un_cv Un l1 -> Un_cv Un l2 -> l1 = l2. +intros Un l1 l2; unfold Un_cv in |- *; unfold R_dist in |- *; intros. +apply cond_eq. +intros; cut (0 < eps / 2); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H (eps / 2) H2); intros. +elim (H0 (eps / 2) H2); intros. +pose (N := max x x0). +apply Rle_lt_trans with (Rabs (l1 - Un N) + Rabs (Un N - l2)). +replace (l1 - l2) with (l1 - Un N + (Un N - l2)); + [ apply Rabs_triang | ring ]. +rewrite (double_var eps); apply Rplus_lt_compat. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H3; + unfold ge, N in |- *; apply le_max_l. +apply H4; unfold ge, N in |- *; apply le_max_r. Qed. (**********) -Lemma CV_plus : (An,Bn:nat->R;l1,l2:R) (Un_cv An l1) -> (Un_cv Bn l2) -> (Un_cv [i:nat]``(An i)+(Bn i)`` ``l1+l2``). -Unfold Un_cv; Unfold R_dist; Intros. -Cut ``0<eps/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. -Elim (H ``eps/2`` H2); Intros. -Elim (H0 ``eps/2`` H2); Intros. -Pose N := (max x x0). -Exists N; Intros. -Replace ``(An n)+(Bn n)-(l1+l2)`` with ``((An n)-l1)+((Bn n)-l2)``; [Idtac | Ring]. -Apply Rle_lt_trans with ``(Rabsolu ((An n)-l1))+(Rabsolu ((Bn n)-l2))``. -Apply Rabsolu_triang. -Rewrite (double_var eps); Apply Rplus_lt. -Apply H3; Unfold ge; Apply le_trans with N; [Unfold N; Apply le_max_l | Assumption]. -Apply H4; Unfold ge; Apply le_trans with N; [Unfold N; Apply le_max_r | Assumption]. +Lemma CV_plus : + forall (An Bn:nat -> R) (l1 l2:R), + Un_cv An l1 -> Un_cv Bn l2 -> Un_cv (fun i:nat => An i + Bn i) (l1 + l2). +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +cut (0 < eps / 2); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (H (eps / 2) H2); intros. +elim (H0 (eps / 2) H2); intros. +pose (N := max x x0). +exists N; intros. +replace (An n + Bn n - (l1 + l2)) with (An n - l1 + (Bn n - l2)); + [ idtac | ring ]. +apply Rle_lt_trans with (Rabs (An n - l1) + Rabs (Bn n - l2)). +apply Rabs_triang. +rewrite (double_var eps); apply Rplus_lt_compat. +apply H3; unfold ge in |- *; apply le_trans with N; + [ unfold N in |- *; apply le_max_l | assumption ]. +apply H4; unfold ge in |- *; apply le_trans with N; + [ unfold N in |- *; apply le_max_r | assumption ]. Qed. (**********) -Lemma cv_cvabs : (Un:nat->R;l:R) (Un_cv Un l) -> (Un_cv [i:nat](Rabsolu (Un i)) (Rabsolu l)). -Unfold Un_cv; Unfold R_dist; Intros. -Elim (H eps H0); Intros. -Exists x; Intros. -Apply Rle_lt_trans with ``(Rabsolu ((Un n)-l))``. -Apply Rabsolu_triang_inv2. -Apply H1; Assumption. +Lemma cv_cvabs : + forall (Un:nat -> R) (l:R), + Un_cv Un l -> Un_cv (fun i:nat => Rabs (Un i)) (Rabs l). +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H eps H0); intros. +exists x; intros. +apply Rle_lt_trans with (Rabs (Un n - l)). +apply Rabs_triang_inv2. +apply H1; assumption. Qed. (**********) -Lemma CV_Cauchy : (Un:nat->R) (sigTT R [l:R](Un_cv Un l)) -> (Cauchy_crit Un). -Intros; Elim X; Intros. -Unfold Cauchy_crit; Intros. -Unfold Un_cv in p; Unfold R_dist in p. -Cut ``0<eps/2``; [Intro | Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Sup0]]. -Elim (p ``eps/2`` H0); Intros. -Exists x0; Intros. -Unfold R_dist; Apply Rle_lt_trans with ``(Rabsolu ((Un n)-x))+(Rabsolu (x-(Un m)))``. -Replace ``(Un n)-(Un m)`` with ``((Un n)-x)+(x-(Un m))``; [Apply Rabsolu_triang | Ring]. -Rewrite (double_var eps); Apply Rplus_lt. -Apply H1; Assumption. -Rewrite <- Rabsolu_Ropp; Rewrite Ropp_distr2; Apply H1; Assumption. +Lemma CV_Cauchy : + forall Un:nat -> R, sigT (fun l:R => Un_cv Un l) -> Cauchy_crit Un. +intros; elim X; intros. +unfold Cauchy_crit in |- *; intros. +unfold Un_cv in p; unfold R_dist in p. +cut (0 < eps / 2); + [ intro + | unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; prove_sup0 ] ]. +elim (p (eps / 2) H0); intros. +exists x0; intros. +unfold R_dist in |- *; + apply Rle_lt_trans with (Rabs (Un n - x) + Rabs (x - Un m)). +replace (Un n - Un m) with (Un n - x + (x - Un m)); + [ apply Rabs_triang | ring ]. +rewrite (double_var eps); apply Rplus_lt_compat. +apply H1; assumption. +rewrite <- Rabs_Ropp; rewrite Ropp_minus_distr; apply H1; assumption. Qed. (**********) -Lemma maj_by_pos : (Un:nat->R) (sigTT R [l:R](Un_cv Un l)) -> (EXT l:R | ``0<l``/\((n:nat)``(Rabsolu (Un n))<=l``)). -Intros; Elim X; Intros. -Cut (sigTT R [l:R](Un_cv [k:nat](Rabsolu (Un k)) l)). -Intro. -Assert H := (CV_Cauchy [k:nat](Rabsolu (Un k)) X0). -Assert H0 := (cauchy_bound [k:nat](Rabsolu (Un k)) H). -Elim H0; Intros. -Exists ``x0+1``. -Cut ``0<=x0``. -Intro. -Split. -Apply ge0_plus_gt0_is_gt0; [Assumption | Apply Rlt_R0_R1]. -Intros. -Apply Rle_trans with x0. -Unfold is_upper_bound in H1. -Apply H1. -Exists n; Reflexivity. -Pattern 1 x0; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Apply Rlt_R0_R1. -Apply Rle_trans with (Rabsolu (Un O)). -Apply Rabsolu_pos. -Unfold is_upper_bound in H1. -Apply H1. -Exists O; Reflexivity. -Apply existTT with (Rabsolu x). -Apply cv_cvabs; Assumption. +Lemma maj_by_pos : + forall Un:nat -> R, + sigT (fun l:R => Un_cv Un l) -> + exists l : R | 0 < l /\ (forall n:nat, Rabs (Un n) <= l). +intros; elim X; intros. +cut (sigT (fun l:R => Un_cv (fun k:nat => Rabs (Un k)) l)). +intro. +assert (H := CV_Cauchy (fun k:nat => Rabs (Un k)) X0). +assert (H0 := cauchy_bound (fun k:nat => Rabs (Un k)) H). +elim H0; intros. +exists (x0 + 1). +cut (0 <= x0). +intro. +split. +apply Rplus_le_lt_0_compat; [ assumption | apply Rlt_0_1 ]. +intros. +apply Rle_trans with x0. +unfold is_upper_bound in H1. +apply H1. +exists n; reflexivity. +pattern x0 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + apply Rlt_0_1. +apply Rle_trans with (Rabs (Un 0%nat)). +apply Rabs_pos. +unfold is_upper_bound in H1. +apply H1. +exists 0%nat; reflexivity. +apply existT with (Rabs x). +apply cv_cvabs; assumption. Qed. (**********) -Lemma CV_mult : (An,Bn:nat->R;l1,l2:R) (Un_cv An l1) -> (Un_cv Bn l2) -> (Un_cv [i:nat]``(An i)*(Bn i)`` ``l1*l2``). -Intros. -Cut (sigTT R [l:R](Un_cv An l)). -Intro. -Assert H1 := (maj_by_pos An X). -Elim H1; Intros M H2. -Elim H2; Intros. -Unfold Un_cv; Unfold R_dist; Intros. -Cut ``0<eps/(2*M)``. -Intro. -Case (Req_EM l2 R0); Intro. -Unfold Un_cv in H0; Unfold R_dist in H0. -Elim (H0 ``eps/(2*M)`` H6); Intros. -Exists x; Intros. -Apply Rle_lt_trans with ``(Rabsolu ((An n)*(Bn n)-(An n)*l2))+(Rabsolu ((An n)*l2-l1*l2))``. -Replace ``(An n)*(Bn n)-l1*l2`` with ``((An n)*(Bn n)-(An n)*l2)+((An n)*l2-l1*l2)``; [Apply Rabsolu_triang | Ring]. -Replace ``(Rabsolu ((An n)*(Bn n)-(An n)*l2))`` with ``(Rabsolu (An n))*(Rabsolu ((Bn n)-l2))``. -Replace ``(Rabsolu ((An n)*l2-l1*l2))`` with R0. -Rewrite Rplus_Or. -Apply Rle_lt_trans with ``M*(Rabsolu ((Bn n)-l2))``. -Do 2 Rewrite <- (Rmult_sym ``(Rabsolu ((Bn n)-l2))``). -Apply Rle_monotony. -Apply Rabsolu_pos. -Apply H4. -Apply Rlt_monotony_contra with ``/M``. -Apply Rlt_Rinv; Apply H3. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Rewrite (Rmult_sym ``/M``). -Apply Rlt_trans with ``eps/(2*M)``. -Apply H8; Assumption. -Unfold Rdiv; Rewrite Rinv_Rmult. -Apply Rlt_monotony_contra with ``2``. -Sup0. -Replace ``2*(eps*(/2*/M))`` with ``(2*/2)*(eps*/M)``; [Idtac | Ring]. -Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite double. -Pattern 1 ``eps*/M``; Rewrite <- Rplus_Or. -Apply Rlt_compatibility; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Assumption]. -DiscrR. -DiscrR. -Red; Intro; Rewrite H10 in H3; Elim (Rlt_antirefl ? H3). -Red; Intro; Rewrite H10 in H3; Elim (Rlt_antirefl ? H3). -Rewrite H7; Do 2 Rewrite Rmult_Or; Unfold Rminus; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Reflexivity. -Replace ``(An n)*(Bn n)-(An n)*l2`` with ``(An n)*((Bn n)-l2)``; [Idtac | Ring]. -Symmetry; Apply Rabsolu_mult. -Cut ``0<eps/(2*(Rabsolu l2))``. -Intro. -Unfold Un_cv in H; Unfold R_dist in H; Unfold Un_cv in H0; Unfold R_dist in H0. -Elim (H ``eps/(2*(Rabsolu l2))`` H8); Intros N1 H9. -Elim (H0 ``eps/(2*M)`` H6); Intros N2 H10. -Pose N := (max N1 N2). -Exists N; Intros. -Apply Rle_lt_trans with ``(Rabsolu ((An n)*(Bn n)-(An n)*l2))+(Rabsolu ((An n)*l2-l1*l2))``. -Replace ``(An n)*(Bn n)-l1*l2`` with ``((An n)*(Bn n)-(An n)*l2)+((An n)*l2-l1*l2)``; [Apply Rabsolu_triang | Ring]. -Replace ``(Rabsolu ((An n)*(Bn n)-(An n)*l2))`` with ``(Rabsolu (An n))*(Rabsolu ((Bn n)-l2))``. -Replace ``(Rabsolu ((An n)*l2-l1*l2))`` with ``(Rabsolu l2)*(Rabsolu ((An n)-l1))``. -Rewrite (double_var eps); Apply Rplus_lt. -Apply Rle_lt_trans with ``M*(Rabsolu ((Bn n)-l2))``. -Do 2 Rewrite <- (Rmult_sym ``(Rabsolu ((Bn n)-l2))``). -Apply Rle_monotony. -Apply Rabsolu_pos. -Apply H4. -Apply Rlt_monotony_contra with ``/M``. -Apply Rlt_Rinv; Apply H3. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Rewrite (Rmult_sym ``/M``). -Apply Rlt_le_trans with ``eps/(2*M)``. -Apply H10. -Unfold ge; Apply le_trans with N. -Unfold N; Apply le_max_r. -Assumption. -Unfold Rdiv; Rewrite Rinv_Rmult. -Right; Ring. -DiscrR. -Red; Intro; Rewrite H12 in H3; Elim (Rlt_antirefl ? H3). -Red; Intro; Rewrite H12 in H3; Elim (Rlt_antirefl ? H3). -Apply Rlt_monotony_contra with ``/(Rabsolu l2)``. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Apply Rlt_le_trans with ``eps/(2*(Rabsolu l2))``. -Apply H9. -Unfold ge; Apply le_trans with N. -Unfold N; Apply le_max_l. -Assumption. -Unfold Rdiv; Right; Rewrite Rinv_Rmult. -Ring. -DiscrR. -Apply Rabsolu_no_R0; Assumption. -Apply Rabsolu_no_R0; Assumption. -Replace ``(An n)*l2-l1*l2`` with ``l2*((An n)-l1)``; [Symmetry; Apply Rabsolu_mult | Ring]. -Replace ``(An n)*(Bn n)-(An n)*l2`` with ``(An n)*((Bn n)-l2)``; [Symmetry; Apply Rabsolu_mult | Ring]. -Unfold Rdiv; Apply Rmult_lt_pos. -Assumption. -Apply Rlt_Rinv; Apply Rmult_lt_pos; [Sup0 | Apply Rabsolu_pos_lt; Assumption]. -Unfold Rdiv; Apply Rmult_lt_pos; [Assumption | Apply Rlt_Rinv; Apply Rmult_lt_pos; [Sup0 | Assumption]]. -Apply existTT with l1; Assumption. +Lemma CV_mult : + forall (An Bn:nat -> R) (l1 l2:R), + Un_cv An l1 -> Un_cv Bn l2 -> Un_cv (fun i:nat => An i * Bn i) (l1 * l2). +intros. +cut (sigT (fun l:R => Un_cv An l)). +intro. +assert (H1 := maj_by_pos An X). +elim H1; intros M H2. +elim H2; intros. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +cut (0 < eps / (2 * M)). +intro. +case (Req_dec l2 0); intro. +unfold Un_cv in H0; unfold R_dist in H0. +elim (H0 (eps / (2 * M)) H6); intros. +exists x; intros. +apply Rle_lt_trans with + (Rabs (An n * Bn n - An n * l2) + Rabs (An n * l2 - l1 * l2)). +replace (An n * Bn n - l1 * l2) with + (An n * Bn n - An n * l2 + (An n * l2 - l1 * l2)); + [ apply Rabs_triang | ring ]. +replace (Rabs (An n * Bn n - An n * l2)) with + (Rabs (An n) * Rabs (Bn n - l2)). +replace (Rabs (An n * l2 - l1 * l2)) with 0. +rewrite Rplus_0_r. +apply Rle_lt_trans with (M * Rabs (Bn n - l2)). +do 2 rewrite <- (Rmult_comm (Rabs (Bn n - l2))). +apply Rmult_le_compat_l. +apply Rabs_pos. +apply H4. +apply Rmult_lt_reg_l with (/ M). +apply Rinv_0_lt_compat; apply H3. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite (Rmult_comm (/ M)). +apply Rlt_trans with (eps / (2 * M)). +apply H8; assumption. +unfold Rdiv in |- *; rewrite Rinv_mult_distr. +apply Rmult_lt_reg_l with 2. +prove_sup0. +replace (2 * (eps * (/ 2 * / M))) with (2 * / 2 * (eps * / M)); + [ idtac | ring ]. +rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite double. +pattern (eps * / M) at 1 in |- *; rewrite <- Rplus_0_r. +apply Rplus_lt_compat_l; apply Rmult_lt_0_compat; + [ assumption | apply Rinv_0_lt_compat; assumption ]. +discrR. +discrR. +red in |- *; intro; rewrite H10 in H3; elim (Rlt_irrefl _ H3). +red in |- *; intro; rewrite H10 in H3; elim (Rlt_irrefl _ H3). +rewrite H7; do 2 rewrite Rmult_0_r; unfold Rminus in |- *; + rewrite Rplus_opp_r; rewrite Rabs_R0; reflexivity. +replace (An n * Bn n - An n * l2) with (An n * (Bn n - l2)); [ idtac | ring ]. +symmetry in |- *; apply Rabs_mult. +cut (0 < eps / (2 * Rabs l2)). +intro. +unfold Un_cv in H; unfold R_dist in H; unfold Un_cv in H0; + unfold R_dist in H0. +elim (H (eps / (2 * Rabs l2)) H8); intros N1 H9. +elim (H0 (eps / (2 * M)) H6); intros N2 H10. +pose (N := max N1 N2). +exists N; intros. +apply Rle_lt_trans with + (Rabs (An n * Bn n - An n * l2) + Rabs (An n * l2 - l1 * l2)). +replace (An n * Bn n - l1 * l2) with + (An n * Bn n - An n * l2 + (An n * l2 - l1 * l2)); + [ apply Rabs_triang | ring ]. +replace (Rabs (An n * Bn n - An n * l2)) with + (Rabs (An n) * Rabs (Bn n - l2)). +replace (Rabs (An n * l2 - l1 * l2)) with (Rabs l2 * Rabs (An n - l1)). +rewrite (double_var eps); apply Rplus_lt_compat. +apply Rle_lt_trans with (M * Rabs (Bn n - l2)). +do 2 rewrite <- (Rmult_comm (Rabs (Bn n - l2))). +apply Rmult_le_compat_l. +apply Rabs_pos. +apply H4. +apply Rmult_lt_reg_l with (/ M). +apply Rinv_0_lt_compat; apply H3. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite (Rmult_comm (/ M)). +apply Rlt_le_trans with (eps / (2 * M)). +apply H10. +unfold ge in |- *; apply le_trans with N. +unfold N in |- *; apply le_max_r. +assumption. +unfold Rdiv in |- *; rewrite Rinv_mult_distr. +right; ring. +discrR. +red in |- *; intro; rewrite H12 in H3; elim (Rlt_irrefl _ H3). +red in |- *; intro; rewrite H12 in H3; elim (Rlt_irrefl _ H3). +apply Rmult_lt_reg_l with (/ Rabs l2). +apply Rinv_0_lt_compat; apply Rabs_pos_lt; assumption. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; apply Rlt_le_trans with (eps / (2 * Rabs l2)). +apply H9. +unfold ge in |- *; apply le_trans with N. +unfold N in |- *; apply le_max_l. +assumption. +unfold Rdiv in |- *; right; rewrite Rinv_mult_distr. +ring. +discrR. +apply Rabs_no_R0; assumption. +apply Rabs_no_R0; assumption. +replace (An n * l2 - l1 * l2) with (l2 * (An n - l1)); + [ symmetry in |- *; apply Rabs_mult | ring ]. +replace (An n * Bn n - An n * l2) with (An n * (Bn n - l2)); + [ symmetry in |- *; apply Rabs_mult | ring ]. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +assumption. +apply Rinv_0_lt_compat; apply Rmult_lt_0_compat; + [ prove_sup0 | apply Rabs_pos_lt; assumption ]. +unfold Rdiv in |- *; apply Rmult_lt_0_compat; + [ assumption + | apply Rinv_0_lt_compat; apply Rmult_lt_0_compat; + [ prove_sup0 | assumption ] ]. +apply existT with l1; assumption. Qed. -Lemma tech9 : (Un:nat->R) (Un_growing Un) -> ((m,n:nat)(le m n)->``(Un m)<=(Un n)``). -Intros; Unfold Un_growing in H. -Induction n. -Induction m. -Right; Reflexivity. -Elim (le_Sn_O ? H0). -Cut (le m n)\/m=(S n). -Intro; Elim H1; Intro. -Apply Rle_trans with (Un n). -Apply Hrecn; Assumption. -Apply H. -Rewrite H2; Right; Reflexivity. -Inversion H0. -Right; Reflexivity. -Left; Assumption. +Lemma tech9 : + forall Un:nat -> R, + Un_growing Un -> forall m n:nat, (m <= n)%nat -> Un m <= Un n. +intros; unfold Un_growing in H. +induction n as [| n Hrecn]. +induction m as [| m Hrecm]. +right; reflexivity. +elim (le_Sn_O _ H0). +cut ((m <= n)%nat \/ m = S n). +intro; elim H1; intro. +apply Rle_trans with (Un n). +apply Hrecn; assumption. +apply H. +rewrite H2; right; reflexivity. +inversion H0. +right; reflexivity. +left; assumption. Qed. -Lemma tech10 : (Un:nat->R;x:R) (Un_growing Un) -> (is_lub (EUn Un) x) -> (Un_cv Un x). -Intros; Cut (bound (EUn Un)). -Intro; Assert H2 := (Un_cv_crit ? H H1). -Elim H2; Intros. -Case (total_order_T x x0); Intro. -Elim s; Intro. -Cut (n:nat)``(Un n)<=x``. -Intro; Unfold Un_cv in H3; Cut ``0<x0-x``. -Intro; Elim (H3 ``x0-x`` H5); Intros. -Cut (ge x1 x1). -Intro; Assert H8 := (H6 x1 H7). -Unfold R_dist in H8; Rewrite Rabsolu_left1 in H8. -Rewrite Ropp_distr2 in H8; Unfold Rminus in H8. -Assert H9 := (Rlt_anti_compatibility ``x0`` ? ? H8). -Assert H10 := (Ropp_Rlt ? ? H9). -Assert H11 := (H4 x1). -Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H10 H11)). -Apply Rle_minus; Apply Rle_trans with x. -Apply H4. -Left; Assumption. -Unfold ge; Apply le_n. -Apply Rgt_minus; Assumption. -Intro; Unfold is_lub in H0; Unfold is_upper_bound in H0; Elim H0; Intros. -Apply H4; Unfold EUn; Exists n; Reflexivity. -Rewrite b; Assumption. -Cut ((n:nat)``(Un n)<=x0``). -Intro; Unfold is_lub in H0; Unfold is_upper_bound in H0; Elim H0; Intros. -Cut (y:R)(EUn Un y)->``y<=x0``. -Intro; Assert H8 := (H6 ? H7). -Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H8 r)). -Unfold EUn; Intros; Elim H7; Intros. -Rewrite H8; Apply H4. -Intro; Case (total_order_Rle (Un n) x0); Intro. -Assumption. -Cut (n0:nat)(le n n0) -> ``x0<(Un n0)``. -Intro; Unfold Un_cv in H3; Cut ``0<(Un n)-x0``. -Intro; Elim (H3 ``(Un n)-x0`` H5); Intros. -Cut (ge (max n x1) x1). -Intro; Assert H8 := (H6 (max n x1) H7). -Unfold R_dist in H8. -Rewrite Rabsolu_right in H8. -Unfold Rminus in H8; Do 2 Rewrite <- (Rplus_sym ``-x0``) in H8. -Assert H9 := (Rlt_anti_compatibility ? ? ? H8). -Cut ``(Un n)<=(Un (max n x1))``. -Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H10 H9)). -Apply tech9; [Assumption | Apply le_max_l]. -Apply Rge_trans with ``(Un n)-x0``. -Unfold Rminus; Apply Rle_sym1; Do 2 Rewrite <- (Rplus_sym ``-x0``); Apply Rle_compatibility. -Apply tech9; [Assumption | Apply le_max_l]. -Left; Assumption. -Unfold ge; Apply le_max_r. -Apply Rlt_anti_compatibility with x0. -Rewrite Rplus_Or; Unfold Rminus; Rewrite (Rplus_sym x0); Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Apply H4; Apply le_n. -Intros; Apply Rlt_le_trans with (Un n). -Case (total_order_Rlt_Rle x0 (Un n)); Intro. -Assumption. -Elim n0; Assumption. -Apply tech9; Assumption. -Unfold bound; Exists x; Unfold is_lub in H0; Elim H0; Intros; Assumption. +Lemma tech10 : + forall (Un:nat -> R) (x:R), Un_growing Un -> is_lub (EUn Un) x -> Un_cv Un x. +intros; cut (bound (EUn Un)). +intro; assert (H2 := Un_cv_crit _ H H1). +elim H2; intros. +case (total_order_T x x0); intro. +elim s; intro. +cut (forall n:nat, Un n <= x). +intro; unfold Un_cv in H3; cut (0 < x0 - x). +intro; elim (H3 (x0 - x) H5); intros. +cut (x1 >= x1)%nat. +intro; assert (H8 := H6 x1 H7). +unfold R_dist in H8; rewrite Rabs_left1 in H8. +rewrite Ropp_minus_distr in H8; unfold Rminus in H8. +assert (H9 := Rplus_lt_reg_r x0 _ _ H8). +assert (H10 := Ropp_lt_cancel _ _ H9). +assert (H11 := H4 x1). +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H10 H11)). +apply Rle_minus; apply Rle_trans with x. +apply H4. +left; assumption. +unfold ge in |- *; apply le_n. +apply Rgt_minus; assumption. +intro; unfold is_lub in H0; unfold is_upper_bound in H0; elim H0; intros. +apply H4; unfold EUn in |- *; exists n; reflexivity. +rewrite b; assumption. +cut (forall n:nat, Un n <= x0). +intro; unfold is_lub in H0; unfold is_upper_bound in H0; elim H0; intros. +cut (forall y:R, EUn Un y -> y <= x0). +intro; assert (H8 := H6 _ H7). +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H8 r)). +unfold EUn in |- *; intros; elim H7; intros. +rewrite H8; apply H4. +intro; case (Rle_dec (Un n) x0); intro. +assumption. +cut (forall n0:nat, (n <= n0)%nat -> x0 < Un n0). +intro; unfold Un_cv in H3; cut (0 < Un n - x0). +intro; elim (H3 (Un n - x0) H5); intros. +cut (max n x1 >= x1)%nat. +intro; assert (H8 := H6 (max n x1) H7). +unfold R_dist in H8. +rewrite Rabs_right in H8. +unfold Rminus in H8; do 2 rewrite <- (Rplus_comm (- x0)) in H8. +assert (H9 := Rplus_lt_reg_r _ _ _ H8). +cut (Un n <= Un (max n x1)). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H10 H9)). +apply tech9; [ assumption | apply le_max_l ]. +apply Rge_trans with (Un n - x0). +unfold Rminus in |- *; apply Rle_ge; do 2 rewrite <- (Rplus_comm (- x0)); + apply Rplus_le_compat_l. +apply tech9; [ assumption | apply le_max_l ]. +left; assumption. +unfold ge in |- *; apply le_max_r. +apply Rplus_lt_reg_r with x0. +rewrite Rplus_0_r; unfold Rminus in |- *; rewrite (Rplus_comm x0); + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r; + apply H4; apply le_n. +intros; apply Rlt_le_trans with (Un n). +case (Rlt_le_dec x0 (Un n)); intro. +assumption. +elim n0; assumption. +apply tech9; assumption. +unfold bound in |- *; exists x; unfold is_lub in H0; elim H0; intros; + assumption. Qed. -Lemma tech13 : (An:nat->R;k:R) ``0<=k<1`` -> (Un_cv [n:nat](Rabsolu ``(An (S n))/(An n)``) k) -> (EXT k0 : R | ``k<k0<1`` /\ (EX N:nat | (n:nat) (le N n)->``(Rabsolu ((An (S n))/(An n)))<k0``)). -Intros; Exists ``k+(1-k)/2``. -Split. -Split. -Pattern 1 k; Rewrite <- Rplus_Or; Apply Rlt_compatibility. -Unfold Rdiv; Apply Rmult_lt_pos. -Apply Rlt_anti_compatibility with k; Rewrite Rplus_Or; Replace ``k+(1-k)`` with R1; [Elim H; Intros; Assumption | Ring]. -Apply Rlt_Rinv; Sup0. -Apply Rlt_monotony_contra with ``2``. -Sup0. -Unfold Rdiv; Rewrite Rmult_1r; Rewrite Rmult_Rplus_distr; Pattern 1 ``2``; Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym; [Idtac | DiscrR]; Rewrite Rmult_1r; Replace ``2*k+(1-k)`` with ``1+k``; [Idtac | Ring]. -Elim H; Intros. -Apply Rlt_compatibility; Assumption. -Unfold Un_cv in H0; Cut ``0<(1-k)/2``. -Intro; Elim (H0 ``(1-k)/2`` H1); Intros. -Exists x; Intros. -Assert H4 := (H2 n H3). -Unfold R_dist in H4; Rewrite <- Rabsolu_Rabsolu; Replace ``(Rabsolu ((An (S n))/(An n)))`` with ``((Rabsolu ((An (S n))/(An n)))-k)+k``; [Idtac | Ring]; Apply Rle_lt_trans with ``(Rabsolu ((Rabsolu ((An (S n))/(An n)))-k))+(Rabsolu k)``. -Apply Rabsolu_triang. -Rewrite (Rabsolu_right k). -Apply Rlt_anti_compatibility with ``-k``; Rewrite <- (Rplus_sym k); Repeat Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Repeat Rewrite Rplus_Ol; Apply H4. -Apply Rle_sym1; Elim H; Intros; Assumption. -Unfold Rdiv; Apply Rmult_lt_pos. -Apply Rlt_anti_compatibility with k; Rewrite Rplus_Or; Elim H; Intros; Replace ``k+(1-k)`` with R1; [Assumption | Ring]. -Apply Rlt_Rinv; Sup0. +Lemma tech13 : + forall (An:nat -> R) (k:R), + 0 <= k < 1 -> + Un_cv (fun n:nat => Rabs (An (S n) / An n)) k -> + exists k0 : R + | k < k0 < 1 /\ + ( exists N : nat + | (forall n:nat, (N <= n)%nat -> Rabs (An (S n) / An n) < k0)). +intros; exists (k + (1 - k) / 2). +split. +split. +pattern k at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply Rplus_lt_reg_r with k; rewrite Rplus_0_r; replace (k + (1 - k)) with 1; + [ elim H; intros; assumption | ring ]. +apply Rinv_0_lt_compat; prove_sup0. +apply Rmult_lt_reg_l with 2. +prove_sup0. +unfold Rdiv in |- *; rewrite Rmult_1_r; rewrite Rmult_plus_distr_l; + pattern 2 at 1 in |- *; rewrite Rmult_comm; rewrite Rmult_assoc; + rewrite <- Rinv_l_sym; [ idtac | discrR ]; rewrite Rmult_1_r; + replace (2 * k + (1 - k)) with (1 + k); [ idtac | ring ]. +elim H; intros. +apply Rplus_lt_compat_l; assumption. +unfold Un_cv in H0; cut (0 < (1 - k) / 2). +intro; elim (H0 ((1 - k) / 2) H1); intros. +exists x; intros. +assert (H4 := H2 n H3). +unfold R_dist in H4; rewrite <- Rabs_Rabsolu; + replace (Rabs (An (S n) / An n)) with (Rabs (An (S n) / An n) - k + k); + [ idtac | ring ]; + apply Rle_lt_trans with (Rabs (Rabs (An (S n) / An n) - k) + Rabs k). +apply Rabs_triang. +rewrite (Rabs_right k). +apply Rplus_lt_reg_r with (- k); rewrite <- (Rplus_comm k); + repeat rewrite <- Rplus_assoc; rewrite Rplus_opp_l; + repeat rewrite Rplus_0_l; apply H4. +apply Rle_ge; elim H; intros; assumption. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply Rplus_lt_reg_r with k; rewrite Rplus_0_r; elim H; intros; + replace (k + (1 - k)) with 1; [ assumption | ring ]. +apply Rinv_0_lt_compat; prove_sup0. Qed. (**********) -Lemma growing_ineq : (Un:nat->R;l:R) (Un_growing Un) -> (Un_cv Un l) -> ((n:nat)``(Un n)<=l``). -Intros; Case (total_order_T (Un n) l); Intro. -Elim s; Intro. -Left; Assumption. -Right; Assumption. -Cut ``0<(Un n)-l``. -Intro; Unfold Un_cv in H0; Unfold R_dist in H0. -Elim (H0 ``(Un n)-l`` H1); Intros N1 H2. -Pose N := (max n N1). -Cut ``(Un n)-l<=(Un N)-l``. -Intro; Cut ``(Un N)-l<(Un n)-l``. -Intro; Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H3 H4)). -Apply Rle_lt_trans with ``(Rabsolu ((Un N)-l))``. -Apply Rle_Rabsolu. -Apply H2. -Unfold ge N; Apply le_max_r. -Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-l``); Apply Rle_compatibility. -Apply tech9. -Assumption. -Unfold N; Apply le_max_l. -Apply Rlt_anti_compatibility with l. -Rewrite Rplus_Or. -Replace ``l+((Un n)-l)`` with (Un n); [Assumption | Ring]. +Lemma growing_ineq : + forall (Un:nat -> R) (l:R), + Un_growing Un -> Un_cv Un l -> forall n:nat, Un n <= l. +intros; case (total_order_T (Un n) l); intro. +elim s; intro. +left; assumption. +right; assumption. +cut (0 < Un n - l). +intro; unfold Un_cv in H0; unfold R_dist in H0. +elim (H0 (Un n - l) H1); intros N1 H2. +pose (N := max n N1). +cut (Un n - l <= Un N - l). +intro; cut (Un N - l < Un n - l). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H3 H4)). +apply Rle_lt_trans with (Rabs (Un N - l)). +apply RRle_abs. +apply H2. +unfold ge, N in |- *; apply le_max_r. +unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- l)); + apply Rplus_le_compat_l. +apply tech9. +assumption. +unfold N in |- *; apply le_max_l. +apply Rplus_lt_reg_r with l. +rewrite Rplus_0_r. +replace (l + (Un n - l)) with (Un n); [ assumption | ring ]. Qed. (* Un->l => (-Un) -> (-l) *) -Lemma CV_opp : (An:nat->R;l:R) (Un_cv An l) -> (Un_cv (opp_seq An) ``-l``). -Intros An l. -Unfold Un_cv; Unfold R_dist; Intros. -Elim (H eps H0); Intros. -Exists x; Intros. -Unfold opp_seq; Replace ``-(An n)- (-l)`` with ``-((An n)-l)``; [Rewrite Rabsolu_Ropp | Ring]. -Apply H1; Assumption. +Lemma CV_opp : + forall (An:nat -> R) (l:R), Un_cv An l -> Un_cv (opp_seq An) (- l). +intros An l. +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H eps H0); intros. +exists x; intros. +unfold opp_seq in |- *; replace (- An n - - l) with (- (An n - l)); + [ rewrite Rabs_Ropp | ring ]. +apply H1; assumption. Qed. (**********) -Lemma decreasing_ineq : (Un:nat->R;l:R) (Un_decreasing Un) -> (Un_cv Un l) -> ((n:nat)``l<=(Un n)``). -Intros. -Assert H1 := (decreasing_growing ? H). -Assert H2 := (CV_opp ? ? H0). -Assert H3 := (growing_ineq ? ? H1 H2). -Apply Ropp_Rle. -Unfold opp_seq in H3; Apply H3. +Lemma decreasing_ineq : + forall (Un:nat -> R) (l:R), + Un_decreasing Un -> Un_cv Un l -> forall n:nat, l <= Un n. +intros. +assert (H1 := decreasing_growing _ H). +assert (H2 := CV_opp _ _ H0). +assert (H3 := growing_ineq _ _ H1 H2). +apply Ropp_le_cancel. +unfold opp_seq in H3; apply H3. Qed. (**********) -Lemma CV_minus : (An,Bn:nat->R;l1,l2:R) (Un_cv An l1) -> (Un_cv Bn l2) -> (Un_cv [i:nat]``(An i)-(Bn i)`` ``l1-l2``). -Intros. -Replace [i:nat]``(An i)-(Bn i)`` with [i:nat]``(An i)+((opp_seq Bn) i)``. -Unfold Rminus; Apply CV_plus. -Assumption. -Apply CV_opp; Assumption. -Unfold Rminus opp_seq; Reflexivity. +Lemma CV_minus : + forall (An Bn:nat -> R) (l1 l2:R), + Un_cv An l1 -> Un_cv Bn l2 -> Un_cv (fun i:nat => An i - Bn i) (l1 - l2). +intros. +replace (fun i:nat => An i - Bn i) with (fun i:nat => An i + opp_seq Bn i). +unfold Rminus in |- *; apply CV_plus. +assumption. +apply CV_opp; assumption. +unfold Rminus, opp_seq in |- *; reflexivity. Qed. (* Un -> +oo *) -Definition cv_infty [Un:nat->R] : Prop := (M:R)(EXT N:nat | (n:nat) (le N n) -> ``M<(Un n)``). +Definition cv_infty (Un:nat -> R) : Prop := + forall M:R, exists N : nat | (forall n:nat, (N <= n)%nat -> M < Un n). (* Un -> +oo => /Un -> O *) -Lemma cv_infty_cv_R0 : (Un:nat->R) ((n:nat)``(Un n)<>0``) -> (cv_infty Un) -> (Un_cv [n:nat]``/(Un n)`` R0). -Unfold cv_infty Un_cv; Unfold R_dist; Intros. -Elim (H0 ``/eps``); Intros N0 H2. -Exists N0; Intros. -Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite (Rabsolu_Rinv ? (H n)). -Apply Rlt_monotony_contra with (Rabsolu (Un n)). -Apply Rabsolu_pos_lt; Apply H. -Rewrite <- Rinv_r_sym. -Apply Rlt_monotony_contra with ``/eps``. -Apply Rlt_Rinv; Assumption. -Rewrite Rmult_1r; Rewrite (Rmult_sym ``/eps``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Apply Rlt_le_trans with (Un n). -Apply H2; Assumption. -Apply Rle_Rabsolu. -Red; Intro; Rewrite H4 in H1; Elim (Rlt_antirefl ? H1). -Apply Rabsolu_no_R0; Apply H. +Lemma cv_infty_cv_R0 : + forall Un:nat -> R, + (forall n:nat, Un n <> 0) -> cv_infty Un -> Un_cv (fun n:nat => / Un n) 0. +unfold cv_infty, Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H0 (/ eps)); intros N0 H2. +exists N0; intros. +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + rewrite (Rabs_Rinv _ (H n)). +apply Rmult_lt_reg_l with (Rabs (Un n)). +apply Rabs_pos_lt; apply H. +rewrite <- Rinv_r_sym. +apply Rmult_lt_reg_l with (/ eps). +apply Rinv_0_lt_compat; assumption. +rewrite Rmult_1_r; rewrite (Rmult_comm (/ eps)); rewrite Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; apply Rlt_le_trans with (Un n). +apply H2; assumption. +apply RRle_abs. +red in |- *; intro; rewrite H4 in H1; elim (Rlt_irrefl _ H1). +apply Rabs_no_R0; apply H. Qed. (**********) -Lemma decreasing_prop : (Un:nat->R;m,n:nat) (Un_decreasing Un) -> (le m n) -> ``(Un n)<=(Un m)``. -Unfold Un_decreasing; Intros. -Induction n. -Induction m. -Right; Reflexivity. -Elim (le_Sn_O ? H0). -Cut (le m n)\/m=(S n). -Intro; Elim H1; Intro. -Apply Rle_trans with (Un n). -Apply H. -Apply Hrecn; Assumption. -Rewrite H2; Right; Reflexivity. -Inversion H0; [Right; Reflexivity | Left; Assumption]. +Lemma decreasing_prop : + forall (Un:nat -> R) (m n:nat), + Un_decreasing Un -> (m <= n)%nat -> Un n <= Un m. +unfold Un_decreasing in |- *; intros. +induction n as [| n Hrecn]. +induction m as [| m Hrecm]. +right; reflexivity. +elim (le_Sn_O _ H0). +cut ((m <= n)%nat \/ m = S n). +intro; elim H1; intro. +apply Rle_trans with (Un n). +apply H. +apply Hrecn; assumption. +rewrite H2; right; reflexivity. +inversion H0; [ right; reflexivity | left; assumption ]. Qed. (* |x|^n/n! -> 0 *) -Lemma cv_speed_pow_fact : (x:R) (Un_cv [n:nat]``(pow x n)/(INR (fact n))`` R0). -Intro; Cut (Un_cv [n:nat]``(pow (Rabsolu x) n)/(INR (fact n))`` R0) -> (Un_cv [n:nat]``(pow x n)/(INR (fact n))`` ``0``). -Intro; Apply H. -Unfold Un_cv; Unfold R_dist; Intros; Case (Req_EM x R0); Intro. -Exists (S O); Intros. -Rewrite H1; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite Rabsolu_R0; Rewrite pow_ne_zero; [Unfold Rdiv; Rewrite Rmult_Ol; Rewrite Rabsolu_R0; Assumption | Red; Intro; Rewrite H3 in H2; Elim (le_Sn_n ? H2)]. -Assert H2 := (Rabsolu_pos_lt x H1); Pose M := (up (Rabsolu x)); Cut `0<=M`. -Intro; Elim (IZN M H3); Intros M_nat H4. -Pose Un := [n:nat]``(pow (Rabsolu x) (plus M_nat n))/(INR (fact (plus M_nat n)))``. -Cut (Un_cv Un R0); Unfold Un_cv; Unfold R_dist; Intros. -Elim (H5 eps H0); Intros N H6. -Exists (plus M_nat N); Intros; Cut (EX p:nat | (ge p N)/\n=(plus M_nat p)). -Intro; Elim H8; Intros p H9. -Elim H9; Intros; Rewrite H11; Unfold Un in H6; Apply H6; Assumption. -Exists (minus n M_nat). -Split. -Unfold ge; Apply simpl_le_plus_l with M_nat; Rewrite <- le_plus_minus. -Assumption. -Apply le_trans with (plus M_nat N). -Apply le_plus_l. -Assumption. -Apply le_plus_minus; Apply le_trans with (plus M_nat N); [Apply le_plus_l | Assumption]. -Pose Vn := [n:nat]``(Rabsolu x)*(Un O)/(INR (S n))``. -Cut (le (1) M_nat). -Intro; Cut (n:nat)``0<(Un n)``. -Intro; Cut (Un_decreasing Un). -Intro; Cut (n:nat)``(Un (S n))<=(Vn n)``. -Intro; Cut (Un_cv Vn R0). -Unfold Un_cv; Unfold R_dist; Intros. -Elim (H10 eps0 H5); Intros N1 H11. -Exists (S N1); Intros. -Cut (n:nat)``0<(Vn n)``. -Intro; Apply Rle_lt_trans with ``(Rabsolu ((Vn (pred n))-0))``. -Repeat Rewrite Rabsolu_right. -Unfold Rminus; Rewrite Ropp_O; Do 2 Rewrite Rplus_Or; Replace n with (S (pred n)). -Apply H9. -Inversion H12; Simpl; Reflexivity. -Apply Rle_sym1; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Left; Apply H13. -Apply Rle_sym1; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Left; Apply H7. -Apply H11; Unfold ge; Apply le_S_n; Replace (S (pred n)) with n; [Unfold ge in H12; Exact H12 | Inversion H12; Simpl; Reflexivity]. -Intro; Apply Rlt_le_trans with (Un (S n0)); [Apply H7 | Apply H9]. -Cut (cv_infty [n:nat](INR (S n))). -Intro; Cut (Un_cv [n:nat]``/(INR (S n))`` R0). -Unfold Un_cv R_dist; Intros; Unfold Vn. -Cut ``0<eps1/((Rabsolu x)*(Un O))``. -Intro; Elim (H11 ? H13); Intros N H14. -Exists N; Intros; Replace ``(Rabsolu x)*(Un O)/(INR (S n))-0`` with ``((Rabsolu x)*(Un O))*(/(INR (S n))-0)``; [Idtac | Unfold Rdiv; Ring]. -Rewrite Rabsolu_mult; Apply Rlt_monotony_contra with ``/(Rabsolu ((Rabsolu x)*(Un O)))``. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt. -Apply prod_neq_R0. -Apply Rabsolu_no_R0; Assumption. -Assert H16 := (H7 O); Red; Intro; Rewrite H17 in H16; Elim (Rlt_antirefl ? H16). -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Replace ``/(Rabsolu ((Rabsolu x)*(Un O)))*eps1`` with ``eps1/((Rabsolu x)*(Un O))``. -Apply H14; Assumption. -Unfold Rdiv; Rewrite (Rabsolu_right ``(Rabsolu x)*(Un O)``). -Apply Rmult_sym. -Apply Rle_sym1; Apply Rmult_le_pos. -Apply Rabsolu_pos. -Left; Apply H7. -Apply Rabsolu_no_R0. -Apply prod_neq_R0; [Apply Rabsolu_no_R0; Assumption | Assert H16 := (H7 O); Red; Intro; Rewrite H17 in H16; Elim (Rlt_antirefl ? H16)]. -Unfold Rdiv; Apply Rmult_lt_pos. -Assumption. -Apply Rlt_Rinv; Apply Rmult_lt_pos. -Apply Rabsolu_pos_lt; Assumption. -Apply H7. -Apply (cv_infty_cv_R0 [n:nat]``(INR (S n))``). -Intro; Apply not_O_INR; Discriminate. -Assumption. -Unfold cv_infty; Intro; Case (total_order_T M0 R0); Intro. -Elim s; Intro. -Exists O; Intros. -Apply Rlt_trans with R0; [Assumption | Apply lt_INR_0; Apply lt_O_Sn]. -Exists O; Intros; Rewrite b; Apply lt_INR_0; Apply lt_O_Sn. -Pose M0_z := (up M0). -Assert H10 := (archimed M0). -Cut `0<=M0_z`. -Intro; Elim (IZN ? H11); Intros M0_nat H12. -Exists M0_nat; Intros. -Apply Rlt_le_trans with (IZR M0_z). -Elim H10; Intros; Assumption. -Rewrite H12; Rewrite <- INR_IZR_INZ; Apply le_INR. -Apply le_trans with n; [Assumption | Apply le_n_Sn]. -Apply le_IZR; Left; Simpl; Unfold M0_z; Apply Rlt_trans with M0; [Assumption | Elim H10; Intros; Assumption]. -Intro; Apply Rle_trans with ``(Rabsolu x)*(Un n)*/(INR (S n))``. -Unfold Un; Replace (plus M_nat (S n)) with (plus (plus M_nat n) (1)). -Rewrite pow_add; Replace (pow (Rabsolu x) (S O)) with (Rabsolu x); [Idtac | Simpl; Ring]. -Unfold Rdiv; Rewrite <- (Rmult_sym (Rabsolu x)); Repeat Rewrite Rmult_assoc; Repeat Apply Rle_monotony. -Apply Rabsolu_pos. -Left; Apply pow_lt; Assumption. -Replace (plus (plus M_nat n) (S O)) with (S (plus M_nat n)). -Rewrite fact_simpl; Rewrite mult_sym; Rewrite mult_INR; Rewrite Rinv_Rmult. -Apply Rle_monotony. -Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H10 := (sym_eq ? ? ? H9); Elim (fact_neq_0 ? H10). -Left; Apply Rinv_lt. -Apply Rmult_lt_pos; Apply lt_INR_0; Apply lt_O_Sn. -Apply lt_INR; Apply lt_n_S. -Pattern 1 n; Replace n with (plus O n); [Idtac | Reflexivity]. -Apply lt_reg_r. -Apply lt_le_trans with (S O); [Apply lt_O_Sn | Assumption]. -Apply INR_fact_neq_0. -Apply not_O_INR; Discriminate. -Apply INR_eq; Rewrite S_INR; Do 3 Rewrite plus_INR; Reflexivity. -Apply INR_eq; Do 3 Rewrite plus_INR; Do 2 Rewrite S_INR; Ring. -Unfold Vn; Rewrite Rmult_assoc; Unfold Rdiv; Rewrite (Rmult_sym (Un O)); Rewrite (Rmult_sym (Un n)). -Repeat Apply Rle_monotony. -Apply Rabsolu_pos. -Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply lt_O_Sn. -Apply decreasing_prop; [Assumption | Apply le_O_n]. -Unfold Un_decreasing; Intro; Unfold Un. -Replace (plus M_nat (S n)) with (plus (plus M_nat n) (1)). -Rewrite pow_add; Unfold Rdiv; Rewrite Rmult_assoc; Apply Rle_monotony. -Left; Apply pow_lt; Assumption. -Replace (pow (Rabsolu x) (S O)) with (Rabsolu x); [Idtac | Simpl; Ring]. -Replace (plus (plus M_nat n) (S O)) with (S (plus M_nat n)). -Apply Rle_monotony_contra with (INR (fact (S (plus M_nat n)))). -Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H9 := (sym_eq ? ? ? H8); Elim (fact_neq_0 ? H9). -Rewrite (Rmult_sym (Rabsolu x)); Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l. -Rewrite fact_simpl; Rewrite mult_INR; Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Apply Rle_trans with (INR M_nat). -Left; Rewrite INR_IZR_INZ. -Rewrite <- H4; Assert H8 := (archimed (Rabsolu x)); Elim H8; Intros; Assumption. -Apply le_INR; Apply le_trans with (S M_nat); [Apply le_n_Sn | Apply le_n_S; Apply le_plus_l]. -Apply INR_fact_neq_0. -Apply INR_fact_neq_0. -Apply INR_eq; Rewrite S_INR; Do 3 Rewrite plus_INR; Reflexivity. -Apply INR_eq; Do 3 Rewrite plus_INR; Do 2 Rewrite S_INR; Ring. -Intro; Unfold Un; Unfold Rdiv; Apply Rmult_lt_pos. -Apply pow_lt; Assumption. -Apply Rlt_Rinv; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H8 := (sym_eq ? ? ? H7); Elim (fact_neq_0 ? H8). -Clear Un Vn; Apply INR_le; Simpl. -Induction M_nat. -Assert H6 := (archimed (Rabsolu x)); Fold M in H6; Elim H6; Intros. -Rewrite H4 in H7; Rewrite <- INR_IZR_INZ in H7. -Simpl in H7; Elim (Rlt_antirefl ? (Rlt_trans ? ? ? H2 H7)). -Replace R1 with (INR (S O)); [Apply le_INR | Reflexivity]; Apply le_n_S; Apply le_O_n. -Apply le_IZR; Simpl; Left; Apply Rlt_trans with (Rabsolu x). -Assumption. -Elim (archimed (Rabsolu x)); Intros; Assumption. -Unfold Un_cv; Unfold R_dist; Intros; Elim (H eps H0); Intros. -Exists x0; Intros; Apply Rle_lt_trans with ``(Rabsolu ((pow (Rabsolu x) n)/(INR (fact n))-0))``. -Unfold Rminus; Rewrite Ropp_O; Do 2 Rewrite Rplus_Or; Rewrite (Rabsolu_right ``(pow (Rabsolu x) n)/(INR (fact n))``). -Unfold Rdiv; Rewrite Rabsolu_mult; Rewrite (Rabsolu_right ``/(INR (fact n))``). -Rewrite Pow_Rabsolu; Right; Reflexivity. -Apply Rle_sym1; Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H4 := (sym_eq ? ? ? H3); Elim (fact_neq_0 ? H4). -Apply Rle_sym1; Unfold Rdiv; Apply Rmult_le_pos. -Case (Req_EM x R0); Intro. -Rewrite H3; Rewrite Rabsolu_R0. -Induction n; [Simpl; Left; Apply Rlt_R0_R1 | Simpl; Rewrite Rmult_Ol; Right; Reflexivity]. -Left; Apply pow_lt; Apply Rabsolu_pos_lt; Assumption. -Left; Apply Rlt_Rinv; Apply lt_INR_0; Apply neq_O_lt; Red; Intro; Assert H4 := (sym_eq ? ? ? H3); Elim (fact_neq_0 ? H4). -Apply H1; Assumption. -Qed. +Lemma cv_speed_pow_fact : + forall x:R, Un_cv (fun n:nat => x ^ n / INR (fact n)) 0. +intro; + cut + (Un_cv (fun n:nat => Rabs x ^ n / INR (fact n)) 0 -> + Un_cv (fun n:nat => x ^ n / INR (fact n)) 0). +intro; apply H. +unfold Un_cv in |- *; unfold R_dist in |- *; intros; case (Req_dec x 0); + intro. +exists 1%nat; intros. +rewrite H1; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + rewrite Rabs_R0; rewrite pow_ne_zero; + [ unfold Rdiv in |- *; rewrite Rmult_0_l; rewrite Rabs_R0; assumption + | red in |- *; intro; rewrite H3 in H2; elim (le_Sn_n _ H2) ]. +assert (H2 := Rabs_pos_lt x H1); pose (M := up (Rabs x)); cut (0 <= M)%Z. +intro; elim (IZN M H3); intros M_nat H4. +pose (Un := fun n:nat => Rabs x ^ (M_nat + n) / INR (fact (M_nat + n))). +cut (Un_cv Un 0); unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H5 eps H0); intros N H6. +exists (M_nat + N)%nat; intros; + cut ( exists p : nat | (p >= N)%nat /\ n = (M_nat + p)%nat). +intro; elim H8; intros p H9. +elim H9; intros; rewrite H11; unfold Un in H6; apply H6; assumption. +exists (n - M_nat)%nat. +split. +unfold ge in |- *; apply (fun p n m:nat => plus_le_reg_l n m p) with M_nat; + rewrite <- le_plus_minus. +assumption. +apply le_trans with (M_nat + N)%nat. +apply le_plus_l. +assumption. +apply le_plus_minus; apply le_trans with (M_nat + N)%nat; + [ apply le_plus_l | assumption ]. +pose (Vn := fun n:nat => Rabs x * (Un 0%nat / INR (S n))). +cut (1 <= M_nat)%nat. +intro; cut (forall n:nat, 0 < Un n). +intro; cut (Un_decreasing Un). +intro; cut (forall n:nat, Un (S n) <= Vn n). +intro; cut (Un_cv Vn 0). +unfold Un_cv in |- *; unfold R_dist in |- *; intros. +elim (H10 eps0 H5); intros N1 H11. +exists (S N1); intros. +cut (forall n:nat, 0 < Vn n). +intro; apply Rle_lt_trans with (Rabs (Vn (pred n) - 0)). +repeat rewrite Rabs_right. +unfold Rminus in |- *; rewrite Ropp_0; do 2 rewrite Rplus_0_r; + replace n with (S (pred n)). +apply H9. +inversion H12; simpl in |- *; reflexivity. +apply Rle_ge; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; left; + apply H13. +apply Rle_ge; unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; left; + apply H7. +apply H11; unfold ge in |- *; apply le_S_n; replace (S (pred n)) with n; + [ unfold ge in H12; exact H12 | inversion H12; simpl in |- *; reflexivity ]. +intro; apply Rlt_le_trans with (Un (S n0)); [ apply H7 | apply H9 ]. +cut (cv_infty (fun n:nat => INR (S n))). +intro; cut (Un_cv (fun n:nat => / INR (S n)) 0). +unfold Un_cv, R_dist in |- *; intros; unfold Vn in |- *. +cut (0 < eps1 / (Rabs x * Un 0%nat)). +intro; elim (H11 _ H13); intros N H14. +exists N; intros; + replace (Rabs x * (Un 0%nat / INR (S n)) - 0) with + (Rabs x * Un 0%nat * (/ INR (S n) - 0)); + [ idtac | unfold Rdiv in |- *; ring ]. +rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs (Rabs x * Un 0%nat)). +apply Rinv_0_lt_compat; apply Rabs_pos_lt. +apply prod_neq_R0. +apply Rabs_no_R0; assumption. +assert (H16 := H7 0%nat); red in |- *; intro; rewrite H17 in H16; + elim (Rlt_irrefl _ H16). +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l. +replace (/ Rabs (Rabs x * Un 0%nat) * eps1) with (eps1 / (Rabs x * Un 0%nat)). +apply H14; assumption. +unfold Rdiv in |- *; rewrite (Rabs_right (Rabs x * Un 0%nat)). +apply Rmult_comm. +apply Rle_ge; apply Rmult_le_pos. +apply Rabs_pos. +left; apply H7. +apply Rabs_no_R0. +apply prod_neq_R0; + [ apply Rabs_no_R0; assumption + | assert (H16 := H7 0%nat); red in |- *; intro; rewrite H17 in H16; + elim (Rlt_irrefl _ H16) ]. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +assumption. +apply Rinv_0_lt_compat; apply Rmult_lt_0_compat. +apply Rabs_pos_lt; assumption. +apply H7. +apply (cv_infty_cv_R0 (fun n:nat => INR (S n))). +intro; apply not_O_INR; discriminate. +assumption. +unfold cv_infty in |- *; intro; case (total_order_T M0 0); intro. +elim s; intro. +exists 0%nat; intros. +apply Rlt_trans with 0; [ assumption | apply lt_INR_0; apply lt_O_Sn ]. +exists 0%nat; intros; rewrite b; apply lt_INR_0; apply lt_O_Sn. +pose (M0_z := up M0). +assert (H10 := archimed M0). +cut (0 <= M0_z)%Z. +intro; elim (IZN _ H11); intros M0_nat H12. +exists M0_nat; intros. +apply Rlt_le_trans with (IZR M0_z). +elim H10; intros; assumption. +rewrite H12; rewrite <- INR_IZR_INZ; apply le_INR. +apply le_trans with n; [ assumption | apply le_n_Sn ]. +apply le_IZR; left; simpl in |- *; unfold M0_z in |- *; + apply Rlt_trans with M0; [ assumption | elim H10; intros; assumption ]. +intro; apply Rle_trans with (Rabs x * Un n * / INR (S n)). +unfold Un in |- *; replace (M_nat + S n)%nat with (M_nat + n + 1)%nat. +rewrite pow_add; replace (Rabs x ^ 1) with (Rabs x); + [ idtac | simpl in |- *; ring ]. +unfold Rdiv in |- *; rewrite <- (Rmult_comm (Rabs x)); + repeat rewrite Rmult_assoc; repeat apply Rmult_le_compat_l. +apply Rabs_pos. +left; apply pow_lt; assumption. +replace (M_nat + n + 1)%nat with (S (M_nat + n)). +rewrite fact_simpl; rewrite mult_comm; rewrite mult_INR; + rewrite Rinv_mult_distr. +apply Rmult_le_compat_l. +left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red in |- *; + intro; assert (H10 := sym_eq H9); elim (fact_neq_0 _ H10). +left; apply Rinv_lt_contravar. +apply Rmult_lt_0_compat; apply lt_INR_0; apply lt_O_Sn. +apply lt_INR; apply lt_n_S. +pattern n at 1 in |- *; replace n with (0 + n)%nat; [ idtac | reflexivity ]. +apply plus_lt_compat_r. +apply lt_le_trans with 1%nat; [ apply lt_O_Sn | assumption ]. +apply INR_fact_neq_0. +apply not_O_INR; discriminate. +apply INR_eq; rewrite S_INR; do 3 rewrite plus_INR; reflexivity. +apply INR_eq; do 3 rewrite plus_INR; do 2 rewrite S_INR; ring. +unfold Vn in |- *; rewrite Rmult_assoc; unfold Rdiv in |- *; + rewrite (Rmult_comm (Un 0%nat)); rewrite (Rmult_comm (Un n)). +repeat apply Rmult_le_compat_l. +apply Rabs_pos. +left; apply Rinv_0_lt_compat; apply lt_INR_0; apply lt_O_Sn. +apply decreasing_prop; [ assumption | apply le_O_n ]. +unfold Un_decreasing in |- *; intro; unfold Un in |- *. +replace (M_nat + S n)%nat with (M_nat + n + 1)%nat. +rewrite pow_add; unfold Rdiv in |- *; rewrite Rmult_assoc; + apply Rmult_le_compat_l. +left; apply pow_lt; assumption. +replace (Rabs x ^ 1) with (Rabs x); [ idtac | simpl in |- *; ring ]. +replace (M_nat + n + 1)%nat with (S (M_nat + n)). +apply Rmult_le_reg_l with (INR (fact (S (M_nat + n)))). +apply lt_INR_0; apply neq_O_lt; red in |- *; intro; assert (H9 := sym_eq H8); + elim (fact_neq_0 _ H9). +rewrite (Rmult_comm (Rabs x)); rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_l. +rewrite fact_simpl; rewrite mult_INR; rewrite Rmult_assoc; + rewrite <- Rinv_r_sym. +rewrite Rmult_1_r; apply Rle_trans with (INR M_nat). +left; rewrite INR_IZR_INZ. +rewrite <- H4; assert (H8 := archimed (Rabs x)); elim H8; intros; assumption. +apply le_INR; apply le_trans with (S M_nat); + [ apply le_n_Sn | apply le_n_S; apply le_plus_l ]. +apply INR_fact_neq_0. +apply INR_fact_neq_0. +apply INR_eq; rewrite S_INR; do 3 rewrite plus_INR; reflexivity. +apply INR_eq; do 3 rewrite plus_INR; do 2 rewrite S_INR; ring. +intro; unfold Un in |- *; unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply pow_lt; assumption. +apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red in |- *; intro; + assert (H8 := sym_eq H7); elim (fact_neq_0 _ H8). +clear Un Vn; apply INR_le; simpl in |- *. +induction M_nat as [| M_nat HrecM_nat]. +assert (H6 := archimed (Rabs x)); fold M in H6; elim H6; intros. +rewrite H4 in H7; rewrite <- INR_IZR_INZ in H7. +simpl in H7; elim (Rlt_irrefl _ (Rlt_trans _ _ _ H2 H7)). +replace 1 with (INR 1); [ apply le_INR | reflexivity ]; apply le_n_S; + apply le_O_n. +apply le_IZR; simpl in |- *; left; apply Rlt_trans with (Rabs x). +assumption. +elim (archimed (Rabs x)); intros; assumption. +unfold Un_cv in |- *; unfold R_dist in |- *; intros; elim (H eps H0); intros. +exists x0; intros; + apply Rle_lt_trans with (Rabs (Rabs x ^ n / INR (fact n) - 0)). +unfold Rminus in |- *; rewrite Ropp_0; do 2 rewrite Rplus_0_r; + rewrite (Rabs_right (Rabs x ^ n / INR (fact n))). +unfold Rdiv in |- *; rewrite Rabs_mult; rewrite (Rabs_right (/ INR (fact n))). +rewrite RPow_abs; right; reflexivity. +apply Rle_ge; left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; + red in |- *; intro; assert (H4 := sym_eq H3); elim (fact_neq_0 _ H4). +apply Rle_ge; unfold Rdiv in |- *; apply Rmult_le_pos. +case (Req_dec x 0); intro. +rewrite H3; rewrite Rabs_R0. +induction n as [| n Hrecn]; + [ simpl in |- *; left; apply Rlt_0_1 + | simpl in |- *; rewrite Rmult_0_l; right; reflexivity ]. +left; apply pow_lt; apply Rabs_pos_lt; assumption. +left; apply Rinv_0_lt_compat; apply lt_INR_0; apply neq_O_lt; red in |- *; + intro; assert (H4 := sym_eq H3); elim (fact_neq_0 _ H4). +apply H1; assumption. +Qed.
\ No newline at end of file diff --git a/theories/Reals/SeqSeries.v b/theories/Reals/SeqSeries.v index 03963fc4d..ffac3df29 100644 --- a/theories/Reals/SeqSeries.v +++ b/theories/Reals/SeqSeries.v @@ -8,9 +8,9 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Max. +Require Import Rbase. +Require Import Rfunctions. +Require Import Max. Require Export Rseries. Require Export SeqProp. Require Export Rcomplete. @@ -21,287 +21,397 @@ Require Export Rsigma. Require Export Rprod. Require Export Cauchy_prod. Require Export Alembert. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. Open Local Scope R_scope. (**********) -Lemma sum_maj1 : (fn:nat->R->R;An:nat->R;x,l1,l2:R;N:nat) (Un_cv [n:nat](SP fn n x) l1) -> (Un_cv [n:nat](sum_f_R0 An n) l2) -> ((n:nat)``(Rabsolu (fn n x))<=(An n)``) -> ``(Rabsolu (l1-(SP fn N x)))<=l2-(sum_f_R0 An N)``. -Intros; Cut (sigTT R [l:R](Un_cv [n:nat](sum_f_R0 [l:nat](fn (plus (S N) l) x) n) l)). -Intro; Cut (sigTT R [l:R](Un_cv [n:nat](sum_f_R0 [l:nat](An (plus (S N) l)) n) l)). -Intro; Elim X; Intros l1N H2. -Elim X0; Intros l2N H3. -Cut ``l1-(SP fn N x)==l1N``. -Intro; Cut ``l2-(sum_f_R0 An N)==l2N``. -Intro; Rewrite H4; Rewrite H5. -Apply sum_cv_maj with [l:nat](An (plus (S N) l)) [l:nat][x:R](fn (plus (S N) l) x) x. -Unfold SP; Apply H2. -Apply H3. -Intros; Apply H1. -Symmetry; EApply UL_sequence. -Apply H3. -Unfold Un_cv in H0; Unfold Un_cv; Intros; Elim (H0 eps H5); Intros N0 H6. -Unfold R_dist in H6; Exists N0; Intros. -Unfold R_dist; Replace (Rminus (sum_f_R0 [l:nat](An (plus (S N) l)) n) (Rminus l2 (sum_f_R0 An N))) with (Rminus (Rplus (sum_f_R0 An N) (sum_f_R0 [l:nat](An (plus (S N) l)) n)) l2); [Idtac | Ring]. -Replace (Rplus (sum_f_R0 An N) (sum_f_R0 [l:nat](An (plus (S N) l)) n)) with (sum_f_R0 An (S (plus N n))). -Apply H6; Unfold ge; Apply le_trans with n. -Apply H7. -Apply le_trans with (plus N n). -Apply le_plus_r. -Apply le_n_Sn. -Cut (le O N). -Cut (lt N (S (plus N n))). -Intros; Assert H10 := (sigma_split An H9 H8). -Unfold sigma in H10. -Do 2 Rewrite <- minus_n_O in H10. -Replace (sum_f_R0 An (S (plus N n))) with (sum_f_R0 [k:nat](An (plus (0) k)) (S (plus N n))). -Replace (sum_f_R0 An N) with (sum_f_R0 [k:nat](An (plus (0) k)) N). -Cut (minus (S (plus N n)) (S N))=n. -Intro; Rewrite H11 in H10. -Apply H10. -Apply INR_eq; Rewrite minus_INR. -Do 2 Rewrite S_INR; Rewrite plus_INR; Ring. -Apply le_n_S; Apply le_plus_l. -Apply sum_eq; Intros. -Reflexivity. -Apply sum_eq; Intros. -Reflexivity. -Apply le_lt_n_Sm; Apply le_plus_l. -Apply le_O_n. -Symmetry; EApply UL_sequence. -Apply H2. -Unfold Un_cv in H; Unfold Un_cv; Intros. -Elim (H eps H4); Intros N0 H5. -Unfold R_dist in H5; Exists N0; Intros. -Unfold R_dist SP; Replace (Rminus (sum_f_R0 [l:nat](fn (plus (S N) l) x) n) (Rminus l1 (sum_f_R0 [k:nat](fn k x) N))) with (Rminus (Rplus (sum_f_R0 [k:nat](fn k x) N) (sum_f_R0 [l:nat](fn (plus (S N) l) x) n)) l1); [Idtac | Ring]. -Replace (Rplus (sum_f_R0 [k:nat](fn k x) N) (sum_f_R0 [l:nat](fn (plus (S N) l) x) n)) with (sum_f_R0 [k:nat](fn k x) (S (plus N n))). -Unfold SP in H5; Apply H5; Unfold ge; Apply le_trans with n. -Apply H6. -Apply le_trans with (plus N n). -Apply le_plus_r. -Apply le_n_Sn. -Cut (le O N). -Cut (lt N (S (plus N n))). -Intros; Assert H9 := (sigma_split [k:nat](fn k x) H8 H7). -Unfold sigma in H9. -Do 2 Rewrite <- minus_n_O in H9. -Replace (sum_f_R0 [k:nat](fn k x) (S (plus N n))) with (sum_f_R0 [k:nat](fn (plus (0) k) x) (S (plus N n))). -Replace (sum_f_R0 [k:nat](fn k x) N) with (sum_f_R0 [k:nat](fn (plus (0) k) x) N). -Cut (minus (S (plus N n)) (S N))=n. -Intro; Rewrite H10 in H9. -Apply H9. -Apply INR_eq; Rewrite minus_INR. -Do 2 Rewrite S_INR; Rewrite plus_INR; Ring. -Apply le_n_S; Apply le_plus_l. -Apply sum_eq; Intros. -Reflexivity. -Apply sum_eq; Intros. -Reflexivity. -Apply le_lt_n_Sm. -Apply le_plus_l. -Apply le_O_n. -Apply existTT with ``l2-(sum_f_R0 An N)``. -Unfold Un_cv in H0; Unfold Un_cv; Intros. -Elim (H0 eps H2); Intros N0 H3. -Unfold R_dist in H3; Exists N0; Intros. -Unfold R_dist; Replace (Rminus (sum_f_R0 [l:nat](An (plus (S N) l)) n) (Rminus l2 (sum_f_R0 An N))) with (Rminus (Rplus (sum_f_R0 An N) (sum_f_R0 [l:nat](An (plus (S N) l)) n)) l2); [Idtac | Ring]. -Replace (Rplus (sum_f_R0 An N) (sum_f_R0 [l:nat](An (plus (S N) l)) n)) with (sum_f_R0 An (S (plus N n))). -Apply H3; Unfold ge; Apply le_trans with n. -Apply H4. -Apply le_trans with (plus N n). -Apply le_plus_r. -Apply le_n_Sn. -Cut (le O N). -Cut (lt N (S (plus N n))). -Intros; Assert H7 := (sigma_split An H6 H5). -Unfold sigma in H7. -Do 2 Rewrite <- minus_n_O in H7. -Replace (sum_f_R0 An (S (plus N n))) with (sum_f_R0 [k:nat](An (plus (0) k)) (S (plus N n))). -Replace (sum_f_R0 An N) with (sum_f_R0 [k:nat](An (plus (0) k)) N). -Cut (minus (S (plus N n)) (S N))=n. -Intro; Rewrite H8 in H7. -Apply H7. -Apply INR_eq; Rewrite minus_INR. -Do 2 Rewrite S_INR; Rewrite plus_INR; Ring. -Apply le_n_S; Apply le_plus_l. -Apply sum_eq; Intros. -Reflexivity. -Apply sum_eq; Intros. -Reflexivity. -Apply le_lt_n_Sm. -Apply le_plus_l. -Apply le_O_n. -Apply existTT with ``l1-(SP fn N x)``. -Unfold Un_cv in H; Unfold Un_cv; Intros. -Elim (H eps H2); Intros N0 H3. -Unfold R_dist in H3; Exists N0; Intros. -Unfold R_dist SP. -Replace (Rminus (sum_f_R0 [l:nat](fn (plus (S N) l) x) n) (Rminus l1 (sum_f_R0 [k:nat](fn k x) N))) with (Rminus (Rplus (sum_f_R0 [k:nat](fn k x) N) (sum_f_R0 [l:nat](fn (plus (S N) l) x) n)) l1); [Idtac | Ring]. -Replace (Rplus (sum_f_R0 [k:nat](fn k x) N) (sum_f_R0 [l:nat](fn (plus (S N) l) x) n)) with (sum_f_R0 [k:nat](fn k x) (S (plus N n))). -Unfold SP in H3; Apply H3. -Unfold ge; Apply le_trans with n. -Apply H4. -Apply le_trans with (plus N n). -Apply le_plus_r. -Apply le_n_Sn. -Cut (le O N). -Cut (lt N (S (plus N n))). -Intros; Assert H7 := (sigma_split [k:nat](fn k x) H6 H5). -Unfold sigma in H7. -Do 2 Rewrite <- minus_n_O in H7. -Replace (sum_f_R0 [k:nat](fn k x) (S (plus N n))) with (sum_f_R0 [k:nat](fn (plus (0) k) x) (S (plus N n))). -Replace (sum_f_R0 [k:nat](fn k x) N) with (sum_f_R0 [k:nat](fn (plus (0) k) x) N). -Cut (minus (S (plus N n)) (S N))=n. -Intro; Rewrite H8 in H7. -Apply H7. -Apply INR_eq; Rewrite minus_INR. -Do 2 Rewrite S_INR; Rewrite plus_INR; Ring. -Apply le_n_S; Apply le_plus_l. -Apply sum_eq; Intros. -Reflexivity. -Apply sum_eq; Intros. -Reflexivity. -Apply le_lt_n_Sm. -Apply le_plus_l. -Apply le_O_n. +Lemma sum_maj1 : + forall (fn:nat -> R -> R) (An:nat -> R) (x l1 l2:R) + (N:nat), + Un_cv (fun n:nat => SP fn n x) l1 -> + Un_cv (fun n:nat => sum_f_R0 An n) l2 -> + (forall n:nat, Rabs (fn n x) <= An n) -> + Rabs (l1 - SP fn N x) <= l2 - sum_f_R0 An N. +intros; + cut + (sigT + (fun l:R => + Un_cv (fun n:nat => sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) l)). +intro; + cut + (sigT + (fun l:R => + Un_cv (fun n:nat => sum_f_R0 (fun l:nat => An (S N + l)%nat) n) l)). +intro; elim X; intros l1N H2. +elim X0; intros l2N H3. +cut (l1 - SP fn N x = l1N). +intro; cut (l2 - sum_f_R0 An N = l2N). +intro; rewrite H4; rewrite H5. +apply sum_cv_maj with + (fun l:nat => An (S N + l)%nat) (fun (l:nat) (x:R) => fn (S N + l)%nat x) x. +unfold SP in |- *; apply H2. +apply H3. +intros; apply H1. +symmetry in |- *; eapply UL_sequence. +apply H3. +unfold Un_cv in H0; unfold Un_cv in |- *; intros; elim (H0 eps H5); + intros N0 H6. +unfold R_dist in H6; exists N0; intros. +unfold R_dist in |- *; + replace (sum_f_R0 (fun l:nat => An (S N + l)%nat) n - (l2 - sum_f_R0 An N)) + with (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n - l2); + [ idtac | ring ]. +replace (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n) with + (sum_f_R0 An (S (N + n))). +apply H6; unfold ge in |- *; apply le_trans with n. +apply H7. +apply le_trans with (N + n)%nat. +apply le_plus_r. +apply le_n_Sn. +cut (0 <= N)%nat. +cut (N < S (N + n))%nat. +intros; assert (H10 := sigma_split An H9 H8). +unfold sigma in H10. +do 2 rewrite <- minus_n_O in H10. +replace (sum_f_R0 An (S (N + n))) with + (sum_f_R0 (fun k:nat => An (0 + k)%nat) (S (N + n))). +replace (sum_f_R0 An N) with (sum_f_R0 (fun k:nat => An (0 + k)%nat) N). +cut ((S (N + n) - S N)%nat = n). +intro; rewrite H11 in H10. +apply H10. +apply INR_eq; rewrite minus_INR. +do 2 rewrite S_INR; rewrite plus_INR; ring. +apply le_n_S; apply le_plus_l. +apply sum_eq; intros. +reflexivity. +apply sum_eq; intros. +reflexivity. +apply le_lt_n_Sm; apply le_plus_l. +apply le_O_n. +symmetry in |- *; eapply UL_sequence. +apply H2. +unfold Un_cv in H; unfold Un_cv in |- *; intros. +elim (H eps H4); intros N0 H5. +unfold R_dist in H5; exists N0; intros. +unfold R_dist, SP in |- *; + replace + (sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n - + (l1 - sum_f_R0 (fun k:nat => fn k x) N)) with + (sum_f_R0 (fun k:nat => fn k x) N + + sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n - l1); + [ idtac | ring ]. +replace + (sum_f_R0 (fun k:nat => fn k x) N + + sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) with + (sum_f_R0 (fun k:nat => fn k x) (S (N + n))). +unfold SP in H5; apply H5; unfold ge in |- *; apply le_trans with n. +apply H6. +apply le_trans with (N + n)%nat. +apply le_plus_r. +apply le_n_Sn. +cut (0 <= N)%nat. +cut (N < S (N + n))%nat. +intros; assert (H9 := sigma_split (fun k:nat => fn k x) H8 H7). +unfold sigma in H9. +do 2 rewrite <- minus_n_O in H9. +replace (sum_f_R0 (fun k:nat => fn k x) (S (N + n))) with + (sum_f_R0 (fun k:nat => fn (0 + k)%nat x) (S (N + n))). +replace (sum_f_R0 (fun k:nat => fn k x) N) with + (sum_f_R0 (fun k:nat => fn (0 + k)%nat x) N). +cut ((S (N + n) - S N)%nat = n). +intro; rewrite H10 in H9. +apply H9. +apply INR_eq; rewrite minus_INR. +do 2 rewrite S_INR; rewrite plus_INR; ring. +apply le_n_S; apply le_plus_l. +apply sum_eq; intros. +reflexivity. +apply sum_eq; intros. +reflexivity. +apply le_lt_n_Sm. +apply le_plus_l. +apply le_O_n. +apply existT with (l2 - sum_f_R0 An N). +unfold Un_cv in H0; unfold Un_cv in |- *; intros. +elim (H0 eps H2); intros N0 H3. +unfold R_dist in H3; exists N0; intros. +unfold R_dist in |- *; + replace (sum_f_R0 (fun l:nat => An (S N + l)%nat) n - (l2 - sum_f_R0 An N)) + with (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n - l2); + [ idtac | ring ]. +replace (sum_f_R0 An N + sum_f_R0 (fun l:nat => An (S N + l)%nat) n) with + (sum_f_R0 An (S (N + n))). +apply H3; unfold ge in |- *; apply le_trans with n. +apply H4. +apply le_trans with (N + n)%nat. +apply le_plus_r. +apply le_n_Sn. +cut (0 <= N)%nat. +cut (N < S (N + n))%nat. +intros; assert (H7 := sigma_split An H6 H5). +unfold sigma in H7. +do 2 rewrite <- minus_n_O in H7. +replace (sum_f_R0 An (S (N + n))) with + (sum_f_R0 (fun k:nat => An (0 + k)%nat) (S (N + n))). +replace (sum_f_R0 An N) with (sum_f_R0 (fun k:nat => An (0 + k)%nat) N). +cut ((S (N + n) - S N)%nat = n). +intro; rewrite H8 in H7. +apply H7. +apply INR_eq; rewrite minus_INR. +do 2 rewrite S_INR; rewrite plus_INR; ring. +apply le_n_S; apply le_plus_l. +apply sum_eq; intros. +reflexivity. +apply sum_eq; intros. +reflexivity. +apply le_lt_n_Sm. +apply le_plus_l. +apply le_O_n. +apply existT with (l1 - SP fn N x). +unfold Un_cv in H; unfold Un_cv in |- *; intros. +elim (H eps H2); intros N0 H3. +unfold R_dist in H3; exists N0; intros. +unfold R_dist, SP in |- *. +replace + (sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n - + (l1 - sum_f_R0 (fun k:nat => fn k x) N)) with + (sum_f_R0 (fun k:nat => fn k x) N + + sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n - l1); + [ idtac | ring ]. +replace + (sum_f_R0 (fun k:nat => fn k x) N + + sum_f_R0 (fun l:nat => fn (S N + l)%nat x) n) with + (sum_f_R0 (fun k:nat => fn k x) (S (N + n))). +unfold SP in H3; apply H3. +unfold ge in |- *; apply le_trans with n. +apply H4. +apply le_trans with (N + n)%nat. +apply le_plus_r. +apply le_n_Sn. +cut (0 <= N)%nat. +cut (N < S (N + n))%nat. +intros; assert (H7 := sigma_split (fun k:nat => fn k x) H6 H5). +unfold sigma in H7. +do 2 rewrite <- minus_n_O in H7. +replace (sum_f_R0 (fun k:nat => fn k x) (S (N + n))) with + (sum_f_R0 (fun k:nat => fn (0 + k)%nat x) (S (N + n))). +replace (sum_f_R0 (fun k:nat => fn k x) N) with + (sum_f_R0 (fun k:nat => fn (0 + k)%nat x) N). +cut ((S (N + n) - S N)%nat = n). +intro; rewrite H8 in H7. +apply H7. +apply INR_eq; rewrite minus_INR. +do 2 rewrite S_INR; rewrite plus_INR; ring. +apply le_n_S; apply le_plus_l. +apply sum_eq; intros. +reflexivity. +apply sum_eq; intros. +reflexivity. +apply le_lt_n_Sm. +apply le_plus_l. +apply le_O_n. Qed. (* Comparaison of convergence for series *) -Lemma Rseries_CV_comp : (An,Bn:nat->R) ((n:nat)``0<=(An n)<=(Bn n)``) -> (sigTT ? [l:R](Un_cv [N:nat](sum_f_R0 Bn N) l)) -> (sigTT ? [l:R](Un_cv [N:nat](sum_f_R0 An N) l)). -Intros; Apply cv_cauchy_2. -Assert H0 := (cv_cauchy_1 ? X). -Unfold Cauchy_crit_series; Unfold Cauchy_crit. -Intros; Elim (H0 eps H1); Intros. -Exists x; Intros. -Cut (Rle (R_dist (sum_f_R0 An n) (sum_f_R0 An m)) (R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m))). -Intro; Apply Rle_lt_trans with (R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)). -Assumption. -Apply H2; Assumption. -Assert H5 := (lt_eq_lt_dec n m). -Elim H5; Intro. -Elim a; Intro. -Rewrite (tech2 An n m); [Idtac | Assumption]. -Rewrite (tech2 Bn n m); [Idtac | Assumption]. -Unfold R_dist; Unfold Rminus; Do 2 Rewrite Ropp_distr1; Do 2 Rewrite <- Rplus_assoc; Do 2 Rewrite Rplus_Ropp_r; Do 2 Rewrite Rplus_Ol; Do 2 Rewrite Rabsolu_Ropp; Repeat Rewrite Rabsolu_right. -Apply sum_Rle; Intros. -Elim (H (plus (S n) n0)); Intros. -Apply H8. -Apply Rle_sym1; Apply cond_pos_sum; Intro. -Elim (H (plus (S n) n0)); Intros. -Apply Rle_trans with (An (plus (S n) n0)); Assumption. -Apply Rle_sym1; Apply cond_pos_sum; Intro. -Elim (H (plus (S n) n0)); Intros; Assumption. -Rewrite b; Unfold R_dist; Unfold Rminus; Do 2 Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0; Right; Reflexivity. -Rewrite (tech2 An m n); [Idtac | Assumption]. -Rewrite (tech2 Bn m n); [Idtac | Assumption]. -Unfold R_dist; Unfold Rminus; Do 2 Rewrite Rplus_assoc; Rewrite (Rplus_sym (sum_f_R0 An m)); Rewrite (Rplus_sym (sum_f_R0 Bn m)); Do 2 Rewrite Rplus_assoc; Do 2 Rewrite Rplus_Ropp_l; Do 2 Rewrite Rplus_Or; Repeat Rewrite Rabsolu_right. -Apply sum_Rle; Intros. -Elim (H (plus (S m) n0)); Intros; Apply H8. -Apply Rle_sym1; Apply cond_pos_sum; Intro. -Elim (H (plus (S m) n0)); Intros. -Apply Rle_trans with (An (plus (S m) n0)); Assumption. -Apply Rle_sym1. -Apply cond_pos_sum; Intro. -Elim (H (plus (S m) n0)); Intros; Assumption. +Lemma Rseries_CV_comp : + forall An Bn:nat -> R, + (forall n:nat, 0 <= An n <= Bn n) -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 Bn N) l) -> + sigT (fun l:R => Un_cv (fun N:nat => sum_f_R0 An N) l). +intros; apply cv_cauchy_2. +assert (H0 := cv_cauchy_1 _ X). +unfold Cauchy_crit_series in |- *; unfold Cauchy_crit in |- *. +intros; elim (H0 eps H1); intros. +exists x; intros. +cut + (R_dist (sum_f_R0 An n) (sum_f_R0 An m) <= + R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)). +intro; apply Rle_lt_trans with (R_dist (sum_f_R0 Bn n) (sum_f_R0 Bn m)). +assumption. +apply H2; assumption. +assert (H5 := lt_eq_lt_dec n m). +elim H5; intro. +elim a; intro. +rewrite (tech2 An n m); [ idtac | assumption ]. +rewrite (tech2 Bn n m); [ idtac | assumption ]. +unfold R_dist in |- *; unfold Rminus in |- *; do 2 rewrite Ropp_plus_distr; + do 2 rewrite <- Rplus_assoc; do 2 rewrite Rplus_opp_r; + do 2 rewrite Rplus_0_l; do 2 rewrite Rabs_Ropp; repeat rewrite Rabs_right. +apply sum_Rle; intros. +elim (H (S n + n0)%nat); intros. +apply H8. +apply Rle_ge; apply cond_pos_sum; intro. +elim (H (S n + n0)%nat); intros. +apply Rle_trans with (An (S n + n0)%nat); assumption. +apply Rle_ge; apply cond_pos_sum; intro. +elim (H (S n + n0)%nat); intros; assumption. +rewrite b; unfold R_dist in |- *; unfold Rminus in |- *; + do 2 rewrite Rplus_opp_r; rewrite Rabs_R0; right; + reflexivity. +rewrite (tech2 An m n); [ idtac | assumption ]. +rewrite (tech2 Bn m n); [ idtac | assumption ]. +unfold R_dist in |- *; unfold Rminus in |- *; do 2 rewrite Rplus_assoc; + rewrite (Rplus_comm (sum_f_R0 An m)); rewrite (Rplus_comm (sum_f_R0 Bn m)); + do 2 rewrite Rplus_assoc; do 2 rewrite Rplus_opp_l; + do 2 rewrite Rplus_0_r; repeat rewrite Rabs_right. +apply sum_Rle; intros. +elim (H (S m + n0)%nat); intros; apply H8. +apply Rle_ge; apply cond_pos_sum; intro. +elim (H (S m + n0)%nat); intros. +apply Rle_trans with (An (S m + n0)%nat); assumption. +apply Rle_ge. +apply cond_pos_sum; intro. +elim (H (S m + n0)%nat); intros; assumption. Qed. (* Cesaro's theorem *) -Lemma Cesaro : (An,Bn:nat->R;l:R) (Un_cv Bn l) -> ((n:nat)``0<(An n)``) -> (cv_infty [n:nat](sum_f_R0 An n)) -> (Un_cv [n:nat](Rdiv (sum_f_R0 [k:nat]``(An k)*(Bn k)`` n) (sum_f_R0 An n)) l). -Proof with Trivial. -Unfold Un_cv; Intros; Assert H3 : (n:nat)``0<(sum_f_R0 An n)``. -Intro; Apply tech1. -Assert H4 : (n:nat) ``(sum_f_R0 An n)<>0``. -Intro; Red; Intro; Assert H5 := (H3 n); Rewrite H4 in H5; Elim (Rlt_antirefl ? H5). -Assert H5 := (cv_infty_cv_R0 ? H4 H1); Assert H6 : ``0<eps/2``. -Unfold Rdiv; Apply Rmult_lt_pos. -Apply Rlt_Rinv; Sup. -Elim (H ? H6); Clear H; Intros N1 H; Pose C := (Rabsolu (sum_f_R0 [k:nat]``(An k)*((Bn k)-l)`` N1)); Assert H7 : (EX N:nat | (n:nat) (le N n) -> ``C/(sum_f_R0 An n)<eps/2``). -Case (Req_EM C R0); Intro. -Exists O; Intros. -Rewrite H7; Unfold Rdiv; Rewrite Rmult_Ol; Apply Rmult_lt_pos. -Apply Rlt_Rinv; Sup. -Assert H8 : ``0<eps/(2*(Rabsolu C))``. -Unfold Rdiv; Apply Rmult_lt_pos. -Apply Rlt_Rinv; Apply Rmult_lt_pos. -Sup. -Apply Rabsolu_pos_lt. -Elim (H5 ? H8); Intros; Exists x; Intros; Assert H11 := (H9 ? H10); Unfold R_dist in H11; Unfold Rminus in H11; Rewrite Ropp_O in H11; Rewrite Rplus_Or in H11. -Apply Rle_lt_trans with (Rabsolu ``C/(sum_f_R0 An n)``). -Apply Rle_Rabsolu. -Unfold Rdiv; Rewrite Rabsolu_mult; Apply Rlt_monotony_contra with ``/(Rabsolu C)``. -Apply Rlt_Rinv; Apply Rabsolu_pos_lt. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Replace ``/(Rabsolu C)*(eps*/2)`` with ``eps/(2*(Rabsolu C))``. -Unfold Rdiv; Rewrite Rinv_Rmult. -Ring. -DiscrR. -Apply Rabsolu_no_R0. -Apply Rabsolu_no_R0. -Elim H7; Clear H7; Intros N2 H7; Pose N := (max N1 N2); Exists (S N); Intros; Unfold R_dist; Replace (Rminus (Rdiv (sum_f_R0 [k:nat]``(An k)*(Bn k)`` n) (sum_f_R0 An n)) l) with (Rdiv (sum_f_R0 [k:nat]``(An k)*((Bn k)-l)`` n) (sum_f_R0 An n)). -Assert H9 : (lt N1 n). -Apply lt_le_trans with (S N). -Apply le_lt_n_Sm; Unfold N; Apply le_max_l. -Rewrite (tech2 [k:nat]``(An k)*((Bn k)-l)`` ? ? H9); Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Apply Rle_lt_trans with (Rplus (Rabsolu (Rdiv (sum_f_R0 [k:nat]``(An k)*((Bn k)-l)`` N1) (sum_f_R0 An n))) (Rabsolu (Rdiv (sum_f_R0 [i:nat]``(An (plus (S N1) i))*((Bn (plus (S N1) i))-l)`` (minus n (S N1))) (sum_f_R0 An n)))). -Apply Rabsolu_triang. -Rewrite (double_var eps); Apply Rplus_lt. -Unfold Rdiv; Rewrite Rabsolu_mult; Fold C; Rewrite Rabsolu_right. -Apply (H7 n); Apply le_trans with (S N). -Apply le_trans with N; [Unfold N; Apply le_max_r | Apply le_n_Sn]. -Apply Rle_sym1; Left; Apply Rlt_Rinv. +Lemma Cesaro : + forall (An Bn:nat -> R) (l:R), + Un_cv Bn l -> + (forall n:nat, 0 < An n) -> + cv_infty (fun n:nat => sum_f_R0 An n) -> + Un_cv (fun n:nat => sum_f_R0 (fun k:nat => An k * Bn k) n / sum_f_R0 An n) + l. +Proof with trivial. +unfold Un_cv in |- *; intros; assert (H3 : forall n:nat, 0 < sum_f_R0 An n)... +intro; apply tech1... +assert (H4 : forall n:nat, sum_f_R0 An n <> 0)... +intro; red in |- *; intro; assert (H5 := H3 n); rewrite H4 in H5; + elim (Rlt_irrefl _ H5)... +assert (H5 := cv_infty_cv_R0 _ H4 H1); assert (H6 : 0 < eps / 2)... +unfold Rdiv in |- *; apply Rmult_lt_0_compat... +apply Rinv_0_lt_compat; prove_sup... +elim (H _ H6); clear H; intros N1 H; + pose (C := Rabs (sum_f_R0 (fun k:nat => An k * (Bn k - l)) N1)); + assert + (H7 : + exists N : nat + | (forall n:nat, (N <= n)%nat -> C / sum_f_R0 An n < eps / 2))... +case (Req_dec C 0); intro... +exists 0%nat; intros... +rewrite H7; unfold Rdiv in |- *; rewrite Rmult_0_l; apply Rmult_lt_0_compat... +apply Rinv_0_lt_compat; prove_sup... +assert (H8 : 0 < eps / (2 * Rabs C))... +unfold Rdiv in |- *; apply Rmult_lt_0_compat... +apply Rinv_0_lt_compat; apply Rmult_lt_0_compat... +prove_sup... +apply Rabs_pos_lt... +elim (H5 _ H8); intros; exists x; intros; assert (H11 := H9 _ H10); + unfold R_dist in H11; unfold Rminus in H11; rewrite Ropp_0 in H11; + rewrite Rplus_0_r in H11... +apply Rle_lt_trans with (Rabs (C / sum_f_R0 An n))... +apply RRle_abs... +unfold Rdiv in |- *; rewrite Rabs_mult; apply Rmult_lt_reg_l with (/ Rabs C)... +apply Rinv_0_lt_compat; apply Rabs_pos_lt... +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym... +rewrite Rmult_1_l; replace (/ Rabs C * (eps * / 2)) with (eps / (2 * Rabs C))... +unfold Rdiv in |- *; rewrite Rinv_mult_distr... +ring... +discrR... +apply Rabs_no_R0... +apply Rabs_no_R0... +elim H7; clear H7; intros N2 H7; pose (N := max N1 N2); exists (S N); intros; + unfold R_dist in |- *; + replace (sum_f_R0 (fun k:nat => An k * Bn k) n / sum_f_R0 An n - l) with + (sum_f_R0 (fun k:nat => An k * (Bn k - l)) n / sum_f_R0 An n)... +assert (H9 : (N1 < n)%nat)... +apply lt_le_trans with (S N)... +apply le_lt_n_Sm; unfold N in |- *; apply le_max_l... +rewrite (tech2 (fun k:nat => An k * (Bn k - l)) _ _ H9); unfold Rdiv in |- *; + rewrite Rmult_plus_distr_r; + apply Rle_lt_trans with + (Rabs (sum_f_R0 (fun k:nat => An k * (Bn k - l)) N1 / sum_f_R0 An n) + + Rabs + (sum_f_R0 (fun i:nat => An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)) + (n - S N1) / sum_f_R0 An n))... +apply Rabs_triang... +rewrite (double_var eps); apply Rplus_lt_compat... +unfold Rdiv in |- *; rewrite Rabs_mult; fold C in |- *; rewrite Rabs_right... +apply (H7 n); apply le_trans with (S N)... +apply le_trans with N; [ unfold N in |- *; apply le_max_r | apply le_n_Sn ]... +apply Rle_ge; left; apply Rinv_0_lt_compat... -Unfold R_dist in H; Unfold Rdiv; Rewrite Rabsolu_mult; Rewrite (Rabsolu_right ``/(sum_f_R0 An n)``). -Apply Rle_lt_trans with (Rmult (sum_f_R0 [i:nat](Rabsolu ``(An (plus (S N1) i))*((Bn (plus (S N1) i))-l)``) (minus n (S N1))) ``/(sum_f_R0 An n)``). -Do 2 Rewrite <- (Rmult_sym ``/(sum_f_R0 An n)``); Apply Rle_monotony. -Left; Apply Rlt_Rinv. -Apply (sum_Rabsolu [i:nat]``(An (plus (S N1) i))*((Bn (plus (S N1) i))-l)`` (minus n (S N1))). -Apply Rle_lt_trans with (Rmult (sum_f_R0 [i:nat]``(An (plus (S N1) i))*eps/2`` (minus n (S N1))) ``/(sum_f_R0 An n)``). -Do 2 Rewrite <- (Rmult_sym ``/(sum_f_R0 An n)``); Apply Rle_monotony. -Left; Apply Rlt_Rinv. -Apply sum_Rle; Intros; Rewrite Rabsolu_mult; Pattern 2 (An (plus (S N1) n0)); Rewrite <- (Rabsolu_right (An (plus (S N1) n0))). -Apply Rle_monotony. -Apply Rabsolu_pos. -Left; Apply H; Unfold ge; Apply le_trans with (S N1); [Apply le_n_Sn | Apply le_plus_l]. -Apply Rle_sym1; Left. -Rewrite <- (scal_sum [i:nat](An (plus (S N1) i)) (minus n (S N1)) ``eps/2``); Unfold Rdiv; Repeat Rewrite Rmult_assoc; Apply Rlt_monotony. -Pattern 2 ``/2``; Rewrite <- Rmult_1r; Apply Rlt_monotony. -Apply Rlt_Rinv; Sup. -Rewrite Rmult_sym; Apply Rlt_monotony_contra with (sum_f_R0 An n). -Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite Rmult_1r; Rewrite (tech2 An N1 n). -Rewrite Rplus_sym; Pattern 1 (sum_f_R0 [i:nat](An (plus (S N1) i)) (minus n (S N1))); Rewrite <- Rplus_Or; Apply Rlt_compatibility. -Apply Rle_sym1; Left; Apply Rlt_Rinv. -Replace (sum_f_R0 [k:nat]``(An k)*((Bn k)-l)`` n) with (Rplus (sum_f_R0 [k:nat]``(An k)*(Bn k)`` n) (sum_f_R0 [k:nat]``(An k)*-l`` n)). -Rewrite <- (scal_sum An n ``-l``); Field. -Rewrite <- plus_sum; Apply sum_eq; Intros; Ring. +unfold R_dist in H; unfold Rdiv in |- *; rewrite Rabs_mult; + rewrite (Rabs_right (/ sum_f_R0 An n))... +apply Rle_lt_trans with + (sum_f_R0 (fun i:nat => Rabs (An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l))) + (n - S N1) * / sum_f_R0 An n)... +do 2 rewrite <- (Rmult_comm (/ sum_f_R0 An n)); apply Rmult_le_compat_l... +left; apply Rinv_0_lt_compat... +apply + (Rsum_abs (fun i:nat => An (S N1 + i)%nat * (Bn (S N1 + i)%nat - l)) + (n - S N1))... +apply Rle_lt_trans with + (sum_f_R0 (fun i:nat => An (S N1 + i)%nat * (eps / 2)) (n - S N1) * + / sum_f_R0 An n)... +do 2 rewrite <- (Rmult_comm (/ sum_f_R0 An n)); apply Rmult_le_compat_l... +left; apply Rinv_0_lt_compat... +apply sum_Rle; intros; rewrite Rabs_mult; + pattern (An (S N1 + n0)%nat) at 2 in |- *; + rewrite <- (Rabs_right (An (S N1 + n0)%nat))... +apply Rmult_le_compat_l... +apply Rabs_pos... +left; apply H; unfold ge in |- *; apply le_trans with (S N1); + [ apply le_n_Sn | apply le_plus_l ]... +apply Rle_ge; left... +rewrite <- (scal_sum (fun i:nat => An (S N1 + i)%nat) (n - S N1) (eps / 2)); + unfold Rdiv in |- *; repeat rewrite Rmult_assoc; apply Rmult_lt_compat_l... +pattern (/ 2) at 2 in |- *; rewrite <- Rmult_1_r; apply Rmult_lt_compat_l... +apply Rinv_0_lt_compat; prove_sup... +rewrite Rmult_comm; apply Rmult_lt_reg_l with (sum_f_R0 An n)... +rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym... +rewrite Rmult_1_l; rewrite Rmult_1_r; rewrite (tech2 An N1 n)... +rewrite Rplus_comm; + pattern (sum_f_R0 (fun i:nat => An (S N1 + i)%nat) (n - S N1)) at 1 in |- *; + rewrite <- Rplus_0_r; apply Rplus_lt_compat_l... +apply Rle_ge; left; apply Rinv_0_lt_compat... +replace (sum_f_R0 (fun k:nat => An k * (Bn k - l)) n) with + (sum_f_R0 (fun k:nat => An k * Bn k) n + + sum_f_R0 (fun k:nat => An k * - l) n)... +rewrite <- (scal_sum An n (- l)); field... +rewrite <- plus_sum; apply sum_eq; intros; ring... Qed. -Lemma Cesaro_1 : (An:nat->R;l:R) (Un_cv An l) -> (Un_cv [n:nat]``(sum_f_R0 An (pred n))/(INR n)`` l). -Proof with Trivial. -Intros Bn l H; Pose An := [_:nat]R1. -Assert H0 : (n:nat) ``0<(An n)``. -Intro; Unfold An; Apply Rlt_R0_R1. -Assert H1 : (n:nat)``0<(sum_f_R0 An n)``. -Intro; Apply tech1. -Assert H2 : (cv_infty [n:nat](sum_f_R0 An n)). -Unfold cv_infty; Intro; Case (total_order_Rle M R0); Intro. -Exists O; Intros; Apply Rle_lt_trans with R0. -Assert H2 : ``0<M``. -Auto with real. -Clear n; Pose m := (up M); Elim (archimed M); Intros; Assert H5 : `0<=m`. -Apply le_IZR; Unfold m; Simpl; Left; Apply Rlt_trans with M. -Elim (IZN ? H5); Intros; Exists x; Intros; Unfold An; Rewrite sum_cte; Rewrite Rmult_1l; Apply Rlt_trans with (IZR (up M)). -Apply Rle_lt_trans with (INR x). -Rewrite INR_IZR_INZ; Fold m; Rewrite <- H6; Right. -Apply lt_INR; Apply le_lt_n_Sm. -Assert H3 := (Cesaro ? ? ? H H0 H2). -Unfold Un_cv; Unfold Un_cv in H3; Intros; Elim (H3 ? H4); Intros; Exists (S x); Intros; Unfold R_dist; Unfold R_dist in H5; Apply Rle_lt_trans with (Rabsolu (Rminus (Rdiv (sum_f_R0 [k:nat]``(An k)*(Bn k)`` (pred n)) (sum_f_R0 An (pred n))) l)). -Right; Replace ``(sum_f_R0 Bn (pred n))/(INR n)-l`` with (Rminus (Rdiv (sum_f_R0 [k:nat]``(An k)*(Bn k)`` (pred n)) (sum_f_R0 An (pred n))) l). -Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-l``); Apply Rplus_plus_r. -Unfold An; Replace (sum_f_R0 [k:nat]``1*(Bn k)`` (pred n)) with (sum_f_R0 Bn (pred n)). -Rewrite sum_cte; Rewrite Rmult_1l; Replace (S (pred n)) with n. -Apply S_pred with O; Apply lt_le_trans with (S x). -Apply lt_O_Sn. -Apply sum_eq; Intros; Ring. -Apply H5; Unfold ge; Apply le_S_n; Replace (S (pred n)) with n. -Apply S_pred with O; Apply lt_le_trans with (S x). -Apply lt_O_Sn. -Qed. +Lemma Cesaro_1 : + forall (An:nat -> R) (l:R), + Un_cv An l -> Un_cv (fun n:nat => sum_f_R0 An (pred n) / INR n) l. +Proof with trivial. +intros Bn l H; pose (An := fun _:nat => 1)... +assert (H0 : forall n:nat, 0 < An n)... +intro; unfold An in |- *; apply Rlt_0_1... +assert (H1 : forall n:nat, 0 < sum_f_R0 An n)... +intro; apply tech1... +assert (H2 : cv_infty (fun n:nat => sum_f_R0 An n))... +unfold cv_infty in |- *; intro; case (Rle_dec M 0); intro... +exists 0%nat; intros; apply Rle_lt_trans with 0... +assert (H2 : 0 < M)... +auto with real... +clear n; pose (m := up M); elim (archimed M); intros; + assert (H5 : (0 <= m)%Z)... +apply le_IZR; unfold m in |- *; simpl in |- *; left; apply Rlt_trans with M... +elim (IZN _ H5); intros; exists x; intros; unfold An in |- *; rewrite sum_cte; + rewrite Rmult_1_l; apply Rlt_trans with (IZR (up M))... +apply Rle_lt_trans with (INR x)... +rewrite INR_IZR_INZ; fold m in |- *; rewrite <- H6; right... +apply lt_INR; apply le_lt_n_Sm... +assert (H3 := Cesaro _ _ _ H H0 H2)... +unfold Un_cv in |- *; unfold Un_cv in H3; intros; elim (H3 _ H4); intros; + exists (S x); intros; unfold R_dist in |- *; unfold R_dist in H5; + apply Rle_lt_trans with + (Rabs + (sum_f_R0 (fun k:nat => An k * Bn k) (pred n) / sum_f_R0 An (pred n) - l))... +right; + replace (sum_f_R0 Bn (pred n) / INR n - l) with + (sum_f_R0 (fun k:nat => An k * Bn k) (pred n) / sum_f_R0 An (pred n) - l)... +unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (- l)); + apply Rplus_eq_compat_l... +unfold An in |- *; + replace (sum_f_R0 (fun k:nat => 1 * Bn k) (pred n)) with + (sum_f_R0 Bn (pred n))... +rewrite sum_cte; rewrite Rmult_1_l; replace (S (pred n)) with n... +apply S_pred with 0%nat; apply lt_le_trans with (S x)... +apply lt_O_Sn... +apply sum_eq; intros; ring... +apply H5; unfold ge in |- *; apply le_S_n; replace (S (pred n)) with n... +apply S_pred with 0%nat; apply lt_le_trans with (S x)... +apply lt_O_Sn... +Qed.
\ No newline at end of file diff --git a/theories/Reals/SplitAbsolu.v b/theories/Reals/SplitAbsolu.v index bc876692d..5ea76696a 100644 --- a/theories/Reals/SplitAbsolu.v +++ b/theories/Reals/SplitAbsolu.v @@ -8,15 +8,18 @@ (*i $Id$ i*) -Require Rbasic_fun. +Require Import Rbasic_fun. -Recursive Tactic Definition SplitAbs := - Match Context With - | [ |- [(case_Rabsolu ?1)] ] -> - Case (case_Rabsolu ?1); Try SplitAbs. +Ltac split_case_Rabs := + match goal with + | |- context [(Rcase_abs ?X1)] => + case (Rcase_abs X1); try split_case_Rabs + end. -Recursive Tactic Definition SplitAbsolu := - Match Context With - | [ id:[(Rabsolu ?)] |- ? ] -> Generalize id; Clear id;Try SplitAbsolu - | [ |- [(Rabsolu ?1)] ] -> Unfold Rabsolu; Try SplitAbs;Intros. +Ltac split_Rabs := + match goal with + | id:context [(Rabs _)] |- _ => generalize id; clear id; try split_Rabs + | |- context [(Rabs ?X1)] => + unfold Rabs in |- *; try split_case_Rabs; intros + end.
\ No newline at end of file diff --git a/theories/Reals/SplitRmult.v b/theories/Reals/SplitRmult.v index 71b2ebf21..281745a11 100644 --- a/theories/Reals/SplitRmult.v +++ b/theories/Reals/SplitRmult.v @@ -11,9 +11,10 @@ (*i Lemma mult_non_zero :(r1,r2:R)``r1<>0`` /\ ``r2<>0`` -> ``r1*r2<>0``. i*) -Require Rbase. - -Recursive Tactic Definition SplitRmult := - Match Context With - | [ |- ~(Rmult ?1 ?2)==R0 ] -> Apply mult_non_zero; Split;Try SplitRmult. +Require Import Rbase. +Ltac split_Rmult := + match goal with + | |- ((?X1 * ?X2)%R <> 0%R) => + apply Rmult_integral_contrapositive; split; try split_Rmult + end. diff --git a/theories/Reals/Sqrt_reg.v b/theories/Reals/Sqrt_reg.v index 35f6d0f32..def3cd0a4 100644 --- a/theories/Reals/Sqrt_reg.v +++ b/theories/Reals/Sqrt_reg.v @@ -8,290 +8,344 @@ (*i $Id$ i*) -Require Rbase. -Require Rfunctions. -Require Ranalysis1. -Require R_sqrt. -V7only [Import R_scope.]. Open Local Scope R_scope. +Require Import Rbase. +Require Import Rfunctions. +Require Import Ranalysis1. +Require Import R_sqrt. Open Local Scope R_scope. (**********) -Lemma sqrt_var_maj : (h:R) ``(Rabsolu h) <= 1`` -> ``(Rabsolu ((sqrt (1+h))-1))<=(Rabsolu h)``. -Intros; Cut ``0<=1+h``. -Intro; Apply Rle_trans with ``(Rabsolu ((sqrt (Rsqr (1+h)))-1))``. -Case (total_order_T h R0); Intro. -Elim s; Intro. -Repeat Rewrite Rabsolu_left. -Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-1``). -Do 2 Rewrite Ropp_distr1;Rewrite Ropp_Ropp; Apply Rle_compatibility. -Apply Rle_Ropp1; Apply sqrt_le_1. -Apply pos_Rsqr. -Apply H0. -Pattern 2 ``1+h``; Rewrite <- Rmult_1r; Unfold Rsqr; Apply Rle_monotony. -Apply H0. -Pattern 2 R1; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Assumption. -Apply Rlt_anti_compatibility with R1; Rewrite Rplus_Or; Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or. -Pattern 2 R1; Rewrite <- sqrt_1; Apply sqrt_lt_1. -Apply pos_Rsqr. -Left; Apply Rlt_R0_R1. -Pattern 2 R1; Rewrite <- Rsqr_1; Apply Rsqr_incrst_1. -Pattern 2 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Assumption. -Apply H0. -Left; Apply Rlt_R0_R1. -Apply Rlt_anti_compatibility with R1; Rewrite Rplus_Or; Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or. -Pattern 2 R1; Rewrite <- sqrt_1; Apply sqrt_lt_1. -Apply H0. -Left; Apply Rlt_R0_R1. -Pattern 2 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Assumption. -Rewrite b; Rewrite Rplus_Or; Rewrite Rsqr_1; Rewrite sqrt_1; Right; Reflexivity. -Repeat Rewrite Rabsolu_right. -Unfold Rminus; Do 2 Rewrite <- (Rplus_sym ``-1``); Apply Rle_compatibility. -Apply sqrt_le_1. -Apply H0. -Apply pos_Rsqr. -Pattern 1 ``1+h``; Rewrite <- Rmult_1r; Unfold Rsqr; Apply Rle_monotony. -Apply H0. -Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Assumption. -Apply Rle_sym1; Apply Rle_anti_compatibility with R1. -Rewrite Rplus_Or; Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or. -Pattern 1 R1; Rewrite <- sqrt_1; Apply sqrt_le_1. -Left; Apply Rlt_R0_R1. -Apply pos_Rsqr. -Pattern 1 R1; Rewrite <- Rsqr_1; Apply Rsqr_incr_1. -Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rle_compatibility; Left; Assumption. -Left; Apply Rlt_R0_R1. -Apply H0. -Apply Rle_sym1; Left; Apply Rlt_anti_compatibility with R1. -Rewrite Rplus_Or; Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or. -Pattern 1 R1; Rewrite <- sqrt_1; Apply sqrt_lt_1. -Left; Apply Rlt_R0_R1. -Apply H0. -Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Assumption. -Rewrite sqrt_Rsqr. -Replace ``(1+h)-1`` with h; [Right; Reflexivity | Ring]. -Apply H0. -Case (total_order_T h R0); Intro. -Elim s; Intro. -Rewrite (Rabsolu_left h a) in H. -Apply Rle_anti_compatibility with ``-h``. -Rewrite Rplus_Or; Rewrite Rplus_sym; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Exact H. -Left; Rewrite b; Rewrite Rplus_Or; Apply Rlt_R0_R1. -Left; Apply gt0_plus_gt0_is_gt0. -Apply Rlt_R0_R1. -Apply r. +Lemma sqrt_var_maj : + forall h:R, Rabs h <= 1 -> Rabs (sqrt (1 + h) - 1) <= Rabs h. +intros; cut (0 <= 1 + h). +intro; apply Rle_trans with (Rabs (sqrt (Rsqr (1 + h)) - 1)). +case (total_order_T h 0); intro. +elim s; intro. +repeat rewrite Rabs_left. +unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (-1)). +do 2 rewrite Ropp_plus_distr; rewrite Ropp_involutive; + apply Rplus_le_compat_l. +apply Ropp_le_contravar; apply sqrt_le_1. +apply Rle_0_sqr. +apply H0. +pattern (1 + h) at 2 in |- *; rewrite <- Rmult_1_r; unfold Rsqr in |- *; + apply Rmult_le_compat_l. +apply H0. +pattern 1 at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + assumption. +apply Rplus_lt_reg_r with 1; rewrite Rplus_0_r; rewrite Rplus_comm; + unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_r. +pattern 1 at 2 in |- *; rewrite <- sqrt_1; apply sqrt_lt_1. +apply Rle_0_sqr. +left; apply Rlt_0_1. +pattern 1 at 2 in |- *; rewrite <- Rsqr_1; apply Rsqr_incrst_1. +pattern 1 at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + assumption. +apply H0. +left; apply Rlt_0_1. +apply Rplus_lt_reg_r with 1; rewrite Rplus_0_r; rewrite Rplus_comm; + unfold Rminus in |- *; rewrite Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_r. +pattern 1 at 2 in |- *; rewrite <- sqrt_1; apply sqrt_lt_1. +apply H0. +left; apply Rlt_0_1. +pattern 1 at 2 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + assumption. +rewrite b; rewrite Rplus_0_r; rewrite Rsqr_1; rewrite sqrt_1; right; + reflexivity. +repeat rewrite Rabs_right. +unfold Rminus in |- *; do 2 rewrite <- (Rplus_comm (-1)); + apply Rplus_le_compat_l. +apply sqrt_le_1. +apply H0. +apply Rle_0_sqr. +pattern (1 + h) at 1 in |- *; rewrite <- Rmult_1_r; unfold Rsqr in |- *; + apply Rmult_le_compat_l. +apply H0. +pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + assumption. +apply Rle_ge; apply Rplus_le_reg_l with 1. +rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *; + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r. +pattern 1 at 1 in |- *; rewrite <- sqrt_1; apply sqrt_le_1. +left; apply Rlt_0_1. +apply Rle_0_sqr. +pattern 1 at 1 in |- *; rewrite <- Rsqr_1; apply Rsqr_incr_1. +pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_le_compat_l; left; + assumption. +left; apply Rlt_0_1. +apply H0. +apply Rle_ge; left; apply Rplus_lt_reg_r with 1. +rewrite Rplus_0_r; rewrite Rplus_comm; unfold Rminus in |- *; + rewrite Rplus_assoc; rewrite Rplus_opp_l; rewrite Rplus_0_r. +pattern 1 at 1 in |- *; rewrite <- sqrt_1; apply sqrt_lt_1. +left; apply Rlt_0_1. +apply H0. +pattern 1 at 1 in |- *; rewrite <- Rplus_0_r; apply Rplus_lt_compat_l; + assumption. +rewrite sqrt_Rsqr. +replace (1 + h - 1) with h; [ right; reflexivity | ring ]. +apply H0. +case (total_order_T h 0); intro. +elim s; intro. +rewrite (Rabs_left h a) in H. +apply Rplus_le_reg_l with (- h). +rewrite Rplus_0_r; rewrite Rplus_comm; rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; exact H. +left; rewrite b; rewrite Rplus_0_r; apply Rlt_0_1. +left; apply Rplus_lt_0_compat. +apply Rlt_0_1. +apply r. Qed. (* sqrt is continuous in 1 *) -Lemma sqrt_continuity_pt_R1 : (continuity_pt sqrt R1). -Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. -Pose alpha := (Rmin eps R1). -Exists alpha; Intros. -Split. -Unfold alpha; Unfold Rmin; Case (total_order_Rle eps R1); Intro. -Assumption. -Apply Rlt_R0_R1. -Intros; Elim H0; Intros. -Rewrite sqrt_1; Replace x with ``1+(x-1)``; [Idtac | Ring]; Apply Rle_lt_trans with ``(Rabsolu (x-1))``. -Apply sqrt_var_maj. -Apply Rle_trans with alpha. -Left; Apply H2. -Unfold alpha; Apply Rmin_r. -Apply Rlt_le_trans with alpha; [Apply H2 | Unfold alpha; Apply Rmin_l]. +Lemma sqrt_continuity_pt_R1 : continuity_pt sqrt 1. +unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + intros. +pose (alpha := Rmin eps 1). +exists alpha; intros. +split. +unfold alpha in |- *; unfold Rmin in |- *; case (Rle_dec eps 1); intro. +assumption. +apply Rlt_0_1. +intros; elim H0; intros. +rewrite sqrt_1; replace x with (1 + (x - 1)); [ idtac | ring ]; + apply Rle_lt_trans with (Rabs (x - 1)). +apply sqrt_var_maj. +apply Rle_trans with alpha. +left; apply H2. +unfold alpha in |- *; apply Rmin_r. +apply Rlt_le_trans with alpha; + [ apply H2 | unfold alpha in |- *; apply Rmin_l ]. Qed. (* sqrt is continuous forall x>0 *) -Lemma sqrt_continuity_pt : (x:R) ``0<x`` -> (continuity_pt sqrt x). -Intros; Generalize sqrt_continuity_pt_R1. -Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Unfold dist; Simpl; Unfold R_dist; Intros. -Cut ``0<eps/(sqrt x)``. -Intro; Elim (H0 ? H2); Intros alp_1 H3. -Elim H3; Intros. -Pose alpha := ``alp_1*x``. -Exists (Rmin alpha x); Intros. -Split. -Change ``0<(Rmin alpha x)``; Unfold Rmin; Case (total_order_Rle alpha x); Intro. -Unfold alpha; Apply Rmult_lt_pos; Assumption. -Apply H. -Intros; Replace x0 with ``x+(x0-x)``; [Idtac | Ring]; Replace ``(sqrt (x+(x0-x)))-(sqrt x)`` with ``(sqrt x)*((sqrt (1+(x0-x)/x))-(sqrt 1))``. -Rewrite Rabsolu_mult; Rewrite (Rabsolu_right (sqrt x)). -Apply Rlt_monotony_contra with ``/(sqrt x)``. -Apply Rlt_Rinv; Apply sqrt_lt_R0; Assumption. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Rewrite Rmult_sym. -Unfold Rdiv in H5. -Case (Req_EM x x0); Intro. -Rewrite H7; Unfold Rminus Rdiv; Rewrite Rplus_Ropp_r; Rewrite Rmult_Ol; Rewrite Rplus_Or; Rewrite Rplus_Ropp_r; Rewrite Rabsolu_R0. -Apply Rmult_lt_pos. -Assumption. -Apply Rlt_Rinv; Rewrite <- H7; Apply sqrt_lt_R0; Assumption. -Apply H5. -Split. -Unfold D_x no_cond. -Split. -Trivial. -Red; Intro. -Cut ``(x0-x)*/x==0``. -Intro. -Elim (without_div_Od ? ? H9); Intro. -Elim H7. -Apply (Rminus_eq_right ? ? H10). -Assert H11 := (without_div_Oi1 ? x H10). -Rewrite <- Rinv_l_sym in H11. -Elim R1_neq_R0; Exact H11. -Red; Intro; Rewrite H12 in H; Elim (Rlt_antirefl ? H). -Symmetry; Apply r_Rplus_plus with R1; Rewrite Rplus_Or; Unfold Rdiv in H8; Exact H8. -Unfold Rminus; Rewrite Rplus_sym; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Elim H6; Intros. -Unfold Rdiv; Rewrite Rabsolu_mult. -Rewrite Rabsolu_Rinv. -Rewrite (Rabsolu_right x). -Rewrite Rmult_sym; Apply Rlt_monotony_contra with x. -Apply H. -Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite Rmult_sym; Fold alpha. -Apply Rlt_le_trans with (Rmin alpha x). -Apply H9. -Apply Rmin_l. -Red; Intro; Rewrite H10 in H; Elim (Rlt_antirefl ? H). -Apply Rle_sym1; Left; Apply H. -Red; Intro; Rewrite H10 in H; Elim (Rlt_antirefl ? H). -Assert H7 := (sqrt_lt_R0 x H). -Red; Intro; Rewrite H8 in H7; Elim (Rlt_antirefl ? H7). -Apply Rle_sym1; Apply sqrt_positivity. -Left; Apply H. -Unfold Rminus; Rewrite Rmult_Rplus_distr; Rewrite Ropp_mul3; Repeat Rewrite <- sqrt_times. -Rewrite Rmult_1r; Rewrite Rmult_Rplus_distr; Rewrite Rmult_1r; Unfold Rdiv; Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Reflexivity. -Red; Intro; Rewrite H7 in H; Elim (Rlt_antirefl ? H). -Left; Apply H. -Left; Apply Rlt_R0_R1. -Left; Apply H. -Elim H6; Intros. -Case (case_Rabsolu ``x0-x``); Intro. -Rewrite (Rabsolu_left ``x0-x`` r) in H8. -Rewrite Rplus_sym. -Apply Rle_anti_compatibility with ``-((x0-x)/x)``. -Rewrite Rplus_Or; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Unfold Rdiv; Rewrite <- Ropp_mul1. -Apply Rle_monotony_contra with x. -Apply H. -Rewrite Rmult_1r; Rewrite Rmult_sym; Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Left; Apply Rlt_le_trans with (Rmin alpha x). -Apply H8. -Apply Rmin_r. -Red; Intro; Rewrite H9 in H; Elim (Rlt_antirefl ? H). -Apply ge0_plus_ge0_is_ge0. -Left; Apply Rlt_R0_R1. -Unfold Rdiv; Apply Rmult_le_pos. -Apply Rle_sym2; Exact r. -Left; Apply Rlt_Rinv; Apply H. -Unfold Rdiv; Apply Rmult_lt_pos. -Apply H1. -Apply Rlt_Rinv; Apply sqrt_lt_R0; Apply H. +Lemma sqrt_continuity_pt : forall x:R, 0 < x -> continuity_pt sqrt x. +intros; generalize sqrt_continuity_pt_R1. +unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + unfold dist in |- *; simpl in |- *; unfold R_dist in |- *; + intros. +cut (0 < eps / sqrt x). +intro; elim (H0 _ H2); intros alp_1 H3. +elim H3; intros. +pose (alpha := alp_1 * x). +exists (Rmin alpha x); intros. +split. +change (0 < Rmin alpha x) in |- *; unfold Rmin in |- *; + case (Rle_dec alpha x); intro. +unfold alpha in |- *; apply Rmult_lt_0_compat; assumption. +apply H. +intros; replace x0 with (x + (x0 - x)); [ idtac | ring ]; + replace (sqrt (x + (x0 - x)) - sqrt x) with + (sqrt x * (sqrt (1 + (x0 - x) / x) - sqrt 1)). +rewrite Rabs_mult; rewrite (Rabs_right (sqrt x)). +apply Rmult_lt_reg_l with (/ sqrt x). +apply Rinv_0_lt_compat; apply sqrt_lt_R0; assumption. +rewrite <- Rmult_assoc; rewrite <- Rinv_l_sym. +rewrite Rmult_1_l; rewrite Rmult_comm. +unfold Rdiv in H5. +case (Req_dec x x0); intro. +rewrite H7; unfold Rminus, Rdiv in |- *; rewrite Rplus_opp_r; + rewrite Rmult_0_l; rewrite Rplus_0_r; rewrite Rplus_opp_r; + rewrite Rabs_R0. +apply Rmult_lt_0_compat. +assumption. +apply Rinv_0_lt_compat; rewrite <- H7; apply sqrt_lt_R0; assumption. +apply H5. +split. +unfold D_x, no_cond in |- *. +split. +trivial. +red in |- *; intro. +cut ((x0 - x) * / x = 0). +intro. +elim (Rmult_integral _ _ H9); intro. +elim H7. +apply (Rminus_diag_uniq_sym _ _ H10). +assert (H11 := Rmult_eq_0_compat_r _ x H10). +rewrite <- Rinv_l_sym in H11. +elim R1_neq_R0; exact H11. +red in |- *; intro; rewrite H12 in H; elim (Rlt_irrefl _ H). +symmetry in |- *; apply Rplus_eq_reg_l with 1; rewrite Rplus_0_r; + unfold Rdiv in H8; exact H8. +unfold Rminus in |- *; rewrite Rplus_comm; rewrite <- Rplus_assoc; + rewrite Rplus_opp_l; rewrite Rplus_0_l; elim H6; intros. +unfold Rdiv in |- *; rewrite Rabs_mult. +rewrite Rabs_Rinv. +rewrite (Rabs_right x). +rewrite Rmult_comm; apply Rmult_lt_reg_l with x. +apply H. +rewrite <- Rmult_assoc; rewrite <- Rinv_r_sym. +rewrite Rmult_1_l; rewrite Rmult_comm; fold alpha in |- *. +apply Rlt_le_trans with (Rmin alpha x). +apply H9. +apply Rmin_l. +red in |- *; intro; rewrite H10 in H; elim (Rlt_irrefl _ H). +apply Rle_ge; left; apply H. +red in |- *; intro; rewrite H10 in H; elim (Rlt_irrefl _ H). +assert (H7 := sqrt_lt_R0 x H). +red in |- *; intro; rewrite H8 in H7; elim (Rlt_irrefl _ H7). +apply Rle_ge; apply sqrt_positivity. +left; apply H. +unfold Rminus in |- *; rewrite Rmult_plus_distr_l; + rewrite Ropp_mult_distr_r_reverse; repeat rewrite <- sqrt_mult. +rewrite Rmult_1_r; rewrite Rmult_plus_distr_l; rewrite Rmult_1_r; + unfold Rdiv in |- *; rewrite Rmult_comm; rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; reflexivity. +red in |- *; intro; rewrite H7 in H; elim (Rlt_irrefl _ H). +left; apply H. +left; apply Rlt_0_1. +left; apply H. +elim H6; intros. +case (Rcase_abs (x0 - x)); intro. +rewrite (Rabs_left (x0 - x) r) in H8. +rewrite Rplus_comm. +apply Rplus_le_reg_l with (- ((x0 - x) / x)). +rewrite Rplus_0_r; rewrite <- Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_l; unfold Rdiv in |- *; rewrite <- Ropp_mult_distr_l_reverse. +apply Rmult_le_reg_l with x. +apply H. +rewrite Rmult_1_r; rewrite Rmult_comm; rewrite Rmult_assoc; + rewrite <- Rinv_l_sym. +rewrite Rmult_1_r; left; apply Rlt_le_trans with (Rmin alpha x). +apply H8. +apply Rmin_r. +red in |- *; intro; rewrite H9 in H; elim (Rlt_irrefl _ H). +apply Rplus_le_le_0_compat. +left; apply Rlt_0_1. +unfold Rdiv in |- *; apply Rmult_le_pos. +apply Rge_le; exact r. +left; apply Rinv_0_lt_compat; apply H. +unfold Rdiv in |- *; apply Rmult_lt_0_compat. +apply H1. +apply Rinv_0_lt_compat; apply sqrt_lt_R0; apply H. Qed. (* sqrt is derivable for all x>0 *) -Lemma derivable_pt_lim_sqrt : (x:R) ``0<x`` -> (derivable_pt_lim sqrt x ``/(2*(sqrt x))``). -Intros; Pose g := [h:R]``(sqrt x)+(sqrt (x+h))``. -Cut (continuity_pt g R0). -Intro; Cut ``(g 0)<>0``. -Intro; Assert H2 := (continuity_pt_inv g R0 H0 H1). -Unfold derivable_pt_lim; Intros; Unfold continuity_pt in H2; Unfold continue_in in H2; Unfold limit1_in in H2; Unfold limit_in in H2; Simpl in H2; Unfold R_dist in H2. -Elim (H2 eps H3); Intros alpha H4. -Elim H4; Intros. -Pose alpha1 := (Rmin alpha x). -Cut ``0<alpha1``. -Intro; Exists (mkposreal alpha1 H7); Intros. -Replace ``((sqrt (x+h))-(sqrt x))/h`` with ``/((sqrt x)+(sqrt (x+h)))``. -Unfold inv_fct g in H6; Replace ``2*(sqrt x)`` with ``(sqrt x)+(sqrt (x+0))``. -Apply H6. -Split. -Unfold D_x no_cond. -Split. -Trivial. -Apply not_sym; Exact H8. -Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Apply Rlt_le_trans with alpha1. -Exact H9. -Unfold alpha1; Apply Rmin_l. -Rewrite Rplus_Or; Ring. -Cut ``0<=x+h``. -Intro; Cut ``0<(sqrt x)+(sqrt (x+h))``. -Intro; Apply r_Rmult_mult with ``((sqrt x)+(sqrt (x+h)))``. -Rewrite <- Rinv_r_sym. -Rewrite Rplus_sym; Unfold Rdiv; Rewrite <- Rmult_assoc; Rewrite Rsqr_plus_minus; Repeat Rewrite Rsqr_sqrt. -Rewrite Rplus_sym; Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Rewrite <- Rinv_r_sym. -Reflexivity. -Apply H8. -Left; Apply H. -Assumption. -Red; Intro; Rewrite H12 in H11; Elim (Rlt_antirefl ? H11). -Red; Intro; Rewrite H12 in H11; Elim (Rlt_antirefl ? H11). -Apply gt0_plus_ge0_is_gt0. -Apply sqrt_lt_R0; Apply H. -Apply sqrt_positivity; Apply H10. -Case (case_Rabsolu h); Intro. -Rewrite (Rabsolu_left h r) in H9. -Apply Rle_anti_compatibility with ``-h``. -Rewrite Rplus_Or; Rewrite Rplus_sym; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Or; Left; Apply Rlt_le_trans with alpha1. -Apply H9. -Unfold alpha1; Apply Rmin_r. -Apply ge0_plus_ge0_is_ge0. -Left; Assumption. -Apply Rle_sym2; Apply r. -Unfold alpha1; Unfold Rmin; Case (total_order_Rle alpha x); Intro. -Apply H5. -Apply H. -Unfold g; Rewrite Rplus_Or. -Cut ``0<(sqrt x)+(sqrt x)``. -Intro; Red; Intro; Rewrite H2 in H1; Elim (Rlt_antirefl ? H1). -Apply gt0_plus_gt0_is_gt0; Apply sqrt_lt_R0; Apply H. -Replace g with (plus_fct (fct_cte (sqrt x)) (comp sqrt (plus_fct (fct_cte x) id))); [Idtac | Reflexivity]. -Apply continuity_pt_plus. -Apply continuity_pt_const; Unfold constant fct_cte; Intro; Reflexivity. -Apply continuity_pt_comp. -Apply continuity_pt_plus. -Apply continuity_pt_const; Unfold constant fct_cte; Intro; Reflexivity. -Apply derivable_continuous_pt; Apply derivable_pt_id. -Apply sqrt_continuity_pt. -Unfold plus_fct fct_cte id; Rewrite Rplus_Or; Apply H. +Lemma derivable_pt_lim_sqrt : + forall x:R, 0 < x -> derivable_pt_lim sqrt x (/ (2 * sqrt x)). +intros; pose (g := fun h:R => sqrt x + sqrt (x + h)). +cut (continuity_pt g 0). +intro; cut (g 0 <> 0). +intro; assert (H2 := continuity_pt_inv g 0 H0 H1). +unfold derivable_pt_lim in |- *; intros; unfold continuity_pt in H2; + unfold continue_in in H2; unfold limit1_in in H2; + unfold limit_in in H2; simpl in H2; unfold R_dist in H2. +elim (H2 eps H3); intros alpha H4. +elim H4; intros. +pose (alpha1 := Rmin alpha x). +cut (0 < alpha1). +intro; exists (mkposreal alpha1 H7); intros. +replace ((sqrt (x + h) - sqrt x) / h) with (/ (sqrt x + sqrt (x + h))). +unfold inv_fct, g in H6; replace (2 * sqrt x) with (sqrt x + sqrt (x + 0)). +apply H6. +split. +unfold D_x, no_cond in |- *. +split. +trivial. +apply (sym_not_eq (A:=R)); exact H8. +unfold Rminus in |- *; rewrite Ropp_0; rewrite Rplus_0_r; + apply Rlt_le_trans with alpha1. +exact H9. +unfold alpha1 in |- *; apply Rmin_l. +rewrite Rplus_0_r; ring. +cut (0 <= x + h). +intro; cut (0 < sqrt x + sqrt (x + h)). +intro; apply Rmult_eq_reg_l with (sqrt x + sqrt (x + h)). +rewrite <- Rinv_r_sym. +rewrite Rplus_comm; unfold Rdiv in |- *; rewrite <- Rmult_assoc; + rewrite Rsqr_plus_minus; repeat rewrite Rsqr_sqrt. +rewrite Rplus_comm; unfold Rminus in |- *; rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; rewrite <- Rinv_r_sym. +reflexivity. +apply H8. +left; apply H. +assumption. +red in |- *; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11). +red in |- *; intro; rewrite H12 in H11; elim (Rlt_irrefl _ H11). +apply Rplus_lt_le_0_compat. +apply sqrt_lt_R0; apply H. +apply sqrt_positivity; apply H10. +case (Rcase_abs h); intro. +rewrite (Rabs_left h r) in H9. +apply Rplus_le_reg_l with (- h). +rewrite Rplus_0_r; rewrite Rplus_comm; rewrite Rplus_assoc; + rewrite Rplus_opp_r; rewrite Rplus_0_r; left; apply Rlt_le_trans with alpha1. +apply H9. +unfold alpha1 in |- *; apply Rmin_r. +apply Rplus_le_le_0_compat. +left; assumption. +apply Rge_le; apply r. +unfold alpha1 in |- *; unfold Rmin in |- *; case (Rle_dec alpha x); intro. +apply H5. +apply H. +unfold g in |- *; rewrite Rplus_0_r. +cut (0 < sqrt x + sqrt x). +intro; red in |- *; intro; rewrite H2 in H1; elim (Rlt_irrefl _ H1). +apply Rplus_lt_0_compat; apply sqrt_lt_R0; apply H. +replace g with (fct_cte (sqrt x) + comp sqrt (fct_cte x + id))%F; + [ idtac | reflexivity ]. +apply continuity_pt_plus. +apply continuity_pt_const; unfold constant, fct_cte in |- *; intro; + reflexivity. +apply continuity_pt_comp. +apply continuity_pt_plus. +apply continuity_pt_const; unfold constant, fct_cte in |- *; intro; + reflexivity. +apply derivable_continuous_pt; apply derivable_pt_id. +apply sqrt_continuity_pt. +unfold plus_fct, fct_cte, id in |- *; rewrite Rplus_0_r; apply H. Qed. (**********) -Lemma derivable_pt_sqrt : (x:R) ``0<x`` -> (derivable_pt sqrt x). -Unfold derivable_pt; Intros. -Apply Specif.existT with ``/(2*(sqrt x))``. -Apply derivable_pt_lim_sqrt; Assumption. +Lemma derivable_pt_sqrt : forall x:R, 0 < x -> derivable_pt sqrt x. +unfold derivable_pt in |- *; intros. +apply existT with (/ (2 * sqrt x)). +apply derivable_pt_lim_sqrt; assumption. Qed. (**********) -Lemma derive_pt_sqrt : (x:R;pr:``0<x``) ``(derive_pt sqrt x (derivable_pt_sqrt ? pr)) == /(2*(sqrt x))``. -Intros. -Apply derive_pt_eq_0. -Apply derivable_pt_lim_sqrt; Assumption. +Lemma derive_pt_sqrt : + forall (x:R) (pr:0 < x), + derive_pt sqrt x (derivable_pt_sqrt _ pr) = / (2 * sqrt x). +intros. +apply derive_pt_eq_0. +apply derivable_pt_lim_sqrt; assumption. Qed. (* We show that sqrt is continuous for all x>=0 *) (* Remark : by definition of sqrt (as extension of Rsqrt on |R), *) (* we could also show that sqrt is continuous for all x *) -Lemma continuity_pt_sqrt : (x:R) ``0<=x`` -> (continuity_pt sqrt x). -Intros; Case (total_order R0 x); Intro. -Apply (sqrt_continuity_pt x H0). -Elim H0; Intro. -Unfold continuity_pt; Unfold continue_in; Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros. -Exists (Rsqr eps); Intros. -Split. -Change ``0<(Rsqr eps)``; Apply Rsqr_pos_lt. -Red; Intro; Rewrite H3 in H2; Elim (Rlt_antirefl ? H2). -Intros; Elim H3; Intros. -Rewrite <- H1; Rewrite sqrt_0; Unfold Rminus; Rewrite Ropp_O; Rewrite Rplus_Or; Rewrite <- H1 in H5; Unfold Rminus in H5; Rewrite Ropp_O in H5; Rewrite Rplus_Or in H5. -Case (case_Rabsolu x0); Intro. -Unfold sqrt; Case (case_Rabsolu x0); Intro. -Rewrite Rabsolu_R0; Apply H2. -Assert H6 := (Rle_sym2 ? ? r0); Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H6 r)). -Rewrite Rabsolu_right. -Apply Rsqr_incrst_0. -Rewrite Rsqr_sqrt. -Rewrite (Rabsolu_right x0 r) in H5; Apply H5. -Apply Rle_sym2; Exact r. -Apply sqrt_positivity; Apply Rle_sym2; Exact r. -Left; Exact H2. -Apply Rle_sym1; Apply sqrt_positivity; Apply Rle_sym2; Exact r. -Elim (Rlt_antirefl ? (Rlt_le_trans ? ? ? H1 H)). -Qed. +Lemma continuity_pt_sqrt : forall x:R, 0 <= x -> continuity_pt sqrt x. +intros; case (Rtotal_order 0 x); intro. +apply (sqrt_continuity_pt x H0). +elim H0; intro. +unfold continuity_pt in |- *; unfold continue_in in |- *; + unfold limit1_in in |- *; unfold limit_in in |- *; + simpl in |- *; unfold R_dist in |- *; intros. +exists (Rsqr eps); intros. +split. +change (0 < Rsqr eps) in |- *; apply Rsqr_pos_lt. +red in |- *; intro; rewrite H3 in H2; elim (Rlt_irrefl _ H2). +intros; elim H3; intros. +rewrite <- H1; rewrite sqrt_0; unfold Rminus in |- *; rewrite Ropp_0; + rewrite Rplus_0_r; rewrite <- H1 in H5; unfold Rminus in H5; + rewrite Ropp_0 in H5; rewrite Rplus_0_r in H5. +case (Rcase_abs x0); intro. +unfold sqrt in |- *; case (Rcase_abs x0); intro. +rewrite Rabs_R0; apply H2. +assert (H6 := Rge_le _ _ r0); elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H6 r)). +rewrite Rabs_right. +apply Rsqr_incrst_0. +rewrite Rsqr_sqrt. +rewrite (Rabs_right x0 r) in H5; apply H5. +apply Rge_le; exact r. +apply sqrt_positivity; apply Rge_le; exact r. +left; exact H2. +apply Rle_ge; apply sqrt_positivity; apply Rge_le; exact r. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H1 H)). +Qed.
\ No newline at end of file |