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Diffstat (limited to 'theories7/Reals/Sqrt_reg.v')
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diff --git a/theories7/Reals/Sqrt_reg.v b/theories7/Reals/Sqrt_reg.v deleted file mode 100644 index d2068e5d..00000000 --- a/theories7/Reals/Sqrt_reg.v +++ /dev/null @@ -1,297 +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: 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. |