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Diffstat (limited to 'theories7/Reals')
54 files changed, 25174 insertions, 0 deletions
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. diff --git a/theories7/Reals/AltSeries.v b/theories7/Reals/AltSeries.v new file mode 100644 index 00000000..af4b558a --- /dev/null +++ b/theories7/Reals/AltSeries.v @@ -0,0 +1,362 @@ +(************************************************************************) +(* 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: AltSeries.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. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +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)``. + +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. +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. +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. +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. +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. +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. +Qed. + +(************************************************) +(* Convergence of alternated series *) +(* *) +(* 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. +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. +Qed. + +(************************************) +(* Application : construction of PI *) +(************************************) + +Definition PI_tg := [n:nat]``/(INR (plus (mult (S (S O)) n) (S O)))``. + +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]. +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]. +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. +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. +Qed. + +(* Now, PI is defined *) +Definition PI : R := (Rmult ``4`` (Cases exist_PI of (existTT 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]. +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. diff --git a/theories7/Reals/ArithProp.v b/theories7/Reals/ArithProp.v new file mode 100644 index 00000000..468675ca --- /dev/null +++ b/theories7/Reals/ArithProp.v @@ -0,0 +1,134 @@ +(************************************************************************) +(* 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: ArithProp.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) + +Require Rbase. +Require Rbasic_fun. +Require Even. +Require Div2. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +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. +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. +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]. +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. +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. +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)). +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. diff --git a/theories7/Reals/Binomial.v b/theories7/Reals/Binomial.v new file mode 100644 index 00000000..1dfd2ec0 --- /dev/null +++ b/theories7/Reals/Binomial.v @@ -0,0 +1,181 @@ +(************************************************************************) +(* 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: Binomial.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require PartSum. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Definition C [n,p:nat] : R := ``(INR (fact n))/((INR (fact p))*(INR (fact (minus 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. +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. +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. +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. +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. +(* 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. diff --git a/theories7/Reals/Cauchy_prod.v b/theories7/Reals/Cauchy_prod.v new file mode 100644 index 00000000..9442eff0 --- /dev/null +++ b/theories7/Reals/Cauchy_prod.v @@ -0,0 +1,347 @@ +(************************************************************************) +(* 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: Cauchy_prod.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rseries. +Require PartSum. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +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. +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. +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. diff --git a/theories7/Reals/Cos_plus.v b/theories7/Reals/Cos_plus.v new file mode 100644 index 00000000..481e51bf --- /dev/null +++ b/theories7/Reals/Cos_plus.v @@ -0,0 +1,1017 @@ +(************************************************************************) +(* 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: Cos_plus.v,v 1.1.2.1 2004/07/16 19:31:31 herbelin Exp $ 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. + +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))). + +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. +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. +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. +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]. +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. +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. +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. diff --git a/theories7/Reals/Cos_rel.v b/theories7/Reals/Cos_rel.v new file mode 100644 index 00000000..e29825ab --- /dev/null +++ b/theories7/Reals/Cos_rel.v @@ -0,0 +1,360 @@ +(************************************************************************) +(* 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: Cos_rel.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo_def. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +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 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 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 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 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 Reste [x,y:R] : nat -> R := [N:nat]``(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. +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. +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. +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. +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. diff --git a/theories7/Reals/DiscrR.v b/theories7/Reals/DiscrR.v new file mode 100644 index 00000000..31c90727 --- /dev/null +++ b/theories7/Reals/DiscrR.v @@ -0,0 +1,58 @@ +(************************************************************************) +(* 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: DiscrR.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*) + +Require RIneq. +Require Omega. +V7only [Import R_scope.]. 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]. +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. +Qed. + +Lemma IZR_eq : (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. +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]. + +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. + +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]. + +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]. diff --git a/theories7/Reals/Exp_prop.v b/theories7/Reals/Exp_prop.v new file mode 100644 index 00000000..6ed9c00b --- /dev/null +++ b/theories7/Reals/Exp_prop.v @@ -0,0 +1,890 @@ +(************************************************************************) +(* 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: Exp_prop.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo. +Require Ranalysis1. +Require PSeries_reg. +Require Div2. +Require Even. +Require Max. +V7only [Import R_scope.]. +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). + +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. +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)). + +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. +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)))))``. + +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). +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. +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. +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]. +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. +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. +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. +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. +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. +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. +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). +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. diff --git a/theories7/Reals/Integration.v b/theories7/Reals/Integration.v new file mode 100644 index 00000000..410429ed --- /dev/null +++ b/theories7/Reals/Integration.v @@ -0,0 +1,13 @@ +(************************************************************************) +(* 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: Integration.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*) + +Require Export NewtonInt. +Require Export RiemannInt_SF. +Require Export RiemannInt.
\ No newline at end of file diff --git a/theories7/Reals/MVT.v b/theories7/Reals/MVT.v new file mode 100644 index 00000000..eae414b1 --- /dev/null +++ b/theories7/Reals/MVT.v @@ -0,0 +1,517 @@ +(************************************************************************) +(* 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: MVT.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Ranalysis1. +Require Rtopology. +V7only [Import R_scope.]. 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``. +(*** 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``. +(*** 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``. +(*** 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). +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. +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. +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]]]. +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)]. +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)). +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]. +Qed. + +(**********) +Lemma increasing_decreasing_opp : (f:R->R) (increasing f) -> (decreasing (opp_fct f)). +Unfold increasing decreasing opp_fct; Intros; Generalize (H x y H0); Intro; Apply Rge_Ropp; Apply Rle_sym1; Assumption. +Qed. + +(**********) +Lemma 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. +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]. +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. +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. +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. +Qed. + +(**********) +Lemma increasing_decreasing : (f:R->R) (increasing f) -> (decreasing f) -> (constant f). +Unfold increasing decreasing constant; Intros; Case (total_order x y); Intro. +Generalize (Rlt_le x y H1); Intro; Apply (Rle_antisym (f x) (f y) (H x y H2) (H0 x y H2)). +Elim H1; Intro. +Rewrite H2; Reflexivity. +Generalize (Rlt_le y x H2); Intro; Symmetry; Apply (Rle_antisym (f y) (f x) (H y x H3) (H0 y x H3)). +Qed. + +(**********) +Lemma null_derivative_1 : (f:R->R;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). +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]. +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). +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). +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)). +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. +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)). +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. diff --git a/theories7/Reals/NewtonInt.v b/theories7/Reals/NewtonInt.v new file mode 100644 index 00000000..56e5f15e --- /dev/null +++ b/theories7/Reals/NewtonInt.v @@ -0,0 +1,600 @@ +(************************************************************************) +(* 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: NewtonInt.v,v 1.1.2.1 2004/07/16 19:31:32 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo. +Require Ranalysis. +V7only [Import R_scope.]. 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 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)``. + +(* 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]]. +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. +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. +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. +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. +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. +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. +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. +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. +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. +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. +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. +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)). +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. +(* 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)). +(* a<b & b=c *) +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)). +(* a=b *) +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. +(*****************) +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)). +(* a>b & b=c *) +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. +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. +(* 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)). +(* 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)). +(* 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)). +(* 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]. +(* 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)). +(* 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]. +(* 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]. +Qed. + diff --git a/theories7/Reals/PSeries_reg.v b/theories7/Reals/PSeries_reg.v new file mode 100644 index 00000000..68645379 --- /dev/null +++ b/theories7/Reals/PSeries_reg.v @@ -0,0 +1,194 @@ +(************************************************************************) +(* 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: PSeries_reg.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Ranalysis1. +Require Max. +Require Even. +V7only [Import R_scope.]. Open Local Scope R_scope. + +Definition Boule [x:R;r:posreal] : R -> Prop := [y:R]``(Rabsolu (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``). + +(* 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] : Type := (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). + +(* 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. +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]. +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. +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. +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. +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. diff --git a/theories7/Reals/PartSum.v b/theories7/Reals/PartSum.v new file mode 100644 index 00000000..ee5fa498 --- /dev/null +++ b/theories7/Reals/PartSum.v @@ -0,0 +1,476 @@ +(************************************************************************) +(* 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 1.1.2.1 2004/07/16 19:31:33 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; 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. +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. diff --git a/theories7/Reals/RIneq.v b/theories7/Reals/RIneq.v new file mode 100644 index 00000000..00d41c70 --- /dev/null +++ b/theories7/Reals/RIneq.v @@ -0,0 +1,1631 @@ +(************************************************************************) +(* 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: RIneq.v,v 1.2.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +(***************************************************************************) +(** Basic lemmas for the classical reals numbers *) +(***************************************************************************) + +Require Export Raxioms. +Require Export ZArithRing. +Require Omega. +Require Export Field. + +Open Local Scope Z_scope. +Open Local Scope R_scope. + +Implicit Variable 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. +Defined. + +Add Field R Rplus Rmult R1 R0 Ropp [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. +Qed. +Hints Resolve Rlt_antirefl : real. + +Lemma Rle_refl : (x:R) ``x<=x``. +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. +Qed. + +Lemma Rgt_not_eq:(r1,r2:R)``r1>r2``->``r1<>r2``. +Intros; Apply sym_not_eqT; 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. +Qed. +Hints Resolve imp_not_Req : 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. +Qed. +Hints Resolve Req_EM : real. + +(**********) +Lemma total_order:(r1,r2:R)``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. +Qed. + + +(*********************************************************************************) +(** Order Lemma : relating [<], [>], [<=] and [>=] *) +(*********************************************************************************) + +(**********) +Lemma Rlt_le:(r1,r2:R)``r1<r2``-> ``r1<=r2``. +Intros ; Red ; Tauto. +Qed. +Hints Resolve Rlt_le : real. + +(**********) +Lemma Rle_ge : (r1,r2:R)``r1<=r2`` -> ``r2>=r1``. +NewDestruct 1; Red; Auto with real. +Qed. + +Hints Immediate Rle_ge : real. + +(**********) +Lemma Rge_le : (r1,r2:R)``r1>=r2`` -> ``r2<=r1``. +NewDestruct 1; Red; Auto with real. +Qed. + +Hints 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. +Qed. + +Hints Immediate not_Rle : real. + +Lemma not_Rge:(r1,r2:R)~``r1>=r2`` -> ``r1<r2``. +Intros; Apply not_Rle; 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. +Qed. + +Lemma Rle_not:(r1,r2:R)``r1>r2`` -> ~``r1<=r2``. +Proof Rlt_le_not. + +Hints Immediate Rlt_le_not : 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. +Qed. + +(**********) +Lemma Rlt_ge_not:(r1,r2:R)``r1<r2`` -> ~``r1>=r2``. +Generalize Rlt_le_not. Unfold Rle Rge. Intuition EAuto 3. +Qed. + +Hints Immediate Rlt_ge_not : real. + +(**********) +Lemma eq_Rle:(r1,r2:R)r1==r2->``r1<=r2``. +Unfold Rle; Tauto. +Qed. +Hints Immediate eq_Rle : real. + +Lemma eq_Rge:(r1,r2:R)r1==r2->``r1>=r2``. +Unfold Rge; Tauto. +Qed. +Hints Immediate eq_Rge : real. + +Lemma eq_Rle_sym:(r1,r2:R)r2==r1->``r1<=r2``. +Unfold Rle; Auto. +Qed. +Hints Immediate eq_Rle_sym : real. + +Lemma eq_Rge_sym:(r1,r2:R)r2==r1->``r1>=r2``. +Unfold Rge; Auto. +Qed. +Hints Immediate eq_Rge_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. +Qed. +Hints Resolve Rle_antisym : real. + +(**********) +Lemma Rle_le_eq:(r1,r2:R)(``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. +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. +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. +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. +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. +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. +Qed. + +(**********) +Lemma total_order_Rgt:(r1,r2:R)(sumboolT ``r1>r2`` ~(``r1>r2``)). +Intros;Unfold Rgt;Intros;Apply total_order_Rlt. +Qed. + +(**********) +Lemma total_order_Rge:(r1,r2:R)(sumboolT (``r1>=r2``) ~(``r1>=r2``)). +Intros;Generalize (total_order_Rle 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. +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. +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. +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. +Qed. + +(****************************************************************) +(** Field Lemmas *) +(* This part contains lemma involving the Fields operations *) +(****************************************************************) +(*********************************************************) +(** Addition *) +(*********************************************************) + +Lemma Rplus_ne:(r:R)``r+0==r``/\``0+r==r``. +Intro;Split;Ring. +Qed. +Hints Resolve Rplus_ne : real v62. + +Lemma Rplus_Or:(r:R)``r+0==r``. +Intro; Ring. +Qed. +Hints Resolve Rplus_Or : real. + +(**********) +Lemma Rplus_Ropp_l:(r:R)``(-r)+r==0``. + Intro; Ring. +Qed. +Hints Resolve Rplus_Ropp_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 ]. +Qed. + +(*i New i*) +Hint eqT_R_congr : real := Resolve (congr_eqT R). + +Lemma Rplus_plus_r:(r,r1,r2:R)(r1==r2)->``r+r1==r+r2``. + Auto with real. +Qed. + +(*i Old i*)Hints Resolve Rplus_plus_r : 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. +Qed. +Hints Resolve r_Rplus_plus : 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. +Qed. + +(***********************************************************) +(** Multiplication *) +(***********************************************************) + +(**********) +Lemma Rinv_r:(r:R)``r<>0``->``r* (/r)==1``. + Intros; Rewrite -> Rmult_sym; Auto with real. +Qed. +Hints Resolve Rinv_r : real. + +Lemma Rinv_l_sym:(r:R)``r<>0``->``1==(/r) * r``. + Symmetry; Auto with real. +Qed. + +Lemma Rinv_r_sym:(r:R)``r<>0``->``1==r* (/r)``. + Symmetry; Auto with real. +Qed. +Hints Resolve Rinv_l_sym Rinv_r_sym : real. + + +(**********) +Lemma Rmult_Or :(r:R) ``r*0==0``. +Intro; Ring. +Qed. +Hints Resolve Rmult_Or : real v62. + +(**********) +Lemma Rmult_Ol:(r:R) ``0*r==0``. +Intro; Ring. +Qed. +Hints Resolve Rmult_Ol : real v62. + +(**********) +Lemma Rmult_ne:(r:R)``r*1==r``/\``1*r==r``. +Intro;Split;Ring. +Qed. +Hints Resolve Rmult_ne : real v62. + +(**********) +Lemma Rmult_1r:(r:R)(``r*1==r``). +Intro; Ring. +Qed. +Hints Resolve Rmult_1r : real. + +(**********) +Lemma Rmult_mult_r:(r,r1,r2:R)r1==r2->``r*r1==r*r2``. + Auto with real. +Qed. + +(*i OLD i*)Hints Resolve Rmult_mult_r : 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. +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. +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. +Qed. + +Hints Resolve without_div_Oi : real. + +(**********) +Lemma without_div_Oi1:(r1,r2:R) ``r1==0`` -> ``r1*r2==0``. + Auto with real. +Qed. + +(**********) +Lemma without_div_Oi2:(r1,r2:R) ``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. +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. +Qed. +Hints Resolve mult_non_zero : real. + +(**********) +Lemma Rmult_Rplus_distrl: + (r1,r2,r3:R) ``(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).]. + +(***********) +Lemma Rsqr_O:(Rsqr ``0``)==``0``. + Unfold Rsqr; 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. +Qed. + +(*********************************************************) +(** Opposite *) +(*********************************************************) + +(**********) +Lemma eq_Ropp:(r1,r2:R)(r1==r2)->``-r1 == -r2``. + Auto with real. +Qed. +Hints Resolve eq_Ropp : real. + +(**********) +Lemma Ropp_O:``-0==0``. + Ring. +Qed. +Hints Resolve Ropp_O : real v62. + +(**********) +Lemma eq_RoppO:(r:R)``r==0``-> ``-r==0``. + Intros; Rewrite -> H; Auto with real. +Qed. +Hints Resolve eq_RoppO : real. + +(**********) +Lemma Ropp_Ropp:(r:R)``-(-r)==r``. + Intro; Ring. +Qed. +Hints Resolve Ropp_Ropp : real. + +(*********) +Lemma Ropp_neq:(r:R)``r<>0``->``-r<>0``. +Red;Intros r H H0. +Apply H. +Transitivity ``-(-r)``; Auto with real. +Qed. +Hints Resolve Ropp_neq : real. + +(**********) +Lemma Ropp_distr1:(r1,r2:R)``-(r1+r2)==(-r1 + -r2)``. + Intros; Ring. +Qed. +Hints Resolve Ropp_distr1 : real. + +(** Opposite and multiplication *) + +Lemma Ropp_mul1:(r1,r2:R)``(-r1)*r2 == -(r1*r2)``. + Intros; Ring. +Qed. +Hints Resolve Ropp_mul1 : real. + +(**********) +Lemma Ropp_mul2:(r1,r2:R)``(-r1)*(-r2)==r1*r2``. + Intros; Ring. +Qed. +Hints Resolve Ropp_mul2 : real. + +Lemma Ropp_mul3 : (r1,r2:R) ``r1*(-r2) == -(r1*r2)``. +Intros; Rewrite <- Ropp_mul1; Ring. +Qed. + +(** Substraction *) + +Lemma minus_R0:(r:R)``r-0==r``. +Intro;Ring. +Qed. +Hints Resolve minus_R0 : real. + +Lemma Rminus_Ropp:(r:R)``0-r==-r``. +Intro;Ring. +Qed. +Hints Resolve Rminus_Ropp : real. + +(**********) +Lemma Ropp_distr2:(r1,r2:R)``-(r1-r2)==r2-r1``. + Intros; Ring. +Qed. +Hints Resolve Ropp_distr2 : real. + +Lemma Ropp_distr3:(r1,r2:R)``-(r2-r1)==r1-r2``. +Intros; Ring. +Qed. +Hints Resolve Ropp_distr3 : real. + +(**********) +Lemma eq_Rminus:(r1,r2:R)(r1==r2)->``r1-r2==0``. + Intros; Rewrite H; Ring. +Qed. +Hints Resolve eq_Rminus : 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). +Qed. +Hints Immediate Rminus_eq : 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. +Qed. +Hints Immediate Rminus_eq_right : real. + +Lemma Rplus_Rminus: (p,q:R)``p+(q-p)``==q. +Intros; Ring. +Qed. +Hints Resolve Rplus_Rminus:real. + +(**********) +Lemma Rminus_eq_contra:(r1,r2:R)``r1<>r2``->``r1-r2<>0``. +Red; Intros r1 r2 H H0. +Apply H; Auto with real. +Qed. +Hints 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. +Qed. +Hints 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. +Qed. +Hints 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. +Qed. + +(** Inverse *) +Lemma Rinv_R1:``/1==1``. +Field;Auto with real. +Qed. +Hints Resolve Rinv_R1 : 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. +Qed. +Hints Resolve Rinv_neq_R0 : real. + +(*********) +Lemma Rinv_Rinv:(r:R)``r<>0``->``/(/r)==r``. +Intros;Field;Auto with real. +Qed. +Hints Resolve Rinv_Rinv : real. + +(*********) +Lemma Rinv_Rmult:(r1,r2:R)``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. +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. +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. +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. +Qed. +Hints 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. +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. +Qed. + +Hints Resolve Rlt_compatibility_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). +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. +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. +Qed. + +Hints Resolve Rle_compatibility Rle_compatibility_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). +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. +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. +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. +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. +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. +Qed. + +Hints Immediate Rplus_lt Rplus_le Rplus_lt_le_lt Rplus_le_lt_lt : 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. +Qed. +Hints Resolve Rgt_Ropp. + +(**********) +Lemma Rlt_Ropp:(r1,r2:R) ``r1 < r2`` -> ``-r1 > -r2``. +Unfold Rgt; Auto with real. +Qed. +Hints Resolve Rlt_Ropp : 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. +Qed. +Hints Immediate Ropp_Rlt : real. + +Lemma Rlt_Ropp1:(r1,r2:R) ``r2 < r1`` -> ``-r1 < -r2``. +Auto with real. +Qed. +Hints Resolve Rlt_Ropp1 : real. + +(**********) +Lemma Rle_Ropp:(r1,r2:R) ``r1 <= r2`` -> ``-r1 >= -r2``. +Unfold Rge; Intros r1 r2 [H|H]; Auto with real. +Qed. +Hints Resolve Rle_Ropp : 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. +Qed. +Hints Immediate Ropp_Rle : real. + +Lemma Rle_Ropp1:(r1,r2:R) ``r2 <= r1`` -> ``-r1 <= -r2``. +Intros r1 r2 H;Elim H;Auto with real. +Qed. +Hints Resolve Rle_Ropp1 : real. + +(**********) +Lemma Rge_Ropp:(r1,r2:R) ``r1 >= r2`` -> ``-r1 <= -r2``. +Unfold Rge; Intros r1 r2 [H|H]; Auto with real. +Qed. +Hints Resolve Rge_Ropp : real. + +(**********) +Lemma Rlt_RO_Ropp:(r:R) ``0 < r`` -> ``0 > -r``. +Intros; Replace ``0`` with ``-0``; Auto with real. +Qed. +Hints Resolve Rlt_RO_Ropp : real. + +(**********) +Lemma Rgt_RO_Ropp:(r:R) ``0 > r`` -> ``0 < -r``. +Intros; Replace ``0`` with ``-0``; Auto with real. +Qed. +Hints Resolve Rgt_RO_Ropp : real. + +(**********) +Lemma Rgt_RoppO:(r:R)``r>0``->``(-r)<0``. +Intros; Rewrite <- Ropp_O; Auto with real. +Qed. + +(**********) +Lemma Rlt_RoppO:(r:R)``r<0``->``-r>0``. +Intros; Rewrite <- Ropp_O; Auto with real. +Qed. +Hints Resolve Rgt_RoppO Rlt_RoppO: real. + +(**********) +Lemma Rle_RO_Ropp:(r:R) ``0 <= r`` -> ``0 >= -r``. +Intros; Replace ``0`` with ``-0``; Auto with real. +Qed. +Hints Resolve Rle_RO_Ropp : real. + +(**********) +Lemma Rge_RO_Ropp:(r:R) ``0 >= r`` -> ``0 <= -r``. +Intros; Replace ``0`` with ``-0``; Auto with real. +Qed. +Hints Resolve Rge_RO_Ropp : 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. +Qed. +Hints Resolve Rlt_monotony_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. +Qed. + +V7only [ +Notation Rlt_monotony_rev := Rlt_monotony_contra. +Notation "'Rlt_monotony_contra' a b c" := (Rlt_monotony_contra c a b) + (at level 10, a,b,c at level 9, only parsing). +]. + +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. +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. +Qed. +Hints Resolve Rle_monotony : 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. +Qed. +Hints Resolve Rle_monotony_r : real. + +Lemma Rmult_le_reg_l: + (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. +Qed. + +V7only [ +Notation "'Rle_monotony_contra' a b c" := (Rmult_le_reg_l c a b) + (at level 10, a,b,c at level 9, only parsing). +Notation Rle_monotony_contra := Rmult_le_reg_l. +]. + + +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. +Qed. +Hints Resolve Rle_anti_monotony1 : real. + +Lemma Rle_anti_monotony + :(r,r1,r2:R)``r <= 0`` -> ``r1 <= r2`` -> ``r*r1 >= r*r2``. +Intros; Apply Rle_ge; Auto with real. +Qed. +Hints Resolve Rle_anti_monotony : 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. +Qed. +Hints Resolve Rle_Rmult_comp :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. +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. +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. +Qed. +Hints Resolve Rlt_minus : real. + +(**********) +Lemma Rle_minus:(r1,r2:R)``r1 <= r2`` -> ``r1-r2 <= 0``. +NewDestruct 1; Unfold Rle; 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. +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. +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. +Qed. +Hints 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. +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. +Qed. +Hints Resolve pos_Rsqr pos_Rsqr1 : 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. +Qed. +Hints Resolve Rlt_R0_R1 : real. + +Lemma Rle_R0_R1:``0<=1``. +Left. +Exact Rlt_R0_R1. +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. +Qed. +Hints Resolve Rlt_Rinv : 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. +Qed. +Hints Resolve Rlt_Rinv2 : 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. + +(*********************************************************) +(** Greater *) +(*********************************************************) + +(**********) +Lemma Rge_ge_eq:(r1,r2:R)``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. +Qed. + +(**********) +Lemma Rnot_lt_le:(r1,r2:R)~(``r1<r2``)->``r2<=r1``. +Intros; Apply Rge_le; Apply Rlt_not_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). +Qed. + +(**********) +Lemma Rgt_ge:(r1,r2:R)``r1>r2`` -> ``r1 >= r2``. +Red; 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" := (Rge_le b a) + (at level 10, a,b at next level). +Notation "'Rle_sym2' a" := [b:R](Rge_le b a) + (at level 10, a at next level). +Notation Rle_sym2 := Rge_le. +(* +(**********) +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. +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. +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. +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. +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. +Qed. +Hints Resolve Rlt_r_plus_R1: real. + +(**********) +Lemma Rlt_r_r_plus_R1:(r:R)``r<r+1``. +Intros. +Pattern 1 r; Replace r with ``r+0``; Auto with real. +Qed. +Hints Resolve Rlt_r_r_plus_R1: 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. +Qed. + +(***********) +Lemma Rgt_plus_plus_r:(r,r1,r2:R)``r1>r2``->``r+r1 > r+r2``. +Unfold Rgt; Auto with real. +Qed. +Hints Resolve Rgt_plus_plus_r : 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). +Qed. + +(***********) +Lemma Rge_plus_plus_r:(r,r1,r2:R)``r1>=r2`` -> ``r+r1 >= r+r2``. +Intros; Apply Rle_ge; Auto with real. +Qed. +Hints Resolve Rge_plus_plus_r : 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. +Qed. + +(***********) +Lemma Rmult_ge_compat_r: + (z,x,y:R) ``z>=0`` -> ``x>=y`` -> ``x*z >= y*z``. +Intros z x y; Intros; Apply Rle_ge; Apply Rle_monotony_r; Apply Rge_le; Assumption. +Qed. + +V7only [ +Notation "'Rge_monotony' a b c" := (Rmult_ge_compat_r c a b) + (at level 10, a,b,c at level 9, only parsing). +Notation Rge_monotony := Rmult_ge_compat_r. +]. + +(***********) +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. +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. +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). +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. +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. +Qed. + +(*********) +Lemma Rmult_lt_pos:(x,y:R)``0<x`` -> ``0<y`` -> ``0<x*y``. +Proof Rmult_gt. + +(***********) +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. +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. +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. +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. +Qed. + + +(**********************************************************) +(** Injection from [N] to [R] *) +(**********************************************************) + +(**********) +Lemma S_INR:(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. +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. +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. +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. +Qed. + +Hints 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. +Qed. +Hints 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. +Qed. +Hints 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. +Qed. +Hints 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. +Qed. +Hints Resolve INR_pos : real. + +(**********) +Lemma pos_INR:(n:nat)``0 <= (INR n)``. +Intro n; Case n. +Simpl; Auto with real. +Auto with arith real. +Qed. +Hints 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. +Qed. +Hints 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. +Qed. +Hints 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. +Qed. +Hints 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. +Qed. +Hints 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. +Qed. +Hints 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. +Qed. +Hints 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. +Qed. +Hints 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. +Qed. +Hints 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. +Qed. + +(**********) +Lemma INR_IZR_INZ:(n:nat)(INR n)==(IZR (INZ n)). +Induction n; Auto with real. +Intros; Simpl; Rewrite bij1; 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. +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. +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. +Qed. + +(**********) +Lemma Ropp_Ropp_IZR:(z:Z)(IZR (`-z`))==``-(IZR z)``. +Intro z; Case z; Simpl; 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. +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. +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). +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. +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. +Qed. + +(**********) +Lemma not_O_IZR:(z:Z)`z<>0`->``(IZR z)<>0``. +Intros z H; Red; 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. +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. +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. +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. +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. +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. +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. +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. +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. +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. +Qed. + +(*****************************************************************) +(** Definitions of new types *) +(*****************************************************************) + +Record nonnegreal : Type := mknonnegreal { +nonneg :> R; +cond_nonneg : ``0<=nonneg`` }. + +Record posreal : Type := mkposreal { +pos :> R; +cond_pos : ``0<pos`` }. + +Record nonposreal : Type := mknonposreal { +nonpos :> R; +cond_nonpos : ``nonpos<=0`` }. + +Record negreal : Type := mknegreal { +neg :> R; +cond_neg : ``neg<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. +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). +Qed. + +Lemma double : (x:R) ``2*x==x+x``. +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]. +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. diff --git a/theories7/Reals/RList.v b/theories7/Reals/RList.v new file mode 100644 index 00000000..b89296fb --- /dev/null +++ b/theories7/Reals/RList.v @@ -0,0 +1,427 @@ +(************************************************************************) +(* 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: RList.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +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. diff --git a/theories7/Reals/R_Ifp.v b/theories7/Reals/R_Ifp.v new file mode 100644 index 00000000..621cca64 --- /dev/null +++ b/theories7/Reals/R_Ifp.v @@ -0,0 +1,552 @@ +(************************************************************************) +(* 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: R_Ifp.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +(**********************************************************) +(** Complements for the reals.Integer and fractional part *) +(* *) +(**********************************************************) + +Require Rbase. +Require Omega. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(*********************************************************) +(** Fractional part *) +(*********************************************************) + +(**********) +Definition Int_part:R->Z:=[r:R](`(up r)-1`). + +(**********) +Definition frac_part:R->R:=[r:R](Rminus 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. +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. +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. +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. +Qed. + +(**********) +Lemma base_fp:(r:R)(Rge (frac_part r) R0)/\(Rlt (frac_part r) R1). +Intro; Unfold frac_part; Unfold Int_part; 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. + (*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. +Qed. + +(*********************************************************) +(** Properties *) +(*********************************************************) + +(**********) +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. +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. +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. +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. +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. +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. +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. +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. +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. +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. +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. +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. diff --git a/theories7/Reals/R_sqr.v b/theories7/Reals/R_sqr.v new file mode 100644 index 00000000..fc01a164 --- /dev/null +++ b/theories7/Reals/R_sqr.v @@ -0,0 +1,232 @@ +(************************************************************************) +(* 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: R_sqr.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rbasic_fun. +V7only [Import R_scope.]. Open Local Scope R_scope. + +(****************************************************) +(* Rsqr : some results *) +(****************************************************) + +Tactic Definition SqRing := Unfold Rsqr; Ring. + +Lemma Rsqr_neg : (x:R) ``(Rsqr x)==(Rsqr (-x))``. +Intros; SqRing. +Qed. + +Lemma Rsqr_times : (x,y:R) ``(Rsqr (x*y))==(Rsqr x)*(Rsqr y)``. +Intros; SqRing. +Qed. + +Lemma Rsqr_plus : (x,y:R) ``(Rsqr (x+y))==(Rsqr x)+(Rsqr y)+2*x*y``. +Intros; SqRing. +Qed. + +Lemma Rsqr_minus : (x,y:R) ``(Rsqr (x-y))==(Rsqr x)+(Rsqr y)-2*x*y``. +Intros; SqRing. +Qed. + +Lemma Rsqr_neg_minus : (x,y:R) ``(Rsqr (x-y))==(Rsqr (y-x))``. +Intros; SqRing. +Qed. + +Lemma Rsqr_1 : ``(Rsqr 1)==1``. +SqRing. +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). +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]]. +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. +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. +Qed. + +Lemma Rsqr_minus_plus : (a,b:R) ``(a-b)*(a+b)==(Rsqr a)-(Rsqr b)``. +Intros; SqRing. +Qed. + +Lemma Rsqr_plus_minus : (a,b:R) ``(a+b)*(a-b)==(Rsqr a)-(Rsqr b)``. +Intros; SqRing. +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]]. +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]]. +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. +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)]]. +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. +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. diff --git a/theories7/Reals/R_sqrt.v b/theories7/Reals/R_sqrt.v new file mode 100644 index 00000000..8c87659b --- /dev/null +++ b/theories7/Reals/R_sqrt.v @@ -0,0 +1,251 @@ +(************************************************************************) +(* 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: R_sqrt.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rsqrt_def. +V7only [Import R_scope.]. 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. +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. +Qed. + +Lemma sqrt_0 : ``(sqrt 0)==0``. +Apply Rsqr_eq_0; Unfold Rsqr; 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. +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. +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). +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)]. +Qed. + +Lemma sqrt_def : (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)). +Qed. + +Lemma sqrt_Rsqr : (x:R) ``0<=x``->``(sqrt (Rsqr x))==x``. +Intros; Unfold Rsqr; 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. +Qed. + +Lemma Rsqr_sqrt : (x:R) ``0<=x``->(Rsqr (sqrt x))==x. +Intros x H1; Unfold Rsqr; 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)]. +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))]. +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)]]. +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. +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)]. +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. +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)]. +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. +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). +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). +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]. +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. +Qed. diff --git a/theories7/Reals/Ranalysis.v b/theories7/Reals/Ranalysis.v new file mode 100644 index 00000000..d5d84f50 --- /dev/null +++ b/theories7/Reals/Ranalysis.v @@ -0,0 +1,477 @@ +(************************************************************************) +(* 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: Ranalysis.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rtrigo. +Require SeqSeries. +Require Export Ranalysis1. +Require Export Ranalysis2. +Require Export Ranalysis3. +Require Export Rtopology. +Require Export MVT. +Require Export PSeries_reg. +Require Export Exp_prop. +Require Export Rtrigo_reg. +Require Export Rsqrt_def. +Require Export R_sqrt. +Require Export Rtrigo_calc. +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. + +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). + +(**********) +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). + +(**********) +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. + +(**********) +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. + +(**********) +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. + +(**********) +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. + +(**********) +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). + +(**********) +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). + +(**********) +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. + +(**********) +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. diff --git a/theories7/Reals/Ranalysis1.v b/theories7/Reals/Ranalysis1.v new file mode 100644 index 00000000..8cb4c358 --- /dev/null +++ b/theories7/Reals/Ranalysis1.v @@ -0,0 +1,1046 @@ +(************************************************************************) +(* 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: Ranalysis1.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Export Rlimit. +Require Export Rderiv. +V7only [Import R_scope.]. Open Local Scope R_scope. +Implicit Variable 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. + +(****************************************************) +(** 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 no_cond : R->Prop := [x:R] True. + +(**********) +Definition constant_D_eq [f:R->R;D:R->Prop;c:R] : Prop := (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). + +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. +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. +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. +Qed. + +Lemma div_eq_inv : (f1,f2:R->R) (div_fct f1 f2)==(mult_fct f1 (inv_fct f2)). +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. +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)). +Qed. + +Lemma continuity_opp : (f:R->R) (continuity f)->(continuity (opp_fct f)). +Unfold continuity; 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)). +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)). +Qed. + +Lemma continuity_const : (f:R->R) (constant f) -> (continuity f). +Unfold continuity; 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)). +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)). +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)). +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))). +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_abs [f:R->R;x:R] : R -> Prop := [l:R](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 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)). + +Arguments Scope derivable_pt_lim [Rfun_scope R_scope]. +Arguments Scope derivable_pt_abs [Rfun_scope R_scope R_scope]. +Arguments Scope derivable_pt [Rfun_scope R_scope]. +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``. +(************************************) +(** Class of differential functions *) +(************************************) +Record Differential : Type := mkDifferential { +d1 :> R->R; +cond_diff : (derivable d1) }. + +Record Differential_D2 : Type := mkDifferential_D2 { +d2 :> R->R; +cond_D1 : (derivable d2); +cond_D2 : (derivable (derive d2 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. +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). +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). +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. +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. +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. +Qed. + +Theorem derivable_continuous : (f:R->R) (derivable f) -> (continuity f). +Unfold derivable continuity; 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. +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). +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). +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). +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. diff --git a/theories7/Reals/Ranalysis2.v b/theories7/Reals/Ranalysis2.v new file mode 100644 index 00000000..35fa58d5 --- /dev/null +++ b/theories7/Reals/Ranalysis2.v @@ -0,0 +1,302 @@ +(************************************************************************) +(* 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: Ranalysis2.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Ranalysis1. +V7only [Import R_scope.]. 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. +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. +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. +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. +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. +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. +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. +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. +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. +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. +Qed. + +Lemma quadruple : (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. +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. diff --git a/theories7/Reals/Ranalysis3.v b/theories7/Reals/Ranalysis3.v new file mode 100644 index 00000000..6ce63bbc --- /dev/null +++ b/theories7/Reals/Ranalysis3.v @@ -0,0 +1,617 @@ +(************************************************************************) +(* 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: Ranalysis3.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Ranalysis1. +Require Ranalysis2. +V7only [Import R_scope.]. 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. +(***********************************) +(* 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. +(***********************************) +(* 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]. +(***********************************) +(* 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). +(***********************************) +(* 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. +(***********************************) +(* 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)). +(***********************************) +(* 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. +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. +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)). +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. diff --git a/theories7/Reals/Ranalysis4.v b/theories7/Reals/Ranalysis4.v new file mode 100644 index 00000000..061854dc --- /dev/null +++ b/theories7/Reals/Ranalysis4.v @@ -0,0 +1,313 @@ +(************************************************************************) +(* 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: Ranalysis4.v,v 1.1.2.1 2004/07/16 19:31:33 herbelin Exp $ 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. + +(**********) +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. +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. +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. +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). +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. +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. +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. +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). +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). +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. +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]. +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. +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. +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. +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. +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. +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. +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. +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. +Qed. + +Lemma derivable_exp : (derivable exp). +Unfold derivable; Apply derivable_pt_exp. +Qed. + +Lemma derivable_cosh : (derivable cosh). +Unfold derivable; Apply derivable_pt_cosh. +Qed. + +Lemma derivable_sinh : (derivable sinh). +Unfold derivable; 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. +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. +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. diff --git a/theories7/Reals/Raxioms.v b/theories7/Reals/Raxioms.v new file mode 100644 index 00000000..caf8524c --- /dev/null +++ b/theories7/Reals/Raxioms.v @@ -0,0 +1,172 @@ +(************************************************************************) +(* 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: Raxioms.v,v 1.2.2.1 2004/07/16 19:31:33 herbelin Exp $ i*) + +(*********************************************************) +(** Axiomatisation of the classical reals *) +(*********************************************************) + +Require Export ZArith_base. +V7only [ +Require Export Rsyntax. +Import R_scope. +]. +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 *) +(*********************************************************) + +(*********************************************************) +(** Addition *) +(*********************************************************) + +(**********) +Axiom Rplus_sym:(r1,r2:R)``r1+r2==r2+r1``. +Hints Resolve Rplus_sym : real. + +(**********) +Axiom Rplus_assoc:(r1,r2,r3:R)``(r1+r2)+r3==r1+(r2+r3)``. +Hints Resolve Rplus_assoc : real. + +(**********) +Axiom Rplus_Ropp_r:(r:R)``r+(-r)==0``. +Hints Resolve Rplus_Ropp_r : real v62. + +(**********) +Axiom Rplus_Ol:(r:R)``0+r==r``. +Hints Resolve Rplus_Ol : real. + +(***********************************************************) +(** Multiplication *) +(***********************************************************) + +(**********) +Axiom Rmult_sym:(r1,r2:R)``r1*r2==r2*r1``. +Hints Resolve Rmult_sym : real v62. + +(**********) +Axiom Rmult_assoc:(r1,r2,r3:R)``(r1*r2)*r3==r1*(r2*r3)``. +Hints Resolve Rmult_assoc : real v62. + +(**********) +Axiom Rinv_l:(r:R)``r<>0``->``(/r)*r==1``. +Hints Resolve Rinv_l : real. + +(**********) +Axiom Rmult_1l:(r:R)``1*r==r``. +Hints Resolve Rmult_1l : real. + +(**********) +Axiom R1_neq_R0:``1<>0``. +Hints 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. + +(*********************************************************) +(** Order axioms *) +(*********************************************************) +(*********************************************************) +(** Total Order *) +(*********************************************************) + +(**********) +Axiom total_order_T:(r1,r2:R)(sumorT (sumboolT ``r1<r2`` r1==r2) ``r1>r2``). + +(*********************************************************) +(** Lower *) +(*********************************************************) + +(**********) +Axiom Rlt_antisym:(r1,r2:R)``r1<r2`` -> ~ ``r2<r1``. + +(**********) +Axiom Rlt_trans:(r1,r2,r3:R) + ``r1<r2``->``r2<r3``->``r1<r3``. + +(**********) +Axiom Rlt_compatibility:(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``. + +Hints Resolve Rlt_antisym Rlt_compatibility Rlt_monotony : 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). +Arguments Scope INR [nat_scope]. + + +(**********************************************************) +(** Injection from [Z] to [R] *) +(**********************************************************) + +(**********) +Definition IZR:Z->R:=[z:Z](Cases z of + ZERO => ``0`` + |(POS n) => (INR (convert n)) + |(NEG n) => ``-(INR (convert n))`` + end). +Arguments Scope IZR [Z_scope]. + +(**********************************************************) +(** [R] Archimedian *) +(**********************************************************) + +(**********) +Axiom archimed:(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 bound:=[E:R->Prop](ExT [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``. + +(**********) +Axiom complet:(E:R->Prop)(bound E)-> + (ExT [x:R] (E x))-> + (sigTT R [m:R](is_lub E m)). + diff --git a/theories7/Reals/Rbase.v b/theories7/Reals/Rbase.v new file mode 100644 index 00000000..54226206 --- /dev/null +++ b/theories7/Reals/Rbase.v @@ -0,0 +1,14 @@ +(************************************************************************) +(* 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: Rbase.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +Require Export Rdefinitions. +Require Export Raxioms. +Require Export RIneq. +Require Export DiscrR. diff --git a/theories7/Reals/Rbasic_fun.v b/theories7/Reals/Rbasic_fun.v new file mode 100644 index 00000000..3d143e34 --- /dev/null +++ b/theories7/Reals/Rbasic_fun.v @@ -0,0 +1,476 @@ +(************************************************************************) +(* 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: Rbasic_fun.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +(*********************************************************) +(** Complements for the real numbers *) +(* *) +(*********************************************************) + +Require Rbase. +Require R_Ifp. +Require Fourier. +V7only [Import R_scope.]. Open Local Scope R_scope. + +Implicit Variable Type r:R. + +(*******************************) +(** Rmin *) +(*******************************) + +(*********) +Definition Rmin :R->R->R:=[x,y:R] + Cases (total_order_Rle x y) of + (leftT _) => x + | (rightT _) => 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. +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. +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). +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]. +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]. +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). +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)]. +Qed. + +(*******************************) +(** Rmax *) +(*******************************) + +(*********) +Definition Rmax :R->R->R:=[x,y:R] + Cases (total_order_Rle x y) of + (leftT _) => y + | (rightT _) => 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)). +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. +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. +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. +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. +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)]. +Qed. + +(*******************************) +(** Rabsolu *) +(*******************************) + +(*********) +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). +Qed. + +(*********) +Definition Rabsolu:R->R:= + [r:R](Cases (case_Rabsolu r) of + (leftT _) => (Ropp r) + |(rightT _) => r + end). + +(*********) +Lemma Rabsolu_R0:(Rabsolu R0)==R0. +Unfold Rabsolu;Case (case_Rabsolu R0);Auto;Intro. +Generalize (Rlt_antirefl R0);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. +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. +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. +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. +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. +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. +Qed. + +Lemma Rle_Rabsolu: + (x:R) (Rle x (Rabsolu x)). +Intro; Unfold Rabsolu;Case (case_Rabsolu 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]. +Qed. + +(*********) +Lemma Rabsolu_Rabsolu:(x:R)(Rabsolu (Rabsolu x))==(Rabsolu x). +Intro;Apply (Rabsolu_pos_eq (Rabsolu x) (Rabsolu_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. +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. +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. +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. +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. +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. +(**) +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_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. +(**) +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 (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. +(**) +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). +(**) +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). +(**) +Unfold Rle;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. +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]]. +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. +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. + diff --git a/theories7/Reals/Rcomplete.v b/theories7/Reals/Rcomplete.v new file mode 100644 index 00000000..5985a382 --- /dev/null +++ b/theories7/Reals/Rcomplete.v @@ -0,0 +1,175 @@ +(************************************************************************) +(* 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: Rcomplete.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rseries. +Require SeqProp. +Require Max. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(****************************************************) +(* R is complete : *) +(* Each sequence which satisfies *) +(* the Cauchy's criterion converges *) +(* *) +(* 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. diff --git a/theories7/Reals/Rdefinitions.v b/theories7/Reals/Rdefinitions.v new file mode 100644 index 00000000..79be0176 --- /dev/null +++ b/theories7/Reals/Rdefinitions.v @@ -0,0 +1,69 @@ +(************************************************************************) +(* 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: Rdefinitions.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + + +(*********************************************************) +(** Definitions for the axiomatization *) +(* *) +(*********************************************************) + +Require Export ZArith_base. + +Parameter R:Set. + +(* Declare Scope positive_scope with Key R *) +Delimits 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. + +V8Infix "+" Rplus : R_scope. +V8Infix "*" Rmult : R_scope. +V8Notation "- x" := (Ropp x) : R_scope. +V8Notation "/ x" := (Rinv x) : R_scope. + +V8Infix "<" Rlt : R_scope. + +(*i*******************************************************i*) + +(**********) +Definition Rgt:R->R->Prop:=[r1,r2:R](Rlt r2 r1). + +(**********) +Definition Rle:R->R->Prop:=[r1,r2:R]((Rlt r1 r2)\/(r1==r2)). + +(**********) +Definition Rge:R->R->Prop:=[r1,r2:R]((Rgt r1 r2)\/(r1==r2)). + +(**********) +Definition Rminus:R->R->R:=[r1,r2:R](Rplus r1 (Ropp r2)). + +(**********) +Definition Rdiv:R->R->R:=[r1,r2:R](Rmult r1 (Rinv r2)). + +V8Infix "-" Rminus : R_scope. +V8Infix "/" Rdiv : R_scope. + +V8Infix "<=" Rle : R_scope. +V8Infix ">=" Rge : R_scope. +V8Infix ">" 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. diff --git a/theories7/Reals/Rderiv.v b/theories7/Reals/Rderiv.v new file mode 100644 index 00000000..b55aa6ea --- /dev/null +++ b/theories7/Reals/Rderiv.v @@ -0,0 +1,453 @@ +(************************************************************************) +(* 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: Rderiv.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +(*********************************************************) +(** Definition of the derivative,continuity *) +(* *) +(*********************************************************) + +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. + +(*********) +Definition D_x:(R->Prop)->R->R->Prop:=[D:R->Prop][y:R][x:R] + (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 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). + +(*********) +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. +(**) + 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. +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. +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. +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. +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. +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. +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. +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. +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. +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)). +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]. +Qed. + diff --git a/theories7/Reals/Reals.v b/theories7/Reals/Reals.v new file mode 100644 index 00000000..d0f879ab --- /dev/null +++ b/theories7/Reals/Reals.v @@ -0,0 +1,32 @@ +(************************************************************************) +(* 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: Reals.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +(* The library REALS is divided in 6 parts : + - Rbase: basic lemmas on R + equalities and inequalities + Ring and Field are instantiated on R + - Rfunctions: some useful functions (Rabsolu, Rmin, Rmax, fact...) + - SeqSeries: theory of sequences and series + - Rtrigo: theory of trigonometric functions + - Ranalysis: some topology and general results of real analysis (mean value theorem, intermediate value theorem,...) + - Integration: Newton and Riemann' integrals + + Tactics are: + - DiscrR: for goals like ``?1<>0`` + - Sup: for goals like ``?1<?2`` + - RCompute: for equalities with constants like ``10*10==100`` + - Reg: for goals like (continuity_pt ?1 ?2) or (derivable_pt ?1 ?2) *) + +Require Export Rbase. +Require Export Rfunctions. +Require Export SeqSeries. +Require Export Rtrigo. +Require Export Ranalysis. +Require Export Integration. diff --git a/theories7/Reals/Rfunctions.v b/theories7/Reals/Rfunctions.v new file mode 100644 index 00000000..fe6ccd96 --- /dev/null +++ b/theories7/Reals/Rfunctions.v @@ -0,0 +1,832 @@ +(************************************************************************) +(* 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: Rfunctions.v,v 1.2.2.1 2004/07/16 19:31:34 herbelin Exp $ i*) + +(*i Some properties about pow and sum have been made with John Harrison i*) +(*i Some Lemmas (about pow and powerRZ) have been done by Laurent Thery i*) + +(********************************************************) +(** Definition of the sum functions *) +(* *) +(********************************************************) + +Require 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. ]. +Open Local Scope nat_scope. +Open Local Scope R_scope. + +(*******************************) +(** Lemmas about factorial *) +(*******************************) +(*********) +Lemma INR_fact_neq_0:(n:nat)~(INR (fact n))==R0. +Proof. +Intro;Red;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)). +Proof. +Intro; Reflexivity. +Qed. + +(*********) +Lemma simpl_fact:(n:nat)(Rmult (Rinv (INR (fact (S n)))) + (Rinv (Rinv (INR (fact n)))))== + (Rinv (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. +Qed. + +(*******************************) +(* Power *) +(*******************************) +(*********) +Fixpoint pow [r:R;n:nat]:R:= + Cases n of + O => R1 + |(S n) => (Rmult r (pow r n)) + end. + +V8Infix "^" pow : R_scope. + +Lemma pow_O: (x : R) (pow x O) == R1. +Proof. +Reflexivity. +Qed. + +Lemma pow_1: (x : R) (pow x (1)) == x. +Proof. +Simpl; Auto with real. +Qed. + +Lemma pow_add: + (x : R) (n, m : nat) (pow x (plus n m)) == (Rmult (pow x n) (pow x m)). +Proof. +Intros x n; Elim n; Simpl; 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). +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. +Qed. + +Hints 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 tech_pow_Rmult:(x:R)(n:nat)(Rmult x (pow x n))==(pow x (S n)). +Proof. +Induction n; Simpl; 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. +Qed. + +(**********) +Lemma pow_1_even : (n:nat) ``(pow (-1) (mult (S (S O)) 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]. +Qed. + +(**********) +Lemma pow_1_odd : (n:nat) ``(pow (-1) (S (mult (S (S O)) 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. +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]. +Qed. + +(*******************************) +(** PowerRZ *) +(*******************************) +(*i Due to L.Thery i*) + +Tactic Definition CaseEqk name := +Generalize (refl_equal ? name); Pattern -1 name; 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. + +Infix Local "^Z" powerRZ (at level 2, left associativity) : R_scope. + +Lemma Zpower_NR0: + (x : Z) (n : nat) (Zle ZERO x) -> (Zle ZERO (Zpower_nat x n)). +Proof. +NewInduction n; Unfold Zpower_nat; Simpl; Auto with zarith. +Qed. + +Lemma powerRZ_O: (x : R) (powerRZ x ZERO) == R1. +Proof. +Reflexivity. +Qed. + +Lemma powerRZ_1: (x : R) (powerRZ x (Zs ZERO)) == x. +Proof. +Simpl; Auto with real. +Qed. + +Lemma powerRZ_NOR: (x : R) (z : Z) ~ x == R0 -> ~ (powerRZ x z) == R0. +Proof. +NewDestruct z; Simpl; 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)). +Proof. +Intro x; NewDestruct n as [|n1|n1]; NewDestruct m as [|m1|m1]; Simpl; + Auto with real. +(* POS/POS *) +Rewrite convert_add; 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. +(* 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. +(* 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. +Qed. +Hints 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. +Qed. + +Lemma powerRZ_lt: (x : R) (z : Z) (Rlt R0 x) -> (Rlt R0 (powerRZ x z)). +Proof. +Intros x z; Case z; Simpl; Auto with real. +Qed. +Hints Resolve powerRZ_lt :real. + +Lemma powerRZ_le: (x : R) (z : Z) (Rlt R0 x) -> (Rle R0 (powerRZ x z)). +Proof. +Intros x z H'; Apply Rlt_le; Auto with real. +Qed. +Hints 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). +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. +Qed. + +Lemma powerRZ_R1: (n : Z) (powerRZ R1 n) == R1. +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. +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'))) + 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_O [n:nat]:nat:= + (sum_nat_f_O [x:nat]x n). + +(*********) +Definition sum_nat [s,n:nat]:nat:= + (sum_nat_f s n [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))) + 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. +Qed. + +(*******************************) +(* Distance in R *) +(*******************************) + +(*********) +Definition R_dist:R->R->R:=[x,y:R](Rabsolu (Rminus x y)). + +(*********) +Lemma R_dist_pos:(x,y:R)(Rge (R_dist x y) R0). +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. +Qed. + +(*********) +Lemma R_dist_sym:(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. +Qed. + +(*********) +Lemma R_dist_refl:(x,y:R)((R_dist x y)==R0<->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). +Qed. + +Lemma R_dist_eq:(x:R)(R_dist x x)==R0. +Proof. +Unfold R_dist;Intros;SplitAbsolu;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))). +Proof. +Intros;Unfold R_dist; Replace ``x-y`` with ``(x-z)+(z-y)``; + [Apply (Rabsolu_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))). +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. +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)). diff --git a/theories7/Reals/Rgeom.v b/theories7/Reals/Rgeom.v new file mode 100644 index 00000000..12c52e37 --- /dev/null +++ b/theories7/Reals/Rgeom.v @@ -0,0 +1,84 @@ +(************************************************************************) +(* 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: Rgeom.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ 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]]]. +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. +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. +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]. +Qed. + +(******************************************************************) +(** Translation *) +(******************************************************************) + +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. +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. +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)``. + +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. +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. +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]. +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. +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. +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. diff --git a/theories7/Reals/RiemannInt.v b/theories7/Reals/RiemannInt.v new file mode 100644 index 00000000..c9301b56 --- /dev/null +++ b/theories7/Reals/RiemannInt.v @@ -0,0 +1,1699 @@ +(************************************************************************) +(* 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: RiemannInt.v,v 1.1.2.1 2004/07/16 19:31:34 herbelin Exp $ 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. + +(********************************************) +(* 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))). +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. +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. +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. +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))]. +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]. +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]. +Qed. + +Definition RinvN : nat->posreal := [N:nat](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). +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]. +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)]]. +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. +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)). +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. +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]]. +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]. +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]]]. +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. +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. +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). +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]]. +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]. +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. +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). +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))]. +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)). +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]. +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)]. +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]. +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]. +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]. +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]]. +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)]. +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. +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]. +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]. +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. +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. +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]. +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]. +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]. +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]]. +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. +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. +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. +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)). +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]. +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]]. +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]. +(*****) +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]. +(*****) +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. +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]. +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. +Qed. + +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. +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)]]. +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]]. +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. diff --git a/theories7/Reals/RiemannInt_SF.v b/theories7/Reals/RiemannInt_SF.v new file mode 100644 index 00000000..3e2cc457 --- /dev/null +++ b/theories7/Reals/RiemannInt_SF.v @@ -0,0 +1,1400 @@ +(************************************************************************) +(* 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: RiemannInt_SF.v,v 1.2.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Ranalysis. +Require Classical_Prop. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Implicit Arguments On. + +(**************************************************) +(* 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)). + +Lemma IZN_var:(z:Z)(`0<=z`)->{ n:nat | z=(INZ n)}. +Intros; Apply inject_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]. +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 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_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 is_subdivision [f:R->R;a,b:R;l:Rlist] : Type := (sigTT ? [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)). + +(* 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. + +(* 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. + +(********************************) +(* 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. +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. +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)]. +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. +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. +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. +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. +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]. +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]]. +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]]. +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. +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. +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]. +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]. +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]]]. +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]]. +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. +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)))]]. +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]]. +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]. +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. +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]. +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]]. +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]]. +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]]. +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. +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]. +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. +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. +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)))]]). +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]. +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. +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]. +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. +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. +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. +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. +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]]. +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; Pose 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; Pose 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))]]. +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. +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)). +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]]. +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. +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]. +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]. +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. diff --git a/theories7/Reals/Rlimit.v b/theories7/Reals/Rlimit.v new file mode 100644 index 00000000..3308b2e3 --- /dev/null +++ b/theories7/Reals/Rlimit.v @@ -0,0 +1,539 @@ +(************************************************************************) +(* v * The Coq Proof Assistant / The Coq Development Team *) +(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) +(* \VV/ **************************************************************) +(* // * This file is distributed under the terms of the *) +(* * GNU Lesser General Public License Version 2.1 *) +(************************************************************************) + +(*i $Id: Rlimit.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +(*********************************************************) +(* Definition of the limit *) +(* *) +(*********************************************************) + +Require Rbase. +Require Rfunctions. +Require Classical_Prop. +Require Fourier. +V7only [Import R_scope.]. Open Local Scope R_scope. + +(*******************************) +(* Calculus *) +(*******************************) +(*********) +Lemma eps2_Rgt_R0:(eps:R)(Rgt eps R0)-> + (Rgt (Rmult eps (Rinv (Rplus R1 R1))) R0). +Intros;Fourier. +Qed. + +(*********) +Lemma eps2:(eps:R)(Rplus (Rmult eps (Rinv (Rplus R1 R1))) + (Rmult eps (Rinv (Rplus R1 R1))))==eps. +Intro esp. +Assert H := (double_var esp). +Unfold Rdiv in H. +Symmetry; Exact H. +Qed. + +(*********) +Lemma eps4:(eps:R) + (Rplus (Rmult eps (Rinv (Rplus (Rplus R1 R1) (Rplus R1 R1) ))) + (Rmult eps (Rinv (Rplus (Rplus R1 R1) (Rplus R1 R1) ))))== + (Rmult eps (Rinv (Rplus R1 R1))). +Intro eps. +Replace ``2+2`` with ``2*2``. +Pattern 3 eps; Rewrite double_var. +Rewrite (Rmult_Rplus_distrl ``eps/2`` ``eps/2`` ``/2``). +Unfold Rdiv. +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_Rmult. +Reflexivity. +DiscrR. +DiscrR. +Ring. +Qed. + +(*********) +Lemma Rlt_eps2_eps:(eps:R)(Rgt eps R0)-> + (Rlt (Rmult eps (Rinv (Rplus R1 R1))) eps). +Intros. +Pattern 2 eps; Rewrite <- Rmult_1r. +Repeat Rewrite (Rmult_sym eps). +Apply Rlt_monotony_r. +Exact H. +Apply Rlt_monotony_contra with ``2``. +Fourier. +Rewrite Rmult_1r; Rewrite <- Rinv_r_sym. +Fourier. +DiscrR. +Qed. + +(*********) +Lemma Rlt_eps4_eps:(eps:R)(Rgt eps R0)-> + (Rlt (Rmult eps (Rinv (Rplus (Rplus R1 R1) (Rplus R1 R1)))) eps). +Intros. +Replace ``2+2`` with ``4``. +Pattern 2 eps; Rewrite <- Rmult_1r. +Repeat Rewrite (Rmult_sym eps). +Apply Rlt_monotony_r. +Exact H. +Apply Rlt_monotony_contra with ``4``. +Replace ``4`` with ``2*2``. +Apply Rmult_lt_pos; Fourier. +Ring. +Rewrite Rmult_1r; Rewrite <- Rinv_r_sym. +Fourier. +DiscrR. +Ring. +Qed. + +(*********) +Lemma prop_eps:(r:R)((eps:R)(Rgt eps R0)->(Rlt r eps))->(Rle r R0). +Intros;Elim (total_order r R0); Intro. +Apply Rlt_le; Assumption. +Elim H0; Intro. +Apply eq_Rle; Assumption. +Clear H0;Generalize (H r H1); Intro;Generalize (Rlt_antirefl r); + Intro;ElimType False; Auto. +Qed. + +(*********) +Definition mul_factor := [l,l':R](Rinv (Rplus R1 (Rplus (Rabsolu l) + (Rabsolu l')))). + +(*********) +Lemma mul_factor_wd : (l,l':R) + ~(Rplus R1 (Rplus (Rabsolu l) (Rabsolu l')))==R0. +Intros;Rewrite (Rplus_sym R1 (Rplus (Rabsolu l) (Rabsolu l'))); + Apply tech_Rplus. +Cut (Rle (Rabsolu (Rplus l l')) (Rplus (Rabsolu l) (Rabsolu l'))). +Cut (Rle R0 (Rabsolu (Rplus l l'))). +Exact (Rle_trans ? ? ?). +Exact (Rabsolu_pos (Rplus l l')). +Exact (Rabsolu_triang ? ?). +Exact Rlt_R0_R1. +Qed. + +(*********) +Lemma mul_factor_gt:(eps:R)(l,l':R)(Rgt eps R0)-> + (Rgt (Rmult eps (mul_factor l l')) R0). +Intros;Unfold Rgt;Rewrite <- (Rmult_Or eps);Apply Rlt_monotony. +Assumption. +Unfold mul_factor;Apply Rlt_Rinv; + Cut (Rle R1 (Rplus R1 (Rplus (Rabsolu l) (Rabsolu l')))). +Cut (Rlt R0 R1). +Exact (Rlt_le_trans ? ? ?). +Exact Rlt_R0_R1. +Replace (Rle R1 (Rplus R1 (Rplus (Rabsolu l) (Rabsolu l')))) + with (Rle (Rplus R1 R0) (Rplus R1 (Rplus (Rabsolu l) (Rabsolu l')))). +Apply Rle_compatibility. +Cut (Rle (Rabsolu (Rplus l l')) (Rplus (Rabsolu l) (Rabsolu l'))). +Cut (Rle R0 (Rabsolu (Rplus l l'))). +Exact (Rle_trans ? ? ?). +Exact (Rabsolu_pos ?). +Exact (Rabsolu_triang ? ?). +Rewrite (proj1 ? ? (Rplus_ne R1));Trivial. +Qed. + +(*********) +Lemma mul_factor_gt_f:(eps:R)(l,l':R)(Rgt eps R0)-> + (Rgt (Rmin R1 (Rmult eps (mul_factor l l'))) R0). +Intros;Apply Rmin_Rgt_r;Split. +Exact Rlt_R0_R1. +Exact (mul_factor_gt eps l l' H). +Qed. + + +(*******************************) +(* Metric space *) +(*******************************) + +(*********) +Record Metric_Space:Type:= { + Base:Type; + dist:Base->Base->R; + dist_pos:(x,y:Base)(Rge (dist x y) R0); + dist_sym:(x,y:Base)(dist x y)==(dist y x); + dist_refl:(x,y:Base)((dist x y)==R0<->x==y); + dist_tri:(x,y,z:Base)(Rle (dist x y) + (Rplus (dist x z) (dist z y))) }. + +(*******************************) +(* Limit in Metric space *) +(*******************************) + +(*********) +Definition limit_in:= + [X:Metric_Space; X':Metric_Space; f:(Base X)->(Base X'); + D:(Base X)->Prop; x0:(Base X); l:(Base X')] + (eps:R)(Rgt eps R0)-> + (EXT alp:R | (Rgt alp R0)/\(x:(Base X))(D x)/\ + (Rlt (dist X x x0) alp)-> + (Rlt (dist X' (f x) l) eps)). + +(*******************************) +(* R is a metric space *) +(*******************************) + +(*********) +Definition R_met:Metric_Space:=(Build_Metric_Space R R_dist + R_dist_pos R_dist_sym R_dist_refl R_dist_tri). + +(*******************************) +(* Limit 1 arg *) +(*******************************) +(*********) +Definition Dgf:=[Df,Dg:R->Prop][f:R->R][x:R](Df x)/\(Dg (f x)). + +(*********) +Definition limit1_in:(R->R)->(R->Prop)->R->R->Prop:= + [f:R->R; D:R->Prop; l:R; x0:R](limit_in R_met R_met f D x0 l). + +(*********) +Lemma tech_limit:(f:R->R)(D:R->Prop)(l:R)(x0:R)(D x0)-> + (limit1_in f D l x0)->l==(f x0). +Intros f D l x0 H H0. +Case (Rabsolu_pos (Rminus (f x0) l)); Intros H1. +Absurd (Rlt (dist R_met (f x0) l) (dist R_met (f x0) l)). +Apply Rlt_antirefl. +Case (H0 (dist R_met (f x0) l)); Auto. +Intros alpha1 (H2, H3); Apply H3; Auto; Split; Auto. +Case (dist_refl R_met x0 x0); Intros Hr1 Hr2; Rewrite Hr2; Auto. +Case (dist_refl R_met (f x0) l); Intros Hr1 Hr2; Apply sym_eqT; Auto. +Qed. + +(*********) +Lemma tech_limit_contr:(f:R->R)(D:R->Prop)(l:R)(x0:R)(D x0)->~l==(f x0) + ->~(limit1_in f D l x0). +Intros;Generalize (tech_limit f D l x0);Tauto. +Qed. + +(*********) +Lemma lim_x:(D:R->Prop)(x0:R)(limit1_in [x:R]x D x0 x0). +Unfold limit1_in; Unfold limit_in; Simpl; Intros;Split with eps; + Split; Auto;Intros;Elim H0; Intros; Auto. +Qed. + +(*********) +Lemma limit_plus:(f,g:R->R)(D:R->Prop)(l,l':R)(x0:R) + (limit1_in f D l x0)->(limit1_in g D l' x0)-> + (limit1_in [x:R](Rplus (f x) (g x)) D (Rplus l l') x0). +Intros;Unfold limit1_in; Unfold limit_in; Simpl; Intros; + Elim (H (Rmult eps (Rinv (Rplus R1 R1))) (eps2_Rgt_R0 eps H1)); + Elim (H0 (Rmult eps (Rinv (Rplus R1 R1))) (eps2_Rgt_R0 eps H1)); + Simpl;Clear H H0; Intros; Elim H; Elim H0; Clear H H0; Intros; + Split with (Rmin x1 x); Split. +Exact (Rmin_Rgt_r x1 x R0 (conj ? ? H H2)). +Intros;Elim H4; Clear H4; Intros; + Cut (Rlt (Rplus (R_dist (f x2) l) (R_dist (g x2) l')) eps). + Cut (Rle (R_dist (Rplus (f x2) (g x2)) (Rplus l l')) + (Rplus (R_dist (f x2) l) (R_dist (g x2) l'))). +Exact (Rle_lt_trans ? ? ?). +Exact (R_dist_plus ? ? ? ?). +Elim (Rmin_Rgt_l x1 x (R_dist x2 x0) H5); Clear H5; Intros. +Generalize (H3 x2 (conj (D x2) (Rlt (R_dist x2 x0) x) H4 H6)); + Generalize (H0 x2 (conj (D x2) (Rlt (R_dist x2 x0) x1) H4 H5)); + Intros; + Replace eps + with (Rplus (Rmult eps (Rinv (Rplus R1 R1))) + (Rmult eps (Rinv (Rplus R1 R1)))). +Exact (Rplus_lt ? ? ? ? H7 H8). +Exact (eps2 eps). +Qed. + +(*********) +Lemma limit_Ropp:(f:R->R)(D:R->Prop)(l:R)(x0:R) + (limit1_in f D l x0)->(limit1_in [x:R](Ropp (f x)) D (Ropp l) x0). +Unfold limit1_in;Unfold limit_in;Simpl;Intros;Elim (H eps H0);Clear H; + Intros;Elim H;Clear H;Intros;Split with x;Split;Auto;Intros; + Generalize (H1 x1 H2);Clear H1;Intro;Unfold R_dist;Unfold Rminus; + Rewrite (Ropp_Ropp l);Rewrite (Rplus_sym (Ropp (f x1)) l); + Fold (Rminus l (f x1));Fold (R_dist l (f x1));Rewrite R_dist_sym; + Assumption. +Qed. + +(*********) +Lemma limit_minus:(f,g:R->R)(D:R->Prop)(l,l':R)(x0:R) + (limit1_in f D l x0)->(limit1_in g D l' x0)-> + (limit1_in [x:R](Rminus (f x) (g x)) D (Rminus l l') x0). +Intros;Unfold Rminus;Generalize (limit_Ropp g D l' x0 H0);Intro; + Exact (limit_plus f [x:R](Ropp (g x)) D l (Ropp l') x0 H H1). +Qed. + +(*********) +Lemma limit_free:(f:R->R)(D:R->Prop)(x:R)(x0:R) + (limit1_in [h:R](f x) D (f x) x0). +Unfold limit1_in;Unfold limit_in;Simpl;Intros;Split with eps;Split; + Auto;Intros;Elim (R_dist_refl (f x) (f x));Intros a b; + Rewrite (b (refl_eqT R (f x)));Unfold Rgt in H;Assumption. +Qed. + +(*********) +Lemma limit_mul:(f,g:R->R)(D:R->Prop)(l,l':R)(x0:R) + (limit1_in f D l x0)->(limit1_in g D l' x0)-> + (limit1_in [x:R](Rmult (f x) (g x)) D (Rmult l l') x0). +Intros;Unfold limit1_in; Unfold limit_in; Simpl; Intros; + Elim (H (Rmin R1 (Rmult eps (mul_factor l l'))) + (mul_factor_gt_f eps l l' H1)); + Elim (H0 (Rmult eps (mul_factor l l')) (mul_factor_gt eps l l' H1)); + Clear H H0; Simpl; Intros; Elim H; Elim H0; Clear H H0; Intros; + Split with (Rmin x1 x); Split. +Exact (Rmin_Rgt_r x1 x R0 (conj ? ? H H2)). +Intros; Elim H4; Clear H4; Intros;Unfold R_dist; + Replace (Rminus (Rmult (f x2) (g x2)) (Rmult l l')) with + (Rplus (Rmult (f x2) (Rminus (g x2) l')) (Rmult l' (Rminus (f x2) l))). +Cut (Rlt (Rplus (Rabsolu (Rmult (f x2) (Rminus (g x2) l'))) (Rabsolu (Rmult l' + (Rminus (f x2) l)))) eps). +Cut (Rle (Rabsolu (Rplus (Rmult (f x2) (Rminus (g x2) l')) (Rmult l' (Rminus + (f x2) l)))) (Rplus (Rabsolu (Rmult (f x2) (Rminus (g x2) l'))) (Rabsolu + (Rmult l' (Rminus (f x2) l))))). +Exact (Rle_lt_trans ? ? ?). +Exact (Rabsolu_triang ? ?). +Rewrite (Rabsolu_mult (f x2) (Rminus (g x2) l')); + Rewrite (Rabsolu_mult l' (Rminus (f x2) l)); + Cut (Rle (Rplus (Rmult (Rplus R1 (Rabsolu l)) (Rmult eps (mul_factor l l'))) + (Rmult (Rabsolu l') (Rmult eps (mul_factor l l')))) eps). +Cut (Rlt (Rplus (Rmult (Rabsolu (f x2)) (Rabsolu (Rminus (g x2) l'))) (Rmult + (Rabsolu l') (Rabsolu (Rminus (f x2) l)))) (Rplus (Rmult (Rplus R1 (Rabsolu + l)) (Rmult eps (mul_factor l l'))) (Rmult (Rabsolu l') (Rmult eps + (mul_factor l l'))))). +Exact (Rlt_le_trans ? ? ?). +Elim (Rmin_Rgt_l x1 x (R_dist x2 x0) H5); Clear H5; Intros; + Generalize (H0 x2 (conj (D x2) (Rlt (R_dist x2 x0) x1) H4 H5));Intro; + Generalize (Rmin_Rgt_l ? ? ? H7);Intro;Elim H8;Intros;Clear H0 H8; + Apply Rplus_lt_le_lt. +Apply Rmult_lt_0. +Apply Rle_sym1. +Exact (Rabsolu_pos (Rminus (g x2) l')). +Rewrite (Rplus_sym R1 (Rabsolu l));Unfold Rgt;Apply Rlt_r_plus_R1; + Exact (Rabsolu_pos l). +Unfold R_dist in H9; + Apply (Rlt_anti_compatibility (Ropp (Rabsolu l)) (Rabsolu (f x2)) + (Rplus R1 (Rabsolu l))). +Rewrite <- (Rplus_assoc (Ropp (Rabsolu l)) R1 (Rabsolu l)); + Rewrite (Rplus_sym (Ropp (Rabsolu l)) R1); + Rewrite (Rplus_assoc R1 (Ropp (Rabsolu l)) (Rabsolu l)); + Rewrite (Rplus_Ropp_l (Rabsolu l)); + Rewrite (proj1 ? ? (Rplus_ne R1)); + Rewrite (Rplus_sym (Ropp (Rabsolu l)) (Rabsolu (f x2))); + Generalize H9; +Cut (Rle (Rminus (Rabsolu (f x2)) (Rabsolu l)) (Rabsolu (Rminus (f x2) l))). +Exact (Rle_lt_trans ? ? ?). +Exact (Rabsolu_triang_inv ? ?). +Generalize (H3 x2 (conj (D x2) (Rlt (R_dist x2 x0) x) H4 H6));Trivial. +Apply Rle_monotony. +Exact (Rabsolu_pos l'). +Unfold Rle;Left;Assumption. +Rewrite (Rmult_sym (Rplus R1 (Rabsolu l)) (Rmult eps (mul_factor l l'))); + Rewrite (Rmult_sym (Rabsolu l') (Rmult eps (mul_factor l l'))); + Rewrite <- (Rmult_Rplus_distr + (Rmult eps (mul_factor l l')) + (Rplus R1 (Rabsolu l)) + (Rabsolu l')); + Rewrite (Rmult_assoc eps (mul_factor l l') (Rplus (Rplus R1 (Rabsolu l)) + (Rabsolu l'))); + Rewrite (Rplus_assoc R1 (Rabsolu l) (Rabsolu l'));Unfold mul_factor; + Rewrite (Rinv_l (Rplus R1 (Rplus (Rabsolu l) (Rabsolu l'))) + (mul_factor_wd l l')); + Rewrite (proj1 ? ? (Rmult_ne eps));Apply eq_Rle;Trivial. +Ring. +Qed. + +(*********) +Definition adhDa:(R->Prop)->R->Prop:=[D:R->Prop][a:R] + (alp:R)(Rgt alp R0)->(EXT x:R | (D x)/\(Rlt (R_dist x a) alp)). + +(*********) +Lemma single_limit:(f:R->R)(D:R->Prop)(l:R)(l':R)(x0:R) + (adhDa D x0)->(limit1_in f D l x0)->(limit1_in f D l' x0)->l==l'. +Unfold limit1_in; Unfold limit_in; Intros. +Cut (eps:R)(Rgt eps R0)->(Rlt (dist R_met l l') + (Rmult (Rplus R1 R1) eps)). +Clear H0 H1;Unfold dist; Unfold R_met; Unfold R_dist; + Unfold Rabsolu;Case (case_Rabsolu (Rminus l l')); Intros. +Cut (eps:R)(Rgt eps R0)->(Rlt (Ropp (Rminus l l')) eps). +Intro;Generalize (prop_eps (Ropp (Rminus l l')) H1);Intro; + Generalize (Rlt_RoppO (Rminus l l') r); Intro;Unfold Rgt in H3; + Generalize (Rle_not (Ropp (Rminus l l')) R0 H3); Intro; + ElimType False; Auto. +Intros;Cut (Rgt (Rmult eps (Rinv (Rplus R1 R1))) R0). +Intro;Generalize (H0 (Rmult eps (Rinv (Rplus R1 R1))) H2); + Rewrite (Rmult_sym eps (Rinv (Rplus R1 R1))); + Rewrite <- (Rmult_assoc (Rplus R1 R1) (Rinv (Rplus R1 R1)) eps); + Rewrite (Rinv_r (Rplus R1 R1)). +Elim (Rmult_ne eps);Intros a b;Rewrite b;Clear a b;Trivial. +Apply (imp_not_Req (Rplus R1 R1) R0);Right;Generalize Rlt_R0_R1;Intro; + Unfold Rgt;Generalize (Rlt_compatibility R1 R0 R1 H3);Intro; + Elim (Rplus_ne R1);Intros a b;Rewrite a in H4;Clear a b; + Apply (Rlt_trans R0 R1 (Rplus R1 R1) H3 H4). +Unfold Rgt;Unfold Rgt in H1; + Rewrite (Rmult_sym eps(Rinv (Rplus R1 R1))); + Rewrite <-(Rmult_Or (Rinv (Rplus R1 R1))); + Apply (Rlt_monotony (Rinv (Rplus R1 R1)) R0 eps);Auto. +Apply (Rlt_Rinv (Rplus R1 R1));Cut (Rlt R1 (Rplus R1 R1)). +Intro;Apply (Rlt_trans R0 R1 (Rplus R1 R1) Rlt_R0_R1 H2). +Generalize (Rlt_compatibility R1 R0 R1 Rlt_R0_R1);Elim (Rplus_ne R1); + Intros a b;Rewrite a;Clear a b;Trivial. +(**) +Cut (eps:R)(Rgt eps R0)->(Rlt (Rminus l l') eps). +Intro;Generalize (prop_eps (Rminus l l') H1);Intro; + Elim (Rle_le_eq (Rminus l l') R0);Intros a b;Clear b; + Apply (Rminus_eq l l');Apply a;Split. +Assumption. +Apply (Rle_sym2 R0 (Rminus l l') r). +Intros;Cut (Rgt (Rmult eps (Rinv (Rplus R1 R1))) R0). +Intro;Generalize (H0 (Rmult eps (Rinv (Rplus R1 R1))) H2); + Rewrite (Rmult_sym eps (Rinv (Rplus R1 R1))); + Rewrite <- (Rmult_assoc (Rplus R1 R1) (Rinv (Rplus R1 R1)) eps); + Rewrite (Rinv_r (Rplus R1 R1)). +Elim (Rmult_ne eps);Intros a b;Rewrite b;Clear a b;Trivial. +Apply (imp_not_Req (Rplus R1 R1) R0);Right;Generalize Rlt_R0_R1;Intro; + Unfold Rgt;Generalize (Rlt_compatibility R1 R0 R1 H3);Intro; + Elim (Rplus_ne R1);Intros a b;Rewrite a in H4;Clear a b; + Apply (Rlt_trans R0 R1 (Rplus R1 R1) H3 H4). +Unfold Rgt;Unfold Rgt in H1; + Rewrite (Rmult_sym eps(Rinv (Rplus R1 R1))); + Rewrite <-(Rmult_Or (Rinv (Rplus R1 R1))); + Apply (Rlt_monotony (Rinv (Rplus R1 R1)) R0 eps);Auto. +Apply (Rlt_Rinv (Rplus R1 R1));Cut (Rlt R1 (Rplus R1 R1)). +Intro;Apply (Rlt_trans R0 R1 (Rplus R1 R1) Rlt_R0_R1 H2). +Generalize (Rlt_compatibility R1 R0 R1 Rlt_R0_R1);Elim (Rplus_ne R1); + Intros a b;Rewrite a;Clear a b;Trivial. +(**) +Intros;Unfold adhDa in H;Elim (H0 eps H2);Intros;Elim (H1 eps H2); + Intros;Clear H0 H1;Elim H3;Elim H4;Clear H3 H4;Intros; + Simpl;Simpl in H1 H4;Generalize (Rmin_Rgt x x1 R0);Intro;Elim H5; + Intros;Clear H5; + Elim (H (Rmin x x1) (H7 (conj (Rgt x R0) (Rgt x1 R0) H3 H0))); + Intros; Elim H5;Intros;Clear H5 H H6 H7; + Generalize (Rmin_Rgt x x1 (R_dist x2 x0));Intro;Elim H; + Intros;Clear H H6;Unfold Rgt in H5;Elim (H5 H9);Intros;Clear H5 H9; + Generalize (H1 x2 (conj (D x2) (Rlt (R_dist x2 x0) x1) H8 H6)); + Generalize (H4 x2 (conj (D x2) (Rlt (R_dist x2 x0) x) H8 H)); + Clear H8 H H6 H1 H4 H0 H3;Intros; + Generalize (Rplus_lt (R_dist (f x2) l) eps (R_dist (f x2) l') eps + H H0); Unfold R_dist;Intros; + Rewrite (Rabsolu_minus_sym (f x2) l) in H1; + Rewrite (Rmult_sym (Rplus R1 R1) eps);Rewrite (Rmult_Rplus_distr eps R1 R1); + Elim (Rmult_ne eps);Intros a b;Rewrite a;Clear a b; + Generalize (R_dist_tri l l' (f x2));Unfold R_dist;Intros; + Apply (Rle_lt_trans (Rabsolu (Rminus l l')) + (Rplus (Rabsolu (Rminus l (f x2))) (Rabsolu (Rminus (f x2) l'))) + (Rplus eps eps) H3 H1). +Qed. + +(*********) +Lemma limit_comp:(f,g:R->R)(Df,Dg:R->Prop)(l,l':R)(x0:R) + (limit1_in f Df l x0)->(limit1_in g Dg l' l)-> + (limit1_in [x:R](g (f x)) (Dgf Df Dg f) l' x0). +Unfold limit1_in limit_in Dgf;Simpl. +Intros f g Df Dg l l' x0 Hf Hg eps eps_pos. +Elim (Hg eps eps_pos). +Intros alpg lg. +Elim (Hf alpg). +2: Tauto. +Intros alpf lf. +Exists alpf. +Intuition. +Qed. + +(*********) + +Lemma limit_inv : (f:R->R)(D:R->Prop)(l:R)(x0:R) (limit1_in f D l x0)->~(l==R0)->(limit1_in [x:R](Rinv (f x)) D (Rinv l) x0). +Unfold limit1_in; Unfold limit_in; Simpl; Unfold R_dist; Intros; Elim (H ``(Rabsolu l)/2``). +Intros delta1 H2; Elim (H ``eps*((Rsqr l)/2)``). +Intros delta2 H3; Elim H2; Elim H3; Intros; Exists (Rmin delta1 delta2); Split. +Unfold Rmin; Case (total_order_Rle delta1 delta2); Intro; Assumption. +Intro; Generalize (H5 x); Clear H5; Intro H5; Generalize (H7 x); Clear H7; Intro H7; Intro H10; Elim H10; Intros; Cut (D x)/\``(Rabsolu (x-x0))<delta1``. +Cut (D x)/\``(Rabsolu (x-x0))<delta2``. +Intros; Generalize (H5 H11); Clear H5; Intro H5; Generalize (H7 H12); Clear H7; Intro H7; Generalize (Rabsolu_triang_inv l (f x)); Intro; Rewrite Rabsolu_minus_sym in H7; Generalize (Rle_lt_trans ``(Rabsolu l)-(Rabsolu (f x))`` ``(Rabsolu (l-(f x)))`` ``(Rabsolu l)/2`` H13 H7); Intro; Generalize (Rlt_compatibility ``(Rabsolu (f x))-(Rabsolu l)/2`` ``(Rabsolu l)-(Rabsolu (f x))`` ``(Rabsolu l)/2`` H14); Replace ``(Rabsolu (f x))-(Rabsolu l)/2+((Rabsolu l)-(Rabsolu (f x)))`` with ``(Rabsolu l)/2``. +Unfold Rminus; Rewrite Rplus_assoc; Rewrite Rplus_Ropp_l; Rewrite Rplus_Or; Intro; Cut ~``(f x)==0``. +Intro; Replace ``/(f x)+ -/l`` with ``(l-(f x))*/(l*(f x))``. +Rewrite Rabsolu_mult; Rewrite Rabsolu_Rinv. +Cut ``/(Rabsolu (l*(f x)))<2/(Rsqr l)``. +Intro; Rewrite Rabsolu_minus_sym in H5; Cut ``0<=/(Rabsolu (l*(f x)))``. +Intro; Generalize (Rmult_lt2 ``(Rabsolu (l-(f x)))`` ``eps*(Rsqr l)/2`` ``/(Rabsolu (l*(f x)))`` ``2/(Rsqr l)`` (Rabsolu_pos ``l-(f x)``) H18 H5 H17); Replace ``eps*(Rsqr l)/2*2/(Rsqr l)`` with ``eps``. +Intro; Assumption. +Unfold Rdiv; Unfold Rsqr; Rewrite Rinv_Rmult. +Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym l). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite (Rmult_sym l). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Reflexivity. +DiscrR. +Exact H0. +Exact H0. +Exact H0. +Exact H0. +Left; Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Apply prod_neq_R0; Assumption. +Rewrite Rmult_sym; Rewrite Rabsolu_mult; Rewrite Rinv_Rmult. +Rewrite (Rsqr_abs l); Unfold Rsqr; Unfold Rdiv; Rewrite Rinv_Rmult. +Repeat Rewrite <- Rmult_assoc; Apply Rlt_monotony_r. +Apply Rlt_Rinv; Apply Rabsolu_pos_lt; Assumption. +Apply Rlt_monotony_contra with ``(Rabsolu (f x))*(Rabsolu l)*/2``. +Repeat Apply Rmult_lt_pos. +Apply Rabsolu_pos_lt; Assumption. +Apply Rabsolu_pos_lt; Assumption. +Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H17; Generalize (lt_INR_0 (2) (neq_O_lt (2) H17)); Unfold INR; Intro H18; Assumption | Discriminate]. +Replace ``(Rabsolu (f x))*(Rabsolu l)*/2*/(Rabsolu (f x))`` with ``(Rabsolu l)/2``. +Replace ``(Rabsolu (f x))*(Rabsolu l)*/2*(2*/(Rabsolu l))`` with ``(Rabsolu (f x))``. +Assumption. +Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym (Rabsolu l)). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r; Reflexivity. +DiscrR. +Apply Rabsolu_no_R0. +Assumption. +Unfold Rdiv. +Repeat Rewrite Rmult_assoc. +Rewrite (Rmult_sym (Rabsolu (f x))). +Repeat Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Reflexivity. +Apply Rabsolu_no_R0; Assumption. +Apply Rabsolu_no_R0; Assumption. +Apply Rabsolu_no_R0; Assumption. +Apply Rabsolu_no_R0; Assumption. +Apply Rabsolu_no_R0; Assumption. +Apply prod_neq_R0; Assumption. +Rewrite (Rinv_Rmult ? ? H0 H16). +Unfold Rminus; Rewrite Rmult_Rplus_distrl. +Rewrite <- Rmult_assoc. +Rewrite <- Rinv_r_sym. +Rewrite Rmult_1l. +Rewrite Ropp_mul1. +Rewrite (Rmult_sym (f x)). +Rewrite Rmult_assoc. +Rewrite <- Rinv_l_sym. +Rewrite Rmult_1r. +Reflexivity. +Assumption. +Assumption. +Red; Intro; Rewrite H16 in H15; Rewrite Rabsolu_R0 in H15; Cut ``0<(Rabsolu l)/2``. +Intro; Elim (Rlt_antirefl ``0`` (Rlt_trans ``0`` ``(Rabsolu l)/2`` ``0`` H17 H15)). +Unfold Rdiv; Apply Rmult_lt_pos. +Apply Rabsolu_pos_lt; Assumption. +Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H17; Generalize (lt_INR_0 (2) (neq_O_lt (2) H17)); Unfold INR; Intro; Assumption | Discriminate]. +Pattern 3 (Rabsolu l); Rewrite double_var. +Ring. +Split; [Assumption | Apply Rlt_le_trans with (Rmin delta1 delta2); [Assumption | Apply Rmin_r]]. +Split; [Assumption | Apply Rlt_le_trans with (Rmin delta1 delta2); [Assumption | Apply Rmin_l]]. +Change ``0<eps*(Rsqr l)/2``; Unfold Rdiv; Repeat Rewrite Rmult_assoc; Repeat Apply Rmult_lt_pos. +Assumption. +Apply Rsqr_pos_lt; Assumption. +Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H3; Generalize (lt_INR_0 (2) (neq_O_lt (2) H3)); Unfold INR; Intro; Assumption | Discriminate]. +Change ``0<(Rabsolu l)/2``; Unfold Rdiv; Apply Rmult_lt_pos; [Apply Rabsolu_pos_lt; Assumption | Apply Rlt_Rinv; Cut ~(O=(2)); [Intro H3; Generalize (lt_INR_0 (2) (neq_O_lt (2) H3)); Unfold INR; Intro; Assumption | Discriminate]]. +Qed. diff --git a/theories7/Reals/Rpower.v b/theories7/Reals/Rpower.v new file mode 100644 index 00000000..0acfa8d2 --- /dev/null +++ b/theories7/Reals/Rpower.v @@ -0,0 +1,560 @@ +(************************************************************************) +(* 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: Rpower.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) +(*i Due to L.Thery i*) + +(************************************************************) +(* 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. + +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. +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. +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. +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)). +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. +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]. +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). +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. + +(* 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). + +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. +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). +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. +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. +Qed. + +Theorem ln_exp: (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. +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. +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). +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. +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. +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. +Qed. + +(******************************************************************) +(* Definition of Rpower *) +(******************************************************************) + +Definition Rpower := [x : R] [y : R] ``(exp (y*(ln x)))``. + +Infix Local "^R" Rpower (at level 2, 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. +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. +Qed. + +Theorem Rpower_O: (x : R) ``0<x`` -> ``(Rpower x 0) == 1``. +Intros x H; Unfold Rpower. +Rewrite Rmult_Ol; 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. +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. +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. +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. +Qed. + +Theorem Rpower_Ropp: (x, y : R) ``(Rpower x (-y)) == /(Rpower x y)``. +Unfold Rpower. +Intros x y; Rewrite Ropp_mul1. +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. +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. +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. +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. +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. +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). +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]. +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. +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. +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. +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. diff --git a/theories7/Reals/Rprod.v b/theories7/Reals/Rprod.v new file mode 100644 index 00000000..a524a915 --- /dev/null +++ b/theories7/Reals/Rprod.v @@ -0,0 +1,164 @@ +(************************************************************************) +(* 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: Rprod.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Compare. +Require Rbase. +Require Rfunctions. +Require Rseries. +Require PartSum. +Require Binomial. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +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. + +(**********) +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]. +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. +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. +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. +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. +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]. +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. +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. diff --git a/theories7/Reals/Rseries.v b/theories7/Reals/Rseries.v new file mode 100644 index 00000000..a38099dd --- /dev/null +++ b/theories7/Reals/Rseries.v @@ -0,0 +1,279 @@ +(************************************************************************) +(* 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: Rseries.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Classical. +Require Compare. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Implicit Variable Type r:R. + +(* classical is needed for [Un_cv_crit] *) +(*********************************************************) +(* Definition of sequence and properties *) +(* *) +(*********************************************************) + +Section sequence. + +(*********) +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. + +(*********) +Definition EUn:R->Prop:=[r:R](Ex [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 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 Un_growing:Prop:=(n:nat)(Rle (Un n) (Un (S n))). + +(*********) +Lemma EUn_noempty:(ExT [r:R] (EUn r)). +Unfold EUn;Split with (Un O);Split with O;Trivial. +Qed. + +(*********) +Lemma Un_in_EUn:(n:nat)(EUn (Un n)). +Intro;Unfold EUn;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. +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). +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)). +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))). +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. +Qed. + +End sequence. + +(*****************************************************************) +(* Definition of Power Series and properties *) +(* *) +(*****************************************************************) + +Section Isequence. + +(*********) +Variable An:nat->R. + +(*********) +Definition Pser:R->R->Prop:=[x,l:R] + (infinit_sum [n:nat](Rmult (An n) (pow 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. diff --git a/theories7/Reals/Rsigma.v b/theories7/Reals/Rsigma.v new file mode 100644 index 00000000..f9e8e92b --- /dev/null +++ b/theories7/Reals/Rsigma.v @@ -0,0 +1,117 @@ +(************************************************************************) +(* 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: Rsigma.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rseries. +Require PartSum. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +Set Implicit Arguments. + +Section Sigma. + +Variable f : nat->R. + +Definition sigma [low,high:nat] : R := (sum_f_R0 [k:nat](f (plus low k)) (minus 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]. +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. +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. +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]. +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]. +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]. +Qed. + +End Sigma. diff --git a/theories7/Reals/Rsqrt_def.v b/theories7/Reals/Rsqrt_def.v new file mode 100644 index 00000000..17367dce --- /dev/null +++ b/theories7/Reals/Rsqrt_def.v @@ -0,0 +1,688 @@ +(************************************************************************) +(* 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: Rsqrt_def.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Sumbool. +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Ranalysis1. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +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. + +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). + +(**********) +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. +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]. +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. +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. +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. +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. +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. +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. +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. +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. +Qed. + +Definition pow_2_n := [n:nat](pow ``2`` n). + +Lemma pow_2_n_neq_R0 : (n:nat) ``(pow_2_n n)<>0``. +Intro. +Unfold pow_2_n. +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. +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. +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)). +Qed. + +Definition cond_positivity [x:R] : bool := Cases (total_order_Rle R0 x) of + (leftT _) => true +| (rightT _) => 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. +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. +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. +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. +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. +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. +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. + +(**********) +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. +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. diff --git a/theories7/Reals/Rsyntax.v b/theories7/Reals/Rsyntax.v new file mode 100644 index 00000000..7b1b6266 --- /dev/null +++ b/theories7/Reals/Rsyntax.v @@ -0,0 +1,236 @@ +(************************************************************************) +(* 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: Rsyntax.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +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. +]. diff --git a/theories7/Reals/Rtopology.v b/theories7/Reals/Rtopology.v new file mode 100644 index 00000000..f2ae19b9 --- /dev/null +++ b/theories7/Reals/Rtopology.v @@ -0,0 +1,1178 @@ +(************************************************************************) +(* 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: Rtopology.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ 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). +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. +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. +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. +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. +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. +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. +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]. +Qed. + +Definition eq_Dom [D1,D2:R->Prop] : Prop := (included D1 D2)/\(included D2 D1). + +Infix "=_D" eq_Dom (at level 5, 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). +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). +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)). +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. +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). +Qed. + +Lemma open_set_P4 : (open_set [x:R]False). +Unfold open_set; 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. +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. +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. +Qed. + +Definition image_rec [f:R->R;D:R->Prop] : R->Prop := [x:R](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)). +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. +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. +Qed. + +Record family : Type := mkfamily { + ind : R->Prop; + f :> R->R->Prop; + cond_fam : (x:R)(EXT y:R|(f x y))->(ind x) }. + +Definition family_open_set [f:family] : Prop := (x:R) (open_set (f x)). + +Definition domain_finite [D:R->Prop] : Prop := (EXT l:Rlist | (x:R)(D x)<->(In x l)). + +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_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). + +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. +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 compact [X:R->Prop] : Prop := (f:family) (covering_open_set X f) -> (EXT 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. +Qed. + +Definition bounded [D:R->Prop] : Prop := (EXT m:R | (EXT M:R | (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. +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))]. +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. +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. +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. +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. +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. +(* 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. +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). +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). +Qed. + +Definition image_dir [f:R->R;D:R->Prop] : R->Prop := [x:R](EXT 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. +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]. +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. +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. +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). +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. +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. +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)]. +Qed. + +Definition family_closed_set [f:family] : Prop := (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_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)]. +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. +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``). + +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)]. +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. +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. +(* X est vide *) +Unfold uniform_continuity; Intros; Exists (mkposreal ? Rlt_R0_R1); 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). +(* 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. diff --git a/theories7/Reals/Rtrigo.v b/theories7/Reals/Rtrigo.v new file mode 100644 index 00000000..2b19a00a --- /dev/null +++ b/theories7/Reals/Rtrigo.v @@ -0,0 +1,1111 @@ +(************************************************************************) +(* 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: Rtrigo.v,v 1.1.2.1 2004/07/16 19:31:35 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require 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.]. +Open Local Scope nat_scope. +Open Local Scope R_scope. + +(** sin_PI2 is the only remaining axiom **) +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). +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. +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. +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. +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. +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. +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. +Qed. + +(**********) +Lemma neg_cos : (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. +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. +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. +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. +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. +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. +Qed. + +Lemma cos_2a : (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. +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. +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. +Qed. + +Lemma sin_neg : (x:R) ``(sin (-x))==-(sin x)``. +Apply sin_antisym. +Qed. + +Lemma cos_neg : (x:R) ``(cos (-x))==(cos x)``. +Intro; Symmetry; Apply cos_sym. +Qed. + +Lemma tan_0 : ``(tan 0)==0``. +Unfold tan; Rewrite -> sin_0; Rewrite -> cos_0. +Unfold Rdiv; Apply Rmult_Ol. +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. +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. +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. +Qed. + +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. +Qed. + +Lemma neg_sin : (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. +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]. +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]. +Qed. + +Lemma sin_shift : (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. +Qed. + +Lemma cos_sin : (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]. +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). +Qed. + +Lemma cos_sin_0_var : (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). +Qed. + +(**********) +Lemma _PI2_RLT_0 : ``-(PI/2)<0``. +Rewrite <- Ropp_O; Apply Rlt_Ropp1; 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]. +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. +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))]]. +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. diff --git a/theories7/Reals/Rtrigo_alt.v b/theories7/Reals/Rtrigo_alt.v new file mode 100644 index 00000000..db0e2fea --- /dev/null +++ b/theories7/Reals/Rtrigo_alt.v @@ -0,0 +1,294 @@ +(************************************************************************) +(* 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: Rtrigo_alt.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo_def. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +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 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 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). + +(**********) +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]. +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]. +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. diff --git a/theories7/Reals/Rtrigo_calc.v b/theories7/Reals/Rtrigo_calc.v new file mode 100644 index 00000000..ab181106 --- /dev/null +++ b/theories7/Reals/Rtrigo_calc.v @@ -0,0 +1,350 @@ +(************************************************************************) +(* 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: Rtrigo_calc.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo. +Require R_sqrt. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +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. +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. +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``. + +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. +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. +Qed. + +Lemma deg_rad : (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)). + +Lemma Rsqr_sin_cos_d_one : (x:R) ``(Rsqr (sind x))+(Rsqr (cosd x))==1``. +Intro x; Unfold sind; Unfold cosd; 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. diff --git a/theories7/Reals/Rtrigo_def.v b/theories7/Reals/Rtrigo_def.v new file mode 100644 index 00000000..0897416b --- /dev/null +++ b/theories7/Reals/Rtrigo_def.v @@ -0,0 +1,357 @@ +(************************************************************************) +(* 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: Rtrigo_def.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo_fun. +Require Max. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +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). + +Lemma exp_cof_no_R0 : (n:nat) ``/(INR (fact n))<>0``. +Intro. +Apply Rinv_neq_R0. +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. +Defined. + +Definition exp : R -> R := [x: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). +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. +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)). +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)``. + +Lemma cosh_0 : ``(cosh 0)==1``. +Unfold cosh; Rewrite Ropp_O; Rewrite exp_0. +Unfold Rdiv; 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. +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. +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. +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]. +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. +Qed. + +(**********) +Definition cos_in:R->R->Prop := [x,l:R](infinit_sum [i:nat]``(cos_n i)*(pow 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. +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. +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]. +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. +Qed. + +(**********) +Definition sin_in:R->R->Prop := [x,l:R](infinit_sum [i:nat]``(sin_n i)*(pow 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. +Qed. + +(***********************) +(* Definition of sinus *) +Definition sin : R -> R := [x:R](Cases (exist_sin (Rsqr x)) of (Specif.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. +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. +Qed. + +Lemma sin_0 : ``(sin 0)==0``. +Unfold sin; Case (exist_sin (Rsqr R0)). +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. +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. diff --git a/theories7/Reals/Rtrigo_fun.v b/theories7/Reals/Rtrigo_fun.v new file mode 100644 index 00000000..bc72c0e1 --- /dev/null +++ b/theories7/Reals/Rtrigo_fun.v @@ -0,0 +1,118 @@ +(************************************************************************) +(* 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: Rtrigo_fun.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +Open Local Scope R_scope. + +(*****************************************************************) +(* To define transcendental functions *) +(* *) +(*****************************************************************) +(*****************************************************************) +(* For exponential function *) +(* *) +(*****************************************************************) + +(*********) +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. +(**) +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. +Qed. + + + + + + diff --git a/theories7/Reals/Rtrigo_reg.v b/theories7/Reals/Rtrigo_reg.v new file mode 100644 index 00000000..02e40caf --- /dev/null +++ b/theories7/Reals/Rtrigo_reg.v @@ -0,0 +1,497 @@ +(************************************************************************) +(* 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: Rtrigo_reg.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require SeqSeries. +Require Rtrigo. +Require Ranalysis1. +Require PSeries_reg. +V7only [Import nat_scope. Import Z_scope. Import R_scope.]. +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). +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. +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. +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). +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. +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]. +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). +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. +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. +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. +Qed. + +Lemma derivable_sin : (derivable sin). +Unfold derivable; Intro; Apply derivable_pt_sin. +Qed. + +Lemma derivable_cos : (derivable cos). +Unfold derivable; 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. +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. diff --git a/theories7/Reals/SeqProp.v b/theories7/Reals/SeqProp.v new file mode 100644 index 00000000..b34fa339 --- /dev/null +++ b/theories7/Reals/SeqProp.v @@ -0,0 +1,1089 @@ +(************************************************************************) +(* 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: SeqProp.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Rseries. +Require Classical. +Require Max. +V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. +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))). + +(**********) +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. +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. +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]. +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. +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. +Qed. + +Definition majorant [Un:nat->R;pr:(has_ub Un)] : R := Cases (maj_sup Un pr) of (existTT 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. + +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. +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. +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_minorant [Un:nat->R;pr:(has_lb Un)] : nat -> R := [i:nat](minorant [k:nat](Un (plus i k)) (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. +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. +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. +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. +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. +Qed. + +(**********) +Lemma cauchy_maj : (Un:nat->R) (Cauchy_crit Un) -> (has_ub Un). +Intros. +Unfold has_ub. +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]. +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. +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. +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. +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]. +Qed. + +Lemma not_Rlt : (r1,r2:R)~(``r1<r2``)->``r1>=r2``. +Intros r1 r2 ; Generalize (total_order r1 r2) ; Unfold Rge. +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. +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. +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. +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]. +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. +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. +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. +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. +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. +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. +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. +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]. +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. +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. +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. +Qed. + +(* Un -> +oo *) +Definition cv_infty [Un:nat->R] : Prop := (M:R)(EXT N:nat | (n:nat) (le N n) -> ``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. +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]. +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. diff --git a/theories7/Reals/SeqSeries.v b/theories7/Reals/SeqSeries.v new file mode 100644 index 00000000..dd93c304 --- /dev/null +++ b/theories7/Reals/SeqSeries.v @@ -0,0 +1,307 @@ +(************************************************************************) +(* 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: SeqSeries.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Max. +Require Export Rseries. +Require Export SeqProp. +Require Export Rcomplete. +Require Export PartSum. +Require Export AltSeries. +Require Export Binomial. +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. +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. +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. + +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. +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. diff --git a/theories7/Reals/SplitAbsolu.v b/theories7/Reals/SplitAbsolu.v new file mode 100644 index 00000000..30580a0c --- /dev/null +++ b/theories7/Reals/SplitAbsolu.v @@ -0,0 +1,22 @@ +(************************************************************************) +(* 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: SplitAbsolu.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbasic_fun. + +Recursive Tactic Definition SplitAbs := + Match Context With + | [ |- [(case_Rabsolu ?1)] ] -> + Case (case_Rabsolu ?1); Try SplitAbs. + + +Recursive Tactic Definition SplitAbsolu := + Match Context With + | [ id:[(Rabsolu ?)] |- ? ] -> Generalize id; Clear id;Try SplitAbsolu + | [ |- [(Rabsolu ?1)] ] -> Unfold Rabsolu; Try SplitAbs;Intros. diff --git a/theories7/Reals/SplitRmult.v b/theories7/Reals/SplitRmult.v new file mode 100644 index 00000000..392675c3 --- /dev/null +++ b/theories7/Reals/SplitRmult.v @@ -0,0 +1,19 @@ +(************************************************************************) +(* 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: SplitRmult.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +(*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. + diff --git a/theories7/Reals/Sqrt_reg.v b/theories7/Reals/Sqrt_reg.v new file mode 100644 index 00000000..d2068e5d --- /dev/null +++ b/theories7/Reals/Sqrt_reg.v @@ -0,0 +1,297 @@ +(************************************************************************) +(* 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: Sqrt_reg.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) + +Require Rbase. +Require Rfunctions. +Require Ranalysis1. +Require R_sqrt. +V7only [Import R_scope.]. 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. +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]. +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. +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. +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. +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. +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. |