src/HOL/Transcendental.thy
 changeset 59869 3b5b53eb78ba parent 59867 58043346ca64 child 60017 b785d6d06430
```     1.1 --- a/src/HOL/Transcendental.thy	Tue Mar 31 21:54:32 2015 +0200
1.2 +++ b/src/HOL/Transcendental.thy	Wed Apr 01 14:08:17 2015 +0100
1.3 @@ -4013,6 +4013,27 @@
1.4    apply (rule sin_total, auto)
1.5    done
1.6
1.7 +lemma arcsin_0 [simp]: "arcsin 0 = 0"
1.8 +  using arcsin_sin [of 0]
1.9 +  by simp
1.10 +
1.11 +lemma arcsin_1 [simp]: "arcsin 1 = pi/2"
1.12 +  using arcsin_sin [of "pi/2"]
1.13 +  by simp
1.14 +
1.15 +lemma arcsin_minus_1 [simp]: "arcsin (-1) = - (pi/2)"
1.16 +  using arcsin_sin [of "-pi/2"]
1.17 +  by simp
1.18 +
1.19 +lemma arcsin_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin(-x) = -arcsin x"
1.20 +  by (metis (no_types, hide_lams) arcsin arcsin_sin minus_minus neg_le_iff_le sin_minus)
1.21 +
1.22 +lemma arcsin_eq_iff: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> (arcsin x = arcsin y \<longleftrightarrow> x = y)"
1.23 +  by (metis abs_le_interval_iff arcsin)
1.24 +
1.25 +lemma cos_arcsin_nonzero: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> cos(arcsin x) \<noteq> 0"
1.26 +  using arcsin_lt_bounded cos_gt_zero_pi by force
1.27 +
1.28  lemma arccos:
1.29       "\<lbrakk>-1 \<le> y; y \<le> 1\<rbrakk>
1.30        \<Longrightarrow> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
1.31 @@ -4031,8 +4052,7 @@
1.32    by (blast dest: arccos)
1.33
1.34  lemma arccos_lt_bounded:
1.35 -     "\<lbrakk>-1 < y; y < 1\<rbrakk>
1.36 -      \<Longrightarrow> 0 < arccos y & arccos y < pi"
1.37 +     "\<lbrakk>-1 < y; y < 1\<rbrakk> \<Longrightarrow> 0 < arccos y & arccos y < pi"
1.38    apply (frule order_less_imp_le)
1.39    apply (frule_tac y = y in order_less_imp_le)
1.40    apply (frule arccos_bounded, auto)
1.41 @@ -4081,17 +4101,27 @@
1.42  lemma arccos_1 [simp]: "arccos 1 = 0"
1.43    using arccos_cos by force
1.44
1.45 -lemma arctan [simp]: "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
1.46 +lemma arccos_minus_1 [simp]: "arccos(-1) = pi"
1.47 +  by (metis arccos_cos cos_pi order_refl pi_ge_zero)
1.48 +
1.49 +lemma arccos_minus: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos(-x) = pi - arccos x"
1.50 +  by (metis arccos_cos arccos_cos2 cos_minus_pi cos_total diff_le_0_iff_le le_add_same_cancel1
1.52 +
1.53 +lemma sin_arccos_nonzero: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> ~(sin(arccos x) = 0)"
1.54 +  using arccos_lt_bounded sin_gt_zero by force
1.55 +
1.56 +lemma arctan: "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
1.57    unfolding arctan_def by (rule theI' [OF tan_total])
1.58
1.59  lemma tan_arctan: "tan (arctan y) = y"
1.60 -  by auto
1.61 +  by (simp add: arctan)
1.62
1.63  lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
1.64    by (auto simp only: arctan)
1.65
1.66  lemma arctan_lbound: "- (pi/2) < arctan y"
1.67 -  by auto
1.68 +  by (simp add: arctan)
1.69
1.70  lemma arctan_ubound: "arctan y < pi/2"
1.71    by (auto simp only: arctan)
1.72 @@ -4112,7 +4142,7 @@
1.73  lemma arctan_minus: "arctan (- x) = - arctan x"
1.74    apply (rule arctan_unique)
1.75    apply (simp only: neg_less_iff_less arctan_ubound)
1.76 -  apply (metis minus_less_iff arctan_lbound, simp)
1.77 +  apply (metis minus_less_iff arctan_lbound, simp add: arctan)
1.78    done
1.79
1.80  lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
1.81 @@ -4128,7 +4158,7 @@
1.82    have "(cos (arctan x))\<^sup>2 * (1 + (tan (arctan x))\<^sup>2) = 1"
1.83      unfolding tan_def by (simp add: distrib_left power_divide)
1.84    thus "(cos (arctan x))\<^sup>2 = (1 / sqrt (1 + x\<^sup>2))\<^sup>2"
1.85 -    using `0 < 1 + x\<^sup>2` by (simp add: power_divide eq_divide_eq)
1.86 +    using `0 < 1 + x\<^sup>2` by (simp add: arctan power_divide eq_divide_eq)
1.87  qed
1.88
1.89  lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<^sup>2)"
1.90 @@ -4221,7 +4251,7 @@
1.91  lemma isCont_arctan: "isCont arctan x"
1.92    apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
1.93    apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
1.94 -  apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
1.95 +  apply (subgoal_tac "isCont arctan (tan (arctan x))", simp add: arctan)
1.96    apply (erule (1) isCont_inverse_function2 [where f=tan])
1.97    apply (metis arctan_tan order_le_less_trans order_less_le_trans)
1.98    apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
1.99 @@ -4262,10 +4292,9 @@
1.100    apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
1.101    apply (rule DERIV_cong [OF DERIV_tan])
1.102    apply (rule cos_arctan_not_zero)
1.103 -  apply (simp add: power_inverse tan_sec [symmetric])
1.104 +  apply (simp add: arctan power_inverse tan_sec [symmetric])
1.105    apply (subgoal_tac "0 < 1 + x\<^sup>2", simp)
1.107 -  apply (simp, simp, simp, rule isCont_arctan)
1.109    done
1.110
1.111  declare
1.112 @@ -4275,12 +4304,12 @@
1.113
1.114  lemma filterlim_tan_at_right: "filterlim tan at_bot (at_right (- pi/2))"
1.115    by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. - pi/2 < x \<and> x < pi/2" and P="\<lambda>x. True" and g=arctan])
1.116 -     (auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
1.117 +     (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
1.118             intro!: tan_monotone exI[of _ "pi/2"])
1.119
1.120  lemma filterlim_tan_at_left: "filterlim tan at_top (at_left (pi/2))"
1.121    by (rule filterlim_at_top_at_left[where Q="\<lambda>x. - pi/2 < x \<and> x < pi/2" and P="\<lambda>x. True" and g=arctan])
1.122 -     (auto simp: le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
1.123 +     (auto simp: arctan le_less eventually_at dist_real_def simp del: less_divide_eq_numeral1
1.124             intro!: tan_monotone exI[of _ "pi/2"])
1.125
1.126  lemma tendsto_arctan_at_top: "(arctan ---> (pi/2)) at_top"
1.127 @@ -4312,6 +4341,12 @@
1.128
1.129  subsection{* Prove Totality of the Trigonometric Functions *}
1.130
1.131 +lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
1.132 +  by (simp add: abs_le_iff)
1.133 +
1.134 +lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
1.135 +  by (simp add: sin_arccos abs_le_iff)
1.136 +
1.137  lemma sin_mono_less_eq: "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2; -(pi/2) \<le> y; y \<le> pi/2\<rbrakk>
1.138           \<Longrightarrow> (sin(x) < sin(y) \<longleftrightarrow> x < y)"
1.139  by (metis not_less_iff_gr_or_eq sin_monotone_2pi)
1.140 @@ -4320,12 +4355,12 @@
1.141           \<Longrightarrow> (sin(x) \<le> sin(y) \<longleftrightarrow> x \<le> y)"
1.142  by (meson leD le_less_linear sin_monotone_2pi sin_monotone_2pi_le)
1.143
1.144 -lemma sin_inj_pi: "-(pi/2) \<le> x ==> x \<le> pi/2 ==>
1.145 -         -(pi/2) \<le> y ==> y \<le> pi/2 ==> sin(x) = sin(y) \<Longrightarrow> x = y"
1.146 +lemma sin_inj_pi:
1.147 +    "\<lbrakk>-(pi/2) \<le> x; x \<le> pi/2;-(pi/2) \<le> y; y \<le> pi/2; sin(x) = sin(y)\<rbrakk> \<Longrightarrow> x = y"
1.148  by (metis arcsin_sin)
1.149
1.150 -lemma cos_mono_lt_eq: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi
1.151 -         \<Longrightarrow> (cos(x) < cos(y) \<longleftrightarrow> y < x)"
1.152 +lemma cos_mono_less_eq:
1.153 +    "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi \<Longrightarrow> (cos(x) < cos(y) \<longleftrightarrow> y < x)"
1.154  by (meson cos_monotone_0_pi cos_monotone_0_pi_le leD le_less_linear)
1.155
1.156  lemma cos_mono_le_eq: "0 \<le> x ==> x \<le> pi ==> 0 \<le> y ==> y \<le> pi
1.157 @@ -4407,6 +4442,40 @@
1.158      done
1.159  qed
1.160
1.161 +lemma arcsin_less_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> arcsin x < arcsin y \<longleftrightarrow> x < y"
1.162 +  apply (rule trans [OF sin_mono_less_eq [symmetric]])
1.163 +  using arcsin_ubound arcsin_lbound
1.164 +  apply (auto simp: )
1.165 +  done
1.166 +
1.167 +lemma arcsin_le_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y \<longleftrightarrow> x \<le> y"
1.168 +  using arcsin_less_mono not_le by blast
1.169 +
1.170 +lemma arcsin_less_arcsin: "-1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x < arcsin y"
1.171 +  using arcsin_less_mono by auto
1.172 +
1.173 +lemma arcsin_le_arcsin: "-1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin x \<le> arcsin y"
1.174 +  using arcsin_le_mono by auto
1.175 +
1.176 +lemma arccos_less_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> (arccos x < arccos y \<longleftrightarrow> y < x)"
1.177 +  apply (rule trans [OF cos_mono_less_eq [symmetric]])
1.178 +  using arccos_ubound arccos_lbound
1.179 +  apply (auto simp: )
1.180 +  done
1.181 +
1.182 +lemma arccos_le_mono: "abs x \<le> 1 \<Longrightarrow> abs y \<le> 1 \<Longrightarrow> arccos x \<le> arccos y \<longleftrightarrow> y \<le> x"
1.183 +  using arccos_less_mono [of y x]
1.184 +  by (simp add: not_le [symmetric])
1.185 +
1.186 +lemma arccos_less_arccos: "-1 \<le> x \<Longrightarrow> x < y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y < arccos x"
1.187 +  using arccos_less_mono by auto
1.188 +
1.189 +lemma arccos_le_arccos: "-1 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arccos y \<le> arccos x"
1.190 +  using arccos_le_mono by auto
1.191 +
1.192 +lemma arccos_eq_iff: "abs x \<le> 1 & abs y \<le> 1 \<Longrightarrow> (arccos x = arccos y \<longleftrightarrow> x = y)"
1.193 +  using cos_arccos_abs by fastforce
1.194 +
1.195  subsection {* Machins formula *}
1.196
1.197  lemma arctan_one: "arctan 1 = pi / 4"
1.198 @@ -4418,7 +4487,8 @@
1.199  proof
1.200    show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
1.201      unfolding arctan_one [symmetric] arctan_minus [symmetric]
1.202 -    unfolding arctan_less_iff using assms by auto
1.203 +    unfolding arctan_less_iff using assms  by (auto simp add: arctan)
1.204 +
1.205  qed
1.206
1.208 @@ -4436,7 +4506,7 @@
1.210    show 2: "arctan x + arctan y < pi / 2" by simp
1.211    show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
1.213 +    using cos_gt_zero_pi [OF 1 2] by (simp add: arctan tan_add)
1.214  qed
1.215
1.216  theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
1.217 @@ -4882,7 +4952,7 @@
1.218    show "- (pi / 2) < sgn x * pi / 2 - arctan x"
1.219      using arctan_bounded [of x] assms
1.220      unfolding sgn_real_def
1.221 -    apply (auto simp add: algebra_simps)
1.222 +    apply (auto simp add: arctan algebra_simps)
1.223      apply (drule zero_less_arctan_iff [THEN iffD2])
1.224      apply arith
1.225      done
1.226 @@ -4911,12 +4981,6 @@
1.227    apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
1.228    done
1.229
1.230 -lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
1.231 -  by (simp add: abs_le_iff)
1.232 -
1.233 -lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
1.234 -  by (simp add: sin_arccos abs_le_iff)
1.235 -
1.236  lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
1.237
1.238  lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
```