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| author |  desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2002-07-03 14:11:56 +0000 |
|---|---|---|
| committer | desmettr <desmettr@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2002-07-03 14:11:56 +0000 |
| commit | 707990edb94cbb66abe025d32c2135bc517ef790 (patch) | |
| tree | 48c2500117d954ec852085afe29ec4ed5818d665 /theories/Reals/Rtrigo.v | |
| parent | 134d8e5a392a5c8f525606c7d102fff1f92da9d7 (diff) | |
sin_eq_0 est maintenant prouve
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2835 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Reals/Rtrigo.v')
| -rw-r--r-- | theories/Reals/Rtrigo.v | 679 |
1 files changed, 392 insertions, 287 deletions
diff --git a/theories/Reals/Rtrigo.v b/theories/Reals/Rtrigo.v index 13b44a3ce..4a614bbf8 100644 --- a/theories/Reals/Rtrigo.v +++ b/theories/Reals/Rtrigo.v @@ -9,6 +9,7 @@ (*i $Id$ i*) Require ZArith_base. +Require Zcomplements. Require Classical_Prop. Require DiscrR. Require Rbase. @@ -19,10 +20,61 @@ Require Rlimit. Require Binome. Require Export Rtrigo_def. +Lemma Ropp_mul3 : (r1,r2:R) ``r1*(-r2) == -(r1*r2)``. +Intros; Rewrite <- Ropp_mul1; Ring. +Qed. + +(* Here, we have the euclidian division *) +(* This lemma is used in the proof of sin_eq_0 : (sin x)=0<->x=2kPI *) +Lemma euclidian_division : (x,y:R) ``y<>0`` -> (EXT k:Z | (EXT r : R | ``x==(IZR k)*y+r``/\``0<=r<(Rabsolu y)``)). +Intros. +Pose k0 := Cases (case_Rabsolu y) of + (leftT _) => (Zminus `1` (up ``x/-y``)) + | (rightT _) => (Zminus (up ``x/y``) `1`) end. +Exists k0. +Exists ``x-(IZR k0)*y``. +Split. +Ring. +Unfold k0; Case (case_Rabsolu y); Intro. +Assert H0 := (archimed ``x/-y``); Rewrite <- Z_R_minus; Simpl; Unfold Rminus. +Replace ``-((1+ -(IZR (up (x/( -y)))))*y)`` with ``((IZR (up (x/-y)))-1)*y``; [Idtac | Ring]. +Split. +Apply Rle_monotony_contra with ``/-y``. +Apply Rlt_Rinv; Apply Rgt_RO_Ropp; Exact r. +Rewrite Rmult_Or; Rewrite (Rmult_sym ``/-y``); Rewrite Rmult_Rplus_distrl; Rewrite <- Ropp_Rinv; [Idtac | Assumption]. +Rewrite Rmult_assoc; Repeat Rewrite Ropp_mul3; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r | Assumption]. +Apply Rle_anti_compatibility with ``(IZR (up (x/( -y))))-x/( -y)``. +Rewrite Rplus_Or; Unfold Rdiv; Pattern 4 ``/-y``; Rewrite <- Ropp_Rinv; [Idtac | Assumption]. +Replace ``(IZR (up (x*/ -y)))-x* -/y+( -(x*/y)+ -((IZR (up (x*/ -y)))-1))`` with R1; [Idtac | Ring]. +Elim H0; Intros _ H1; Unfold Rdiv in H1; Exact H1. +Rewrite (Rabsolu_left ? r); Apply Rlt_monotony_contra with ``/-y``. +Apply Rlt_Rinv; Apply Rgt_RO_Ropp; Exact r. +Rewrite <- Rinv_l_sym. +Rewrite (Rmult_sym ``/-y``); Rewrite Rmult_Rplus_distrl; Rewrite <- Ropp_Rinv; [Idtac | Assumption]. +Rewrite Rmult_assoc; Repeat Rewrite Ropp_mul3; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r | Assumption]; Apply Rlt_anti_compatibility with ``((IZR (up (x/( -y))))-1)``. +Replace ``(IZR (up (x/( -y))))-1+1`` with ``(IZR (up (x/( -y))))``; [Idtac | Ring]. +Replace ``(IZR (up (x/( -y))))-1+( -(x*/y)+ -((IZR (up (x/( -y))))-1))`` with ``-(x*/y)``; [Idtac | Ring]. +Rewrite <- Ropp_mul3; Rewrite (Ropp_Rinv ? H); Elim H0; Unfold Rdiv; Intros H1 _; Exact H1. +Apply Ropp_neq; Assumption. +Assert H0 := (archimed ``x/y``); Rewrite <- Z_R_minus; Simpl; Cut ``0<y``. +Intro; Unfold Rminus; Replace ``-(((IZR (up (x/y)))+ -1)*y)`` with ``(1-(IZR (up (x/y))))*y``; [Idtac | Ring]. +Split. +Apply Rle_monotony_contra with ``/y``. +Apply Rlt_Rinv; Assumption. +Rewrite Rmult_Or; Rewrite (Rmult_sym ``/y``); Rewrite Rmult_Rplus_distrl; Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r | Assumption]; Apply Rle_anti_compatibility with ``(IZR (up (x/y)))-x/y``; Rewrite Rplus_Or; Unfold Rdiv; Replace ``(IZR (up (x*/y)))-x*/y+(x*/y+(1-(IZR (up (x*/y)))))`` with R1; [Idtac | Ring]; Elim H0; Intros _ H2; Unfold Rdiv in H2; Exact H2. +Rewrite (Rabsolu_right ? r); Apply Rlt_monotony_contra with ``/y``. +Apply Rlt_Rinv; Assumption. +Rewrite <- (Rinv_l_sym ? H); Rewrite (Rmult_sym ``/y``); Rewrite Rmult_Rplus_distrl; Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym; [Rewrite Rmult_1r | Assumption]; Apply Rlt_anti_compatibility with ``((IZR (up (x/y)))-1)``; Replace ``(IZR (up (x/y)))-1+1`` with ``(IZR (up (x/y)))``; [Idtac | Ring]; Replace ``(IZR (up (x/y)))-1+(x*/y+(1-(IZR (up (x/y)))))`` with ``x*/y``; [Idtac | Ring]; Elim H0; Unfold Rdiv; Intros H2 _; Exact H2. +Case (total_order_T R0 y); Intro. +Elim s; Intro. +Assumption. +Elim H; Symmetry; Exact b. +Assert H1 := (Rle_sym2 ? ? r); Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H1 r0)). +Qed. + +(**********) Lemma PI_neq0 : ~``PI==0``. -Red; Intro. -Generalize PI_RGT_0; Intro; Rewrite H in H0. -Elim (Rlt_antirefl ``0`` H0). +Red; Intro; Assert H0 := PI_RGT_0; Rewrite H in H0; Elim (Rlt_antirefl ? H0). Qed. (**********) @@ -30,14 +82,9 @@ Lemma sin2_cos2 : (x:R) ``(Rsqr (sin x)) + (Rsqr (cos x))==1``. Intro; Unfold Rsqr; Rewrite Rplus_sym; Rewrite <- (cos_minus x x); Unfold Rminus; Rewrite Rplus_Ropp_r; Apply cos_0. Qed. - (**********) Definition tan [x:R] : R := ``(sin x)/(cos x)``. -Lemma Ropp_mul3 : (r1,r2:R) ``r1*(-r2) == -(r1*r2)``. -Intros; Rewrite <- Ropp_mul1; Ring. -Qed. - Lemma tan_plus : (x,y:R) ~``(cos x)==0`` -> ~``(cos y)==0`` -> ~``(cos (x+y))==0`` -> ~``1-(tan x)*(tan y)==0`` -> ``(tan (x+y))==((tan x)+(tan y))/(1-(tan x)*(tan y))``. Intros; Unfold tan; Rewrite sin_plus; Rewrite cos_plus; Unfold Rdiv; Replace ``((cos x)*(cos y)-(sin x)*(sin y))`` with ``((cos x)*(cos y))*(1-(sin x)*/(cos x)*((sin y)*/(cos y)))``. Rewrite Rinv_Rmult. @@ -70,10 +117,6 @@ Intro x; Generalize (cos2 x); Intro H1; Rewrite -> H1. Unfold Rminus; Rewrite Ropp_distr1; Rewrite <- Rplus_assoc; Rewrite Rplus_Ropp_r; Rewrite Rplus_Ol; Symmetry; Apply Ropp_Ropp. Qed. -Lemma aze : ``2<>0``. -DiscrR. -Qed. - Lemma sin_2a : (x:R) ``(sin (2*x))==2*(sin x)*(cos x)``. Intro x; Rewrite double; Rewrite sin_plus. Rewrite <- (Rmult_sym (sin x)); Symmetry; Rewrite Rmult_assoc; Apply double. @@ -96,16 +139,12 @@ Lemma tan_2a : (x:R) ~``(cos x)==0`` -> ~``(cos (2*x))==0`` -> ~``1-(tan x)*(tan Repeat Rewrite double; Intros; Repeat Rewrite double; Rewrite double in H0; Apply tan_plus; Assumption. Qed. -Lemma sin_0 : ``(sin 0)==0``. -Apply Rsqr_eq_0; Rewrite sin2; Rewrite cos_0; SqRing. -Qed. - Lemma sin_neg : (x:R) ``(sin (-x))==-(sin x)``. -Intro x; Replace ``-x`` with ``0-x``; Ring; Replace `` -(sin x)`` with ``(sin 0)*(cos x)-(cos 0)*(sin x)``; [ Apply sin_minus |Rewrite -> sin_0; Rewrite -> cos_0; Ring ]. +Apply sin_impaire. Qed. Lemma cos_neg : (x:R) ``(cos (-x))==(cos x)``. -Intro x; Replace ``(-x)`` with ``(0-x)``; Ring; Replace ``(cos x)`` with ``(cos 0)*(cos x)+(sin 0)*(sin x)``; [ Apply cos_minus | Rewrite -> cos_0; Rewrite -> sin_0; Ring ]. +Intro; Symmetry; Apply cos_paire. Qed. Lemma tan_0 : ``(tan 0)==0``. @@ -142,12 +181,8 @@ Qed. Lemma cos_PI : ``(cos PI)==(-1)``. Replace ``PI`` with ``2*(PI/2)``. Rewrite -> cos_2a; Rewrite -> sin_PI2; Rewrite -> cos_PI2. -Rewrite Rmult_Ol; Rewrite Rmult_1r. -Apply Rminus_Ropp. -Unfold Rdiv. -Repeat Rewrite <- Rmult_assoc. -Apply Rinv_r_simpl_m. -Apply aze. +Ring. +Unfold Rdiv; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m; Apply aze. Qed. Lemma tan_PI : ``(tan PI)==0``. @@ -170,14 +205,11 @@ Ring. Qed. Lemma sin_2PI : ``(sin (2*PI))==0``. -Rewrite -> sin_2a; Rewrite -> sin_PI. -Rewrite Rmult_Or. -Rewrite Rmult_Ol. -Reflexivity. +Rewrite -> sin_2a; Rewrite -> sin_PI; Ring. Qed. Lemma cos_2PI : ``(cos (2*PI))==1``. -Rewrite -> cos_2a; Rewrite -> sin_PI; Rewrite -> cos_PI; Rewrite Rmult_Or; Rewrite minus_R0; Rewrite Ropp_mul1; Rewrite Rmult_1l; Apply Ropp_Ropp. +Rewrite -> cos_2a; Rewrite -> sin_PI; Rewrite -> cos_PI; Ring. Qed. Lemma tan_2PI : ``(tan (2*PI))==0``. @@ -185,12 +217,11 @@ Unfold tan; Rewrite sin_2PI; Unfold Rdiv; Apply Rmult_Ol. Qed. Lemma neg_cos : (x:R) ``(cos (x+PI))==-(cos x)``. -Intro x; Rewrite -> cos_plus; Rewrite -> sin_PI; Rewrite -> cos_PI; Rewrite Rmult_Or; Rewrite minus_R0; Rewrite Ropp_mul3; Rewrite Rmult_1r; Reflexivity. +Intro x; Rewrite -> cos_plus; Rewrite -> sin_PI; Rewrite -> cos_PI; Ring. Qed. Lemma neg_sin : (x:R) ``(sin (x+PI))==-(sin x)``. -Intro x; Rewrite -> sin_plus; Rewrite -> sin_PI; Rewrite -> cos_PI. -Rewrite Rmult_Or; Rewrite Rplus_Or; Rewrite Ropp_mul3; Rewrite Rmult_1r; Reflexivity. +Intro x; Rewrite -> sin_plus; Rewrite -> sin_PI; Rewrite -> cos_PI; Ring. Qed. Lemma sin_PI_x : (x:R) ``(sin (PI-x))==(sin x)``. @@ -225,178 +256,10 @@ Lemma sin_cos : (x:R) ``(sin x)==-(cos (PI/2+x))``. Intro x; Rewrite -> cos_plus; Rewrite -> sin_PI2; Rewrite -> cos_PI2; Ring. Qed. -Axiom sin_eq_0 : (x:R) (sin x)==R0 <-> (EXT k:Z | x==(Rmult (IZR k) PI)). - -Lemma sin_eq_0_0 : (x:R) (sin x)==R0 -> (EXT k:Z | x==(Rmult (IZR k) PI)). -Intros; Elim (sin_eq_0 x); Intros; Apply (H0 H). -Qed. - -Lemma sin_eq_0_1 : (x:R) (EXT k:Z | x==(Rmult (IZR k) PI)) -> (sin x)==R0. -Intros; Elim (sin_eq_0 x); Intros; Apply (H1 H). -Qed. - -Lemma cos_eq_0_0 : (x:R) (cos x)==R0 -> (EXT k : Z | ``x==(IZR k)*PI+PI/2``). -Intros x H; Rewrite -> cos_sin in H; Generalize (sin_eq_0_0 (Rplus (Rdiv PI (INR (2))) x) H); Intro H2; Elim H2; Intros x0 H3; Exists (Zminus x0 (inject_nat (S O))); Rewrite <- Z_R_minus; Ring; Rewrite Rmult_sym; Rewrite <- H3; Unfold INR. -Rewrite (double_var ``-PI``); Unfold Rdiv; Ring. -Qed. - -Lemma cos_eq_0_1 : (x:R) (EXT k : Z | ``x==(IZR k)*PI+PI/2``) -> ``(cos x)==0``. -Intros x H1; Rewrite cos_sin; Elim H1; Intros x0 H2; Rewrite H2; Replace ``PI/2+((IZR x0)*PI+PI/2)`` with ``(IZR x0)*PI+PI``. -Rewrite neg_sin; Rewrite <- Ropp_O. -Apply eq_Ropp; Apply sin_eq_0_1; Exists x0; Reflexivity. -Pattern 2 PI; Rewrite (double_var PI); Ring. -Qed. - -Lemma sin_eq_O_2PI_0 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``(sin x)==0`` -> ``x==0``\/``x==PI``\/``x==2*PI``. -Intros; Generalize (sin_eq_0_0 x H1); Intro. -Elim H2; Intros k0 H3. -Case (total_order PI x); Intro. -Rewrite H3 in H4; Rewrite H3 in H0. -Right; Right. -Generalize (Rlt_monotony_r ``/PI`` ``PI`` ``(IZR k0)*PI`` (Rlt_Rinv ``PI`` PI_RGT_0) H4); Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Intro; Generalize (Rle_monotony_r ``/PI`` ``(IZR k0)*PI`` ``2*PI`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv ``PI`` PI_RGT_0)) H0); Repeat Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. -Repeat Rewrite Rmult_1r; Intro; Generalize (Rlt_compatibility (IZR `-2`) ``1`` (IZR k0) H5); Rewrite <- plus_IZR. -Replace ``(IZR (NEG (xO xH)))+1`` with ``-1``. -Intro; Generalize (Rle_compatibility (IZR `-2`) (IZR k0) ``2`` H6); Rewrite <- plus_IZR. -Replace ``(IZR (NEG (xO xH)))+2`` with ``0``. -Intro; Cut ``-1 < (IZR (Zplus (NEG (xO xH)) k0)) < 1``. -Intro; Generalize (one_IZR_lt1 (Zplus (NEG (xO xH)) k0) H9); Intro. -Cut k0=`2`. -Intro; Rewrite H11 in H3; Rewrite H3; Simpl. -Reflexivity. -Rewrite <- (Zplus_inverse_l `2`) in H10; Generalize (Zsimpl_plus_l `-2` k0 `2` H10); Intro; Assumption. -Split. -Assumption. -Apply Rle_lt_trans with ``0``. -Assumption. -Apply Rlt_R0_R1. -Simpl; Ring. -Simpl; Ring. -Apply PI_neq0. -Apply PI_neq0. -Elim H4; Intro. -Right; Left. -Symmetry; Assumption. -Left. -Rewrite H3 in H5; Rewrite H3 in H; Generalize (Rlt_monotony_r ``/PI`` ``(IZR k0)*PI`` PI (Rlt_Rinv ``PI`` PI_RGT_0) H5); Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Intro; Generalize (Rle_monotony_r ``/PI`` ``0`` ``(IZR k0)*PI`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv ``PI`` PI_RGT_0)) H); Repeat Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. -Rewrite Rmult_1r; Rewrite Rmult_Ol; Intro. -Cut ``-1 < (IZR (k0)) < 1``. -Intro; Generalize (one_IZR_lt1 k0 H8); Intro; Rewrite H9 in H3; Rewrite H3; Simpl; Apply Rmult_Ol. -Split. -Apply Rlt_le_trans with ``0``. -Rewrite <- Ropp_O; Apply Rgt_Ropp; Apply Rlt_R0_R1. -Assumption. -Assumption. -Apply PI_neq0. -Apply PI_neq0. -Qed. - -Lemma sin_eq_O_2PI_1 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``x==0``\/``x==PI``\/``x==2*PI`` -> ``(sin x)==0``. -Intros x H1 H2 H3; Elim H3; Intro H4; [ Rewrite H4; Rewrite -> sin_0; Reflexivity | Elim H4; Intro H5; [Rewrite H5; Rewrite -> sin_PI; Reflexivity | Rewrite H5; Rewrite -> sin_2PI; Reflexivity]]. -Qed. - Lemma PI2_RGT_0 : ``0<PI/2``. Cut ~(O=(2)); [Intro H; Generalize (lt_INR_0 (2) (neq_O_lt (2) H)); Rewrite INR_eq_INR2; Unfold INR2; Intro H1; Generalize (Rmult_lt_pos PI (Rinv ``2``) PI_RGT_0 (Rlt_Rinv ``2`` H1)); Intro H2; Assumption | Discriminate]. Qed. -Lemma cos_eq_0_2PI_0 : (x:R) ``R0<=x`` -> ``x<=2*PI`` -> ``(cos x)==0`` -> ``x==(PI/2)``\/``x==3*(PI/2)``. -Intros; Case (total_order x ``3*(PI/2)``); Intro. -Rewrite cos_sin in H1. -Cut ``0<=PI/2+x``. -Cut ``PI/2+x<=2*PI``. -Intros; Generalize (sin_eq_O_2PI_0 ``PI/2+x`` H4 H3 H1); Intros. -Decompose [or] H5. -Generalize (Rle_compatibility ``PI/2`` ``0`` x H); Rewrite Rplus_Or; Rewrite H6; Intro. -Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``PI/2`` ``0`` PI2_RGT_0 H7)). -Left. -Generalize (Rplus_plus_r ``-(PI/2)`` ``PI/2+x`` PI H7). -Replace ``-(PI/2)+(PI/2+x)`` with x. -Replace ``-(PI/2)+PI`` with ``PI/2``. -Intro; Assumption. -Pattern 3 PI; Rewrite (double_var PI); Ring. -Ring. -Right. -Generalize (Rplus_plus_r ``-(PI/2)`` ``PI/2+x`` ``2*PI`` H7). -Replace ``-(PI/2)+(PI/2+x)`` with x. -Replace ``-(PI/2)+2*PI`` with ``3*(PI/2)``. -Intro; Assumption. -Rewrite double; Pattern 3 4 PI; Rewrite (double_var PI); Ring. -Ring. -Left; Replace ``2*PI`` with ``PI/2+3*(PI/2)``. -Apply Rlt_compatibility; Assumption. -Rewrite (double PI); Pattern 3 4 PI; Rewrite (double_var PI); Ring. -Apply ge0_plus_ge0_is_ge0. -Left; Unfold Rdiv; Apply Rmult_lt_pos. -Apply PI_RGT_0. -Apply Rlt_Rinv; Apply Rgt_2_0. -Assumption. -Elim H2; Intro. -Right; Assumption. -Generalize (cos_eq_0_0 x H1); Intro; Elim H4; Intros k0 H5. -Rewrite H5 in H3; Rewrite H5 in H0; Generalize (Rlt_compatibility ``-(PI/2)`` ``3*PI/2`` ``(IZR k0)*PI+PI/2`` H3); Generalize (Rle_compatibility ``-(PI/2)`` ``(IZR k0)*PI+PI/2`` ``2*PI`` H0). -Replace ``-(PI/2)+3*PI/2`` with PI. -Replace ``-(PI/2)+((IZR k0)*PI+PI/2)`` with ``(IZR k0)*PI``. -Replace ``-(PI/2)+2*PI`` with ``3*(PI/2)``. -Intros; Generalize (Rlt_monotony ``/PI`` ``PI`` ``(IZR k0)*PI`` (Rlt_Rinv PI PI_RGT_0) H7); Generalize (Rle_monotony ``/PI`` ``(IZR k0)*PI`` ``3*(PI/2)`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv PI PI_RGT_0)) H6). -Replace ``/PI*((IZR k0)*PI)`` with (IZR k0). -Replace ``/PI*(3*PI/2)`` with ``3*/2``. -Rewrite <- Rinv_l_sym. -Intros; Generalize (Rlt_compatibility (IZR `-2`) ``1`` (IZR k0) H9); Rewrite <- plus_IZR. -Replace ``(IZR (NEG (xO xH)))+1`` with ``-1``. -Intro; Generalize (Rle_compatibility (IZR `-2`) (IZR k0) ``3*/2`` H8); Rewrite <- plus_IZR. -Replace ``(IZR (NEG (xO xH)))+2`` with ``0``. -Intro; Cut `` -1 < (IZR (Zplus (NEG (xO xH)) k0)) < 1``. -Intro; Generalize (one_IZR_lt1 (Zplus (NEG (xO xH)) k0) H12); Intro. -Cut k0=`2`. -Intro; Rewrite H14 in H8. -Generalize (Rle_monotony ``2`` ``(IZR (POS (xO xH)))`` ``3*/2`` (Rlt_le ``0`` ``2`` Rgt_2_0) H8); Simpl. -Replace ``2*2`` with ``4``. -Replace ``2*(3*/2)`` with ``3``. -Intro; Cut ``3<4``. -Intro; Elim (Rlt_antirefl ``3`` (Rlt_le_trans ``3`` ``4`` ``3`` H16 H15)). -Generalize (Rlt_compatibility ``3`` ``0`` ``1`` Rlt_R0_R1); Rewrite Rplus_Or. -Replace ``3+1`` with ``4``. -Intro; Assumption. -Ring. -Symmetry; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m. -Apply aze. -Ring. -Rewrite <- (Zplus_inverse_l `2`) in H13; Generalize (Zsimpl_plus_l `-2` k0 `2` H13); Intro; Assumption. -Split. -Assumption. -Apply Rle_lt_trans with ``(IZR (NEG (xO xH)))+3*/2``. -Assumption. -Simpl; Replace ``-2+3*/2`` with ``-(1*/2)``. -Apply Rlt_trans with ``0``. -Rewrite <- Ropp_O; Apply Rlt_Ropp. -Apply Rmult_lt_pos; [Apply Rlt_R0_R1 | Apply Rlt_Rinv; Apply Rgt_2_0]. -Apply Rlt_R0_R1. -Rewrite Rmult_1l; Apply r_Rmult_mult with ``2``. -Rewrite Ropp_mul3; Rewrite <- Rinv_r_sym. -Rewrite Rmult_Rplus_distr; Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m. -Ring. -Apply aze. -Apply aze. -Apply aze. -Simpl; Ring. -Simpl; Ring. -Apply PI_neq0. -Unfold Rdiv; Pattern 1 ``3``; Rewrite (Rmult_sym ``3``); Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. -Rewrite Rmult_1l; Apply Rmult_sym. -Apply PI_neq0. -Symmetry; Rewrite (Rmult_sym ``/PI``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. -Apply Rmult_1r. -Apply PI_neq0. -Rewrite double; Pattern 3 4 PI; Rewrite double_var; Ring. -Ring. -Pattern 1 PI; Rewrite double_var; Ring. -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. - Lemma SIN_bound : (x:R) ``-1<=(sin x)<=1``. Intro; Case (total_order_Rle ``-1`` (sin x)); Intro. Case (total_order_Rle (sin x) ``1``); Intro. @@ -410,15 +273,7 @@ Auto with real. Qed. Lemma COS_bound : (x:R) ``-1<=(cos x)<=1``. -Intro; Case (total_order_Rle ``-1`` (cos x)); Intro. -Case (total_order_Rle (cos x) ``1``); Intro. -Split; Assumption. -Cut ``1<(cos x)``. -Intro; Generalize (Rsqr_incrst_1 ``1`` (cos x) H (Rlt_le ``0`` ``1`` Rlt_R0_R1) (Rlt_le ``0`` (cos x) (Rlt_trans ``0`` ``1`` (cos x) Rlt_R0_R1 H))); Rewrite Rsqr_1; Intro; Rewrite cos2 in H0; Unfold Rminus in H0; Generalize (Rlt_compatibility ``-1`` ``1`` ``1+ -(Rsqr (sin x))`` H0); Repeat Rewrite <- Rplus_assoc; Repeat Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Intro; Rewrite <- Ropp_O in H1; Generalize (Rlt_Ropp ``-0`` ``-(Rsqr (sin x))`` H1); Repeat Rewrite Ropp_Ropp; Intro; Generalize (pos_Rsqr (sin x)); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` (Rsqr (sin x)) ``0`` H3 H2)). -Auto with real. -Cut ``(cos x)< -1``. -Intro; Generalize (Rlt_Ropp (cos x) ``-1`` H); Rewrite Ropp_Ropp; Clear H; Intro; Generalize (Rsqr_incrst_1 ``1`` ``-(cos x)`` H (Rlt_le ``0`` ``1`` Rlt_R0_R1) (Rlt_le ``0`` ``-(cos x)`` (Rlt_trans ``0`` ``1`` ``-(cos x)`` Rlt_R0_R1 H))); Rewrite Rsqr_1; Intro; Rewrite <- Rsqr_neg in H0; Rewrite cos2 in H0; Unfold Rminus in H0; Generalize (Rlt_compatibility ``-1`` ``1`` ``1+ -(Rsqr (sin x))`` H0); Repeat Rewrite <- Rplus_assoc; Repeat Rewrite Rplus_Ropp_l; Rewrite Rplus_Ol; Intro; Rewrite <- Ropp_O in H1; Generalize (Rlt_Ropp ``-0`` ``-(Rsqr (sin x))`` H1); Repeat Rewrite Ropp_Ropp; Intro; Generalize (pos_Rsqr (sin x)); Intro; Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` (Rsqr (sin x)) ``0`` H3 H2)). -Auto with real. +Intro; Rewrite <- sin_shift; Apply SIN_bound. Qed. Lemma cos_sin_0 : (x:R) ~(``(cos x)==0``/\``(sin x)==0``). @@ -523,11 +378,11 @@ Qed. (**********) Lemma PI4_RGT_0 : ``0<PI/4``. -Cut ~(O=(4)); [Intro H; Generalize (lt_INR_0 (4) (neq_O_lt (4) H)); Rewrite INR_eq_INR2; Unfold INR2; Intro H1; Generalize (Rmult_lt_pos PI (Rinv ``4``) PI_RGT_0 (Rlt_Rinv ``4`` H1)); Intro H2; Assumption | Discriminate]. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]. Qed. Lemma PI6_RGT_0 : ``0<PI/6``. -Cut ~(O=(6)); [Intro H; Generalize (lt_INR_0 (6) (neq_O_lt (6) H)); Rewrite INR_eq_INR2; Unfold INR2; Intro H1; Generalize (Rmult_lt_pos PI (Rinv ``6``) PI_RGT_0 (Rlt_Rinv ``6`` H1)); Intro H2; Assumption | Discriminate]. +Unfold Rdiv; Apply Rmult_lt_pos; [Apply PI_RGT_0 | Apply Rlt_Rinv; Sup0]. Qed. Lemma _PI2_RLT_0 : ``-(PI/2)<0``. @@ -535,21 +390,21 @@ Rewrite <- Ropp_O; Apply Rlt_Ropp1; Apply PI2_RGT_0. Qed. Lemma PI4_RLT_PI2 : ``PI/4<PI/2``. -Cut ~(O=(2)). -Intro H; Cut ~(O=(1)). -Intro H0; Generalize (lt_INR_0 (2) (neq_O_lt (2) H)); Rewrite INR_eq_INR2; Unfold INR2; Intro H1; Generalize (Rlt_compatibility ``2`` ``0`` ``2`` H1); Rewrite Rplus_sym. -Rewrite Rplus_Ol. -Replace ``2+2`` with ``4``. -Intro H2; Generalize (lt_INR_0 (1) (neq_O_lt (1) H0)); Unfold INR; Intro H3; Generalize (Rlt_compatibility ``1`` ``0`` ``1`` H3); Rewrite Rplus_sym. -Rewrite Rplus_Ol. -Clear H3; Intro H3; Generalize (Rlt_Rinv_R1 ``2`` ``4`` (Rlt_le ``1`` ``2`` H3) H2); Intro H4; Generalize (Rlt_monotony PI (Rinv ``4``) (Rinv ``2``) PI_RGT_0 H4); Intro H5; Assumption. -Ring. -Discriminate. -Discriminate. +Unfold Rdiv; Apply Rlt_monotony. +Apply PI_RGT_0. +Apply Rinv_lt. +Apply Rmult_lt_pos; Sup0. +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``. -Cut ~(O=(4)); [ Intro H; Cut ~(O=(1)); [Intro H0; Generalize (lt_INR_0 (4) (neq_O_lt (4) H)); Rewrite INR_eq_INR2; Unfold INR2; Intro H1; Generalize (Rlt_compatibility ``2`` ``0`` ``4`` H1); Rewrite Rplus_sym; Rewrite Rplus_Ol; Replace ``2+4`` with ``6``; [Intro H2; Generalize (lt_INR_0 (1) (neq_O_lt (1) H0)); Rewrite INR_eq_INR2; Unfold INR2; Intro H3; Generalize (Rlt_compatibility ``1`` ``0`` ``1`` H3); Rewrite Rplus_sym; Rewrite Rplus_Ol; Clear H3; Intro H3; Generalize (Rlt_Rinv_R1 ``2`` ``6`` (Rlt_le ``1`` ``2`` H3) H2); Intro H4; Generalize (Rlt_monotony PI (Rinv ``6``) (Rinv ``2``) PI_RGT_0 H4); Intro H5; Assumption | Ring] | Discriminate] | Discriminate ]. +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 sqrt2_neq_0 : ~``(sqrt 2)==0``. @@ -573,19 +428,22 @@ Cut ~(O=(1)); [Intro H0; Generalize (Rlt_le ``0`` ``2`` Rgt_2_0); Intro H1; Gene Qed. Lemma PI2_Rlt_PI : ``PI/2<PI``. -Cut ~(O=(1)). -Intro H0; Generalize (lt_INR_0 (1) (neq_O_lt (1) H0)); Unfold INR; Intro H1; Generalize (Rlt_compatibility ``1`` ``0`` ``1`` H1); Rewrite Rplus_sym; Rewrite Rplus_Ol; Intro H2; Cut ``1<=1``. -Intro H3; Generalize (Rlt_Rinv_R1 ``1`` ``2`` H3 H2); Intro H4; Generalize (Rlt_monotony PI (Rinv ``2``) (Rinv ``1``) PI_RGT_0 H4). -Rewrite Rinv_R1. -Rewrite Rmult_1r. -Intro; Assumption. -Right; Reflexivity. -Discriminate. +Unfold Rdiv; Pattern 2 PI; Rewrite <- Rmult_1r. +Apply Rlt_monotony. +Apply PI_RGT_0. +Pattern 3 R1; Rewrite <- Rinv_R1; Apply Rinv_lt. +Rewrite Rmult_1l; Apply Rgt_2_0. +Pattern 1 R1; Rewrite <- Rplus_Or; Apply Rlt_compatibility; Apply Rlt_R0_R1. Qed. +(********************************************) +(* Increasing and decreasing of COS and SIN *) +(********************************************) Theorem sin_gt_0 : (x:R) ``0<x`` -> ``x<PI`` -> ``0<(sin x)``. Intros; Elim (SIN x (Rlt_le R0 x H) (Rlt_le x PI H0)); Intros H1 _; Case (total_order x ``PI/2``); Intro H2. -Generalize (sin_lb_gt_0 x H (Rlt_le x ``PI/2`` H2)); Intro H3; Apply (Rlt_le_trans ``0`` (sin_lb x) (sin x) H3 H1). +Apply Rlt_le_trans with (sin_lb x). +Apply sin_lb_gt_0; [Assumption | Left; Assumption]. +Assumption. Elim H2; Intro H3. Rewrite H3; Rewrite sin_PI2; Apply Rlt_R0_R1. Rewrite <- sin_PI_x; Generalize (Rgt_Ropp x ``PI/2`` H3); Intro H4; Generalize (Rlt_compatibility PI (Ropp x) (Ropp ``PI/2``) H4). @@ -617,7 +475,6 @@ Lemma sin_le_0 : (x:R) ``PI<=x`` -> ``x<=2*PI`` -> ``(sin x)<=0``. Intros x H1 H2; Apply Rle_sym2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (sin x)); Apply Rle_Ropp; Rewrite <- neg_sin; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``; [Rewrite -> (sin_period (Rminus x PI) (S O)); Apply sin_ge_0; [Replace ``x-PI`` with ``x+(-PI)``; [Rewrite Rplus_sym; Replace ``0`` with ``(-PI)+PI``; [Apply Rle_compatibility; Assumption | Ring] | Ring] | Replace ``x-PI`` with ``x+(-PI)``; Rewrite Rplus_sym; [Pattern 2 PI; Replace ``PI`` with ``(-PI)+2*PI``; [Apply Rle_compatibility; Assumption | Ring] | Ring]] |Unfold INR; Ring]. Qed. - Lemma cos_le_0 : (x:R) ``PI/2<=x``->``x<=3*(PI/2)``->``(cos x)<=0``. Intros x H1 H2; Apply Rle_sym2; Rewrite <- Ropp_O; Rewrite <- (Ropp_Ropp (cos x)); Apply Rle_Ropp; Rewrite <- neg_cos; Replace ``x+PI`` with ``(x-PI)+2*(INR (S O))*PI``. Rewrite cos_period; Apply cos_ge_0. @@ -683,57 +540,33 @@ Lemma form1 : (p,q:R) ``(cos p)+(cos q)==2*(cos ((p-q)/2))*(cos ((p+q)/2))``. Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``. Rewrite <- (cos_neg q); Replace``-q`` with ``(p-q)/2-(p+q)/2``. Rewrite cos_plus; Rewrite cos_minus; Ring. -Unfold Rdiv Rminus; Rewrite Rmult_Rplus_distrl; Ring. -Rewrite (Rmult_sym ``/2``); Repeat Rewrite <- Ropp_mul1; Assert H := (double_var ``-q``); Unfold Rdiv in H; Symmetry ; Assumption. -Unfold Rdiv Rminus; Rewrite Rmult_Rplus_distrl; Ring; Rewrite (Rmult_sym ``/2``); Repeat Rewrite <- Ropp_mul1; Assert H := (double_var ``p``); Unfold Rdiv in H; Symmetry ; Assumption. +Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring. +Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring. Qed. Lemma form2 : (p,q:R) ``(cos p)-(cos q)==-2*(sin ((p-q)/2))*(sin ((p+q)/2))``. Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``. Rewrite <- (cos_neg q); Replace``-q`` with ``(p-q)/2-(p+q)/2``. Rewrite cos_plus; Rewrite cos_minus; Ring. -Unfold Rdiv Rminus; Rewrite Rmult_Rplus_distrl; Ring; Rewrite (Rmult_sym ``/2``); Repeat Rewrite <- Ropp_mul1; Assert H := (double_var ``-q``); Unfold Rdiv in H; Symmetry ; Assumption. -Unfold Rdiv Rminus; Rewrite Rmult_Rplus_distrl; Ring; Rewrite (Rmult_sym ``/2``); Repeat Rewrite <- Ropp_mul1; Assert H := (double_var ``p``); Unfold Rdiv in H; Symmetry ; Assumption. +Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring. +Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring. Qed. Lemma form3 : (p,q:R) ``(sin p)+(sin q)==2*(cos ((p-q)/2))*(sin ((p+q)/2))``. Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``. Pattern 3 q; Replace ``q`` with ``(p+q)/2-(p-q)/2``. Rewrite sin_plus; Rewrite sin_minus; Ring. -Unfold Rdiv Rminus. -Rewrite Rmult_Rplus_distrl. -Ring. -Rewrite (Rmult_sym ``/2``). -Repeat Rewrite <- Ropp_mul1. -Assert H := (double_var ``q``). -Unfold Rdiv in H; Symmetry ; Assumption. -Unfold Rdiv Rminus. -Rewrite Rmult_Rplus_distrl. -Ring. -Rewrite (Rmult_sym ``/2``). -Repeat Rewrite <- Ropp_mul1. -Assert H := (double_var ``p``). -Unfold Rdiv in H; Symmetry ; Assumption. +Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring. +Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring. Qed. Lemma form4 : (p,q:R) ``(sin p)-(sin q)==2*(cos ((p+q)/2))*(sin ((p-q)/2))``. Intros p q; Pattern 1 p; Replace ``p`` with ``(p-q)/2+(p+q)/2``. Pattern 3 q; Replace ``q`` with ``(p+q)/2-(p-q)/2``. Rewrite sin_plus; Rewrite sin_minus; Ring. -Unfold Rdiv Rminus. -Rewrite Rmult_Rplus_distrl. -Ring. -Rewrite (Rmult_sym ``/2``). -Repeat Rewrite <- Ropp_mul1. -Assert H := (double_var ``q``). -Unfold Rdiv in H; Symmetry ; Assumption. -Unfold Rdiv Rminus. -Rewrite Rmult_Rplus_distrl. -Ring. -Rewrite (Rmult_sym ``/2``). -Repeat Rewrite <- Ropp_mul1. -Assert H := (double_var ``p``). -Unfold Rdiv in H; Symmetry ; Assumption. +Pattern 3 q; Rewrite double_var; Unfold Rdiv; Ring. +Pattern 3 p; Rewrite double_var; Unfold Rdiv; Ring. + Qed. Lemma sin_increasing_0 : (x,y:R) ``-(PI/2)<=x``->``x<=PI/2``->``-(PI/2)<=y``->``y<=PI/2``->``(sin x)<(sin y)``->``x<y``. @@ -1072,45 +905,37 @@ Intros; Case (total_order x y); Intro H4; [Left; Apply (tan_increasing_1 x y H H Qed. Lemma Rgt_3PI2_0 : ``0<3*(PI/2)``. -Cut ~(O=(3)); [Intro H1; Generalize (lt_INR_0 (3) (neq_O_lt (3) H1)); Rewrite INR_eq_INR2; Unfold INR2; Intro H2; Generalize (Rlt_monotony ``PI/2`` ``0`` ``3`` PI2_RGT_0 H2); Rewrite Rmult_Or; Rewrite Rmult_sym; Intro H3; Assumption | Discriminate]. +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``. -Cut ~(O=(2)); [Intro H1; Generalize (lt_INR_0 (2) (neq_O_lt (2) H1)); Unfold INR; Intro H2; Generalize (Rlt_monotony PI ``0`` ``2`` PI_RGT_0 H2); Rewrite Rmult_Or; Rewrite Rmult_sym; Intro H3; Assumption | Discriminate]. +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. +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. +Rewrite double; Pattern 1 2 PI; Rewrite double_var; Ring. Qed. Lemma sin_cos_PI4 : ``(sin (PI/4)) == (cos (PI/4))``. -Rewrite cos_sin; Replace ``PI/2+PI/4`` with ``-(PI/4)+PI``. +Rewrite cos_sin. +Replace ``PI/2+PI/4`` with ``-(PI/4)+PI``. Rewrite neg_sin; Rewrite sin_neg; Ring. -Pattern 2 3 PI; Replace ``PI`` with ``PI/2+PI/2``. -Pattern 2 3 PI; Replace ``PI`` with ``PI/2+PI/2``. -Unfold Rdiv. -Cut ``2*2==4``. -Intro. -Rewrite Rmult_Rplus_distrl. +Cut ``PI==PI/2+PI/2``; [Intro | Apply double_var]. +Pattern 2 3 PI; Rewrite H. +Pattern 2 3 PI; Rewrite H. +Unfold Rdiv; Cut ``2*2==4``. +Intro; Rewrite Rmult_Rplus_distrl. Repeat Rewrite Rmult_assoc. -Rewrite <- Rinv_Rmult. -Rewrite H. -Ring. -Apply aze. -Apply aze. +Rewrite <- Rinv_Rmult; [Rewrite H0; Ring | Apply aze | Apply aze]. Ring. -Symmetry; Apply double_var. -Symmetry; Apply double_var. Qed. Lemma cos_PI4 : ``(cos (PI/4))==1/(sqrt 2)``. @@ -1486,6 +1311,286 @@ Lemma sin_cos5PI4 : ``(cos (5*(PI/4)))==(sin (5*(PI/4)))``. Rewrite cos_5PI4; Rewrite sin_5PI4; Reflexivity. Qed. +Lemma sin_eq_0_1 : (x:R) (EXT k:Z | x==(Rmult (IZR k) PI)) -> (sin x)==R0. +Intros. +Elim H; Intros. +Apply (Zcase_sign x0). +Intro. +Rewrite H1 in H0. +Simpl in H0. +Rewrite H0; Rewrite Rmult_Ol; Apply sin_0. +Intro. +Cut `0<=x0`. +Intro. +Elim (IZN x0 H2); Intros. +Rewrite H3 in H0. +Rewrite <- INR_IZR_INZ in H0. +Rewrite H0. +Elim (even_odd_cor x1); Intros. +Elim H4; Intro. +Rewrite H5. +Rewrite mult_INR. +Simpl. +Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``). +Rewrite sin_period. +Apply sin_0. +Rewrite H5. +Rewrite S_INR; Rewrite mult_INR. +Simpl. +Rewrite Rmult_Rplus_distrl. +Rewrite Rmult_1l; Rewrite sin_plus. +Rewrite sin_PI. +Rewrite Rmult_Or. +Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``). +Rewrite sin_period. +Rewrite sin_0; Ring. +Apply le_IZR. +Left; Apply IZR_lt. +Assert H2 := Zgt_iff_lt. +Elim (H2 x0 `0`); Intros. +Apply H3; Assumption. +Intro. +Rewrite H0. +Replace ``(sin ((IZR x0)*PI))`` with ``-(sin (-(IZR x0)*PI))``. +Cut `0<=-x0`. +Intro. +Rewrite <- Ropp_Ropp_IZR. +Elim (IZN `-x0` H2); Intros. +Rewrite H3. +Rewrite <- INR_IZR_INZ. +Elim (even_odd_cor x1); Intros. +Elim H4; Intro. +Rewrite H5. +Rewrite mult_INR. +Simpl. +Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``). +Rewrite sin_period. +Rewrite sin_0; Ring. +Rewrite H5. +Rewrite S_INR; Rewrite mult_INR. +Simpl. +Rewrite Rmult_Rplus_distrl. +Rewrite Rmult_1l; Rewrite sin_plus. +Rewrite sin_PI. +Rewrite Rmult_Or. +Rewrite <- (Rplus_Ol ``2*(INR x2)*PI``). +Rewrite sin_period. +Rewrite sin_0; Ring. +Apply le_IZR. +Apply Rle_anti_compatibility with ``(IZR x0)``. +Rewrite Rplus_Or. +Rewrite Ropp_Ropp_IZR. +Rewrite Rplus_Ropp_r. +Left; Replace R0 with (IZR `0`); [Apply IZR_lt | Reflexivity]. +Assumption. +Rewrite <- sin_neg. +Rewrite Ropp_mul1. +Rewrite Ropp_Ropp. +Reflexivity. +Qed. + +Lemma sin_eq_0_0 : (x:R) (sin x)==R0 -> (EXT k:Z | x==(Rmult (IZR k) PI)). +Intros. +Assert H0 := (euclidian_division x PI PI_neq0). +Elim H0; Intros q H1. +Elim H1; Intros r H2. +Exists q. +Cut r==R0. +Intro. +Elim H2; Intros H4 _; Rewrite H4; Rewrite H3. +Apply Rplus_Or. +Elim H2; Intros. +Rewrite H3 in H. +Rewrite sin_plus in H. +Cut ``(sin ((IZR q)*PI))==0``. +Intro. +Rewrite H5 in H. +Rewrite Rmult_Ol in H. +Rewrite Rplus_Ol in H. +Assert H6 := (without_div_Od ? ? H). +Elim H6; Intro. +Assert H8 := (sin2_cos2 ``(IZR q)*PI``). +Rewrite H5 in H8; Rewrite H7 in H8. +Rewrite Rsqr_O in H8. +Rewrite Rplus_Or in H8. +Elim R1_neq_R0; Symmetry; Assumption. +Cut r==R0\/``0<r<PI``. +Intro; Elim H8; Intro. +Assumption. +Elim H9; Intros. +Assert H12 := (sin_gt_0 ? H10 H11). +Rewrite H7 in H12; Elim (Rlt_antirefl ? H12). +Rewrite Rabsolu_right in H4. +Elim H4; Intros. +Case (total_order R0 r); Intro. +Right; Split; Assumption. +Elim H10; Intro. +Left; Symmetry; Assumption. +Elim (Rlt_antirefl ? (Rle_lt_trans ? ? ? H8 H11)). +Apply Rle_sym1. +Left; Apply PI_RGT_0. +Apply sin_eq_0_1. +Exists q; Reflexivity. +Qed. + +Lemma cos_eq_0_0 : (x:R) (cos x)==R0 -> (EXT k : Z | ``x==(IZR k)*PI+PI/2``). +Intros x H; Rewrite -> cos_sin in H; Generalize (sin_eq_0_0 (Rplus (Rdiv PI (INR (2))) x) H); Intro H2; Elim H2; Intros x0 H3; Exists (Zminus x0 (inject_nat (S O))); Rewrite <- Z_R_minus; Ring; Rewrite Rmult_sym; Rewrite <- H3; Unfold INR. +Rewrite (double_var ``-PI``); Unfold Rdiv; Ring. +Qed. + +Lemma cos_eq_0_1 : (x:R) (EXT k : Z | ``x==(IZR k)*PI+PI/2``) -> ``(cos x)==0``. +Intros x H1; Rewrite cos_sin; Elim H1; Intros x0 H2; Rewrite H2; Replace ``PI/2+((IZR x0)*PI+PI/2)`` with ``(IZR x0)*PI+PI``. +Rewrite neg_sin; Rewrite <- Ropp_O. +Apply eq_Ropp; Apply sin_eq_0_1; Exists x0; Reflexivity. +Pattern 2 PI; Rewrite (double_var PI); Ring. +Qed. + +Lemma sin_eq_O_2PI_0 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``(sin x)==0`` -> ``x==0``\/``x==PI``\/``x==2*PI``. +Intros; Generalize (sin_eq_0_0 x H1); Intro. +Elim H2; Intros k0 H3. +Case (total_order PI x); Intro. +Rewrite H3 in H4; Rewrite H3 in H0. +Right; Right. +Generalize (Rlt_monotony_r ``/PI`` ``PI`` ``(IZR k0)*PI`` (Rlt_Rinv ``PI`` PI_RGT_0) H4); Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Intro; Generalize (Rle_monotony_r ``/PI`` ``(IZR k0)*PI`` ``2*PI`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv ``PI`` PI_RGT_0)) H0); Repeat Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. +Repeat Rewrite Rmult_1r; Intro; Generalize (Rlt_compatibility (IZR `-2`) ``1`` (IZR k0) H5); Rewrite <- plus_IZR. +Replace ``(IZR (NEG (xO xH)))+1`` with ``-1``. +Intro; Generalize (Rle_compatibility (IZR `-2`) (IZR k0) ``2`` H6); Rewrite <- plus_IZR. +Replace ``(IZR (NEG (xO xH)))+2`` with ``0``. +Intro; Cut ``-1 < (IZR (Zplus (NEG (xO xH)) k0)) < 1``. +Intro; Generalize (one_IZR_lt1 (Zplus (NEG (xO xH)) k0) H9); Intro. +Cut k0=`2`. +Intro; Rewrite H11 in H3; Rewrite H3; Simpl. +Reflexivity. +Rewrite <- (Zplus_inverse_l `2`) in H10; Generalize (Zsimpl_plus_l `-2` k0 `2` H10); Intro; Assumption. +Split. +Assumption. +Apply Rle_lt_trans with ``0``. +Assumption. +Apply Rlt_R0_R1. +Simpl; Ring. +Simpl; Ring. +Apply PI_neq0. +Apply PI_neq0. +Elim H4; Intro. +Right; Left. +Symmetry; Assumption. +Left. +Rewrite H3 in H5; Rewrite H3 in H; Generalize (Rlt_monotony_r ``/PI`` ``(IZR k0)*PI`` PI (Rlt_Rinv ``PI`` PI_RGT_0) H5); Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Intro; Generalize (Rle_monotony_r ``/PI`` ``0`` ``(IZR k0)*PI`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv ``PI`` PI_RGT_0)) H); Repeat Rewrite Rmult_assoc; Repeat Rewrite <- Rinv_r_sym. +Rewrite Rmult_1r; Rewrite Rmult_Ol; Intro. +Cut ``-1 < (IZR (k0)) < 1``. +Intro; Generalize (one_IZR_lt1 k0 H8); Intro; Rewrite H9 in H3; Rewrite H3; Simpl; Apply Rmult_Ol. +Split. +Apply Rlt_le_trans with ``0``. +Rewrite <- Ropp_O; Apply Rgt_Ropp; Apply Rlt_R0_R1. +Assumption. +Assumption. +Apply PI_neq0. +Apply PI_neq0. +Qed. + +Lemma sin_eq_O_2PI_1 : (x:R) ``0<=x`` -> ``x<=2*PI`` -> ``x==0``\/``x==PI``\/``x==2*PI`` -> ``(sin x)==0``. +Intros x H1 H2 H3; Elim H3; Intro H4; [ Rewrite H4; Rewrite -> sin_0; Reflexivity | Elim H4; Intro H5; [Rewrite H5; Rewrite -> sin_PI; Reflexivity | Rewrite H5; Rewrite -> sin_2PI; Reflexivity]]. +Qed. + +Lemma cos_eq_0_2PI_0 : (x:R) ``R0<=x`` -> ``x<=2*PI`` -> ``(cos x)==0`` -> ``x==(PI/2)``\/``x==3*(PI/2)``. +Intros; Case (total_order x ``3*(PI/2)``); Intro. +Rewrite cos_sin in H1. +Cut ``0<=PI/2+x``. +Cut ``PI/2+x<=2*PI``. +Intros; Generalize (sin_eq_O_2PI_0 ``PI/2+x`` H4 H3 H1); Intros. +Decompose [or] H5. +Generalize (Rle_compatibility ``PI/2`` ``0`` x H); Rewrite Rplus_Or; Rewrite H6; Intro. +Elim (Rlt_antirefl ``0`` (Rlt_le_trans ``0`` ``PI/2`` ``0`` PI2_RGT_0 H7)). +Left. +Generalize (Rplus_plus_r ``-(PI/2)`` ``PI/2+x`` PI H7). +Replace ``-(PI/2)+(PI/2+x)`` with x. +Replace ``-(PI/2)+PI`` with ``PI/2``. +Intro; Assumption. +Pattern 3 PI; Rewrite (double_var PI); Ring. +Ring. +Right. +Generalize (Rplus_plus_r ``-(PI/2)`` ``PI/2+x`` ``2*PI`` H7). +Replace ``-(PI/2)+(PI/2+x)`` with x. +Replace ``-(PI/2)+2*PI`` with ``3*(PI/2)``. +Intro; Assumption. +Rewrite double; Pattern 3 4 PI; Rewrite (double_var PI); Ring. +Ring. +Left; Replace ``2*PI`` with ``PI/2+3*(PI/2)``. +Apply Rlt_compatibility; Assumption. +Rewrite (double PI); Pattern 3 4 PI; Rewrite (double_var PI); Ring. +Apply ge0_plus_ge0_is_ge0. +Left; Unfold Rdiv; Apply Rmult_lt_pos. +Apply PI_RGT_0. +Apply Rlt_Rinv; Apply Rgt_2_0. +Assumption. +Elim H2; Intro. +Right; Assumption. +Generalize (cos_eq_0_0 x H1); Intro; Elim H4; Intros k0 H5. +Rewrite H5 in H3; Rewrite H5 in H0; Generalize (Rlt_compatibility ``-(PI/2)`` ``3*PI/2`` ``(IZR k0)*PI+PI/2`` H3); Generalize (Rle_compatibility ``-(PI/2)`` ``(IZR k0)*PI+PI/2`` ``2*PI`` H0). +Replace ``-(PI/2)+3*PI/2`` with PI. +Replace ``-(PI/2)+((IZR k0)*PI+PI/2)`` with ``(IZR k0)*PI``. +Replace ``-(PI/2)+2*PI`` with ``3*(PI/2)``. +Intros; Generalize (Rlt_monotony ``/PI`` ``PI`` ``(IZR k0)*PI`` (Rlt_Rinv PI PI_RGT_0) H7); Generalize (Rle_monotony ``/PI`` ``(IZR k0)*PI`` ``3*(PI/2)`` (Rlt_le ``0`` ``/PI`` (Rlt_Rinv PI PI_RGT_0)) H6). +Replace ``/PI*((IZR k0)*PI)`` with (IZR k0). +Replace ``/PI*(3*PI/2)`` with ``3*/2``. +Rewrite <- Rinv_l_sym. +Intros; Generalize (Rlt_compatibility (IZR `-2`) ``1`` (IZR k0) H9); Rewrite <- plus_IZR. +Replace ``(IZR (NEG (xO xH)))+1`` with ``-1``. +Intro; Generalize (Rle_compatibility (IZR `-2`) (IZR k0) ``3*/2`` H8); Rewrite <- plus_IZR. +Replace ``(IZR (NEG (xO xH)))+2`` with ``0``. +Intro; Cut `` -1 < (IZR (Zplus (NEG (xO xH)) k0)) < 1``. +Intro; Generalize (one_IZR_lt1 (Zplus (NEG (xO xH)) k0) H12); Intro. +Cut k0=`2`. +Intro; Rewrite H14 in H8. +Generalize (Rle_monotony ``2`` ``(IZR (POS (xO xH)))`` ``3*/2`` (Rlt_le ``0`` ``2`` Rgt_2_0) H8); Simpl. +Replace ``2*2`` with ``4``. +Replace ``2*(3*/2)`` with ``3``. +Intro; Cut ``3<4``. +Intro; Elim (Rlt_antirefl ``3`` (Rlt_le_trans ``3`` ``4`` ``3`` H16 H15)). +Generalize (Rlt_compatibility ``3`` ``0`` ``1`` Rlt_R0_R1); Rewrite Rplus_Or. +Replace ``3+1`` with ``4``. +Intro; Assumption. +Ring. +Symmetry; Rewrite <- Rmult_assoc; Apply Rinv_r_simpl_m. +Apply aze. +Ring. +Rewrite <- (Zplus_inverse_l `2`) in H13; Generalize (Zsimpl_plus_l `-2` k0 `2` H13); Intro; Assumption. +Split. +Assumption. +Apply Rle_lt_trans with ``(IZR (NEG (xO xH)))+3*/2``. +Assumption. +Simpl; Replace ``-2+3*/2`` with ``-(1*/2)``. +Apply Rlt_trans with ``0``. +Rewrite <- Ropp_O; Apply Rlt_Ropp. +Apply Rmult_lt_pos; [Apply Rlt_R0_R1 | Apply Rlt_Rinv; Apply Rgt_2_0]. +Apply Rlt_R0_R1. +Rewrite Rmult_1l; Apply r_Rmult_mult with ``2``. +Rewrite Ropp_mul3; Rewrite <- Rinv_r_sym. +Rewrite Rmult_Rplus_distr; Rewrite <- Rmult_assoc; Rewrite Rinv_r_simpl_m. +Ring. +Apply aze. +Apply aze. +Apply aze. +Simpl; Ring. +Simpl; Ring. +Apply PI_neq0. +Unfold Rdiv; Pattern 1 ``3``; Rewrite (Rmult_sym ``3``); Repeat Rewrite <- Rmult_assoc; Rewrite <- Rinv_l_sym. +Rewrite Rmult_1l; Apply Rmult_sym. +Apply PI_neq0. +Symmetry; Rewrite (Rmult_sym ``/PI``); Rewrite Rmult_assoc; Rewrite <- Rinv_r_sym. +Apply Rmult_1r. +Apply PI_neq0. +Rewrite double; Pattern 3 4 PI; Rewrite double_var; Ring. +Ring. +Pattern 1 PI; Rewrite double_var; Ring. +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. + (***************************************************************) (* Radian -> Degree | Degree -> Radian *) (***************************************************************) @@ -1548,4 +1653,4 @@ Simpl; Discriminate. Simpl; Discriminate. Simpl; Discriminate. Elim (Rlt_antirefl ``0`` (Rle_lt_trans ``0`` a ``0`` H H2)). -Qed. +Qed.
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