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diff --git a/theories7/Reals/PartSum.v b/theories7/Reals/PartSum.v deleted file mode 100644 index 4d28bec8..00000000 --- a/theories7/Reals/PartSum.v +++ /dev/null @@ -1,475 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(*i $Id: PartSum.v,v 1.1.2.2 2005/07/13 23:19:16 herbelin Exp $ i*) - -Require Rbase. -Require Rfunctions. -Require Rseries. -Require Rcomplete. -Require Max. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. -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. -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]. -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. -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. -Qed. - -Lemma tech5 : (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. -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. -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. -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. -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. -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]. -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. -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]. -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. -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. -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. -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]. -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. -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. -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. -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. -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. -Qed. - -(* Cauchy's criterion for series *) -Definition Cauchy_crit_series [An:nat->R] : Prop := (Cauchy_crit [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. -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]. -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. -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]]. -Qed. - -Definition SP [fn:nat->R->R;N:nat] : R->R := [x:R](sum_f_R0 [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; LetTac l1 := (sum_f_R0 An N) 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. -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. |