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Diffstat (limited to 'theories/Reals/NewtonInt.v')
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diff --git a/theories/Reals/NewtonInt.v b/theories/Reals/NewtonInt.v new file mode 100644 index 00000000..97cd4b94 --- /dev/null +++ b/theories/Reals/NewtonInt.v @@ -0,0 +1,788 @@ +(************************************************************************) +(* 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 1.11.2.1 2004/07/16 19:31:10 herbelin Exp $ i*) + +Require Import Rbase. +Require Import Rfunctions. +Require Import SeqSeries. +Require Import Rtrigo. +Require Import Ranalysis. Open Local Scope R_scope. + +(*******************************************) +(* Newton's Integral *) +(*******************************************) + +Definition Newton_integrable (f:R -> R) (a b:R) : Type := + sigT (fun g:R -> R => antiderivative f g a b \/ antiderivative f g b a). + +Definition NewtonInt (f:R -> R) (a b:R) (pr:Newton_integrable f a b) : R := + let g := match pr with + | existT a b => a + end in g b - g a. + +(* If f is differentiable, then f' is Newton integrable (Tautology ?) *) +Lemma FTCN_step1 : + forall (f:Differential) (a b:R), + Newton_integrable (fun x:R => derive_pt f x (cond_diff f x)) a b. +intros f a b; unfold Newton_integrable in |- *; apply existT with (d1 f); + unfold antiderivative in |- *; intros; case (Rle_dec a b); + intro; + [ left; split; [ intros; exists (cond_diff f x); reflexivity | assumption ] + | right; split; + [ intros; exists (cond_diff f x); reflexivity | auto with real ] ]. +Defined. + +(* By definition, we have the Fondamental Theorem of Calculus *) +Lemma FTC_Newton : + forall (f:Differential) (a b:R), + NewtonInt (fun x:R => derive_pt f x (cond_diff f x)) a b + (FTCN_step1 f a b) = f b - f a. +intros; unfold NewtonInt in |- *; reflexivity. +Qed. + +(* $\int_a^a f$ exists forall a:R and f:R->R *) +Lemma NewtonInt_P1 : forall (f:R -> R) (a:R), Newton_integrable f a a. +intros f a; unfold Newton_integrable in |- *; + apply existT with (fct_cte (f a) * id)%F; left; + unfold antiderivative in |- *; split. +intros; assert (H1 : derivable_pt (fct_cte (f a) * id) x). +apply derivable_pt_mult. +apply derivable_pt_const. +apply derivable_pt_id. +exists H1; assert (H2 : x = a). +elim H; intros; apply Rle_antisym; assumption. +symmetry in |- *; apply derive_pt_eq_0; + replace (f x) with (0 * id x + fct_cte (f a) x * 1); + [ apply (derivable_pt_lim_mult (fct_cte (f a)) id x); + [ apply derivable_pt_lim_const | apply derivable_pt_lim_id ] + | unfold id, fct_cte in |- *; rewrite H2; ring ]. +right; reflexivity. +Defined. + +(* $\int_a^a f = 0$ *) +Lemma NewtonInt_P2 : + forall (f:R -> R) (a:R), NewtonInt f a a (NewtonInt_P1 f a) = 0. +intros; unfold NewtonInt in |- *; simpl in |- *; + unfold mult_fct, fct_cte, id in |- *; ring. +Qed. + +(* If $\int_a^b f$ exists, then $\int_b^a f$ exists too *) +Lemma NewtonInt_P3 : + forall (f:R -> R) (a b:R) (X:Newton_integrable f a b), + Newton_integrable f b a. +unfold Newton_integrable in |- *; intros; elim X; intros g H; + apply existT with g; tauto. +Defined. + +(* $\int_a^b f = -\int_b^a f$ *) +Lemma NewtonInt_P4 : + forall (f:R -> R) (a b:R) (pr:Newton_integrable f a b), + NewtonInt f a b pr = - NewtonInt f b a (NewtonInt_P3 f a b pr). +intros; unfold Newton_integrable in pr; elim pr; intros; elim p; intro. +unfold NewtonInt in |- *; + case + (NewtonInt_P3 f a b + (existT + (fun g:R -> R => antiderivative f g a b \/ antiderivative f g b a) x + p)). +intros; elim o; intro. +unfold antiderivative in H0; elim H0; intros; elim H2; intro. +unfold antiderivative in H; elim H; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H5 H3)). +rewrite H3; ring. +assert (H1 := antiderivative_Ucte f x x0 a b H H0); elim H1; intros; + unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +assert (H3 : a <= a <= b). +split; [ right; reflexivity | assumption ]. +assert (H4 : a <= b <= b). +split; [ assumption | right; reflexivity ]. +assert (H5 := H2 _ H3); assert (H6 := H2 _ H4); rewrite H5; rewrite H6; ring. +unfold NewtonInt in |- *; + case + (NewtonInt_P3 f a b + (existT + (fun g:R -> R => antiderivative f g a b \/ antiderivative f g b a) x + p)); intros; elim o; intro. +assert (H1 := antiderivative_Ucte f x x0 b a H H0); elim H1; intros; + unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +assert (H3 : b <= a <= a). +split; [ assumption | right; reflexivity ]. +assert (H4 : b <= b <= a). +split; [ right; reflexivity | assumption ]. +assert (H5 := H2 _ H3); assert (H6 := H2 _ H4); rewrite H5; rewrite H6; ring. +unfold antiderivative in H0; elim H0; intros; elim H2; intro. +unfold antiderivative in H; elim H; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H5 H3)). +rewrite H3; ring. +Qed. + +(* The set of Newton integrable functions is a vectorial space *) +Lemma NewtonInt_P5 : + forall (f g:R -> R) (l a b:R), + Newton_integrable f a b -> + Newton_integrable g a b -> + Newton_integrable (fun x:R => l * f x + g x) a b. +unfold Newton_integrable in |- *; intros; elim X; intros; elim X0; intros; + exists (fun y:R => l * x y + x0 y). +elim p; intro. +elim p0; intro. +left; unfold antiderivative in |- *; unfold antiderivative in H, H0; elim H; + clear H; intros; elim H0; clear H0; intros H0 _. +split. +intros; elim (H _ H2); elim (H0 _ H2); intros. +assert (H5 : derivable_pt (fun y:R => l * x y + x0 y) x1). +reg. +exists H5; symmetry in |- *; reg; rewrite <- H3; rewrite <- H4; reflexivity. +assumption. +unfold antiderivative in H, H0; elim H; elim H0; intros; elim H4; intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H5 H2)). +left; rewrite <- H5; unfold antiderivative in |- *; split. +intros; elim H6; intros; assert (H9 : x1 = a). +apply Rle_antisym; assumption. +assert (H10 : a <= x1 <= b). +split; right; [ symmetry in |- *; assumption | rewrite <- H5; assumption ]. +assert (H11 : b <= x1 <= a). +split; right; [ rewrite <- H5; symmetry in |- *; assumption | assumption ]. +assert (H12 : derivable_pt x x1). +unfold derivable_pt in |- *; exists (f x1); elim (H3 _ H10); intros; + eapply derive_pt_eq_1; symmetry in |- *; apply H12. +assert (H13 : derivable_pt x0 x1). +unfold derivable_pt in |- *; exists (g x1); elim (H1 _ H11); intros; + eapply derive_pt_eq_1; symmetry in |- *; apply H13. +assert (H14 : derivable_pt (fun y:R => l * x y + x0 y) x1). +reg. +exists H14; symmetry in |- *; reg. +assert (H15 : derive_pt x0 x1 H13 = g x1). +elim (H1 _ H11); intros; rewrite H15; apply pr_nu. +assert (H16 : derive_pt x x1 H12 = f x1). +elim (H3 _ H10); intros; rewrite H16; apply pr_nu. +rewrite H15; rewrite H16; ring. +right; reflexivity. +elim p0; intro. +unfold antiderivative in H, H0; elim H; elim H0; intros; elim H4; intro. +elim (Rlt_irrefl _ (Rlt_le_trans _ _ _ H5 H2)). +left; rewrite H5; unfold antiderivative in |- *; split. +intros; elim H6; intros; assert (H9 : x1 = a). +apply Rle_antisym; assumption. +assert (H10 : a <= x1 <= b). +split; right; [ symmetry in |- *; assumption | rewrite H5; assumption ]. +assert (H11 : b <= x1 <= a). +split; right; [ rewrite H5; symmetry in |- *; assumption | assumption ]. +assert (H12 : derivable_pt x x1). +unfold derivable_pt in |- *; exists (f x1); elim (H3 _ H11); intros; + eapply derive_pt_eq_1; symmetry in |- *; apply H12. +assert (H13 : derivable_pt x0 x1). +unfold derivable_pt in |- *; exists (g x1); elim (H1 _ H10); intros; + eapply derive_pt_eq_1; symmetry in |- *; apply H13. +assert (H14 : derivable_pt (fun y:R => l * x y + x0 y) x1). +reg. +exists H14; symmetry in |- *; reg. +assert (H15 : derive_pt x0 x1 H13 = g x1). +elim (H1 _ H10); intros; rewrite H15; apply pr_nu. +assert (H16 : derive_pt x x1 H12 = f x1). +elim (H3 _ H11); intros; rewrite H16; apply pr_nu. +rewrite H15; rewrite H16; ring. +right; reflexivity. +right; unfold antiderivative in |- *; unfold antiderivative in H, H0; elim H; + clear H; intros; elim H0; clear H0; intros H0 _; split. +intros; elim (H _ H2); elim (H0 _ H2); intros. +assert (H5 : derivable_pt (fun y:R => l * x y + x0 y) x1). +reg. +exists H5; symmetry in |- *; reg; rewrite <- H3; rewrite <- H4; reflexivity. +assumption. +Defined. + +(**********) +Lemma antiderivative_P1 : + forall (f g F G:R -> R) (l a b:R), + antiderivative f F a b -> + antiderivative g G a b -> + antiderivative (fun x:R => l * f x + g x) (fun x:R => l * F x + G x) a b. +unfold antiderivative in |- *; intros; elim H; elim H0; clear H H0; intros; + split. +intros; elim (H _ H3); elim (H1 _ H3); intros. +assert (H6 : derivable_pt (fun x:R => l * F x + G x) x). +reg. +exists H6; symmetry in |- *; reg; rewrite <- H4; rewrite <- H5; ring. +assumption. +Qed. + +(* $\int_a^b \lambda f + g = \lambda \int_a^b f + \int_a^b f *) +Lemma NewtonInt_P6 : + forall (f g:R -> R) (l a b:R) (pr1:Newton_integrable f a b) + (pr2:Newton_integrable g a b), + NewtonInt (fun x:R => l * f x + g x) a b (NewtonInt_P5 f g l a b pr1 pr2) = + l * NewtonInt f a b pr1 + NewtonInt g a b pr2. +intros f g l a b pr1 pr2; unfold NewtonInt in |- *; + case (NewtonInt_P5 f g l a b pr1 pr2); intros; case pr1; + intros; case pr2; intros; case (total_order_T a b); + intro. +elim s; intro. +elim o; intro. +elim o0; intro. +elim o1; intro. +assert (H2 := antiderivative_P1 f g x0 x1 l a b H0 H1); + assert (H3 := antiderivative_Ucte _ _ _ _ _ H H2); + elim H3; intros; assert (H5 : a <= a <= b). +split; [ right; reflexivity | left; assumption ]. +assert (H6 : a <= b <= b). +split; [ left; assumption | right; reflexivity ]. +assert (H7 := H4 _ H5); assert (H8 := H4 _ H6); rewrite H7; rewrite H8; ring. +unfold antiderivative in H1; elim H1; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H3 a0)). +unfold antiderivative in H0; elim H0; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a0)). +unfold antiderivative in H; elim H; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 a0)). +rewrite b0; ring. +elim o; intro. +unfold antiderivative in H; elim H; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 r)). +elim o0; intro. +unfold antiderivative in H0; elim H0; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 r)). +elim o1; intro. +unfold antiderivative in H1; elim H1; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H3 r)). +assert (H2 := antiderivative_P1 f g x0 x1 l b a H0 H1); + assert (H3 := antiderivative_Ucte _ _ _ _ _ H H2); + elim H3; intros; assert (H5 : b <= a <= a). +split; [ left; assumption | right; reflexivity ]. +assert (H6 : b <= b <= a). +split; [ right; reflexivity | left; assumption ]. +assert (H7 := H4 _ H5); assert (H8 := H4 _ H6); rewrite H7; rewrite H8; ring. +Qed. + +Lemma antiderivative_P2 : + forall (f F0 F1:R -> R) (a b c:R), + antiderivative f F0 a b -> + antiderivative f F1 b c -> + antiderivative f + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) a c. +unfold antiderivative in |- *; intros; elim H; clear H; intros; elim H0; + clear H0; intros; split. +2: apply Rle_trans with b; assumption. +intros; elim H3; clear H3; intros; case (total_order_T x b); intro. +elim s; intro. +assert (H5 : a <= x <= b). +split; [ assumption | left; assumption ]. +assert (H6 := H _ H5); elim H6; clear H6; intros; + assert + (H7 : + derivable_pt_lim + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) x (f x)). +unfold derivable_pt_lim in |- *; assert (H7 : derive_pt F0 x x0 = f x). +symmetry in |- *; assumption. +assert (H8 := derive_pt_eq_1 F0 x (f x) x0 H7); unfold derivable_pt_lim in H8; + intros; elim (H8 _ H9); intros; set (D := Rmin x1 (b - x)). +assert (H11 : 0 < D). +unfold D in |- *; unfold Rmin in |- *; case (Rle_dec x1 (b - x)); intro. +apply (cond_pos x1). +apply Rlt_Rminus; assumption. +exists (mkposreal _ H11); intros; case (Rle_dec x b); intro. +case (Rle_dec (x + h) b); intro. +apply H10. +assumption. +apply Rlt_le_trans with D; [ assumption | unfold D in |- *; apply Rmin_l ]. +elim n; left; apply Rlt_le_trans with (x + D). +apply Rplus_lt_compat_l; apply Rle_lt_trans with (Rabs h). +apply RRle_abs. +apply H13. +apply Rplus_le_reg_l with (- x); rewrite <- Rplus_assoc; rewrite Rplus_opp_l; + rewrite Rplus_0_l; rewrite Rplus_comm; unfold D in |- *; + apply Rmin_r. +elim n; left; assumption. +assert + (H8 : + derivable_pt + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) x). +unfold derivable_pt in |- *; apply existT with (f x); apply H7. +exists H8; symmetry in |- *; apply derive_pt_eq_0; apply H7. +assert (H5 : a <= x <= b). +split; [ assumption | right; assumption ]. +assert (H6 : b <= x <= c). +split; [ right; symmetry in |- *; assumption | assumption ]. +elim (H _ H5); elim (H0 _ H6); intros; assert (H9 : derive_pt F0 x x1 = f x). +symmetry in |- *; assumption. +assert (H10 : derive_pt F1 x x0 = f x). +symmetry in |- *; assumption. +assert (H11 := derive_pt_eq_1 F0 x (f x) x1 H9); + assert (H12 := derive_pt_eq_1 F1 x (f x) x0 H10); + assert + (H13 : + derivable_pt_lim + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) x (f x)). +unfold derivable_pt_lim in |- *; unfold derivable_pt_lim in H11, H12; intros; + elim (H11 _ H13); elim (H12 _ H13); intros; set (D := Rmin x2 x3); + assert (H16 : 0 < D). +unfold D in |- *; unfold Rmin in |- *; case (Rle_dec x2 x3); intro. +apply (cond_pos x2). +apply (cond_pos x3). +exists (mkposreal _ H16); intros; case (Rle_dec x b); intro. +case (Rle_dec (x + h) b); intro. +apply H15. +assumption. +apply Rlt_le_trans with D; [ assumption | unfold D in |- *; apply Rmin_r ]. +replace (F1 (x + h) + (F0 b - F1 b) - F0 x) with (F1 (x + h) - F1 x). +apply H14. +assumption. +apply Rlt_le_trans with D; [ assumption | unfold D in |- *; apply Rmin_l ]. +rewrite b0; ring. +elim n; right; assumption. +assert + (H14 : + derivable_pt + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) x). +unfold derivable_pt in |- *; apply existT with (f x); apply H13. +exists H14; symmetry in |- *; apply derive_pt_eq_0; apply H13. +assert (H5 : b <= x <= c). +split; [ left; assumption | assumption ]. +assert (H6 := H0 _ H5); elim H6; clear H6; intros; + assert + (H7 : + derivable_pt_lim + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) x (f x)). +unfold derivable_pt_lim in |- *; assert (H7 : derive_pt F1 x x0 = f x). +symmetry in |- *; assumption. +assert (H8 := derive_pt_eq_1 F1 x (f x) x0 H7); unfold derivable_pt_lim in H8; + intros; elim (H8 _ H9); intros; set (D := Rmin x1 (x - b)); + assert (H11 : 0 < D). +unfold D in |- *; unfold Rmin in |- *; case (Rle_dec x1 (x - b)); intro. +apply (cond_pos x1). +apply Rlt_Rminus; assumption. +exists (mkposreal _ H11); intros; case (Rle_dec x b); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 r)). +case (Rle_dec (x + h) b); intro. +cut (b < x + h). +intro; elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 H14)). +apply Rplus_lt_reg_r with (- h - b); replace (- h - b + b) with (- h); + [ idtac | ring ]; replace (- h - b + (x + h)) with (x - b); + [ idtac | ring ]; apply Rle_lt_trans with (Rabs h). +rewrite <- Rabs_Ropp; apply RRle_abs. +apply Rlt_le_trans with D. +apply H13. +unfold D in |- *; apply Rmin_r. +replace (F1 (x + h) + (F0 b - F1 b) - (F1 x + (F0 b - F1 b))) with + (F1 (x + h) - F1 x); [ idtac | ring ]; apply H10. +assumption. +apply Rlt_le_trans with D. +assumption. +unfold D in |- *; apply Rmin_l. +assert + (H8 : + derivable_pt + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end) x). +unfold derivable_pt in |- *; apply existT with (f x); apply H7. +exists H8; symmetry in |- *; apply derive_pt_eq_0; apply H7. +Qed. + +Lemma antiderivative_P3 : + forall (f F0 F1:R -> R) (a b c:R), + antiderivative f F0 a b -> + antiderivative f F1 c b -> + antiderivative f F1 c a \/ antiderivative f F0 a c. +intros; unfold antiderivative in H, H0; elim H; clear H; elim H0; clear H0; + intros; case (total_order_T a c); intro. +elim s; intro. +right; unfold antiderivative in |- *; split. +intros; apply H1; elim H3; intros; split; + [ assumption | apply Rle_trans with c; assumption ]. +left; assumption. +right; unfold antiderivative in |- *; split. +intros; apply H1; elim H3; intros; split; + [ assumption | apply Rle_trans with c; assumption ]. +right; assumption. +left; unfold antiderivative in |- *; split. +intros; apply H; elim H3; intros; split; + [ assumption | apply Rle_trans with a; assumption ]. +left; assumption. +Qed. + +Lemma antiderivative_P4 : + forall (f F0 F1:R -> R) (a b c:R), + antiderivative f F0 a b -> + antiderivative f F1 a c -> + antiderivative f F1 b c \/ antiderivative f F0 c b. +intros; unfold antiderivative in H, H0; elim H; clear H; elim H0; clear H0; + intros; case (total_order_T c b); intro. +elim s; intro. +right; unfold antiderivative in |- *; split. +intros; apply H1; elim H3; intros; split; + [ apply Rle_trans with c; assumption | assumption ]. +left; assumption. +right; unfold antiderivative in |- *; split. +intros; apply H1; elim H3; intros; split; + [ apply Rle_trans with c; assumption | assumption ]. +right; assumption. +left; unfold antiderivative in |- *; split. +intros; apply H; elim H3; intros; split; + [ apply Rle_trans with b; assumption | assumption ]. +left; assumption. +Qed. + +Lemma NewtonInt_P7 : + forall (f:R -> R) (a b c:R), + a < b -> + b < c -> + Newton_integrable f a b -> + Newton_integrable f b c -> Newton_integrable f a c. +unfold Newton_integrable in |- *; intros f a b c Hab Hbc X X0; elim X; + clear X; intros F0 H0; elim X0; clear X0; intros F1 H1; + set + (g := + fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end); apply existT with g; left; unfold g in |- *; + apply antiderivative_P2. +elim H0; intro. +assumption. +unfold antiderivative in H; elim H; clear H; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hab)). +elim H1; intro. +assumption. +unfold antiderivative in H; elim H; clear H; intros; + elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 Hbc)). +Qed. + +Lemma NewtonInt_P8 : + forall (f:R -> R) (a b c:R), + Newton_integrable f a b -> + Newton_integrable f b c -> Newton_integrable f a c. +intros. +elim X; intros F0 H0. +elim X0; intros F1 H1. +case (total_order_T a b); intro. +elim s; intro. +case (total_order_T b c); intro. +elim s0; intro. +(* a<b & b<c *) +unfold Newton_integrable in |- *; + apply existT with + (fun x:R => + match Rle_dec x b with + | left _ => F0 x + | right _ => F1 x + (F0 b - F1 b) + end). +elim H0; intro. +elim H1; intro. +left; apply antiderivative_P2; assumption. +unfold antiderivative in H2; elim H2; clear H2; intros _ H2. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a1)). +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)). +(* a<b & b=c *) +rewrite b0 in X; apply X. +(* a<b & b>c *) +case (total_order_T a c); intro. +elim s0; intro. +unfold Newton_integrable in |- *; apply existT with F0. +left. +elim H1; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim H0; intro. +assert (H3 := antiderivative_P3 f F0 F1 a b c H2 H). +elim H3; intro. +unfold antiderivative in H4; elim H4; clear H4; intros _ H4. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 a1)). +assumption. +unfold antiderivative in H2; elim H2; clear H2; intros _ H2. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a0)). +rewrite b0; apply NewtonInt_P1. +unfold Newton_integrable in |- *; apply existT with F1. +right. +elim H1; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim H0; intro. +assert (H3 := antiderivative_P3 f F0 F1 a b c H2 H). +elim H3; intro. +assumption. +unfold antiderivative in H4; elim H4; clear H4; intros _ H4. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 r0)). +unfold antiderivative in H2; elim H2; clear H2; intros _ H2. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a0)). +(* a=b *) +rewrite b0; apply X0. +case (total_order_T b c); intro. +elim s; intro. +(* a>b & b<c *) +case (total_order_T a c); intro. +elim s0; intro. +unfold Newton_integrable in |- *; apply existT with F1. +left. +elim H1; intro. +(*****************) +elim H0; intro. +unfold antiderivative in H2; elim H2; clear H2; intros _ H2. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 r)). +assert (H3 := antiderivative_P4 f F0 F1 b a c H2 H). +elim H3; intro. +assumption. +unfold antiderivative in H4; elim H4; clear H4; intros _ H4. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 a1)). +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)). +rewrite b0; apply NewtonInt_P1. +unfold Newton_integrable in |- *; apply existT with F0. +right. +elim H0; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim H1; intro. +assert (H3 := antiderivative_P4 f F0 F1 b a c H H2). +elim H3; intro. +unfold antiderivative in H4; elim H4; clear H4; intros _ H4. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H4 r0)). +assumption. +unfold antiderivative in H2; elim H2; clear H2; intros _ H2. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H2 a0)). +(* a>b & b=c *) +rewrite b0 in X; apply X. +(* 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 : + forall (f:R -> R) (a b c:R) (pr1:Newton_integrable f a b) + (pr2:Newton_integrable f b c), + NewtonInt f a c (NewtonInt_P8 f a b c pr1 pr2) = + NewtonInt f a b pr1 + NewtonInt f b c pr2. +intros; unfold NewtonInt in |- *. +case (NewtonInt_P8 f a b c pr1 pr2); intros. +case pr1; intros. +case pr2; intros. +case (total_order_T a b); intro. +elim s; intro. +case (total_order_T b c); intro. +elim s0; intro. +(* a<b & b<c *) +elim o0; intro. +elim o1; intro. +elim o; intro. +assert (H2 := antiderivative_P2 f x0 x1 a b c H H0). +assert + (H3 := + antiderivative_Ucte f x + (fun x:R => + match Rle_dec x b with + | left _ => x0 x + | right _ => x1 x + (x0 b - x1 b) + end) a c H1 H2). +elim H3; intros. +assert (H5 : a <= a <= c). +split; [ right; reflexivity | left; apply Rlt_trans with b; assumption ]. +assert (H6 : a <= c <= c). +split; [ left; apply Rlt_trans with b; assumption | right; reflexivity ]. +rewrite (H4 _ H5); rewrite (H4 _ H6). +case (Rle_dec a b); intro. +case (Rle_dec c b); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 a1)). +ring. +elim n; left; assumption. +unfold antiderivative in H1; elim H1; clear H1; intros _ H1. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 (Rlt_trans _ _ _ a0 a1))). +unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 a1)). +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)). +(* a<b & b=c *) +rewrite <- b0. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r. +rewrite <- b0 in o. +elim o0; intro. +elim o; intro. +assert (H1 := antiderivative_Ucte f x x0 a b H0 H). +elim H1; intros. +rewrite (H2 b). +rewrite (H2 a). +ring. +split; [ right; reflexivity | left; assumption ]. +split; [ left; assumption | right; reflexivity ]. +unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 a0)). +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H a0)). +(* a<b & b>c *) +elim o1; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim o0; intro. +elim o; intro. +assert (H2 := antiderivative_P2 f x x1 a c b H1 H). +assert (H3 := antiderivative_Ucte _ _ _ a b H0 H2). +elim H3; intros. +rewrite (H4 a). +rewrite (H4 b). +case (Rle_dec b c); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 r)). +case (Rle_dec a c); intro. +ring. +elim n0; unfold antiderivative in H1; elim H1; intros; assumption. +split; [ left; assumption | right; reflexivity ]. +split; [ right; reflexivity | left; assumption ]. +assert (H2 := antiderivative_P2 _ _ _ _ _ _ H1 H0). +assert (H3 := antiderivative_Ucte _ _ _ c b H H2). +elim H3; intros. +rewrite (H4 c). +rewrite (H4 b). +case (Rle_dec b a); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r0 a0)). +case (Rle_dec c a); intro. +ring. +elim n0; unfold antiderivative in H1; elim H1; intros; assumption. +split; [ left; assumption | right; reflexivity ]. +split; [ right; reflexivity | left; assumption ]. +unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 a0)). +(* a=b *) +rewrite b0 in o; rewrite b0. +elim o; intro. +elim o1; intro. +assert (H1 := antiderivative_Ucte _ _ _ b c H H0). +elim H1; intros. +assert (H3 : b <= c). +unfold antiderivative in H; elim H; intros; assumption. +rewrite (H2 b). +rewrite (H2 c). +ring. +split; [ assumption | right; reflexivity ]. +split; [ right; reflexivity | assumption ]. +assert (H1 : b = c). +unfold antiderivative in H, H0; elim H; elim H0; intros; apply Rle_antisym; + assumption. +rewrite H1; ring. +elim o1; intro. +assert (H1 : b = c). +unfold antiderivative in H, H0; elim H; elim H0; intros; apply Rle_antisym; + assumption. +rewrite H1; ring. +assert (H1 := antiderivative_Ucte _ _ _ c b H H0). +elim H1; intros. +assert (H3 : c <= b). +unfold antiderivative in H; elim H; intros; assumption. +rewrite (H2 c). +rewrite (H2 b). +ring. +split; [ assumption | right; reflexivity ]. +split; [ right; reflexivity | assumption ]. +(* 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_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim o1; intro. +elim o; intro. +assert (H2 := antiderivative_P2 _ _ _ _ _ _ H H1). +assert (H3 := antiderivative_Ucte _ _ _ b c H0 H2). +elim H3; intros. +rewrite (H4 b). +rewrite (H4 c). +case (Rle_dec b a); intro. +case (Rle_dec c a); intro. +assert (H5 : a = c). +unfold antiderivative in H1; elim H1; intros; apply Rle_antisym; assumption. +rewrite H5; ring. +ring. +elim n; left; assumption. +split; [ left; assumption | right; reflexivity ]. +split; [ right; reflexivity | left; assumption ]. +assert (H2 := antiderivative_P2 _ _ _ _ _ _ H0 H1). +assert (H3 := antiderivative_Ucte _ _ _ b a H H2). +elim H3; intros. +rewrite (H4 a). +rewrite (H4 b). +case (Rle_dec b c); intro. +case (Rle_dec a c); intro. +assert (H5 : a = c). +unfold antiderivative in H1; elim H1; intros; apply Rle_antisym; assumption. +rewrite H5; ring. +ring. +elim n; left; assumption. +split; [ right; reflexivity | left; assumption ]. +split; [ left; assumption | right; reflexivity ]. +unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 a0)). +(* a>b & b=c *) +rewrite <- b0. +unfold Rminus in |- *; rewrite Rplus_opp_r; rewrite Rplus_0_r. +rewrite <- b0 in o. +elim o0; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim o; intro. +unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 r)). +assert (H1 := antiderivative_Ucte f x x0 b a H0 H). +elim H1; intros. +rewrite (H2 b). +rewrite (H2 a). +ring. +split; [ left; assumption | right; reflexivity ]. +split; [ right; reflexivity | left; assumption ]. +(* a>b & b>c *) +elim o0; intro. +unfold antiderivative in H; elim H; clear H; intros _ H. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H r)). +elim o1; intro. +unfold antiderivative in H0; elim H0; clear H0; intros _ H0. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H0 r0)). +elim o; intro. +unfold antiderivative in H1; elim H1; clear H1; intros _ H1. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ H1 (Rlt_trans _ _ _ r0 r))). +assert (H2 := antiderivative_P2 _ _ _ _ _ _ H0 H). +assert (H3 := antiderivative_Ucte _ _ _ c a H1 H2). +elim H3; intros. +assert (H5 : c <= a). +unfold antiderivative in H1; elim H1; intros; assumption. +rewrite (H4 c). +rewrite (H4 a). +case (Rle_dec a b); intro. +elim (Rlt_irrefl _ (Rle_lt_trans _ _ _ r1 r)). +case (Rle_dec c b); intro. +ring. +elim n0; left; assumption. +split; [ assumption | right; reflexivity ]. +split; [ right; reflexivity | assumption ]. +Qed. |