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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.