Proprietes (calculatoires) des fonctions trigonometriques

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2875 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 317 insertions, 0 deletions

diff --git a/theories/Reals/Rtrigo_calc.v b/theories/Reals/Rtrigo_calc.v
new file mode 100644
index 000000000..0bcb76d1f
--- /dev/null
+++ b/theories/Reals/Rtrigo_calc.v
@@ -0,0 +1,317 @@
+(***********************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team    *)
+(* <O___,, *        INRIA-Rocquencourt  &  LRI-CNRS-Orsay              *)
+(*   \VV/  *************************************************************)
+(*    //   *      This file is distributed under the terms of the      *)
+(*         *       GNU Lesser General Public License Version 2.1       *)
+(***********************************************************************)
+
+(*i $Id$ i*)
+
+Require Rbase.
+Require DiscrR.
+Require R_sqr.
+Require Rlimit.
+Require Rtrigo.
+Require R_sqrt.
+
+Lemma sqrt2_neq_0 : ~``(sqrt 2)==0``.
+Generalize (Rlt_le ``0`` ``2`` Rgt_2_0); 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 | Generalize (Rlt_le ``0`` ``3`` Rgt_3_0); 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)``.
+Generalize (sqrt_positivity ``2`` (Rlt_le ``0`` ``2`` Rgt_2_0)); 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; Generalize (Rlt_le ``0`` ``2`` Rgt_2_0); Intro H1; Generalize (Rlt_le ``0`` ``3`` Rgt_3_0); 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 cos_PI4 : ``(cos (PI/4))==1/(sqrt 2)``.
+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))``).
+Apply Rlt_R0_R1.
+Apply Rlt_Rinv.
+Apply Rlt_sqrt2_0.
+Rewrite Rsqr_div.
+Rewrite Rsqr_1; Rewrite Rsqr_sqrt.
+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``).
+Replace ``4`` with ``2*2``.
+Rewrite Rinv_Rmult.
+Repeat Rewrite Rmult_assoc.
+Rewrite <- Rinv_l_sym.
+Rewrite Rmult_1r.
+Reflexivity.
+DiscrR.
+DiscrR.
+DiscrR.
+Ring.
+Unfold Rdiv.
+Rewrite Rmult_1l.
+Repeat Rewrite <- Rmult_assoc.
+Rewrite <- Rinv_l_sym.
+Rewrite Rmult_1l.
+Reflexivity.
+DiscrR.
+Left; Apply Rgt_2_0.
+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.
+Replace ``PI*/4`` with ``PI/4``.
+Rewrite cos_PI4; Apply R1_sqrt2_neq_0.
+Unfold Rdiv; Reflexivity.
+Qed.
+
+Lemma cos3PI4 : ``(cos (3*(PI/4)))==-1/(sqrt 2)``.
+Replace ``3*(PI/4)`` with ``(PI/2)-(-(PI/4))``.
+Rewrite cos_shift; Rewrite sin_neg; Rewrite sin_PI4.
+Unfold Rdiv.
+Rewrite Ropp_mul1.
+Reflexivity.
+Unfold Rminus.
+Rewrite Ropp_Ropp.
+Pattern 1 PI; Rewrite double_var.
+Unfold Rdiv.
+Rewrite Rmult_Rplus_distrl.
+Repeat Rewrite Rmult_assoc.
+Rewrite <- Rinv_Rmult.
+Replace ``2*2`` with ``4``.
+Ring.
+Ring.
+Apply aze.
+Apply aze.
+Qed.
+
+Lemma sin3PI4 : ``(sin (3*(PI/4)))==1/(sqrt 2)``.
+Replace ``3*(PI/4)`` with ``(PI/2)-(-(PI/4))``.
+Rewrite sin_shift; Rewrite cos_neg; Rewrite cos_PI4; Reflexivity.
+Unfold Rminus.
+Rewrite Ropp_Ropp.
+Pattern 1 PI; Rewrite double_var.
+Unfold Rdiv.
+Rewrite Rmult_Rplus_distrl.
+Repeat Rewrite Rmult_assoc.
+Rewrite <- Rinv_Rmult.
+Replace ``2*2`` with ``4``.
+Ring.
+Ring.
+Apply aze.
+Apply aze.
+Qed.
+
+Lemma cos_PI6 : ``(cos (PI/6))==(sqrt 3)/2``.
+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; Apply Rgt_2_0.
+Rewrite Rsqr_div.
+Rewrite cos2; Unfold Rsqr; Rewrite sin_PI6; Rewrite sqrt_def.
+Unfold Rdiv.
+Rewrite Rmult_1l.
+Rewrite <- Rinv_Rmult.
+Replace ``2*2`` with ``4``.
+Apply r_Rmult_mult with ``4``.
+Unfold Rminus.
+Rewrite Rmult_Rplus_distr.
+Rewrite Rmult_1r.
+Rewrite Ropp_mul3.
+Rewrite <- Rinv_r_sym.
+Rewrite (Rmult_sym ``3``).
+Rewrite <- Rmult_assoc.
+Rewrite <- Rinv_r_sym.
+Rewrite Rmult_1l.
+Ring.
+DiscrR.
+DiscrR.
+DiscrR.
+Ring.
+Apply aze.
+Apply aze.
+Left; Apply Rgt_3_0.
+Apply aze.
+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.
+Apply aze.
+Apply aze.
+Red; Intro.
+Assert H1 := Rlt_sqrt3_0.
+Rewrite H in H1; Elim (Rlt_antirefl ``0`` H1).
+Apply Rinv_neq_R0.
+Apply aze.
+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.
+Apply aze.
+Apply aze.
+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``).
+Pattern 3 ``(sqrt 3)``; Rewrite double_var.
+Rewrite Rmult_Rplus_distrl.
+Unfold Rdiv.
+Repeat Rewrite Rmult_assoc.
+Rewrite <- Rinv_Rmult.
+Replace ``2*2`` with ``4``.
+Reflexivity.
+Ring.
+Apply aze.
+Apply aze.
+Qed.
+
+Lemma cos_2PI3 : ``(cos (2*(PI/3)))==-1/2``.
+Rewrite double.
+Rewrite cos_plus; Rewrite sin_PI3; Rewrite cos_PI3.
+Unfold Rdiv.
+Rewrite Rmult_1l.
+Apply r_Rmult_mult with ``2*2``.
+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.
+Replace ``3`` with ``(INR (S (S (S O))))`` .
+Apply pos_INR.
+Rewrite INR_eq_INR2.
+Reflexivity.
+Apply aze.
+Apply aze.
+Apply aze.
+Apply aze.
+Apply aze.
+Apply prod_neq_R0; Apply aze.
+Qed.
+
+Lemma tan_2PI3 : ``(tan (2*(PI/3)))==-(sqrt 3)``.
+Unfold tan; Rewrite sin_2PI3; Rewrite cos_2PI3.
+Unfold Rdiv.
+Rewrite Rinv_Rmult.
+Rewrite Rinv_Rinv.
+Repeat Rewrite Rmult_assoc.
+Rewrite (Rmult_sym ``/2``).
+Repeat Rewrite Rmult_assoc.
+Rewrite <- Rinv_r_sym.
+Rewrite Rmult_1r.
+Rewrite <- Ropp_Rinv.
+Rewrite Ropp_mul3.
+Rewrite Rinv_R1; Rewrite Rmult_1r; Reflexivity.
+DiscrR.
+Apply aze.
+Apply aze.
+DiscrR.
+Apply Rinv_neq_R0; Apply aze.
+Qed.
+
+Lemma cos_5PI4 : ``(cos (5*(PI/4)))==-1/(sqrt 2)``.
+Replace ``5*(PI/4)`` with ``(PI/4)+(PI)``.
+Rewrite neg_cos; Rewrite cos_PI4.
+Unfold Rdiv.
+Symmetry; Apply Ropp_mul1.
+Pattern 2 PI; Rewrite double_var.
+Pattern 2 3 PI; Rewrite double_var.
+Unfold Rdiv.
+Rewrite Rmult_Rplus_distrl.
+Repeat Rewrite Rmult_assoc.
+Rewrite <- Rinv_Rmult.
+Replace ``2*2`` with ``4``.
+Ring.
+Ring.
+Apply aze.
+Apply aze.
+Qed.
+
+Lemma sin_5PI4 : ``(sin (5*(PI/4)))==-1/(sqrt 2)``.
+Replace ``5*(PI/4)`` with ``(PI/4)+(PI)``.
+Rewrite neg_sin; Rewrite sin_PI4.
+Unfold Rdiv; Symmetry; Apply Ropp_mul1.
+Pattern 2 PI; Rewrite double_var.
+Pattern 2 3 PI; Rewrite double_var.
+Unfold Rdiv.
+Rewrite Rmult_Rplus_distrl.
+Repeat Rewrite Rmult_assoc.
+Rewrite <- Rinv_Rmult.
+Replace ``2*2`` with ``4``.
+Ring.
+Ring.
+Apply aze.
+Apply aze.
+Qed.
+
+Lemma sin_cos5PI4 : ``(cos (5*(PI/4)))==(sin (5*(PI/4)))``.
+Rewrite cos_5PI4; Rewrite sin_5PI4; Reflexivity.
+Qed.
\ No newline at end of file