diff options
author | desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2002-07-16 10:20:31 +0000 |
---|---|---|
committer | desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2002-07-16 10:20:31 +0000 |
commit | 6e09985d21aa103db613d25615f4133d5a483094 (patch) | |
tree | bfca273f95febd59930ad4e92a685b61e5a0adce /theories/Reals/Rtrigo.v | |
parent | bf014271e07872d855a6704f66981e108ae2e654 (diff) |
*** empty log message ***
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2884 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Reals/Rtrigo.v')
-rw-r--r-- | theories/Reals/Rtrigo.v | 55 |
1 files changed, 1 insertions, 54 deletions
diff --git a/theories/Reals/Rtrigo.v b/theories/Reals/Rtrigo.v index 6ef075502..86c1e124b 100644 --- a/theories/Reals/Rtrigo.v +++ b/theories/Reals/Rtrigo.v @@ -373,14 +373,6 @@ Intros; Unfold cos_lb cos_ub; Apply (cos_bound a (S O) H H0). Qed. (**********) -Lemma PI4_RGT_0 : ``0<PI/4``. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]. -Qed. - -Lemma PI6_RGT_0 : ``0<PI/6``. -Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]. -Qed. - Lemma _PI2_RLT_0 : ``-(PI/2)<0``. Rewrite <- Ropp_O; Apply Rlt_Ropp1; Apply PI2_RGT_0. Qed. @@ -394,15 +386,6 @@ Pattern 1 ``2``; Rewrite <- Rplus_Or. Replace ``4`` with ``2+2``; [Apply Rlt_compatibility; Apply Rgt_2_0 | Ring]. Qed. -Lemma PI6_RLT_PI2 : ``PI/6<PI/2``. -Unfold Rdiv; Apply Rlt_monotony. -Apply PI_RGT_0. -Apply Rinv_lt. -Apply Rmult_lt_pos; Sup0. -Pattern 1 ``2``; Rewrite <- Rplus_Or. -Replace ``6`` with ``2+4``; [Apply Rlt_compatibility; Sup0 | Ring]. -Qed. - Lemma PI2_Rlt_PI : ``PI/2<PI``. Unfold Rdiv; Pattern 2 PI; Rewrite <- Rmult_1r. Apply Rlt_monotony. @@ -880,26 +863,7 @@ Lemma tan_incr_1 : (x,y:R) ``-(PI/4)<=x``->``x<=PI/4`` ->``-(PI/4)<=y``->``y<=PI Intros; Case (total_order x y); Intro H4; [Left; Apply (tan_increasing_1 x y H H0 H1 H2 H4) | Elim H4; Intro H5; [Case (total_order (tan x) (tan y)); Intro H6; [Left; Assumption | Elim H6; Intro H7; [Right; Assumption | Generalize (tan_increasing_0 y x H1 H2 H H0 H7); Intro H8; Rewrite H5 in H8; Elim (Rlt_antirefl y H8)]] | Elim (Rlt_antirefl x (Rle_lt_trans x y x H3 H5))]]. Qed. -Lemma Rgt_3PI2_0 : ``0<3*(PI/2)``. -Apply Rmult_lt_pos; [Apply Rgt_3_0 | Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Apply Rgt_2_0]]. -Qed. - -Lemma Rgt_2PI_0 : ``0<2*PI``. -Apply Rmult_lt_pos; [Apply Rgt_2_0 | 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. - +(**********) Lemma sin_eq_0_1 : (x:R) (EXT k:Z | x==(Rmult (IZR k) PI)) -> (sin x)==R0. Intros. Elim H; Intros. @@ -1178,21 +1142,4 @@ Qed. Lemma cos_eq_0_2PI_1 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``x==PI/2``\/``x==3*(PI/2)`` -> ``(cos x)==0``. Intros x H1 H2 H3; Elim H3; Intro H4; [ Rewrite H4; Rewrite -> cos_PI2; Reflexivity | Rewrite H4; Rewrite -> cos_3PI2; Reflexivity ]. -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. -Simpl; Discriminate. -Simpl; Discriminate. -Simpl; Discriminate. -Simpl; Discriminate. -Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` a ``0`` H H2)). Qed.
\ No newline at end of file |