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diff --git a/theories7/Reals/NewtonInt.v b/theories7/Reals/NewtonInt.v deleted file mode 100644 index 56e5f15e..00000000 --- a/theories7/Reals/NewtonInt.v +++ /dev/null @@ -1,600 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(*i $Id: 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. - |