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Diffstat (limited to 'theories7/Reals/Rtrigo_calc.v')
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diff --git a/theories7/Reals/Rtrigo_calc.v b/theories7/Reals/Rtrigo_calc.v deleted file mode 100644 index ab181106..00000000 --- a/theories7/Reals/Rtrigo_calc.v +++ /dev/null @@ -1,350 +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: Rtrigo_calc.v,v 1.1.2.1 2004/07/16 19:31:36 herbelin Exp $ i*) - -Require Rbase. -Require Rfunctions. -Require SeqSeries. -Require Rtrigo. -Require R_sqrt. -V7only [ Import nat_scope. Import Z_scope. Import R_scope. ]. -Open Local Scope R_scope. - -Lemma tan_PI : ``(tan PI)==0``. -Unfold tan; Rewrite sin_PI; Rewrite cos_PI; Unfold Rdiv; Apply Rmult_Ol. -Qed. - -Lemma sin_3PI2 : ``(sin (3*(PI/2)))==(-1)``. -Replace ``3*(PI/2)`` with ``PI+(PI/2)``. -Rewrite sin_plus; Rewrite sin_PI; Rewrite cos_PI; Rewrite sin_PI2; Ring. -Pattern 1 PI; Rewrite (double_var PI); Ring. -Qed. - -Lemma tan_2PI : ``(tan (2*PI))==0``. -Unfold tan; Rewrite sin_2PI; Unfold Rdiv; Apply Rmult_Ol. -Qed. - -Lemma sin_cos_PI4 : ``(sin (PI/4)) == (cos (PI/4))``. -Proof with Trivial. -Rewrite cos_sin. -Replace ``PI/2+PI/4`` with ``-(PI/4)+PI``. -Rewrite neg_sin; Rewrite sin_neg; Ring. -Cut ``PI==PI/2+PI/2``; [Intro | Apply double_var]. -Pattern 2 3 PI; Rewrite H; Pattern 2 3 PI; Rewrite H. -Assert H0 : ``2<>0``; [DiscrR | Unfold Rdiv; Rewrite Rinv_Rmult; Try Ring]. -Qed. - -Lemma sin_PI3_cos_PI6 : ``(sin (PI/3))==(cos (PI/6))``. -Proof with Trivial. -Replace ``PI/6`` with ``(PI/2)-(PI/3)``. -Rewrite cos_shift. -Assert H0 : ``6<>0``; [DiscrR | Idtac]. -Assert H1 : ``3<>0``; [DiscrR | Idtac]. -Assert H2 : ``2<>0``; [DiscrR | Idtac]. -Apply r_Rmult_mult with ``6``. -Rewrite Rminus_distr; Repeat Rewrite (Rmult_sym ``6``). -Unfold Rdiv; Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite (Rmult_sym ``/3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Pattern 2 PI; Rewrite (Rmult_sym PI); Repeat Rewrite Rmult_1r; Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Ring. -Qed. - -Lemma sin_PI6_cos_PI3 : ``(cos (PI/3))==(sin (PI/6))``. -Proof with Trivial. -Replace ``PI/6`` with ``(PI/2)-(PI/3)``. -Rewrite sin_shift. -Assert H0 : ``6<>0``; [DiscrR | Idtac]. -Assert H1 : ``3<>0``; [DiscrR | Idtac]. -Assert H2 : ``2<>0``; [DiscrR | Idtac]. -Apply r_Rmult_mult with ``6``. -Rewrite Rminus_distr; Repeat Rewrite (Rmult_sym ``6``). -Unfold Rdiv; Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite (Rmult_sym ``/3``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Pattern 2 PI; Rewrite (Rmult_sym PI); Repeat Rewrite Rmult_1r; Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Ring. -Qed. - -Lemma PI6_RGT_0 : ``0<PI/6``. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]. -Qed. - -Lemma PI6_RLT_PI2 : ``PI/6<PI/2``. -Unfold Rdiv; Apply Rlt_monotony. -Apply PI_RGT_0. -Apply Rinv_lt; Sup. -Qed. - -Lemma sin_PI6 : ``(sin (PI/6))==1/2``. -Proof with Trivial. -Assert H : ``2<>0``; [DiscrR | Idtac]. -Apply r_Rmult_mult with ``2*(cos (PI/6))``. -Replace ``2*(cos (PI/6))*(sin (PI/6))`` with ``2*(sin (PI/6))*(cos (PI/6))``. -Rewrite <- sin_2a; Replace ``2*(PI/6)`` with ``PI/3``. -Rewrite sin_PI3_cos_PI6. -Unfold Rdiv; Rewrite Rmult_1l; Rewrite Rmult_assoc; Pattern 2 ``2``; Rewrite (Rmult_sym ``2``); Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Unfold Rdiv; Rewrite Rinv_Rmult. -Rewrite (Rmult_sym ``/2``); Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -DiscrR. -Ring. -Apply prod_neq_R0. -Cut ``0<(cos (PI/6))``; [Intro H1; Auto with real | Apply cos_gt_0; [Apply (Rlt_trans ``-(PI/2)`` ``0`` ``PI/6`` _PI2_RLT_0 PI6_RGT_0) | Apply PI6_RLT_PI2]]. -Qed. - -Lemma sqrt2_neq_0 : ~``(sqrt 2)==0``. -Assert Hyp:``0<2``; [Sup0 | Generalize (Rlt_le ``0`` ``2`` Hyp); Intro H1; Red; Intro H2; Generalize (sqrt_eq_0 ``2`` H1 H2); Intro H; Absurd ``2==0``; [ DiscrR | Assumption]]. -Qed. - -Lemma R1_sqrt2_neq_0 : ~``1/(sqrt 2)==0``. -Generalize (Rinv_neq_R0 ``(sqrt 2)`` sqrt2_neq_0); Intro H; Generalize (prod_neq_R0 ``1`` ``(Rinv (sqrt 2))`` R1_neq_R0 H); Intro H0; Assumption. -Qed. - -Lemma sqrt3_2_neq_0 : ~``2*(sqrt 3)==0``. -Apply prod_neq_R0; [DiscrR | Assert Hyp:``0<3``; [Sup0 | Generalize (Rlt_le ``0`` ``3`` Hyp); Intro H1; Red; Intro H2; Generalize (sqrt_eq_0 ``3`` H1 H2); Intro H; Absurd ``3==0``; [ DiscrR | Assumption]]]. -Qed. - -Lemma Rlt_sqrt2_0 : ``0<(sqrt 2)``. -Assert Hyp:``0<2``; [Sup0 | Generalize (sqrt_positivity ``2`` (Rlt_le ``0`` ``2`` Hyp)); Intro H1; Elim H1; Intro H2; [Assumption | Absurd ``0 == (sqrt 2)``; [Apply not_sym; Apply sqrt2_neq_0 | Assumption]]]. -Qed. - -Lemma Rlt_sqrt3_0 : ``0<(sqrt 3)``. -Cut ~(O=(1)); [Intro H0; Assert Hyp:``0<2``; [Sup0 | Generalize (Rlt_le ``0`` ``2`` Hyp); Intro H1; Assert Hyp2:``0<3``; [Sup0 | Generalize (Rlt_le ``0`` ``3`` Hyp2); Intro H2; Generalize (lt_INR_0 (1) (neq_O_lt (1) H0)); Unfold INR; Intro H3; Generalize (Rlt_compatibility ``2`` ``0`` ``1`` H3); Rewrite Rplus_sym; Rewrite Rplus_Ol; Replace ``2+1`` with ``3``; [Intro H4; Generalize (sqrt_lt_1 ``2`` ``3`` H1 H2 H4); Clear H3; Intro H3; Apply (Rlt_trans ``0`` ``(sqrt 2)`` ``(sqrt 3)`` Rlt_sqrt2_0 H3) | Ring]]] | Discriminate]. -Qed. - -Lemma PI4_RGT_0 : ``0<PI/4``. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]. -Qed. - -Lemma cos_PI4 : ``(cos (PI/4))==1/(sqrt 2)``. -Proof with Trivial. -Apply Rsqr_inj. -Apply cos_ge_0. -Left; Apply (Rlt_trans ``-(PI/2)`` R0 ``PI/4`` _PI2_RLT_0 PI4_RGT_0). -Left; Apply PI4_RLT_PI2. -Left; Apply (Rmult_lt_pos R1 ``(Rinv (sqrt 2))``). -Sup. -Apply Rlt_Rinv; Apply Rlt_sqrt2_0. -Rewrite Rsqr_div. -Rewrite Rsqr_1; Rewrite Rsqr_sqrt. -Assert H : ``2<>0``; [DiscrR | Idtac]. -Unfold Rsqr; Pattern 1 ``(cos (PI/4))``; Rewrite <- sin_cos_PI4; Replace ``(sin (PI/4))*(cos (PI/4))`` with ``(1/2)*(2*(sin (PI/4))*(cos (PI/4)))``. -Rewrite <- sin_2a; Replace ``2*(PI/4)`` with ``PI/2``. -Rewrite sin_PI2. -Apply Rmult_1r. -Unfold Rdiv; Rewrite (Rmult_sym ``2``); Rewrite Rinv_Rmult. -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r. -Unfold Rdiv; Rewrite Rmult_1l; Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l. -Left; Sup. -Apply sqrt2_neq_0. -Qed. - -Lemma sin_PI4 : ``(sin (PI/4))==1/(sqrt 2)``. -Rewrite sin_cos_PI4; Apply cos_PI4. -Qed. - -Lemma tan_PI4 : ``(tan (PI/4))==1``. -Unfold tan; Rewrite sin_cos_PI4. -Unfold Rdiv; Apply Rinv_r. -Change ``(cos (PI/4))<>0``; Rewrite cos_PI4; Apply R1_sqrt2_neq_0. -Qed. - -Lemma cos3PI4 : ``(cos (3*(PI/4)))==-1/(sqrt 2)``. -Proof with Trivial. -Replace ``3*(PI/4)`` with ``(PI/2)-(-(PI/4))``. -Rewrite cos_shift; Rewrite sin_neg; Rewrite sin_PI4. -Unfold Rdiv; Rewrite Ropp_mul1. -Unfold Rminus; Rewrite Ropp_Ropp; Pattern 1 PI; Rewrite double_var; Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_Rmult; [Ring | DiscrR | DiscrR]. -Qed. - -Lemma sin3PI4 : ``(sin (3*(PI/4)))==1/(sqrt 2)``. -Proof with Trivial. -Replace ``3*(PI/4)`` with ``(PI/2)-(-(PI/4))``. -Rewrite sin_shift; Rewrite cos_neg; Rewrite cos_PI4. -Unfold Rminus; Rewrite Ropp_Ropp; Pattern 1 PI; Rewrite double_var; Unfold Rdiv; Rewrite Rmult_Rplus_distrl; Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_Rmult; [Ring | DiscrR | DiscrR]. -Qed. - -Lemma cos_PI6 : ``(cos (PI/6))==(sqrt 3)/2``. -Proof with Trivial. -Apply Rsqr_inj. -Apply cos_ge_0. -Left; Apply (Rlt_trans ``-(PI/2)`` R0 ``PI/6`` _PI2_RLT_0 PI6_RGT_0). -Left; Apply PI6_RLT_PI2. -Left; Apply (Rmult_lt_pos ``(sqrt 3)`` ``(Rinv 2)``). -Apply Rlt_sqrt3_0. -Apply Rlt_Rinv; Sup0. -Assert H : ``2<>0``; [DiscrR | Idtac]. -Assert H1 : ``4<>0``; [Apply prod_neq_R0 | Idtac]. -Rewrite Rsqr_div. -Rewrite cos2; Unfold Rsqr; Rewrite sin_PI6; Rewrite sqrt_def. -Unfold Rdiv; Rewrite Rmult_1l; Apply r_Rmult_mult with ``4``. -Rewrite Rminus_distr; Rewrite (Rmult_sym ``3``); Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_r_sym. -Rewrite Rmult_1l; Rewrite Rmult_1r. -Rewrite <- (Rmult_sym ``/2``); Repeat Rewrite <- Rmult_assoc. -Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Rewrite <- Rinv_r_sym. -Ring. -Left; Sup0. -Qed. - -Lemma tan_PI6 : ``(tan (PI/6))==1/(sqrt 3)``. -Unfold tan; Rewrite sin_PI6; Rewrite cos_PI6; Unfold Rdiv; Repeat Rewrite Rmult_1l; Rewrite Rinv_Rmult. -Rewrite Rinv_Rinv. -Rewrite (Rmult_sym ``/2``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Apply Rmult_1r. -DiscrR. -DiscrR. -Red; Intro; Assert H1 := Rlt_sqrt3_0; Rewrite H in H1; Elim (Rlt_antirefl ``0`` H1). -Apply Rinv_neq_R0; DiscrR. -Qed. - -Lemma sin_PI3 : ``(sin (PI/3))==(sqrt 3)/2``. -Rewrite sin_PI3_cos_PI6; Apply cos_PI6. -Qed. - -Lemma cos_PI3 : ``(cos (PI/3))==1/2``. -Rewrite sin_PI6_cos_PI3; Apply sin_PI6. -Qed. - -Lemma tan_PI3 : ``(tan (PI/3))==(sqrt 3)``. -Unfold tan; Rewrite sin_PI3; Rewrite cos_PI3; Unfold Rdiv; Rewrite Rmult_1l; Rewrite Rinv_Rinv. -Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Apply Rmult_1r. -DiscrR. -DiscrR. -Qed. - -Lemma sin_2PI3 : ``(sin (2*(PI/3)))==(sqrt 3)/2``. -Rewrite double; Rewrite sin_plus; Rewrite sin_PI3; Rewrite cos_PI3; Unfold Rdiv; Repeat Rewrite Rmult_1l; Rewrite (Rmult_sym ``/2``); Repeat Rewrite <- Rmult_assoc; Rewrite double_var; Reflexivity. -Qed. - -Lemma cos_2PI3 : ``(cos (2*(PI/3)))==-1/2``. -Proof with Trivial. -Assert H : ``2<>0``; [DiscrR | Idtac]. -Assert H0 : ``4<>0``; [Apply prod_neq_R0 | Idtac]. -Rewrite double; Rewrite cos_plus; Rewrite sin_PI3; Rewrite cos_PI3; Unfold Rdiv; Rewrite Rmult_1l; Apply r_Rmult_mult with ``4``. -Rewrite Rminus_distr; Repeat Rewrite Rmult_assoc; Rewrite (Rmult_sym ``2``). -Repeat Rewrite Rmult_assoc; Rewrite <- (Rinv_l_sym). -Rewrite Rmult_1r; Rewrite <- Rinv_r_sym. -Pattern 4 ``2``; Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite Ropp_mul3; Rewrite Rmult_1r. -Rewrite (Rmult_sym ``2``); Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite (Rmult_sym ``2``); Rewrite (Rmult_sym ``/2``). -Repeat Rewrite Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1r; Rewrite sqrt_def. -Ring. -Left; Sup. -Qed. - -Lemma tan_2PI3 : ``(tan (2*(PI/3)))==-(sqrt 3)``. -Proof with Trivial. -Assert H : ``2<>0``; [DiscrR | Idtac]. -Unfold tan; Rewrite sin_2PI3; Rewrite cos_2PI3; Unfold Rdiv; Rewrite Ropp_mul1; Rewrite Rmult_1l; Rewrite <- Ropp_Rinv. -Rewrite Rinv_Rinv. -Rewrite Rmult_assoc; Rewrite Ropp_mul3; Rewrite <- Rinv_l_sym. -Ring. -Apply Rinv_neq_R0. -Qed. - -Lemma cos_5PI4 : ``(cos (5*(PI/4)))==-1/(sqrt 2)``. -Proof with Trivial. -Replace ``5*(PI/4)`` with ``(PI/4)+(PI)``. -Rewrite neg_cos; Rewrite cos_PI4; Unfold Rdiv; Rewrite Ropp_mul1. -Pattern 2 PI; Rewrite double_var; Pattern 2 3 PI; Rewrite double_var; Assert H : ``2<>0``; [DiscrR | Unfold Rdiv; Repeat Rewrite Rinv_Rmult; Try Ring]. -Qed. - -Lemma sin_5PI4 : ``(sin (5*(PI/4)))==-1/(sqrt 2)``. -Proof with Trivial. -Replace ``5*(PI/4)`` with ``(PI/4)+(PI)``. -Rewrite neg_sin; Rewrite sin_PI4; Unfold Rdiv; Rewrite Ropp_mul1. -Pattern 2 PI; Rewrite double_var; Pattern 2 3 PI; Rewrite double_var; Assert H : ``2<>0``; [DiscrR | Unfold Rdiv; Repeat Rewrite Rinv_Rmult; Try Ring]. -Qed. - -Lemma sin_cos5PI4 : ``(cos (5*(PI/4)))==(sin (5*(PI/4)))``. -Rewrite cos_5PI4; Rewrite sin_5PI4; Reflexivity. -Qed. - -Lemma Rgt_3PI2_0 : ``0<3*(PI/2)``. -Apply Rmult_lt_pos; [Sup0 | Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]]. -Qed. - -Lemma Rgt_2PI_0 : ``0<2*PI``. -Apply Rmult_lt_pos; [Sup0 | Apply PI_RGT_0]. -Qed. - -Lemma Rlt_PI_3PI2 : ``PI<3*(PI/2)``. -Generalize PI2_RGT_0; Intro H1; Generalize (Rlt_compatibility PI ``0`` ``PI/2`` H1); Replace ``PI+(PI/2)`` with ``3*(PI/2)``. -Rewrite Rplus_Or; Intro H2; Assumption. -Pattern 2 PI; Rewrite double_var; Ring. -Qed. - -Lemma Rlt_3PI2_2PI : ``3*(PI/2)<2*PI``. -Generalize PI2_RGT_0; Intro H1; Generalize (Rlt_compatibility ``3*(PI/2)`` ``0`` ``PI/2`` H1); Replace ``3*(PI/2)+(PI/2)`` with ``2*PI``. -Rewrite Rplus_Or; Intro H2; Assumption. -Rewrite double; Pattern 1 2 PI; Rewrite double_var; Ring. -Qed. - -(***************************************************************) -(* Radian -> Degree | Degree -> Radian *) -(***************************************************************) - -Definition plat : R := ``180``. -Definition toRad [x:R] : R := ``x*PI*/plat``. -Definition toDeg [x:R] : R := ``x*plat*/PI``. - -Lemma rad_deg : (x:R) (toRad (toDeg x))==x. -Intro; Unfold toRad toDeg; Replace ``x*plat*/PI*PI*/plat`` with ``x*(plat*/plat)*(PI*/PI)``; [Idtac | Ring]. -Repeat Rewrite <- Rinv_r_sym. -Ring. -Apply PI_neq0. -Unfold plat; DiscrR. -Qed. - -Lemma toRad_inj : (x,y:R) (toRad x)==(toRad y) -> x==y. -Intros; Unfold toRad in H; Apply r_Rmult_mult with PI. -Rewrite <- (Rmult_sym x); Rewrite <- (Rmult_sym y). -Apply r_Rmult_mult with ``/plat``. -Rewrite <- (Rmult_sym ``x*PI``); Rewrite <- (Rmult_sym ``y*PI``); Assumption. -Apply Rinv_neq_R0; Unfold plat; DiscrR. -Apply PI_neq0. -Qed. - -Lemma deg_rad : (x:R) (toDeg (toRad x))==x. -Intro x; Apply toRad_inj; Rewrite -> (rad_deg (toRad x)); Reflexivity. -Qed. - -Definition sind [x:R] : R := (sin (toRad x)). -Definition cosd [x:R] : R := (cos (toRad x)). -Definition tand [x:R] : R := (tan (toRad x)). - -Lemma Rsqr_sin_cos_d_one : (x:R) ``(Rsqr (sind x))+(Rsqr (cosd x))==1``. -Intro x; Unfold sind; Unfold cosd; Apply sin2_cos2. -Qed. - -(***************************************************) -(* Other properties *) -(***************************************************) - -Lemma sin_lb_ge_0 : (a:R) ``0<=a``->``a<=PI/2``->``0<=(sin_lb a)``. -Intros; Case (total_order R0 a); Intro. -Left; Apply sin_lb_gt_0; Assumption. -Elim H1; Intro. -Rewrite <- H2; Unfold sin_lb; Unfold sin_approx; Unfold sum_f_R0; Unfold sin_term; Repeat Rewrite pow_ne_zero. -Unfold Rdiv; Repeat Rewrite Rmult_Ol; Repeat Rewrite Rmult_Or; Repeat Rewrite Rplus_Or; Right; Reflexivity. -Discriminate. -Discriminate. -Discriminate. -Discriminate. -Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` a ``0`` H H2)). -Qed. |