Imported Upstream version 8.0pl3+8.1alphaupstream/8.0pl3+8.1alpha

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-(************************************************************************)
-(*  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.