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diff --git a/theories7/Reals/Alembert.v b/theories7/Reals/Alembert.v new file mode 100644 index 00000000..702daffc --- /dev/null +++ b/theories7/Reals/Alembert.v @@ -0,0 +1,549 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Alembert.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rseries. +Require SeqProp. +Require PartSum. +Require Max. + +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]. +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. +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]. +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]. +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). +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]. +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. +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. |