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diff --git a/theories7/Reals/Ranalysis3.v b/theories7/Reals/Ranalysis3.v deleted file mode 100644 index 6ce63bbc..00000000 --- a/theories7/Reals/Ranalysis3.v +++ /dev/null @@ -1,617 +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: 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. |