--- a/src/HOL/Transcendental.thy Sun Aug 18 20:41:47 2013 +0200
+++ b/src/HOL/Transcendental.thy Sun Aug 18 22:44:39 2013 +0200
@@ -2,7 +2,6 @@
Author: Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
Author: Lawrence C Paulson
Author: Jeremy Avigad
-
*)
header{*Power Series, Transcendental Functions etc.*}
@@ -23,42 +22,44 @@
qed
lemma lemma_realpow_diff_sumr:
- fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows
- "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
+ fixes y :: "'a::{comm_semiring_0,monoid_mult}"
+ shows
+ "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
-by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac
- del: setsum_op_ivl_Suc)
+ by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac del: setsum_op_ivl_Suc)
lemma lemma_realpow_diff_sumr2:
- fixes y :: "'a::{comm_ring,monoid_mult}" shows
- "x ^ (Suc n) - y ^ (Suc n) =
+ fixes y :: "'a::{comm_ring,monoid_mult}"
+ shows
+ "x ^ (Suc n) - y ^ (Suc n) =
(x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
-apply (induct n, simp)
-apply (simp del: setsum_op_ivl_Suc)
-apply (subst setsum_op_ivl_Suc)
-apply (subst lemma_realpow_diff_sumr)
-apply (simp add: distrib_left del: setsum_op_ivl_Suc)
-apply (subst mult_left_commute [of "x - y"])
-apply (erule subst)
-apply (simp add: algebra_simps)
-done
+ apply (induct n)
+ apply simp
+ apply (simp del: setsum_op_ivl_Suc)
+ apply (subst setsum_op_ivl_Suc)
+ apply (subst lemma_realpow_diff_sumr)
+ apply (simp add: distrib_left del: setsum_op_ivl_Suc)
+ apply (subst mult_left_commute [of "x - y"])
+ apply (erule subst)
+ apply (simp add: algebra_simps)
+ done
lemma lemma_realpow_rev_sumr:
- "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
- (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
-apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
-apply (rule inj_onI, simp)
-apply auto
-apply (rule_tac x="n - x" in image_eqI, simp, simp)
-done
+ "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
+ (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
+ apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
+ apply (rule inj_onI, simp)
+ apply auto
+ apply (rule_tac x="n - x" in image_eqI, simp, simp)
+ done
text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
-x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
+ x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
lemma powser_insidea:
fixes x z :: "'a::real_normed_field"
assumes 1: "summable (\<lambda>n. f n * x ^ n)"
- assumes 2: "norm z < norm x"
+ and 2: "norm z < norm x"
shows "summable (\<lambda>n. norm (f n * z ^ n))"
proof -
from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
@@ -75,7 +76,8 @@
have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
K * norm (z ^ n) * inverse (norm (x ^ n))"
proof (intro exI allI impI)
- fix n::nat assume "0 \<le> n"
+ fix n::nat
+ assume "0 \<le> n"
have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
norm (f n * x ^ n) * norm (z ^ n)"
by (simp add: norm_mult abs_mult)
@@ -108,43 +110,61 @@
qed
lemma powser_inside:
- fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" shows
- "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]
- ==> summable (%n. f(n) * (z ^ n))"
-by (rule powser_insidea [THEN summable_norm_cancel])
-
-lemma sum_split_even_odd: fixes f :: "nat \<Rightarrow> real" shows
- "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
- (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
+ fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}"
+ shows
+ "summable (\<lambda>n. f n * (x ^ n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
+ summable (\<lambda>n. f n * (z ^ n))"
+ by (rule powser_insidea [THEN summable_norm_cancel])
+
+lemma sum_split_even_odd:
+ fixes f :: "nat \<Rightarrow> real"
+ shows
+ "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
+ (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
proof (induct n)
+ case 0
+ then show ?case by simp
+next
case (Suc n)
have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =
- (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
+ (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
using Suc.hyps unfolding One_nat_def by auto
- also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))" by auto
+ also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))"
+ by auto
finally show ?case .
-qed auto
-
-lemma sums_if': fixes g :: "nat \<Rightarrow> real" assumes "g sums x"
+qed
+
+lemma sums_if':
+ fixes g :: "nat \<Rightarrow> real"
+ assumes "g sums x"
shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
unfolding sums_def
proof (rule LIMSEQ_I)
- fix r :: real assume "0 < r"
+ fix r :: real
+ assume "0 < r"
from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast
let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
- { fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> no" by auto
+ {
+ fix m
+ assume "m \<ge> 2 * no"
+ hence "m div 2 \<ge> no" by auto
have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
using sum_split_even_odd by auto
- hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
+ hence "(norm (?SUM (2 * (m div 2)) - x) < r)"
+ using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
moreover
have "?SUM (2 * (m div 2)) = ?SUM m"
proof (cases "even m")
- case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
+ case True
+ show ?thesis
+ unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
next
- case False hence "even (Suc m)" by auto
- from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]] odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]
+ case False
+ hence "even (Suc m)" by auto
+ from even_nat_div_two_times_two[OF this, unfolded numeral_2_eq_2[symmetric]]
+ odd_nat_plus_one_div_two[OF False, unfolded numeral_2_eq_2[symmetric]]
have eq: "Suc (2 * (m div 2)) = m" by auto
hence "even (2 * (m div 2))" using `odd m` by auto
have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
@@ -156,13 +176,19 @@
thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
qed
-lemma sums_if: fixes g :: "nat \<Rightarrow> real" assumes "g sums x" and "f sums y"
+lemma sums_if:
+ fixes g :: "nat \<Rightarrow> real"
+ assumes "g sums x" and "f sums y"
shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
proof -
let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
- { fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
- by (cases B) auto } note if_sum = this
- have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x" using sums_if'[OF `g sums x`] .
+ {
+ fix B T E
+ have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
+ by (cases B) auto
+ } note if_sum = this
+ have g_sums: "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
+ using sums_if'[OF `g sums x`] .
{
have "?s 0 = 0" by auto
have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
@@ -174,8 +200,8 @@
unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
even_Suc Suc_m1 if_eq .
- } from sums_add[OF g_sums this]
- show ?thesis unfolding if_sum .
+ }
+ from sums_add[OF g_sums this] show ?thesis unfolding if_sum .
qed
subsection {* Alternating series test / Leibniz formula *}
@@ -186,106 +212,140 @@
shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and>
((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. -1^i*a i) ----> l)"
(is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
-proof -
+proof (rule nested_sequence_unique)
have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
- have "\<forall> n. ?f n \<le> ?f (Suc n)"
- proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed
- moreover
- have "\<forall> n. ?g (Suc n) \<le> ?g n"
- proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
- unfolding One_nat_def by auto qed
- moreover
- have "\<forall> n. ?f n \<le> ?g n"
- proof fix n show "?f n \<le> ?g n" using fg_diff a_pos
- unfolding One_nat_def by auto qed
- moreover
- have "(\<lambda> n. ?f n - ?g n) ----> 0" unfolding fg_diff
- proof (rule LIMSEQ_I)
- fix r :: real assume "0 < r"
- with `a ----> 0`[THEN LIMSEQ_D]
- obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" by auto
- hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
- thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
+ show "\<forall>n. ?f n \<le> ?f (Suc n)"
+ proof
+ fix n
+ show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
+ qed
+ show "\<forall>n. ?g (Suc n) \<le> ?g n"
+ proof
+ fix n
+ show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
+ unfolding One_nat_def by auto
+ qed
+ show "\<forall>n. ?f n \<le> ?g n"
+ proof
+ fix n
+ show "?f n \<le> ?g n" using fg_diff a_pos
+ unfolding One_nat_def by auto
qed
- ultimately
- show ?thesis by (rule nested_sequence_unique)
+ show "(\<lambda>n. ?f n - ?g n) ----> 0" unfolding fg_diff
+ proof (rule LIMSEQ_I)
+ fix r :: real
+ assume "0 < r"
+ with `a ----> 0`[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
+ by auto
+ hence "\<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
+ thus "\<exists>N. \<forall>n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
+ qed
qed
-lemma summable_Leibniz': fixes a :: "nat \<Rightarrow> real"
- assumes a_zero: "a ----> 0" and a_pos: "\<And> n. 0 \<le> a n"
- and a_monotone: "\<And> n. a (Suc n) \<le> a n"
+lemma summable_Leibniz':
+ fixes a :: "nat \<Rightarrow> real"
+ assumes a_zero: "a ----> 0"
+ and a_pos: "\<And> n. 0 \<le> a n"
+ and a_monotone: "\<And> n. a (Suc n) \<le> a n"
shows summable: "summable (\<lambda> n. (-1)^n * a n)"
- and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
- and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
- and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
- and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
+ and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
+ and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
+ and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
+ and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
proof -
- let "?S n" = "(-1)^n * a n"
- let "?P n" = "\<Sum>i=0..<n. ?S i"
- let "?f n" = "?P (2 * n)"
- let "?g n" = "?P (2 * n + 1)"
- obtain l :: real where below_l: "\<forall> n. ?f n \<le> l" and "?f ----> l" and above_l: "\<forall> n. l \<le> ?g n" and "?g ----> l"
+ let ?S = "\<lambda>n. (-1)^n * a n"
+ let ?P = "\<lambda>n. \<Sum>i=0..<n. ?S i"
+ let ?f = "\<lambda>n. ?P (2 * n)"
+ let ?g = "\<lambda>n. ?P (2 * n + 1)"
+ obtain l :: real
+ where below_l: "\<forall> n. ?f n \<le> l"
+ and "?f ----> l"
+ and above_l: "\<forall> n. l \<le> ?g n"
+ and "?g ----> l"
using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
- let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n"
+ let ?Sa = "\<lambda>m. \<Sum> n = 0..<m. ?S n"
have "?Sa ----> l"
proof (rule LIMSEQ_I)
- fix r :: real assume "0 < r"
-
+ fix r :: real
+ assume "0 < r"
with `?f ----> l`[THEN LIMSEQ_D]
obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
- { fix n :: nat
- assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
+ {
+ fix n :: nat
+ assume "n \<ge> (max (2 * f_no) (2 * g_no))"
+ hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
have "norm (?Sa n - l) < r"
proof (cases "even n")
- case True from even_nat_div_two_times_two[OF this]
- have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto
- with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" by auto
- from f[OF this]
- show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
+ case True
+ from even_nat_div_two_times_two[OF this]
+ have n_eq: "2 * (n div 2) = n"
+ unfolding numeral_2_eq_2[symmetric] by auto
+ with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no"
+ by auto
+ from f[OF this] show ?thesis
+ unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
next
- case False hence "even (n - 1)" by simp
+ case False
+ hence "even (n - 1)" by simp
from even_nat_div_two_times_two[OF this]
- have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto
- hence range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto
-
- from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" by auto
- from g[OF this]
- show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
+ have n_eq: "2 * ((n - 1) div 2) = n - 1"
+ unfolding numeral_2_eq_2[symmetric] by auto
+ hence range_eq: "n - 1 + 1 = n"
+ using odd_pos[OF False] by auto
+
+ from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no"
+ by auto
+ from g[OF this] show ?thesis
+ unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
qed
}
- thus "\<exists> no. \<forall> n \<ge> no. norm (?Sa n - l) < r" by blast
+ thus "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
qed
- hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
+ hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
+ unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
thus "summable ?S" using summable_def by auto
have "l = suminf ?S" using sums_unique[OF sums_l] .
- { fix n show "suminf ?S \<le> ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto }
- { fix n show "?f n \<le> suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_l by auto }
- show "?g ----> suminf ?S" using `?g ----> l` `l = suminf ?S` by auto
- show "?f ----> suminf ?S" using `?f ----> l` `l = suminf ?S` by auto
+ fix n
+ show "suminf ?S \<le> ?g n"
+ unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
+ show "?f n \<le> suminf ?S"
+ unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
+ show "?g ----> suminf ?S"
+ using `?g ----> l` `l = suminf ?S` by auto
+ show "?f ----> suminf ?S"
+ using `?f ----> l` `l = suminf ?S` by auto
qed
-theorem summable_Leibniz: fixes a :: "nat \<Rightarrow> real"
+theorem summable_Leibniz:
+ fixes a :: "nat \<Rightarrow> real"
assumes a_zero: "a ----> 0" and "monoseq a"
shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
- and "0 < a 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n. -1^i * a i .. \<Sum>i=0..<2*n+1. -1^i * a i})" (is "?pos")
- and "a 0 < 0 \<longrightarrow> (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n+1. -1^i * a i .. \<Sum>i=0..<2*n. -1^i * a i})" (is "?neg")
- and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
- and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
+ and "0 < a 0 \<longrightarrow>
+ (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n. -1^i * a i .. \<Sum>i=0..<2*n+1. -1^i * a i})" (is "?pos")
+ and "a 0 < 0 \<longrightarrow>
+ (\<forall>n. (\<Sum>i. -1^i*a i) \<in> { \<Sum>i=0..<2*n+1. -1^i * a i .. \<Sum>i=0..<2*n. -1^i * a i})" (is "?neg")
+ and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
+ and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
proof -
have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
case True
- hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" by auto
- { fix n have "a (Suc n) \<le> a n" using ord[where n="Suc n" and m=n] by auto }
- note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] and mono = this
+ hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n"
+ by auto
+ {
+ fix n
+ have "a (Suc n) \<le> a n"
+ using ord[where n="Suc n" and m=n] by auto
+ } note mono = this
+ note leibniz = summable_Leibniz'[OF `a ----> 0` ge0]
from leibniz[OF mono]
show ?thesis using `0 \<le> a 0` by auto
next
@@ -293,111 +353,129 @@
case False
with monoseq_le[OF `monoseq a` `a ----> 0`]
have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
- hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" by auto
- { fix n have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] by auto }
- note monotone = this
- note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
- have "summable (\<lambda> n. (-1)^n * ?a n)" using leibniz(1) by auto
- then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" unfolding summable_def by auto
- from this[THEN sums_minus]
- have "(\<lambda> n. (-1)^n * a n) sums -l" by auto
+ hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
+ by auto
+ {
+ fix n
+ have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
+ by auto
+ } note monotone = this
+ note leibniz =
+ summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
+ OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
+ have "summable (\<lambda> n. (-1)^n * ?a n)"
+ using leibniz(1) by auto
+ then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
+ unfolding summable_def by auto
+ from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
+ by auto
hence ?summable unfolding summable_def by auto
moreover
- have "\<And> a b :: real. \<bar> - a - - b \<bar> = \<bar>a - b\<bar>" unfolding minus_diff_minus by auto
+ have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
+ unfolding minus_diff_minus by auto
from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
- have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)" by auto
+ have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)"
+ by auto
have ?pos using `0 \<le> ?a 0` by auto
- moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto
- moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel] by auto
+ moreover have ?neg
+ using leibniz(2,4)
+ unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
+ by auto
+ moreover have ?f and ?g
+ using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
+ by auto
ultimately show ?thesis by auto
qed
- from this[THEN conjunct1] this[THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct1] this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1]
- this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]
+ from this[THEN conjunct1]
+ this[THEN conjunct2, THEN conjunct1]
+ this[THEN conjunct2, THEN conjunct2, THEN conjunct1]
+ this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct1]
+ this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]
show ?summable and ?pos and ?neg and ?f and ?g .
qed
subsection {* Term-by-Term Differentiability of Power Series *}
-definition
- diffs :: "(nat => 'a::ring_1) => nat => 'a" where
- "diffs c = (%n. of_nat (Suc n) * c(Suc n))"
+definition diffs :: "(nat => 'a::ring_1) => nat => 'a"
+ where "diffs c = (\<lambda>n. of_nat (Suc n) * c(Suc n))"
text{*Lemma about distributing negation over it*}
-lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
-by (simp add: diffs_def)
+lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
+ by (simp add: diffs_def)
lemma sums_Suc_imp:
assumes f: "f 0 = 0"
shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
-unfolding sums_def
-apply (rule LIMSEQ_imp_Suc)
-apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
-apply (simp only: setsum_shift_bounds_Suc_ivl)
-done
+ unfolding sums_def
+ apply (rule LIMSEQ_imp_Suc)
+ apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
+ apply (simp only: setsum_shift_bounds_Suc_ivl)
+ done
lemma diffs_equiv:
fixes x :: "'a::{real_normed_vector, ring_1}"
- shows "summable (%n. (diffs c)(n) * (x ^ n)) ==>
- (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
+ shows "summable (\<lambda>n. (diffs c)(n) * (x ^ n)) \<Longrightarrow>
+ (\<lambda>n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
(\<Sum>n. (diffs c)(n) * (x ^ n))"
-unfolding diffs_def
-apply (drule summable_sums)
-apply (rule sums_Suc_imp, simp_all)
-done
+ unfolding diffs_def
+ apply (drule summable_sums)
+ apply (rule sums_Suc_imp, simp_all)
+ done
lemma lemma_termdiff1:
fixes z :: "'a :: {monoid_mult,comm_ring}" shows
"(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
(\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
-by(auto simp add: algebra_simps power_add [symmetric])
+ by (auto simp add: algebra_simps power_add [symmetric])
lemma sumr_diff_mult_const2:
"setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
-by (simp add: setsum_subtractf)
+ by (simp add: setsum_subtractf)
lemma lemma_termdiff2:
fixes h :: "'a :: {field}"
- assumes h: "h \<noteq> 0" shows
- "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
- h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
- (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
-apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
-apply (simp add: right_diff_distrib diff_divide_distrib h)
-apply (simp add: mult_assoc [symmetric])
-apply (cases "n", simp)
-apply (simp add: lemma_realpow_diff_sumr2 h
- right_diff_distrib [symmetric] mult_assoc
- del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
-apply (subst lemma_realpow_rev_sumr)
-apply (subst sumr_diff_mult_const2)
-apply simp
-apply (simp only: lemma_termdiff1 setsum_right_distrib)
-apply (rule setsum_cong [OF refl])
-apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
-apply (clarify)
-apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
- del: setsum_op_ivl_Suc power_Suc)
-apply (subst mult_assoc [symmetric], subst power_add [symmetric])
-apply (simp add: mult_ac)
-done
+ assumes h: "h \<noteq> 0"
+ shows
+ "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
+ h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
+ (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
+ apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
+ apply (simp add: right_diff_distrib diff_divide_distrib h)
+ apply (simp add: mult_assoc [symmetric])
+ apply (cases "n", simp)
+ apply (simp add: lemma_realpow_diff_sumr2 h
+ right_diff_distrib [symmetric] mult_assoc
+ del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
+ apply (subst lemma_realpow_rev_sumr)
+ apply (subst sumr_diff_mult_const2)
+ apply simp
+ apply (simp only: lemma_termdiff1 setsum_right_distrib)
+ apply (rule setsum_cong [OF refl])
+ apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
+ apply (clarify)
+ apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
+ del: setsum_op_ivl_Suc power_Suc)
+ apply (subst mult_assoc [symmetric], subst power_add [symmetric])
+ apply (simp add: mult_ac)
+ done
lemma real_setsum_nat_ivl_bounded2:
fixes K :: "'a::linordered_semidom"
assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
- assumes K: "0 \<le> K"
+ and K: "0 \<le> K"
shows "setsum f {0..<n-k} \<le> of_nat n * K"
-apply (rule order_trans [OF setsum_mono])
-apply (rule f, simp)
-apply (simp add: mult_right_mono K)
-done
+ apply (rule order_trans [OF setsum_mono])
+ apply (rule f, simp)
+ apply (simp add: mult_right_mono K)
+ done
lemma lemma_termdiff3:
fixes h z :: "'a::{real_normed_field}"
assumes 1: "h \<noteq> 0"
- assumes 2: "norm z \<le> K"
- assumes 3: "norm (z + h) \<le> K"
+ and 2: "norm z \<le> K"
+ and 3: "norm (z + h) \<le> K"
shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
\<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
proof -
@@ -410,14 +488,14 @@
done
also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
proof (rule mult_right_mono [OF _ norm_ge_zero])
- from norm_ge_zero 2 have K: "0 \<le> K" by (rule order_trans)
+ from norm_ge_zero 2 have K: "0 \<le> K"
+ by (rule order_trans)
have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
apply (erule subst)
apply (simp only: norm_mult norm_power power_add)
apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
done
- show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
- (z + h) ^ q * z ^ (n - 2 - q))
+ show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))
\<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
apply (intro
order_trans [OF norm_setsum]
@@ -437,19 +515,20 @@
fixes f :: "'a::{real_normed_field} \<Rightarrow>
'b::real_normed_vector"
assumes k: "0 < (k::real)"
- assumes le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
+ and le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
shows "f -- 0 --> 0"
-unfolding LIM_eq diff_0_right
-proof (safe)
+ unfolding LIM_eq diff_0_right
+proof safe
let ?h = "of_real (k / 2)::'a"
have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
- fix r::real assume r: "0 < r"
+ fix r::real
+ assume r: "0 < r"
show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
- proof (cases)
+ proof cases
assume "K = 0"
with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
by simp
@@ -459,7 +538,8 @@
with zero_le_K have K: "0 < K" by simp
show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
proof (rule exI, safe)
- from k r K show "0 < min k (r * inverse K / 2)"
+ from k r K
+ show "0 < min k (r * inverse K / 2)"
by (simp add: mult_pos_pos positive_imp_inverse_positive)
next
fix x::'a
@@ -479,14 +559,14 @@
qed
lemma lemma_termdiff5:
- fixes g :: "'a::{real_normed_field} \<Rightarrow>
- nat \<Rightarrow> 'b::banach"
+ fixes g :: "'a::real_normed_field \<Rightarrow> nat \<Rightarrow> 'b::banach"
assumes k: "0 < (k::real)"
assumes f: "summable f"
assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
proof (rule lemma_termdiff4 [OF k])
- fix h::'a assume "h \<noteq> 0" and "norm h < k"
+ fix h::'a
+ assume "h \<noteq> 0" and "norm h < k"
hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
by (simp add: le)
hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
@@ -510,7 +590,7 @@
lemma termdiffs_aux:
fixes x :: "'a::{real_normed_field,banach}"
assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
- assumes 2: "norm x < norm K"
+ and 2: "norm x < norm K"
shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
- of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
proof -
@@ -522,7 +602,6 @@
show ?thesis
proof (rule lemma_termdiff5)
show "0 < r - norm x" using r1 by simp
- next
from r r2 have "norm (of_real r::'a) < norm K"
by simp
with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
@@ -532,8 +611,8 @@
by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
by (rule diffs_equiv [THEN sums_summable])
- also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))
- = (\<lambda>n. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
+ also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
+ (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
apply (rule ext)
apply (simp add: diffs_def)
apply (case_tac n, simp_all add: r_neq_0)
@@ -550,8 +629,8 @@
apply (case_tac "nat", simp)
apply (simp add: r_neq_0)
done
- finally show
- "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
+ finally
+ show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
next
fix h::'a and n::nat
assume h: "h \<noteq> 0"
@@ -575,11 +654,11 @@
lemma termdiffs:
fixes K x :: "'a::{real_normed_field,banach}"
assumes 1: "summable (\<lambda>n. c n * K ^ n)"
- assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
- assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
- assumes 4: "norm x < norm K"
+ and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
+ and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
+ and 4: "norm x < norm K"
shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
-unfolding deriv_def
+ unfolding deriv_def
proof (rule LIM_zero_cancel)
show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
- suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
@@ -613,24 +692,26 @@
apply (simp add: algebra_simps)
done
next
- show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h -
- of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
- by (rule termdiffs_aux [OF 3 4])
+ show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
+ by (rule termdiffs_aux [OF 3 4])
qed
qed
subsection {* Derivability of power series *}
-lemma DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
+lemma DERIV_series':
+ fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
- and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
- and "summable (f' x0)"
- and "summable L" and L_def: "\<And> n x y. \<lbrakk> x \<in> { a <..< b} ; y \<in> { a <..< b} \<rbrakk> \<Longrightarrow> \<bar> f x n - f y n \<bar> \<le> L n * \<bar> x - y \<bar>"
+ and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
+ and "summable (f' x0)"
+ and "summable L"
+ and L_def: "\<And>n x y. \<lbrakk> x \<in> { a <..< b} ; y \<in> { a <..< b} \<rbrakk> \<Longrightarrow> \<bar>f x n - f y n\<bar> \<le> L n * \<bar>x - y\<bar>"
shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
unfolding deriv_def
proof (rule LIM_I)
- fix r :: real assume "0 < r" hence "0 < r/3" by auto
+ fix r :: real
+ assume "0 < r" hence "0 < r/3" by auto
obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
@@ -642,7 +723,7 @@
have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
- let "?diff i x" = "(f (x0 + x) i - f x0 i) / x"
+ let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
let ?r = "r / (3 * real ?N)"
have "0 < 3 * real ?N" by auto
@@ -654,23 +735,30 @@
have "0 < S'" unfolding S'_def
proof (rule iffD2[OF Min_gr_iff])
- show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x"
- proof (rule ballI)
- fix x assume "x \<in> ?s ` {0..<?N}"
- then obtain n where "x = ?s n" and "n \<in> {0..<?N}" using image_iff[THEN iffD1] by blast
+ show "\<forall>x \<in> (?s ` { 0 ..< ?N }). 0 < x"
+ proof
+ fix x
+ assume "x \<in> ?s ` {0..<?N}"
+ then obtain n where "x = ?s n" and "n \<in> {0..<?N}"
+ using image_iff[THEN iffD1] by blast
from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
- obtain s where s_bound: "0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r)" by auto
- have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)
+ obtain s where s_bound: "0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < s \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r)"
+ by auto
+ have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound)
thus "0 < x" unfolding `x = ?s n` .
qed
qed auto
def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
- hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0" and "S \<le> S'" using x0_in_I and `0 < S'`
+ hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
+ and "S \<le> S'" using x0_in_I and `0 < S'`
by auto
- { fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
- hence x_in_I: "x0 + x \<in> { a <..< b }" using S_a S_b by auto
+ {
+ fix x
+ assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
+ hence x_in_I: "x0 + x \<in> { a <..< b }"
+ using S_a S_b by auto
note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
note div_smbl = summable_divide[OF diff_smbl]
@@ -680,109 +768,164 @@
note div_shft_smbl = summable_divide[OF diff_shft_smbl]
note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
- { fix n
+ {
+ fix n
have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
- using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .
- hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 0` by auto
+ using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
+ unfolding abs_divide .
+ hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)"
+ using `x \<noteq> 0` by auto
} note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
- hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") using L_estimate by auto
-
- have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le> (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..
+ hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3")
+ using L_estimate by auto
+
+ have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le>
+ (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..
also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
proof (rule setsum_strict_mono)
- fix n assume "n \<in> { 0 ..< ?N}"
- have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .
+ fix n
+ assume "n \<in> { 0 ..< ?N}"
+ have "\<bar>x\<bar> < S" using `\<bar>x\<bar> < S` .
also have "S \<le> S'" using `S \<le> S'` .
also have "S' \<le> ?s n" unfolding S'_def
proof (rule Min_le_iff[THEN iffD2])
- have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto
+ have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n"
+ using `n \<in> { 0 ..< ?N}` by auto
thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
qed auto
- finally have "\<bar> x \<bar> < ?s n" .
+ finally have "\<bar>x\<bar> < ?s n" .
from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def diff_0_right, unfolded some_eq_ex[symmetric], THEN conjunct2]
have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
- with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n`
- show "\<bar>?diff n x - f' x0 n\<bar> < ?r" by blast
+ with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n` show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
+ by blast
qed auto
- also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)
- also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto
+ also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r"
+ by (rule setsum_constant)
+ also have "\<dots> = real ?N * ?r"
+ unfolding real_eq_of_nat by auto
also have "\<dots> = r/3" by auto
finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
- have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> =
- \<bar> \<Sum>n. ?diff n x - f' x0 n \<bar>" unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric] using suminf_divide[OF diff_smbl, symmetric] by auto
- also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>" unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"] unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]] by (rule abs_triangle_ineq)
- also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto
+ have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
+ \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
+ unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric]
+ using suminf_divide[OF diff_smbl, symmetric] by auto
+ also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>"
+ unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
+ unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]]
+ by (rule abs_triangle_ineq)
+ also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
+ using abs_triangle_ineq4 by auto
also have "\<dots> < r /3 + r/3 + r/3"
using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
by (rule add_strict_mono [OF add_less_le_mono])
- finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"
+ finally have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
by auto
- } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
- norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`
- unfolding real_norm_def diff_0_right by blast
+ }
+ thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
+ norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
+ using `0 < S` unfolding real_norm_def diff_0_right by blast
qed
-lemma DERIV_power_series': fixes f :: "nat \<Rightarrow> real"
+lemma DERIV_power_series':
+ fixes f :: "nat \<Rightarrow> real"
assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
- and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
+ and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
(is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
proof -
- { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
- hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" by auto
+ {
+ fix R'
+ assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
+ hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
+ by auto
have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
proof (rule DERIV_series')
show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
proof -
- have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto
- hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" using `R' < R` by auto
- have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto
- from powser_insidea[OF converges[OF in_Rball] this] show ?thesis by auto
+ have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
+ using `0 < R'` `0 < R` `R' < R` by auto
+ hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
+ using `R' < R` by auto
+ have "norm R' < norm ((R' + R) / 2)"
+ using `0 < R'` `0 < R` `R' < R` by auto
+ from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
+ by auto
qed
- { fix n x y assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
+ {
+ fix n x y
+ assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
proof -
- have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> = (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
- unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto
+ have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
+ (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
+ unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult
+ by auto
also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
proof (rule mult_left_mono)
- have "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p = 0..<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)" by (rule setsum_abs)
+ have "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> (\<Sum>p = 0..<Suc n. \<bar>x ^ p * y ^ (n - p)\<bar>)"
+ by (rule setsum_abs)
also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
proof (rule setsum_mono)
- fix p assume "p \<in> {0..<Suc n}" hence "p \<le> n" by auto
- { fix n fix x :: real assume "x \<in> {-R'<..<R'}"
+ fix p
+ assume "p \<in> {0..<Suc n}"
+ hence "p \<le> n" by auto
+ {
+ fix n
+ fix x :: real
+ assume "x \<in> {-R'<..<R'}"
hence "\<bar>x\<bar> \<le> R'" by auto
- hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto)
- } from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
- have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" unfolding abs_mult by auto
- thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> n` by auto
+ hence "\<bar>x^n\<bar> \<le> R'^n"
+ unfolding power_abs by (rule power_mono, auto)
+ }
+ from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
+ have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
+ unfolding abs_mult by auto
+ thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
+ unfolding power_add[symmetric] using `p \<le> n` by auto
qed
- also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
- finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>" unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
- show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult[symmetric] by auto
+ also have "\<dots> = real (Suc n) * R' ^ n"
+ unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
+ finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
+ unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
+ show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
+ unfolding abs_mult[symmetric] by auto
qed
- also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult mult_assoc[symmetric] by algebra
+ also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
+ unfolding abs_mult mult_assoc[symmetric] by algebra
finally show ?thesis .
- qed }
- { fix n show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
- by (auto intro!: DERIV_intros simp del: power_Suc) }
- { fix x assume "x \<in> {-R' <..< R'}" hence "R' \<in> {-R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto
+ qed
+ }
+ {
+ fix n
+ show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
+ by (auto intro!: DERIV_intros simp del: power_Suc)
+ }
+ {
+ fix x
+ assume "x \<in> {-R' <..< R'}"
+ hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
+ using assms `R' < R` by auto
have "summable (\<lambda> n. f n * x^n)"
proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
fix n
- have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto)
- show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult
- by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right])
+ have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
+ by (rule mult_left_mono) auto
+ show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)"
+ unfolding real_norm_def abs_mult
+ by (rule mult_right_mono) (auto simp add: le[unfolded mult_1_right])
qed
from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
- show "summable (?f x)" by auto }
- show "summable (?f' x0)" using converges[OF `x0 \<in> {-R <..< R}`] .
- show "x0 \<in> {-R' <..< R'}" using `x0 \<in> {-R' <..< R'}` .
+ show "summable (?f x)" by auto
+ }
+ show "summable (?f' x0)"
+ using converges[OF `x0 \<in> {-R <..< R}`] .
+ show "x0 \<in> {-R' <..< R'}"
+ using `x0 \<in> {-R' <..< R'}` .
qed
} note for_subinterval = this
let ?R = "(R + \<bar>x0\<bar>) / 2"
@@ -798,15 +941,17 @@
also have "\<dots> \<le> x0" using False by auto
finally show ?thesis .
qed
- hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto
+ hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
+ using assms by auto
from for_subinterval[OF this]
show ?thesis .
qed
+
subsection {* Exponential Function *}
-definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" where
- "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
+definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
+ where "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
lemma summable_exp_generic:
fixes x :: "'a::{real_normed_algebra_1,banach}"
@@ -844,34 +989,34 @@
proof (rule summable_norm_comparison_test [OF exI, rule_format])
show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
by (rule summable_exp_generic)
-next
- fix n show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
+ fix n
+ show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
by (simp add: norm_power_ineq)
qed
-lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
-by (insert summable_exp_generic [where x=x], simp)
+lemma summable_exp: "summable (\<lambda>n. inverse (real (fact n)) * x ^ n)"
+ using summable_exp_generic [where x=x] by simp
lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
-unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
+ unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
lemma exp_fdiffs:
- "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
-by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
- del: mult_Suc of_nat_Suc)
+ "diffs (\<lambda>n. inverse(real (fact n))) = (\<lambda>n. inverse(real (fact n)))"
+ by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
+ del: mult_Suc of_nat_Suc)
lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
-by (simp add: diffs_def)
+ by (simp add: diffs_def)
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
-unfolding exp_def scaleR_conv_of_real
-apply (rule DERIV_cong)
-apply (rule termdiffs [where K="of_real (1 + norm x)"])
-apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
-apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
-apply (simp del: of_real_add)
-done
+ unfolding exp_def scaleR_conv_of_real
+ apply (rule DERIV_cong)
+ apply (rule termdiffs [where K="of_real (1 + norm x)"])
+ apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
+ apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
+ apply (simp del: of_real_add)
+ done
declare DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
@@ -885,12 +1030,15 @@
"(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
by (rule isCont_tendsto_compose [OF isCont_exp])
-lemma continuous_exp [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
+lemma continuous_exp [continuous_intros]:
+ "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
unfolding continuous_def by (rule tendsto_exp)
-lemma continuous_on_exp [continuous_on_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
+lemma continuous_on_exp [continuous_on_intros]:
+ "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
unfolding continuous_on_def by (auto intro: tendsto_exp)
+
subsubsection {* Properties of the Exponential Function *}
lemma powser_zero:
@@ -903,12 +1051,12 @@
qed
lemma exp_zero [simp]: "exp 0 = 1"
-unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
+ unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
lemma setsum_cl_ivl_Suc2:
"(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
-by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
- del: setsum_cl_ivl_Suc)
+ by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
+ del: setsum_cl_ivl_Suc)
lemma exp_series_add:
fixes x y :: "'a::{real_field}"
@@ -957,18 +1105,18 @@
qed
lemma exp_add: "exp (x + y) = exp x * exp y"
-unfolding exp_def
-by (simp only: Cauchy_product summable_norm_exp exp_series_add)
+ unfolding exp_def
+ by (simp only: Cauchy_product summable_norm_exp exp_series_add)
lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
-by (rule exp_add [symmetric])
+ by (rule exp_add [symmetric])
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
-unfolding exp_def
-apply (subst suminf_of_real)
-apply (rule summable_exp_generic)
-apply (simp add: scaleR_conv_of_real)
-done
+ unfolding exp_def
+ apply (subst suminf_of_real)
+ apply (rule summable_exp_generic)
+ apply (simp add: scaleR_conv_of_real)
+ done
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
proof
@@ -978,7 +1126,7 @@
qed
lemma exp_minus: "exp (- x) = inverse (exp x)"
-by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
+ by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
lemma exp_diff: "exp (x - y) = exp x / exp y"
unfolding diff_minus divide_inverse
@@ -997,31 +1145,29 @@
qed
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
-by (simp add: order_less_le)
+ by (simp add: order_less_le)
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
-by (simp add: not_less)
+ by (simp add: not_less)
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
-by (simp add: not_le)
+ by (simp add: not_le)
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
-by simp
+ by simp
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
-apply (induct "n")
-apply (auto simp add: real_of_nat_Suc distrib_left exp_add mult_commute)
-done
+ by (induct n) (auto simp add: real_of_nat_Suc distrib_left exp_add mult_commute)
text {* Strict monotonicity of exponential. *}
-lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
-apply (drule order_le_imp_less_or_eq, auto)
-apply (simp add: exp_def)
-apply (rule order_trans)
-apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
-apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
-done
+lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) \<Longrightarrow> (1 + x) \<le> exp(x)"
+ apply (drule order_le_imp_less_or_eq, auto)
+ apply (simp add: exp_def)
+ apply (rule order_trans)
+ apply (rule_tac [2] n = 2 and f = "(\<lambda>n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
+ apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
+ done
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
proof -
@@ -1034,7 +1180,8 @@
lemma exp_less_mono:
fixes x y :: real
- assumes "x < y" shows "exp x < exp y"
+ assumes "x < y"
+ shows "exp x < exp y"
proof -
from `x < y` have "0 < y - x" by simp
hence "1 < exp (y - x)" by (rule exp_gt_one)
@@ -1042,19 +1189,19 @@
thus "exp x < exp y" by simp
qed
-lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
-apply (simp add: linorder_not_le [symmetric])
-apply (auto simp add: order_le_less exp_less_mono)
-done
+lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
+ apply (simp add: linorder_not_le [symmetric])
+ apply (auto simp add: order_le_less exp_less_mono)
+ done
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
-by (auto intro: exp_less_mono exp_less_cancel)
+ by (auto intro: exp_less_mono exp_less_cancel)
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
-by (auto simp add: linorder_not_less [symmetric])
+ by (auto simp add: linorder_not_less [symmetric])
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
-by (simp add: order_eq_iff)
+ by (simp add: order_eq_iff)
text {* Comparisons of @{term "exp x"} with one. *}
@@ -1073,7 +1220,7 @@
lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
using exp_inj_iff [where x=x and y=0] by simp
-lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
+lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
proof (rule IVT)
assume "1 \<le> y"
hence "0 \<le> y - 1" by simp
@@ -1081,10 +1228,10 @@
thus "y \<le> exp (y - 1)" by simp
qed (simp_all add: le_diff_eq)
-lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
+lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y"
proof (rule linorder_le_cases [of 1 y])
- assume "1 \<le> y" thus "\<exists>x. exp x = y"
- by (fast dest: lemma_exp_total)
+ assume "1 \<le> y"
+ thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total)
next
assume "0 < y" and "y \<le> 1"
hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
@@ -1096,8 +1243,8 @@
subsection {* Natural Logarithm *}
-definition ln :: "real \<Rightarrow> real" where
- "ln x = (THE u. exp u = x)"
+definition ln :: "real \<Rightarrow> real"
+ where "ln x = (THE u. exp u = x)"
lemma ln_exp [simp]: "ln (exp x) = x"
by (simp add: ln_def)
@@ -1112,27 +1259,27 @@
by (erule subst, rule ln_exp)
lemma ln_one [simp]: "ln 1 = 0"
- by (rule ln_unique, simp)
-
-lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"
- by (rule ln_unique, simp add: exp_add)
+ by (rule ln_unique) simp
+
+lemma ln_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
+ by (rule ln_unique) (simp add: exp_add)
lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
- by (rule ln_unique, simp add: exp_minus)
-
-lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"
- by (rule ln_unique, simp add: exp_diff)
+ by (rule ln_unique) (simp add: exp_minus)
+
+lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
+ by (rule ln_unique) (simp add: exp_diff)
lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
- by (rule ln_unique, simp add: exp_real_of_nat_mult)
-
-lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
- by (subst exp_less_cancel_iff [symmetric], simp)
-
-lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
+ by (rule ln_unique) (simp add: exp_real_of_nat_mult)
+
+lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
+ by (subst exp_less_cancel_iff [symmetric]) simp
+
+lemma ln_le_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
by (simp add: linorder_not_less [symmetric])
-lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
+lemma ln_inj_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
by (simp add: order_eq_iff)
lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
@@ -1146,28 +1293,28 @@
lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
using ln_le_cancel_iff [of 1 x] by simp
-lemma ln_ge_zero_imp_ge_one: "\<lbrakk>0 \<le> ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 \<le> x"
+lemma ln_ge_zero_imp_ge_one: "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
using ln_le_cancel_iff [of 1 x] by simp
-lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> (0 \<le> ln x) = (1 \<le> x)"
+lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
using ln_le_cancel_iff [of 1 x] by simp
-lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x < 0) = (x < 1)"
+lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
using ln_less_cancel_iff [of x 1] by simp
lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
using ln_less_cancel_iff [of 1 x] by simp
-lemma ln_gt_zero_imp_gt_one: "\<lbrakk>0 < ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 < x"
+lemma ln_gt_zero_imp_gt_one: "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
using ln_less_cancel_iff [of 1 x] by simp
-lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> (0 < ln x) = (1 < x)"
+lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
using ln_less_cancel_iff [of 1 x] by simp
-lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x = 0) = (x = 1)"
+lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
using ln_inj_iff [of x 1] by simp
-lemma ln_less_zero: "\<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> ln x < 0"
+lemma ln_less_zero: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
by simp
lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
@@ -1176,7 +1323,7 @@
done
lemma tendsto_ln [tendsto_intros]:
- "\<lbrakk>(f ---> a) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
+ "(f ---> a) F \<Longrightarrow> 0 < a \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
by (rule isCont_tendsto_compose [OF isCont_ln])
lemma continuous_ln:
@@ -1201,37 +1348,51 @@
apply (simp_all add: abs_if isCont_ln)
done
-lemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x"
+lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
declare DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
-lemma ln_series: assumes "0 < x" and "x < 2"
- shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))")
+lemma ln_series:
+ assumes "0 < x" and "x < 2"
+ shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))"
+ (is "ln x = suminf (?f (x - 1))")
proof -
- let "?f' x n" = "(-1)^n * (x - 1)^n"
+ let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n"
have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
proof (rule DERIV_isconst3[where x=x])
- fix x :: real assume "x \<in> {0 <..< 2}" hence "0 < x" and "x < 2" by auto
- have "norm (1 - x) < 1" using `0 < x` and `x < 2` by auto
+ fix x :: real
+ assume "x \<in> {0 <..< 2}"
+ hence "0 < x" and "x < 2" by auto
+ have "norm (1 - x) < 1"
+ using `0 < x` and `x < 2` by auto
have "1 / x = 1 / (1 - (1 - x))" by auto
- also have "\<dots> = (\<Sum> n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
- also have "\<dots> = suminf (?f' x)" unfolding power_mult_distrib[symmetric] by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
- finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
+ also have "\<dots> = (\<Sum> n. (1 - x)^n)"
+ using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
+ also have "\<dots> = suminf (?f' x)"
+ unfolding power_mult_distrib[symmetric]
+ by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
+ finally have "DERIV ln x :> suminf (?f' x)"
+ using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
moreover
have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
- have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
+ have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :>
+ (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
proof (rule DERIV_power_series')
- show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto
- { fix x :: real assume "x \<in> {- 1<..<1}" hence "norm (-x) < 1" by auto
- show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
- unfolding One_nat_def
- by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
- }
+ show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1"
+ using `0 < x` `x < 2` by auto
+ fix x :: real
+ assume "x \<in> {- 1<..<1}"
+ hence "norm (-x) < 1" by auto
+ show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
+ unfolding One_nat_def
+ by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
qed
- hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" unfolding One_nat_def by auto
- hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos .
+ hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
+ unfolding One_nat_def by auto
+ hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
+ unfolding DERIV_iff repos .
ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
by (rule DERIV_diff)
thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
@@ -1241,7 +1402,7 @@
lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) * (x ^ (n+2)))"
proof -
- have "exp x = suminf (%n. inverse(fact n) * (x ^ n))"
+ have "exp x = suminf (\<lambda>n. inverse(fact n) * (x ^ n))"
by (simp add: exp_def)
also from summable_exp have "... = (\<Sum> n::nat = 0 ..< 2. inverse(fact n) * (x ^ n)) +
(\<Sum> n. inverse(fact(n+2)) * (x ^ (n+2)))" (is "_ = ?a + _")
@@ -1251,13 +1412,14 @@
finally show ?thesis .
qed
-lemma exp_bound: "0 <= (x::real) ==> x <= 1 ==> exp x <= 1 + x + x\<^sup>2"
+lemma exp_bound: "0 <= (x::real) \<Longrightarrow> x <= 1 \<Longrightarrow> exp x <= 1 + x + x\<^sup>2"
proof -
assume a: "0 <= x"
assume b: "x <= 1"
- { fix n :: nat
+ {
+ fix n :: nat
have "2 * 2 ^ n \<le> fact (n + 2)"
- by (induct n, simp, simp)
+ by (induct n) simp_all
hence "real ((2::nat) * 2 ^ n) \<le> real (fact (n + 2))"
by (simp only: real_of_nat_le_iff)
hence "2 * 2 ^ n \<le> real (fact (n + 2))"
@@ -1265,7 +1427,7 @@
hence "inverse (fact (n + 2)) \<le> inverse (2 * 2 ^ n)"
by (rule le_imp_inverse_le) simp
hence "inverse (fact (n + 2)) \<le> 1/2 * (1/2)^n"
- by (simp add: inverse_mult_distrib power_inverse)
+ by (simp add: power_inverse)
hence "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
by (rule mult_mono)
(rule mult_mono, simp_all add: power_le_one a b mult_nonneg_nonneg)
@@ -1276,10 +1438,10 @@
by (intro sums_mult geometric_sums, simp)
hence aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
by simp
- have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2"
+ have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2"
proof -
- have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <=
- suminf (%n. (x\<^sup>2/2) * ((1/2)^n))"
+ have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <=
+ suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
apply (rule summable_le)
apply (rule allI, rule aux1)
apply (rule summable_exp [THEN summable_ignore_initial_segment])
@@ -1291,7 +1453,7 @@
thus ?thesis unfolding exp_first_two_terms by auto
qed
-lemma ln_one_minus_pos_upper_bound: "0 <= x ==> x < 1 ==> ln (1 - x) <= - x"
+lemma ln_one_minus_pos_upper_bound: "0 <= x \<Longrightarrow> x < 1 \<Longrightarrow> ln (1 - x) <= - x"
proof -
assume a: "0 <= (x::real)" and b: "x < 1"
have "(1 - x) * (1 + x + x\<^sup>2) = (1 - x^3)"
@@ -1343,13 +1505,13 @@
apply auto
done
-lemma ln_one_plus_pos_lower_bound: "0 <= x ==> x <= 1 ==> x - x\<^sup>2 <= ln (1 + x)"
+lemma ln_one_plus_pos_lower_bound: "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> x - x\<^sup>2 <= ln (1 + x)"
proof -
assume a: "0 <= x" and b: "x <= 1"
have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
by (rule exp_diff)
also have "... <= (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
- apply (rule divide_right_mono)
+ apply (rule divide_right_mono)
apply (rule exp_bound)
apply (rule a, rule b)
apply simp
@@ -1373,7 +1535,7 @@
thus ?thesis by (auto simp only: exp_le_cancel_iff)
qed
-lemma aux5: "x < 1 ==> ln(1 - x) = - ln(1 + x / (1 - x))"
+lemma aux5: "x < 1 \<Longrightarrow> ln(1 - x) = - ln(1 + x / (1 - x))"
proof -
assume a: "x < 1"
have "ln(1 - x) = - ln(1 / (1 - x))"
@@ -1384,27 +1546,28 @@
by simp
also have "... = ln(1 / (1 - x))"
apply (rule ln_div [THEN sym])
- by (insert a, auto)
+ using a apply auto
+ done
finally show ?thesis .
qed
also have " 1 / (1 - x) = 1 + x / (1 - x)" using a by(simp add:field_simps)
finally show ?thesis .
qed
-lemma ln_one_minus_pos_lower_bound: "0 <= x ==> x <= (1 / 2) ==>
- - x - 2 * x\<^sup>2 <= ln (1 - x)"
+lemma ln_one_minus_pos_lower_bound:
+ "0 <= x \<Longrightarrow> x <= (1 / 2) \<Longrightarrow> - x - 2 * x\<^sup>2 <= ln (1 - x)"
proof -
assume a: "0 <= x" and b: "x <= (1 / 2)"
- from b have c: "x < 1"
- by auto
+ from b have c: "x < 1" by auto
then have "ln (1 - x) = - ln (1 + x / (1 - x))"
by (rule aux5)
also have "- (x / (1 - x)) <= ..."
- proof -
+ proof -
have "ln (1 + x / (1 - x)) <= x / (1 - x)"
apply (rule ln_add_one_self_le_self)
apply (rule divide_nonneg_pos)
- by (insert a c, auto)
+ using a c apply auto
+ done
thus ?thesis
by auto
qed
@@ -1414,29 +1577,29 @@
have "0 < 1 - x" using a b by simp
hence e: "-x - 2 * x\<^sup>2 <= - x / (1 - x)"
using mult_right_le_one_le[of "x*x" "2*x"] a b
- by (simp add:field_simps power2_eq_square)
+ by (simp add: field_simps power2_eq_square)
from e d show "- x - 2 * x\<^sup>2 <= ln (1 - x)"
by (rule order_trans)
qed
-lemma ln_add_one_self_le_self2: "-1 < x ==> ln(1 + x) <= x"
+lemma ln_add_one_self_le_self2: "-1 < x \<Longrightarrow> ln(1 + x) <= x"
apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
apply (subst ln_le_cancel_iff)
apply auto
-done
+ done
lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
- "0 <= x ==> x <= 1 ==> abs(ln (1 + x) - x) <= x\<^sup>2"
+ "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> abs(ln (1 + x) - x) <= x\<^sup>2"
proof -
assume x: "0 <= x"
assume x1: "x <= 1"
from x have "ln (1 + x) <= x"
by (rule ln_add_one_self_le_self)
- then have "ln (1 + x) - x <= 0"
+ then have "ln (1 + x) - x <= 0"
by simp
then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
by (rule abs_of_nonpos)
- also have "... = x - ln (1 + x)"
+ also have "... = x - ln (1 + x)"
by simp
also have "... <= x\<^sup>2"
proof -
@@ -1449,11 +1612,11 @@
qed
lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
- "-(1 / 2) <= x ==> x <= 0 ==> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
+ "-(1 / 2) <= x \<Longrightarrow> x <= 0 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
proof -
assume a: "-(1 / 2) <= x"
assume b: "x <= 0"
- have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
+ have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
apply (subst abs_of_nonpos)
apply simp
apply (rule ln_add_one_self_le_self2)
@@ -1469,16 +1632,16 @@
qed
lemma abs_ln_one_plus_x_minus_x_bound:
- "abs x <= 1 / 2 ==> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
+ "abs x <= 1 / 2 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
apply (case_tac "0 <= x")
apply (rule order_trans)
apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
apply auto
apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
apply auto
-done
-
-lemma ln_x_over_x_mono: "exp 1 <= x ==> x <= y ==> (ln y / y) <= (ln x / x)"
+ done
+
+lemma ln_x_over_x_mono: "exp 1 <= x \<Longrightarrow> x <= y \<Longrightarrow> (ln y / y) <= (ln x / x)"
proof -
assume x: "exp 1 <= x" "x <= y"
moreover have "0 < exp (1::real)" by simp
@@ -1517,14 +1680,14 @@
finally show ?thesis using b by (simp add: field_simps)
qed
-lemma ln_le_minus_one:
- "0 < x \<Longrightarrow> ln x \<le> x - 1"
+lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1"
using exp_ge_add_one_self[of "ln x"] by simp
lemma ln_eq_minus_one:
- assumes "0 < x" "ln x = x - 1" shows "x = 1"
+ assumes "0 < x" "ln x = x - 1"
+ shows "x = 1"
proof -
- let "?l y" = "ln y - y + 1"
+ let ?l = "\<lambda>y. ln y - y + 1"
have D: "\<And>x. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
by (auto intro!: DERIV_intros)
@@ -1534,7 +1697,8 @@
from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast
from `x < a` have "?l x < ?l a"
proof (rule DERIV_pos_imp_increasing, safe)
- fix y assume "x \<le> y" "y \<le> a"
+ fix y
+ assume "x \<le> y" "y \<le> a"
with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y"
by (auto simp: field_simps)
with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
@@ -1545,10 +1709,11 @@
finally show "x = 1" using assms by auto
next
assume "1 < x"
- from dense[OF `1 < x`] obtain a where "1 < a" "a < x" by blast
+ from dense[OF this] obtain a where "1 < a" "a < x" by blast
from `a < x` have "?l x < ?l a"
proof (rule DERIV_neg_imp_decreasing, safe)
- fix y assume "a \<le> y" "y \<le> x"
+ fix y
+ assume "a \<le> y" "y \<le> x"
with `1 < a` have "1 / y - 1 < 0" "0 < y"
by (auto simp: field_simps)
with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
@@ -1557,18 +1722,24 @@
also have "\<dots> \<le> 0"
using ln_le_minus_one `1 < a` by (auto simp: field_simps)
finally show "x = 1" using assms by auto
- qed simp
+ next
+ assume "x = 1"
+ then show ?thesis by simp
+ qed
qed
lemma exp_at_bot: "(exp ---> (0::real)) at_bot"
unfolding tendsto_Zfun_iff
proof (rule ZfunI, simp add: eventually_at_bot_dense)
fix r :: real assume "0 < r"
- { fix x assume "x < ln r"
+ {
+ fix x
+ assume "x < ln r"
then have "exp x < exp (ln r)"
by simp
with `0 < r` have "exp x < r"
- by simp }
+ by simp
+ }
then show "\<exists>k. \<forall>n<k. exp n < r" by auto
qed
@@ -1586,6 +1757,7 @@
lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) ---> (0::real)) at_top"
proof (induct k)
+ case 0
show "((\<lambda>x. x ^ 0 / exp x) ---> (0::real)) at_top"
by (simp add: inverse_eq_divide[symmetric])
(metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
@@ -1607,15 +1779,13 @@
qed
-definition
- powr :: "[real,real] => real" (infixr "powr" 80) where
- --{*exponentation with real exponent*}
- "x powr a = exp(a * ln x)"
-
-definition
- log :: "[real,real] => real" where
- --{*logarithm of @{term x} to base @{term a}*}
- "log a x = ln x / ln a"
+definition powr :: "[real,real] => real" (infixr "powr" 80)
+ -- {*exponentation with real exponent*}
+ where "x powr a = exp(a * ln x)"
+
+definition log :: "[real,real] => real"
+ -- {*logarithm of @{term x} to base @{term a}*}
+ where "log a x = ln x / ln a"
lemma tendsto_log [tendsto_intros]:
@@ -1623,12 +1793,20 @@
unfolding log_def by (intro tendsto_intros) auto
lemma continuous_log:
- assumes "continuous F f" and "continuous F g" and "0 < f (Lim F (\<lambda>x. x))" and "f (Lim F (\<lambda>x. x)) \<noteq> 1" and "0 < g (Lim F (\<lambda>x. x))"
+ assumes "continuous F f"
+ and "continuous F g"
+ and "0 < f (Lim F (\<lambda>x. x))"
+ and "f (Lim F (\<lambda>x. x)) \<noteq> 1"
+ and "0 < g (Lim F (\<lambda>x. x))"
shows "continuous F (\<lambda>x. log (f x) (g x))"
using assms unfolding continuous_def by (rule tendsto_log)
lemma continuous_at_within_log[continuous_intros]:
- assumes "continuous (at a within s) f" "continuous (at a within s) g" and "0 < f a" and "f a \<noteq> 1" and "0 < g a"
+ assumes "continuous (at a within s) f"
+ and "continuous (at a within s) g"
+ and "0 < f a"
+ and "f a \<noteq> 1"
+ and "0 < g a"
shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
using assms unfolding continuous_within by (rule tendsto_log)
@@ -1638,80 +1816,80 @@
using assms unfolding continuous_at by (rule tendsto_log)
lemma continuous_on_log[continuous_on_intros]:
- assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
+ assumes "continuous_on s f" "continuous_on s g"
+ and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
shows "continuous_on s (\<lambda>x. log (f x) (g x))"
using assms unfolding continuous_on_def by (fast intro: tendsto_log)
lemma powr_one_eq_one [simp]: "1 powr a = 1"
-by (simp add: powr_def)
+ by (simp add: powr_def)
lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
-by (simp add: powr_def)
+ by (simp add: powr_def)
lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
-by (simp add: powr_def)
+ by (simp add: powr_def)
declare powr_one_gt_zero_iff [THEN iffD2, simp]
-lemma powr_mult:
- "[| 0 < x; 0 < y |] ==> (x * y) powr a = (x powr a) * (y powr a)"
-by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
+lemma powr_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
+ by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
lemma powr_gt_zero [simp]: "0 < x powr a"
-by (simp add: powr_def)
+ by (simp add: powr_def)
lemma powr_ge_pzero [simp]: "0 <= x powr y"
-by (rule order_less_imp_le, rule powr_gt_zero)
+ by (rule order_less_imp_le, rule powr_gt_zero)
lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
-by (simp add: powr_def)
-
-lemma powr_divide:
- "[| 0 < x; 0 < y |] ==> (x / y) powr a = (x powr a)/(y powr a)"
-apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
-apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
-done
+ by (simp add: powr_def)
+
+lemma powr_divide: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
+ apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
+ apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
+ done
lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
apply (simp add: powr_def)
apply (subst exp_diff [THEN sym])
apply (simp add: left_diff_distrib)
-done
+ done
lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
-by (simp add: powr_def exp_add [symmetric] distrib_right)
-
-lemma powr_mult_base:
- "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
-using assms by (auto simp: powr_add)
+ by (simp add: powr_def exp_add [symmetric] distrib_right)
+
+lemma powr_mult_base: "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
+ using assms by (auto simp: powr_add)
lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
-by (simp add: powr_def)
+ by (simp add: powr_def)
lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
-by (simp add: powr_powr mult_commute)
+ by (simp add: powr_powr mult_commute)
lemma powr_minus: "x powr (-a) = inverse (x powr a)"
-by (simp add: powr_def exp_minus [symmetric])
+ by (simp add: powr_def exp_minus [symmetric])
lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
-by (simp add: divide_inverse powr_minus)
-
-lemma powr_less_mono: "[| a < b; 1 < x |] ==> x powr a < x powr b"
-by (simp add: powr_def)
-
-lemma powr_less_cancel: "[| x powr a < x powr b; 1 < x |] ==> a < b"
-by (simp add: powr_def)
-
-lemma powr_less_cancel_iff [simp]: "1 < x ==> (x powr a < x powr b) = (a < b)"
-by (blast intro: powr_less_cancel powr_less_mono)
-
-lemma powr_le_cancel_iff [simp]: "1 < x ==> (x powr a \<le> x powr b) = (a \<le> b)"
-by (simp add: linorder_not_less [symmetric])
+ by (simp add: divide_inverse powr_minus)
+
+lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
+ by (simp add: powr_def)
+
+lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
+ by (simp add: powr_def)
+
+lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a < x powr b) = (a < b)"
+ by (blast intro: powr_less_cancel powr_less_mono)
+
+lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a \<le> x powr b) = (a \<le> b)"
+ by (simp add: linorder_not_less [symmetric])
lemma log_ln: "ln x = log (exp(1)) x"
-by (simp add: log_def)
-
-lemma DERIV_log: assumes "x > 0" shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
+ by (simp add: log_def)
+
+lemma DERIV_log:
+ assumes "x > 0"
+ shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
proof -
def lb \<equiv> "1 / ln b"
moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
@@ -1722,71 +1900,74 @@
lemmas DERIV_log[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
-lemma powr_log_cancel [simp]:
- "[| 0 < a; a \<noteq> 1; 0 < x |] ==> a powr (log a x) = x"
-by (simp add: powr_def log_def)
-
-lemma log_powr_cancel [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a (a powr y) = y"
-by (simp add: log_def powr_def)
-
-lemma log_mult:
- "[| 0 < a; a \<noteq> 1; 0 < x; 0 < y |]
- ==> log a (x * y) = log a x + log a y"
-by (simp add: log_def ln_mult divide_inverse distrib_right)
-
-lemma log_eq_div_ln_mult_log:
- "[| 0 < a; a \<noteq> 1; 0 < b; b \<noteq> 1; 0 < x |]
- ==> log a x = (ln b/ln a) * log b x"
-by (simp add: log_def divide_inverse)
+lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
+ by (simp add: powr_def log_def)
+
+lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y"
+ by (simp add: log_def powr_def)
+
+lemma log_mult:
+ "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow>
+ log a (x * y) = log a x + log a y"
+ by (simp add: log_def ln_mult divide_inverse distrib_right)
+
+lemma log_eq_div_ln_mult_log:
+ "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow>
+ log a x = (ln b/ln a) * log b x"
+ by (simp add: log_def divide_inverse)
text{*Base 10 logarithms*}
-lemma log_base_10_eq1: "0 < x ==> log 10 x = (ln (exp 1) / ln 10) * ln x"
-by (simp add: log_def)
-
-lemma log_base_10_eq2: "0 < x ==> log 10 x = (log 10 (exp 1)) * ln x"
-by (simp add: log_def)
+lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x"
+ by (simp add: log_def)
+
+lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x"
+ by (simp add: log_def)
lemma log_one [simp]: "log a 1 = 0"
-by (simp add: log_def)
+ by (simp add: log_def)
lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
-by (simp add: log_def)
-
-lemma log_inverse:
- "[| 0 < a; a \<noteq> 1; 0 < x |] ==> log a (inverse x) = - log a x"
-apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
-apply (simp add: log_mult [symmetric])
-done
-
-lemma log_divide:
- "[|0 < a; a \<noteq> 1; 0 < x; 0 < y|] ==> log a (x/y) = log a x - log a y"
-by (simp add: log_mult divide_inverse log_inverse)
+ by (simp add: log_def)
+
+lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x"
+ apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
+ apply (simp add: log_mult [symmetric])
+ done
+
+lemma log_divide: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a (x/y) = log a x - log a y"
+ by (simp add: log_mult divide_inverse log_inverse)
lemma log_less_cancel_iff [simp]:
- "[| 1 < a; 0 < x; 0 < y |] ==> (log a x < log a y) = (x < y)"
-apply safe
-apply (rule_tac [2] powr_less_cancel)
-apply (drule_tac a = "log a x" in powr_less_mono, auto)
-done
-
-lemma log_inj: assumes "1 < b" shows "inj_on (log b) {0 <..}"
+ "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
+ apply safe
+ apply (rule_tac [2] powr_less_cancel)
+ apply (drule_tac a = "log a x" in powr_less_mono, auto)
+ done
+
+lemma log_inj:
+ assumes "1 < b"
+ shows "inj_on (log b) {0 <..}"
proof (rule inj_onI, simp)
- fix x y assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
+ fix x y
+ assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
show "x = y"
proof (cases rule: linorder_cases)
+ assume "x = y"
+ then show ?thesis by simp
+ next
assume "x < y" hence "log b x < log b y"
using log_less_cancel_iff[OF `1 < b`] pos by simp
- thus ?thesis using * by simp
+ then show ?thesis using * by simp
next
assume "y < x" hence "log b y < log b x"
using log_less_cancel_iff[OF `1 < b`] pos by simp
- thus ?thesis using * by simp
- qed simp
+ then show ?thesis using * by simp
+ qed
qed
lemma log_le_cancel_iff [simp]:
- "[| 1 < a; 0 < x; 0 < y |] ==> (log a x \<le> log a y) = (x \<le> y)"
-by (simp add: linorder_not_less [symmetric])
+ "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (log a x \<le> log a y) = (x \<le> y)"
+ by (simp add: linorder_not_less [symmetric])
lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
using log_less_cancel_iff[of a 1 x] by simp
@@ -1813,11 +1994,12 @@
using log_le_cancel_iff[of a x a] by simp
lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
- apply (induct n, simp)
+ apply (induct n)
+ apply simp
apply (subgoal_tac "real(Suc n) = real n + 1")
apply (erule ssubst)
apply (subst powr_add, simp, simp)
-done
+ done
lemma powr_realpow_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x^(numeral n)"
unfolding real_of_nat_numeral[symmetric] by (rule powr_realpow)
@@ -1825,16 +2007,19 @@
lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
apply (case_tac "x = 0", simp, simp)
apply (rule powr_realpow [THEN sym], simp)
-done
+ done
lemma powr_int:
assumes "x > 0"
shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
-proof cases
- assume "i < 0"
+proof (cases "i < 0")
+ case True
have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
show ?thesis using `i < 0` `x > 0` by (simp add: r field_simps powr_realpow[symmetric])
-qed (simp add: assms powr_realpow[symmetric])
+next
+ case False
+ then show ?thesis by (simp add: assms powr_realpow[symmetric])
+qed
lemma powr_numeral: "0 < x \<Longrightarrow> x powr numeral n = x^numeral n"
using powr_realpow[of x "numeral n"] by simp
@@ -1842,44 +2027,42 @@
lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr neg_numeral n = 1 / x^numeral n"
using powr_int[of x "neg_numeral n"] by simp
-lemma root_powr_inverse:
- "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
+lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
lemma ln_powr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
-by (unfold powr_def, simp)
+ unfolding powr_def by simp
lemma log_powr: "0 < x ==> 0 \<le> y ==> log b (x powr y) = y * log b x"
- apply (case_tac "y = 0")
+ apply (cases "y = 0")
apply force
apply (auto simp add: log_def ln_powr field_simps)
-done
+ done
lemma log_nat_power: "0 < x ==> log b (x^n) = real n * log b x"
apply (subst powr_realpow [symmetric])
apply (auto simp add: log_powr)
-done
+ done
lemma ln_bound: "1 <= x ==> ln x <= x"
apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
apply simp
apply (rule ln_add_one_self_le_self, simp)
-done
+ done
lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
- apply (case_tac "x = 1", simp)
- apply (case_tac "a = b", simp)
+ apply (cases "x = 1", simp)
+ apply (cases "a = b", simp)
apply (rule order_less_imp_le)
apply (rule powr_less_mono, auto)
-done
+ done
lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
apply (subst powr_zero_eq_one [THEN sym])
apply (rule powr_mono, assumption+)
-done
-
-lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a <
- y powr a"
+ done
+
+lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a < y powr a"
apply (unfold powr_def)
apply (rule exp_less_mono)
apply (rule mult_strict_left_mono)
@@ -1887,10 +2070,9 @@
apply (rule order_less_trans)
prefer 2
apply assumption+
-done
-
-lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a <
- x powr a"
+ done
+
+lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a < x powr a"
apply (unfold powr_def)
apply (rule exp_less_mono)
apply (rule mult_strict_left_mono_neg)
@@ -1899,17 +2081,16 @@
apply (rule order_less_trans)
prefer 2
apply assumption+
-done
+ done
lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
apply (case_tac "a = 0", simp)
apply (case_tac "x = y", simp)
apply (rule order_less_imp_le)
apply (rule powr_less_mono2, auto)
-done
-
-lemma powr_inj:
- "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
+ done
+
+lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
unfolding powr_def exp_inj_iff by simp
lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
@@ -1921,7 +2102,7 @@
apply (rule ln_bound)
apply (erule ge_one_powr_ge_zero)
apply (erule order_less_imp_le)
-done
+ done
lemma ln_powr_bound2:
assumes "1 < x" and "0 < a"
@@ -1962,12 +2143,16 @@
unfolding powr_def by (intro tendsto_intros)
lemma continuous_powr:
- assumes "continuous F f" and "continuous F g" and "0 < f (Lim F (\<lambda>x. x))"
+ assumes "continuous F f"
+ and "continuous F g"
+ and "0 < f (Lim F (\<lambda>x. x))"
shows "continuous F (\<lambda>x. (f x) powr (g x))"
using assms unfolding continuous_def by (rule tendsto_powr)
lemma continuous_at_within_powr[continuous_intros]:
- assumes "continuous (at a within s) f" "continuous (at a within s) g" and "0 < f a"
+ assumes "continuous (at a within s) f"
+ and "continuous (at a within s) g"
+ and "0 < f a"
shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
using assms unfolding continuous_within by (rule tendsto_powr)
@@ -1984,7 +2169,7 @@
(* FIXME: generalize by replacing d by with g x and g ---> d? *)
lemma tendsto_zero_powrI:
assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> 0) F"
- assumes "0 < d"
+ and "0 < d"
shows "((\<lambda>x. f x powr d) ---> 0) F"
proof (rule tendstoI)
fix e :: real assume "0 < e"
@@ -2002,7 +2187,8 @@
qed
lemma tendsto_neg_powr:
- assumes "s < 0" and "LIM x F. f x :> at_top"
+ assumes "s < 0"
+ and "LIM x F. f x :> at_top"
shows "((\<lambda>x. f x powr s) ---> 0) F"
proof (rule tendstoI)
fix e :: real assume "0 < e"
@@ -2026,11 +2212,11 @@
definition cos_coeff :: "nat \<Rightarrow> real" where
"cos_coeff = (\<lambda>n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"
-definition sin :: "real \<Rightarrow> real" where
- "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
-
-definition cos :: "real \<Rightarrow> real" where
- "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
+definition sin :: "real \<Rightarrow> real"
+ where "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
+
+definition cos :: "real \<Rightarrow> real"
+ where "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
unfolding sin_coeff_def by simp
@@ -2047,22 +2233,22 @@
by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)
lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
-unfolding sin_coeff_def
-apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
-apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
-done
+ unfolding sin_coeff_def
+ apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
+ apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
+ done
lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"
-unfolding cos_coeff_def
-apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
-apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
-done
+ unfolding cos_coeff_def
+ apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
+ apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
+ done
lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
-unfolding sin_def by (rule summable_sin [THEN summable_sums])
+ unfolding sin_def by (rule summable_sin [THEN summable_sums])
lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
-unfolding cos_def by (rule summable_cos [THEN summable_sums])
+ unfolding cos_def by (rule summable_cos [THEN summable_sums])
lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
@@ -2174,24 +2360,24 @@
using abs_cos_le_one [of x] unfolding abs_le_iff by simp
lemma DERIV_fun_pow: "DERIV g x :> m ==>
- DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
+ DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
by (auto intro!: DERIV_intros)
lemma DERIV_fun_exp:
- "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
+ "DERIV g x :> m ==> DERIV (\<lambda>x. exp(g x)) x :> exp(g x) * m"
by (auto intro!: DERIV_intros)
lemma DERIV_fun_sin:
- "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
+ "DERIV g x :> m ==> DERIV (\<lambda>x. sin(g x)) x :> cos(g x) * m"
by (auto intro!: DERIV_intros)
lemma DERIV_fun_cos:
- "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
+ "DERIV g x :> m ==> DERIV (\<lambda>x. cos(g x)) x :> -sin(g x) * m"
by (auto intro!: DERIV_intros)
lemma sin_cos_add_lemma:
- "(sin (x + y) - (sin x * cos y + cos x * sin y))\<^sup>2 +
- (cos (x + y) - (cos x * cos y - sin x * sin y))\<^sup>2 = 0"
+ "(sin (x + y) - (sin x * cos y + cos x * sin y))\<^sup>2 +
+ (cos (x + y) - (cos x * cos y - sin x * sin y))\<^sup>2 = 0"
(is "?f x = 0")
proof -
have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
@@ -2245,15 +2431,14 @@
subsection {* The Constant Pi *}
-definition pi :: "real" where
- "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
+definition pi :: real
+ where "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
hence define pi.*}
lemma sin_paired:
- "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1))
- sums sin x"
+ "(\<lambda>n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) sums sin x"
proof -
have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
by (rule sin_converges [THEN sums_group], simp)
@@ -2261,7 +2446,8 @@
qed
lemma sin_gt_zero:
- assumes "0 < x" and "x < 2" shows "0 < sin x"
+ assumes "0 < x" and "x < 2"
+ shows "0 < sin x"
proof -
let ?f = "\<lambda>n. \<Sum>k = n*2..<n*2+2. -1 ^ k / real (fact (2*k+1)) * x^(2*k+1)"
have pos: "\<forall>n. 0 < ?f n"
@@ -2285,13 +2471,10 @@
by (rule suminf_gt_zero)
qed
-lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
-apply (cut_tac x = x in sin_gt_zero)
-apply (auto simp add: cos_squared_eq cos_double)
-done
-
-lemma cos_paired:
- "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
+lemma cos_double_less_one: "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
+ using sin_gt_zero [where x = x] by (auto simp add: cos_squared_eq cos_double)
+
+lemma cos_paired: "(\<lambda>n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
proof -
have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
by (rule cos_converges [THEN sums_group], simp)
@@ -2301,52 +2484,51 @@
lemma real_mult_inverse_cancel:
"[|(0::real) < x; 0 < x1; x1 * y < x * u |]
==> inverse x * y < inverse x1 * u"
-apply (rule_tac c=x in mult_less_imp_less_left)
-apply (auto simp add: mult_assoc [symmetric])
-apply (simp (no_asm) add: mult_ac)
-apply (rule_tac c=x1 in mult_less_imp_less_right)
-apply (auto simp add: mult_ac)
-done
+ apply (rule_tac c=x in mult_less_imp_less_left)
+ apply (auto simp add: mult_assoc [symmetric])
+ apply (simp (no_asm) add: mult_ac)
+ apply (rule_tac c=x1 in mult_less_imp_less_right)
+ apply (auto simp add: mult_ac)
+ done
lemma real_mult_inverse_cancel2:
"[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
-apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)
-done
+ by (auto dest: real_mult_inverse_cancel simp add: mult_ac)
lemma realpow_num_eq_if:
fixes m :: "'a::power"
shows "m ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
-by (cases n, auto)
+ by (cases n) auto
lemma cos_two_less_zero [simp]: "cos (2) < 0"
-apply (cut_tac x = 2 in cos_paired)
-apply (drule sums_minus)
-apply (rule neg_less_iff_less [THEN iffD1])
-apply (frule sums_unique, auto)
-apply (rule_tac y =
- "\<Sum>n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
- in order_less_trans)
-apply (simp (no_asm) add: fact_num_eq_if_nat realpow_num_eq_if del: fact_Suc)
-apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
-apply (rule sumr_pos_lt_pair)
-apply (erule sums_summable, safe)
-unfolding One_nat_def
-apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
- del: fact_Suc)
-apply (simp add: inverse_eq_divide less_divide_eq del: fact_Suc)
-apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
-apply (simp only: real_of_nat_mult)
-apply (rule mult_strict_mono, force)
- apply (rule_tac [3] real_of_nat_ge_zero)
- prefer 2 apply force
-apply (rule real_of_nat_less_iff [THEN iffD2])
-apply (rule fact_less_mono_nat, auto)
-done
+ apply (cut_tac x = 2 in cos_paired)
+ apply (drule sums_minus)
+ apply (rule neg_less_iff_less [THEN iffD1])
+ apply (frule sums_unique, auto)
+ apply (rule_tac y =
+ "\<Sum>n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
+ in order_less_trans)
+ apply (simp (no_asm) add: fact_num_eq_if_nat realpow_num_eq_if del: fact_Suc)
+ apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
+ apply (rule sumr_pos_lt_pair)
+ apply (erule sums_summable, safe)
+ unfolding One_nat_def
+ apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
+ del: fact_Suc)
+ apply (simp add: inverse_eq_divide less_divide_eq del: fact_Suc)
+ apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
+ apply (simp only: real_of_nat_mult)
+ apply (rule mult_strict_mono, force)
+ apply (rule_tac [3] real_of_nat_ge_zero)
+ prefer 2 apply force
+ apply (rule real_of_nat_less_iff [THEN iffD2])
+ apply (rule fact_less_mono_nat, auto)
+ done
lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
-lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
+lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 \<and> cos x = 0"
proof (rule ex_ex1I)
show "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0"
by (rule IVT2, simp_all)
@@ -2367,87 +2549,84 @@
qed
lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
-by (simp add: pi_def)
+ by (simp add: pi_def)
lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
-by (simp add: pi_half cos_is_zero [THEN theI'])
+ by (simp add: pi_half cos_is_zero [THEN theI'])
lemma pi_half_gt_zero [simp]: "0 < pi / 2"
-apply (rule order_le_neq_trans)
-apply (simp add: pi_half cos_is_zero [THEN theI'])
-apply (rule notI, drule arg_cong [where f=cos], simp)
-done
+ apply (rule order_le_neq_trans)
+ apply (simp add: pi_half cos_is_zero [THEN theI'])
+ apply (rule notI, drule arg_cong [where f=cos], simp)
+ done
lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
lemma pi_half_less_two [simp]: "pi / 2 < 2"
-apply (rule order_le_neq_trans)
-apply (simp add: pi_half cos_is_zero [THEN theI'])
-apply (rule notI, drule arg_cong [where f=cos], simp)
-done
+ apply (rule order_le_neq_trans)
+ apply (simp add: pi_half cos_is_zero [THEN theI'])
+ apply (rule notI, drule arg_cong [where f=cos], simp)
+ done
lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
lemmas pi_half_le_two [simp] = pi_half_less_two [THEN order_less_imp_le]
lemma pi_gt_zero [simp]: "0 < pi"
-by (insert pi_half_gt_zero, simp)
+ using pi_half_gt_zero by simp
lemma pi_ge_zero [simp]: "0 \<le> pi"
-by (rule pi_gt_zero [THEN order_less_imp_le])
+ by (rule pi_gt_zero [THEN order_less_imp_le])
lemma pi_neq_zero [simp]: "pi \<noteq> 0"
-by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym])
+ by (rule pi_gt_zero [THEN less_imp_neq, symmetric])
lemma pi_not_less_zero [simp]: "\<not> pi < 0"
-by (simp add: linorder_not_less)
+ by (simp add: linorder_not_less)
lemma minus_pi_half_less_zero: "-(pi/2) < 0"
-by simp
+ by simp
lemma m2pi_less_pi: "- (2 * pi) < pi"
-by simp
+ by simp
lemma sin_pi_half [simp]: "sin(pi/2) = 1"
-apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
-apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
-apply (simp add: power2_eq_1_iff)
-done
+ using sin_cos_squared_add2 [where x = "pi/2"]
+ using sin_gt_zero [OF pi_half_gt_zero pi_half_less_two]
+ by (simp add: power2_eq_1_iff)
lemma cos_pi [simp]: "cos pi = -1"
-by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
+ using cos_add [where x = "pi/2" and y = "pi/2"] by simp
lemma sin_pi [simp]: "sin pi = 0"
-by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
+ using sin_add [where x = "pi/2" and y = "pi/2"] by simp
lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
-by (simp add: cos_diff)
+ by (simp add: cos_diff)
lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
-by (simp add: cos_add)
+ by (simp add: cos_add)
lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
-by (simp add: sin_diff)
+ by (simp add: sin_diff)
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
-by (simp add: sin_add)
+ by (simp add: sin_add)
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
-by (simp add: sin_add)
+ by (simp add: sin_add)
lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
-by (simp add: cos_add)
+ by (simp add: cos_add)
lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
-by (simp add: sin_add cos_double)
+ by (simp add: sin_add cos_double)
lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
-by (simp add: cos_add cos_double)
+ by (simp add: cos_add cos_double)
lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
-apply (induct "n")
-apply (auto simp add: real_of_nat_Suc distrib_right)
-done
+ by (induct n) (auto simp add: real_of_nat_Suc distrib_right)
lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
proof -
@@ -2457,58 +2636,57 @@
qed
lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
-apply (induct "n")
-apply (auto simp add: real_of_nat_Suc distrib_right)
-done
+ by (induct n) (auto simp add: real_of_nat_Suc distrib_right)
lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
-by (simp add: mult_commute [of pi])
+ by (simp add: mult_commute [of pi])
lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
-by (simp add: cos_double)
+ by (simp add: cos_double)
lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
-by simp
+ by simp
lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
-apply (rule sin_gt_zero, assumption)
-apply (rule order_less_trans, assumption)
-apply (rule pi_half_less_two)
-done
+ apply (rule sin_gt_zero, assumption)
+ apply (rule order_less_trans, assumption)
+ apply (rule pi_half_less_two)
+ done
lemma sin_less_zero:
- assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
+ assumes "- pi/2 < x" and "x < 0"
+ shows "sin x < 0"
proof -
have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
thus ?thesis by simp
qed
lemma pi_less_4: "pi < 4"
-by (cut_tac pi_half_less_two, auto)
+ using pi_half_less_two by auto
lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
-apply (cut_tac pi_less_4)
-apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
-apply (cut_tac cos_is_zero, safe)
-apply (rename_tac y z)
-apply (drule_tac x = y in spec)
-apply (drule_tac x = "pi/2" in spec, simp)
-done
+ apply (cut_tac pi_less_4)
+ apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
+ apply (cut_tac cos_is_zero, safe)
+ apply (rename_tac y z)
+ apply (drule_tac x = y in spec)
+ apply (drule_tac x = "pi/2" in spec, simp)
+ done
lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
-apply (rule_tac x = x and y = 0 in linorder_cases)
-apply (rule cos_minus [THEN subst])
-apply (rule cos_gt_zero)
-apply (auto intro: cos_gt_zero)
-done
+ apply (rule_tac x = x and y = 0 in linorder_cases)
+ apply (rule cos_minus [THEN subst])
+ apply (rule cos_gt_zero)
+ apply (auto intro: cos_gt_zero)
+ done
lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
-apply (auto simp add: order_le_less cos_gt_zero_pi)
-apply (subgoal_tac "x = pi/2", auto)
-done
+ apply (auto simp add: order_le_less cos_gt_zero_pi)
+ apply (subgoal_tac "x = pi/2", auto)
+ done
lemma sin_gt_zero_pi: "[| 0 < x; x < pi |] ==> 0 < sin x"
-by (simp add: sin_cos_eq cos_gt_zero_pi)
+ by (simp add: sin_cos_eq cos_gt_zero_pi)
lemma pi_ge_two: "2 \<le> pi"
proof (rule ccontr)
@@ -2528,7 +2706,7 @@
qed
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
-by (auto simp add: order_le_less sin_gt_zero_pi)
+ by (auto simp add: order_le_less sin_gt_zero_pi)
text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}.
It should be possible to factor out some of the common parts. *}
@@ -2636,54 +2814,74 @@
apply (auto simp add: even_mult_two_ex)
done
-lemma cos_monotone_0_pi: assumes "0 \<le> y" and "y < x" and "x \<le> pi"
+lemma cos_monotone_0_pi:
+ assumes "0 \<le> y" and "y < x" and "x \<le> pi"
shows "cos x < cos y"
proof -
have "- (x - y) < 0" using assms by auto
from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
- obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" by auto
+ obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z"
+ by auto
hence "0 < z" and "z < pi" using assms by auto
hence "0 < sin z" using sin_gt_zero_pi by auto
- hence "cos x - cos y < 0" unfolding cos_diff minus_mult_commute[symmetric] using `- (x - y) < 0` by (rule mult_pos_neg2)
+ hence "cos x - cos y < 0"
+ unfolding cos_diff minus_mult_commute[symmetric]
+ using `- (x - y) < 0` by (rule mult_pos_neg2)
thus ?thesis by auto
qed
-lemma cos_monotone_0_pi': assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" shows "cos x \<le> cos y"
+lemma cos_monotone_0_pi':
+ assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi"
+ shows "cos x \<le> cos y"
proof (cases "y < x")
- case True show ?thesis using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
+ case True
+ show ?thesis
+ using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
next
- case False hence "y = x" using `y \<le> x` by auto
+ case False
+ hence "y = x" using `y \<le> x` by auto
thus ?thesis by auto
qed
-lemma cos_monotone_minus_pi_0: assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
+lemma cos_monotone_minus_pi_0:
+ assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
shows "cos y < cos x"
proof -
- have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" using assms by auto
- from cos_monotone_0_pi[OF this]
- show ?thesis unfolding cos_minus .
+ have "0 \<le> -x" and "-x < -y" and "-y \<le> pi"
+ using assms by auto
+ from cos_monotone_0_pi[OF this] show ?thesis
+ unfolding cos_minus .
qed
-lemma cos_monotone_minus_pi_0': assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0" shows "cos y \<le> cos x"
+lemma cos_monotone_minus_pi_0':
+ assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0"
+ shows "cos y \<le> cos x"
proof (cases "y < x")
- case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] by auto
+ case True
+ show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`]
+ by auto
next
- case False hence "y = x" using `y \<le> x` by auto
+ case False
+ hence "y = x" using `y \<le> x` by auto
thus ?thesis by auto
qed
-lemma sin_monotone_2pi': assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2" shows "sin y \<le> sin x"
+lemma sin_monotone_2pi':
+ assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2"
+ shows "sin y \<le> sin x"
proof -
have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi"
using pi_ge_two and assms by auto
- from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] by auto
+ from cos_monotone_0_pi'[OF this] show ?thesis
+ unfolding minus_sin_cos_eq[symmetric] by auto
qed
+
subsection {* Tangent *}
-definition tan :: "real \<Rightarrow> real" where
- "tan = (\<lambda>x. sin x / cos x)"
+definition tan :: "real \<Rightarrow> real"
+ where "tan = (\<lambda>x. sin x / cos x)"
lemma tan_zero [simp]: "tan 0 = 0"
by (simp add: tan_def)
@@ -2719,10 +2917,11 @@
using tan_add [of x x] by (simp add: power2_eq_square)
lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
-by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
+ by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
lemma tan_less_zero:
- assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
+ assumes lb: "- pi/2 < x" and "x < 0"
+ shows "tan x < 0"
proof -
have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
thus ?thesis by simp
@@ -2763,71 +2962,74 @@
"continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))"
unfolding continuous_on_def by (auto intro: tendsto_tan)
-lemma LIM_cos_div_sin: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
+lemma LIM_cos_div_sin: "(\<lambda>x. cos(x)/sin(x)) -- pi/2 --> 0"
by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
-apply (cut_tac LIM_cos_div_sin)
-apply (simp only: LIM_eq)
-apply (drule_tac x = "inverse y" in spec, safe, force)
-apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
-apply (rule_tac x = "(pi/2) - e" in exI)
-apply (simp (no_asm_simp))
-apply (drule_tac x = "(pi/2) - e" in spec)
-apply (auto simp add: tan_def sin_diff cos_diff)
-apply (rule inverse_less_iff_less [THEN iffD1])
-apply (auto simp add: divide_inverse)
-apply (rule mult_pos_pos)
-apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
-apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
-done
+ apply (cut_tac LIM_cos_div_sin)
+ apply (simp only: LIM_eq)
+ apply (drule_tac x = "inverse y" in spec, safe, force)
+ apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
+ apply (rule_tac x = "(pi/2) - e" in exI)
+ apply (simp (no_asm_simp))
+ apply (drule_tac x = "(pi/2) - e" in spec)
+ apply (auto simp add: tan_def sin_diff cos_diff)
+ apply (rule inverse_less_iff_less [THEN iffD1])
+ apply (auto simp add: divide_inverse)
+ apply (rule mult_pos_pos)
+ apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
+ apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
+ done
lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
-apply (frule order_le_imp_less_or_eq, safe)
- prefer 2 apply force
-apply (drule lemma_tan_total, safe)
-apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
-apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
-apply (drule_tac y = xa in order_le_imp_less_or_eq)
-apply (auto dest: cos_gt_zero)
-done
+ apply (frule order_le_imp_less_or_eq, safe)
+ prefer 2 apply force
+ apply (drule lemma_tan_total, safe)
+ apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
+ apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
+ apply (drule_tac y = xa in order_le_imp_less_or_eq)
+ apply (auto dest: cos_gt_zero)
+ done
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
-apply (cut_tac linorder_linear [of 0 y], safe)
-apply (drule tan_total_pos)
-apply (cut_tac [2] y="-y" in tan_total_pos, safe)
-apply (rule_tac [3] x = "-x" in exI)
-apply (auto del: exI intro!: exI)
-done
+ apply (cut_tac linorder_linear [of 0 y], safe)
+ apply (drule tan_total_pos)
+ apply (cut_tac [2] y="-y" in tan_total_pos, safe)
+ apply (rule_tac [3] x = "-x" in exI)
+ apply (auto del: exI intro!: exI)
+ done
lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
-apply (cut_tac y = y in lemma_tan_total1, auto)
-apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
-apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
-apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
-apply (rule_tac [4] Rolle)
-apply (rule_tac [2] Rolle)
-apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
- simp add: differentiable_def)
-txt{*Now, simulate TRYALL*}
-apply (rule_tac [!] DERIV_tan asm_rl)
-apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
- simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
-done
-
-lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
+ apply (cut_tac y = y in lemma_tan_total1, auto)
+ apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
+ apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
+ apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
+ apply (rule_tac [4] Rolle)
+ apply (rule_tac [2] Rolle)
+ apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
+ simp add: differentiable_def)
+ txt{*Now, simulate TRYALL*}
+ apply (rule_tac [!] DERIV_tan asm_rl)
+ apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
+ simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
+ done
+
+lemma tan_monotone:
+ assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
shows "tan y < tan x"
proof -
- have "\<forall> x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse ((cos x')\<^sup>2)"
+ have "\<forall>x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse ((cos x')\<^sup>2)"
proof (rule allI, rule impI)
- fix x' :: real assume "y \<le> x' \<and> x' \<le> x"
+ fix x' :: real
+ assume "y \<le> x' \<and> x' \<le> x"
hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
from cos_gt_zero_pi[OF this]
have "cos x' \<noteq> 0" by auto
thus "DERIV tan x' :> inverse ((cos x')\<^sup>2)" by (rule DERIV_tan)
qed
from MVT2[OF `y < x` this]
- obtain z where "y < z" and "z < x" and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
+ obtain z where "y < z" and "z < x"
+ and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<^sup>2)" by auto
hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
hence "0 < cos z" using cos_gt_zero_pi by auto
hence inv_pos: "0 < inverse ((cos z)\<^sup>2)" by auto
@@ -2837,10 +3039,16 @@
thus ?thesis by auto
qed
-lemma tan_monotone': assumes "- (pi / 2) < y" and "y < pi / 2" and "- (pi / 2) < x" and "x < pi / 2"
+lemma tan_monotone':
+ assumes "- (pi / 2) < y"
+ and "y < pi / 2"
+ and "- (pi / 2) < x"
+ and "x < pi / 2"
shows "(y < x) = (tan y < tan x)"
proof
- assume "y < x" thus "tan y < tan x" using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
+ assume "y < x"
+ thus "tan y < tan x"
+ using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
next
assume "tan y < tan x"
show "y < x"
@@ -2857,26 +3065,37 @@
qed
qed
-lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)" unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
+lemma tan_inverse: "1 / (tan y) = tan (pi / 2 - y)"
+ unfolding tan_def sin_cos_eq[of y] cos_sin_eq[of y] by auto
lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
by (simp add: tan_def)
-lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x"
+lemma tan_periodic_nat[simp]:
+ fixes n :: nat
+ shows "tan (x + real n * pi) = tan x"
proof (induct n arbitrary: x)
+ case 0
+ then show ?case by simp
+next
case (Suc n)
- have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
+ have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi"
+ unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
show ?case unfolding split_pi_off using Suc by auto
-qed auto
+qed
lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
proof (cases "0 \<le> i")
- case True hence i_nat: "real i = real (nat i)" by auto
+ case True
+ hence i_nat: "real i = real (nat i)" by auto
show ?thesis unfolding i_nat by auto
next
- case False hence i_nat: "real i = - real (nat (-i))" by auto
- have "tan x = tan (x + real i * pi - real i * pi)" by auto
- also have "\<dots> = tan (x + real i * pi)" unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
+ case False
+ hence i_nat: "real i = - real (nat (-i))" by auto
+ have "tan x = tan (x + real i * pi - real i * pi)"
+ by auto
+ also have "\<dots> = tan (x + real i * pi)"
+ unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
finally show ?thesis by auto
qed
@@ -2885,150 +3104,144 @@
subsection {* Inverse Trigonometric Functions *}
-definition
- arcsin :: "real => real" where
- "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
-
-definition
- arccos :: "real => real" where
- "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
-
-definition
- arctan :: "real => real" where
- "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
+definition arcsin :: "real => real"
+ where "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
+
+definition arccos :: "real => real"
+ where "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
+
+definition arctan :: "real => real"
+ where "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
lemma arcsin:
- "[| -1 \<le> y; y \<le> 1 |]
- ==> -(pi/2) \<le> arcsin y &
- arcsin y \<le> pi/2 & sin(arcsin y) = y"
-unfolding arcsin_def by (rule theI' [OF sin_total])
+ "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow>
+ -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2 & sin(arcsin y) = y"
+ unfolding arcsin_def by (rule theI' [OF sin_total])
lemma arcsin_pi:
- "[| -1 \<le> y; y \<le> 1 |]
- ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
-apply (drule (1) arcsin)
-apply (force intro: order_trans)
-done
-
-lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
-by (blast dest: arcsin)
-
-lemma arcsin_bounded:
- "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
-by (blast dest: arcsin)
-
-lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
-by (blast dest: arcsin)
-
-lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
-by (blast dest: arcsin)
+ "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
+ apply (drule (1) arcsin)
+ apply (force intro: order_trans)
+ done
+
+lemma sin_arcsin [simp]: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> sin(arcsin y) = y"
+ by (blast dest: arcsin)
+
+lemma arcsin_bounded: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
+ by (blast dest: arcsin)
+
+lemma arcsin_lbound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> -(pi/2) \<le> arcsin y"
+ by (blast dest: arcsin)
+
+lemma arcsin_ubound: "-1 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> arcsin y \<le> pi/2"
+ by (blast dest: arcsin)
lemma arcsin_lt_bounded:
"[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
-apply (frule order_less_imp_le)
-apply (frule_tac y = y in order_less_imp_le)
-apply (frule arcsin_bounded)
-apply (safe, simp)
-apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
-apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
-apply (drule_tac [!] f = sin in arg_cong, auto)
-done
+ apply (frule order_less_imp_le)
+ apply (frule_tac y = y in order_less_imp_le)
+ apply (frule arcsin_bounded)
+ apply (safe, simp)
+ apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
+ apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
+ apply (drule_tac [!] f = sin in arg_cong, auto)
+ done
lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
-apply (unfold arcsin_def)
-apply (rule the1_equality)
-apply (rule sin_total, auto)
-done
+ apply (unfold arcsin_def)
+ apply (rule the1_equality)
+ apply (rule sin_total, auto)
+ done
lemma arccos:
"[| -1 \<le> y; y \<le> 1 |]
==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
-unfolding arccos_def by (rule theI' [OF cos_total])
+ unfolding arccos_def by (rule theI' [OF cos_total])
lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
-by (blast dest: arccos)
+ by (blast dest: arccos)
lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
-by (blast dest: arccos)
+ by (blast dest: arccos)
lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
-by (blast dest: arccos)
+ by (blast dest: arccos)
lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
-by (blast dest: arccos)
+ by (blast dest: arccos)
lemma arccos_lt_bounded:
"[| -1 < y; y < 1 |]
==> 0 < arccos y & arccos y < pi"
-apply (frule order_less_imp_le)
-apply (frule_tac y = y in order_less_imp_le)
-apply (frule arccos_bounded, auto)
-apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
-apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
-apply (drule_tac [!] f = cos in arg_cong, auto)
-done
+ apply (frule order_less_imp_le)
+ apply (frule_tac y = y in order_less_imp_le)
+ apply (frule arccos_bounded, auto)
+ apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
+ apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
+ apply (drule_tac [!] f = cos in arg_cong, auto)
+ done
lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
-apply (simp add: arccos_def)
-apply (auto intro!: the1_equality cos_total)
-done
+ apply (simp add: arccos_def)
+ apply (auto intro!: the1_equality cos_total)
+ done
lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
-apply (simp add: arccos_def)
-apply (auto intro!: the1_equality cos_total)
-done
+ apply (simp add: arccos_def)
+ apply (auto intro!: the1_equality cos_total)
+ done
lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<^sup>2)"
-apply (subgoal_tac "x\<^sup>2 \<le> 1")
-apply (rule power2_eq_imp_eq)
-apply (simp add: cos_squared_eq)
-apply (rule cos_ge_zero)
-apply (erule (1) arcsin_lbound)
-apply (erule (1) arcsin_ubound)
-apply simp
-apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
-apply (rule power_mono, simp, simp)
-done
+ apply (subgoal_tac "x\<^sup>2 \<le> 1")
+ apply (rule power2_eq_imp_eq)
+ apply (simp add: cos_squared_eq)
+ apply (rule cos_ge_zero)
+ apply (erule (1) arcsin_lbound)
+ apply (erule (1) arcsin_ubound)
+ apply simp
+ apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
+ apply (rule power_mono, simp, simp)
+ done
lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<^sup>2)"
-apply (subgoal_tac "x\<^sup>2 \<le> 1")
-apply (rule power2_eq_imp_eq)
-apply (simp add: sin_squared_eq)
-apply (rule sin_ge_zero)
-apply (erule (1) arccos_lbound)
-apply (erule (1) arccos_ubound)
-apply simp
-apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
-apply (rule power_mono, simp, simp)
-done
-
-lemma arctan [simp]:
- "- (pi/2) < arctan y & arctan y < pi/2 & tan (arctan y) = y"
-unfolding arctan_def by (rule theI' [OF tan_total])
-
-lemma tan_arctan: "tan(arctan y) = y"
-by auto
+ apply (subgoal_tac "x\<^sup>2 \<le> 1")
+ apply (rule power2_eq_imp_eq)
+ apply (simp add: sin_squared_eq)
+ apply (rule sin_ge_zero)
+ apply (erule (1) arccos_lbound)
+ apply (erule (1) arccos_ubound)
+ apply simp
+ apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 \<le> 1\<^sup>2", simp)
+ apply (rule power_mono, simp, simp)
+ done
+
+lemma arctan [simp]: "- (pi/2) < arctan y & arctan y < pi/2 & tan (arctan y) = y"
+ unfolding arctan_def by (rule theI' [OF tan_total])
+
+lemma tan_arctan: "tan (arctan y) = y"
+ by auto
lemma arctan_bounded: "- (pi/2) < arctan y & arctan y < pi/2"
-by (auto simp only: arctan)
+ by (auto simp only: arctan)
lemma arctan_lbound: "- (pi/2) < arctan y"
-by auto
+ by auto
lemma arctan_ubound: "arctan y < pi/2"
-by (auto simp only: arctan)
+ by (auto simp only: arctan)
lemma arctan_unique:
- assumes "-(pi/2) < x" and "x < pi/2" and "tan x = y"
+ assumes "-(pi/2) < x"
+ and "x < pi/2"
+ and "tan x = y"
shows "arctan y = x"
using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
-lemma arctan_tan:
- "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
- by (rule arctan_unique, simp_all)
+lemma arctan_tan: "-(pi/2) < x \<Longrightarrow> x < pi/2 \<Longrightarrow> arctan (tan x) = x"
+ by (rule arctan_unique) simp_all
lemma arctan_zero_zero [simp]: "arctan 0 = 0"
- by (rule arctan_unique, simp_all)
+ by (rule arctan_unique) simp_all
lemma arctan_minus: "arctan (- x) = - arctan x"
apply (rule arctan_unique)
@@ -3059,12 +3272,12 @@
by (simp add: eq_divide_eq)
lemma tan_sec: "cos x \<noteq> 0 ==> 1 + (tan x)\<^sup>2 = (inverse (cos x))\<^sup>2"
-apply (rule power_inverse [THEN subst])
-apply (rule_tac c1 = "(cos x)\<^sup>2" in real_mult_right_cancel [THEN iffD1])
-apply (auto dest: field_power_not_zero
- simp add: power_mult_distrib distrib_right power_divide tan_def
- mult_assoc power_inverse [symmetric])
-done
+ apply (rule power_inverse [THEN subst])
+ apply (rule_tac c1 = "(cos x)\<^sup>2" in real_mult_right_cancel [THEN iffD1])
+ apply (auto dest: field_power_not_zero
+ simp add: power_mult_distrib distrib_right power_divide tan_def
+ mult_assoc power_inverse [symmetric])
+ done
lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
@@ -3096,8 +3309,11 @@
by (rule continuous_on_inv) (auto intro: continuous_on_intros simp: arcsin_sin)
also have "sin ` {- pi / 2 .. pi / 2} = {-1 .. 1}"
proof safe
- fix x :: real assume "x \<in> {-1..1}" then show "x \<in> sin ` {- pi / 2..pi / 2}"
- using arcsin_lbound arcsin_ubound by (intro image_eqI[where x="arcsin x"]) auto
+ fix x :: real
+ assume "x \<in> {-1..1}"
+ then show "x \<in> sin ` {- pi / 2..pi / 2}"
+ using arcsin_lbound arcsin_ubound
+ by (intro image_eqI[where x="arcsin x"]) auto
qed simp
finally show ?thesis .
qed
@@ -3117,8 +3333,11 @@
by (rule continuous_on_inv) (auto intro: continuous_on_intros simp: arccos_cos)
also have "cos ` {0 .. pi} = {-1 .. 1}"
proof safe
- fix x :: real assume "x \<in> {-1..1}" then show "x \<in> cos ` {0..pi}"
- using arccos_lbound arccos_ubound by (intro image_eqI[where x="arccos x"]) auto
+ fix x :: real
+ assume "x \<in> {-1..1}"
+ then show "x \<in> cos ` {0..pi}"
+ using arccos_lbound arccos_ubound
+ by (intro image_eqI[where x="arccos x"]) auto
qed simp
finally show ?thesis .
qed
@@ -3133,13 +3352,13 @@
by (auto simp: continuous_on_eq_continuous_at subset_eq)
lemma isCont_arctan: "isCont arctan x"
-apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
-apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
-apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
-apply (erule (1) isCont_inverse_function2 [where f=tan])
-apply (metis arctan_tan order_le_less_trans order_less_le_trans)
-apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
-done
+ apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
+ apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
+ apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
+ apply (erule (1) isCont_inverse_function2 [where f=tan])
+ apply (metis arctan_tan order_le_less_trans order_less_le_trans)
+ apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
+ done
lemma tendsto_arctan [tendsto_intros]: "(f ---> x) F \<Longrightarrow> ((\<lambda>x. arctan (f x)) ---> arctan x) F"
by (rule isCont_tendsto_compose [OF isCont_arctan])
@@ -3149,42 +3368,42 @@
lemma continuous_on_arctan [continuous_on_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. arctan (f x))"
unfolding continuous_on_def by (auto intro: tendsto_arctan)
-
+
lemma DERIV_arcsin:
"\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<^sup>2))"
-apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
-apply (rule DERIV_cong [OF DERIV_sin])
-apply (simp add: cos_arcsin)
-apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
-apply (rule power_strict_mono, simp, simp, simp)
-apply assumption
-apply assumption
-apply simp
-apply (erule (1) isCont_arcsin)
-done
+ apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
+ apply (rule DERIV_cong [OF DERIV_sin])
+ apply (simp add: cos_arcsin)
+ apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
+ apply (rule power_strict_mono, simp, simp, simp)
+ apply assumption
+ apply assumption
+ apply simp
+ apply (erule (1) isCont_arcsin)
+ done
lemma DERIV_arccos:
"\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<^sup>2))"
-apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
-apply (rule DERIV_cong [OF DERIV_cos])
-apply (simp add: sin_arccos)
-apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
-apply (rule power_strict_mono, simp, simp, simp)
-apply assumption
-apply assumption
-apply simp
-apply (erule (1) isCont_arccos)
-done
+ apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
+ apply (rule DERIV_cong [OF DERIV_cos])
+ apply (simp add: sin_arccos)
+ apply (subgoal_tac "\<bar>x\<bar>\<^sup>2 < 1\<^sup>2", simp)
+ apply (rule power_strict_mono, simp, simp, simp)
+ apply assumption
+ apply assumption
+ apply simp
+ apply (erule (1) isCont_arccos)
+ done
lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<^sup>2)"
-apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
-apply (rule DERIV_cong [OF DERIV_tan])
-apply (rule cos_arctan_not_zero)
-apply (simp add: power_inverse tan_sec [symmetric])
-apply (subgoal_tac "0 < 1 + x\<^sup>2", simp)
-apply (simp add: add_pos_nonneg)
-apply (simp, simp, simp, rule isCont_arctan)
-done
+ apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
+ apply (rule DERIV_cong [OF DERIV_tan])
+ apply (rule cos_arctan_not_zero)
+ apply (simp add: power_inverse tan_sec [symmetric])
+ apply (subgoal_tac "0 < 1 + x\<^sup>2", simp)
+ apply (simp add: add_pos_nonneg)
+ apply (simp, simp, simp, rule isCont_arctan)
+ done
declare
DERIV_arcsin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
@@ -3203,14 +3422,16 @@
lemma tendsto_arctan_at_top: "(arctan ---> (pi/2)) at_top"
proof (rule tendstoI)
- fix e :: real assume "0 < e"
+ fix e :: real
+ assume "0 < e"
def y \<equiv> "pi/2 - min (pi/2) e"
then have y: "0 \<le> y" "y < pi/2" "pi/2 \<le> e + y"
using `0 < e` by auto
show "eventually (\<lambda>x. dist (arctan x) (pi / 2) < e) at_top"
proof (intro eventually_at_top_dense[THEN iffD2] exI allI impI)
- fix x assume "tan y < x"
+ fix x
+ assume "tan y < x"
then have "arctan (tan y) < arctan x"
by (simp add: arctan_less_iff)
with y have "y < arctan x"
@@ -3222,7 +3443,9 @@
qed
lemma tendsto_arctan_at_bot: "(arctan ---> - (pi/2)) at_bot"
- unfolding filterlim_at_bot_mirror arctan_minus by (intro tendsto_minus tendsto_arctan_at_top)
+ unfolding filterlim_at_bot_mirror arctan_minus
+ by (intro tendsto_minus tendsto_arctan_at_top)
+
subsection {* More Theorems about Sin and Cos *}
@@ -3262,28 +3485,28 @@
qed
lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
-by (simp add: sin_cos_eq cos_45)
+ by (simp add: sin_cos_eq cos_45)
lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
-by (simp add: sin_cos_eq cos_30)
+ by (simp add: sin_cos_eq cos_30)
lemma cos_60: "cos (pi / 3) = 1 / 2"
-apply (rule power2_eq_imp_eq)
-apply (simp add: cos_squared_eq sin_60 power_divide)
-apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
-done
+ apply (rule power2_eq_imp_eq)
+ apply (simp add: cos_squared_eq sin_60 power_divide)
+ apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
+ done
lemma sin_30: "sin (pi / 6) = 1 / 2"
-by (simp add: sin_cos_eq cos_60)
+ by (simp add: sin_cos_eq cos_60)
lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
-unfolding tan_def by (simp add: sin_30 cos_30)
+ unfolding tan_def by (simp add: sin_30 cos_30)
lemma tan_45: "tan (pi / 4) = 1"
-unfolding tan_def by (simp add: sin_45 cos_45)
+ unfolding tan_def by (simp add: sin_45 cos_45)
lemma tan_60: "tan (pi / 3) = sqrt 3"
-unfolding tan_def by (simp add: sin_60 cos_60)
+ unfolding tan_def by (simp add: sin_60 cos_60)
lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
proof -
@@ -3295,47 +3518,52 @@
qed
lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
-by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
+ by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
-apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
-apply (subst cos_add, simp)
-done
+ apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
+ apply (subst cos_add, simp)
+ done
lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
-by (auto simp add: mult_assoc)
+ by (auto simp add: mult_assoc)
lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
-apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
-apply (subst sin_add, simp)
-done
+ apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
+ apply (subst sin_add, simp)
+ done
lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
-by (simp only: cos_add sin_add real_of_nat_Suc distrib_right distrib_left add_divide_distrib, auto)
-
-lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
+ apply (simp only: cos_add sin_add real_of_nat_Suc distrib_right distrib_left add_divide_distrib)
+ apply auto
+ done
+
+lemma DERIV_cos_add [simp]: "DERIV (\<lambda>x. cos (x + k)) xa :> - sin (xa + k)"
by (auto intro!: DERIV_intros)
lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
-by (auto simp add: sin_zero_iff even_mult_two_ex)
+ by (auto simp add: sin_zero_iff even_mult_two_ex)
lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
-by (cut_tac x = x in sin_cos_squared_add3, auto)
+ using sin_cos_squared_add3 [where x = x] by auto
+
subsection {* Machins formula *}
lemma arctan_one: "arctan 1 = pi / 4"
by (rule arctan_unique, simp_all add: tan_45 m2pi_less_pi)
-lemma tan_total_pi4: assumes "\<bar>x\<bar> < 1"
- shows "\<exists> z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
+lemma tan_total_pi4:
+ assumes "\<bar>x\<bar> < 1"
+ shows "\<exists>z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
proof
show "- (pi / 4) < arctan x \<and> arctan x < pi / 4 \<and> tan (arctan x) = x"
unfolding arctan_one [symmetric] arctan_minus [symmetric]
unfolding arctan_less_iff using assms by auto
qed
-lemma arctan_add: assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
+lemma arctan_add:
+ assumes "\<bar>x\<bar> \<le> 1" and "\<bar>y\<bar> < 1"
shows "arctan x + arctan y = arctan ((x + y) / (1 - x * y))"
proof (rule arctan_unique [symmetric])
have "- (pi / 4) \<le> arctan x" and "- (pi / 4) < arctan y"
@@ -3369,115 +3597,162 @@
thus ?thesis unfolding arctan_one by algebra
qed
+
subsection {* Introducing the arcus tangens power series *}
-lemma monoseq_arctan_series: fixes x :: real
- assumes "\<bar>x\<bar> \<le> 1" shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
-proof (cases "x = 0") case True thus ?thesis unfolding monoseq_def One_nat_def by auto
+lemma monoseq_arctan_series:
+ fixes x :: real
+ assumes "\<bar>x\<bar> \<le> 1"
+ shows "monoseq (\<lambda> n. 1 / real (n*2+1) * x^(n*2+1))" (is "monoseq ?a")
+proof (cases "x = 0")
+ case True
+ thus ?thesis unfolding monoseq_def One_nat_def by auto
next
case False
have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
show "monoseq ?a"
proof -
- { fix n fix x :: real assume "0 \<le> x" and "x \<le> 1"
- have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le> 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
+ {
+ fix n
+ fix x :: real
+ assume "0 \<le> x" and "x \<le> 1"
+ have "1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<le>
+ 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)"
proof (rule mult_mono)
- show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))" by (rule frac_le) simp_all
- show "0 \<le> 1 / real (Suc (n * 2))" by auto
- show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)" by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`)
- show "0 \<le> x ^ Suc (Suc n * 2)" by (rule zero_le_power) (simp add: `0 \<le> x`)
+ show "1 / real (Suc (Suc n * 2)) \<le> 1 / real (Suc (n * 2))"
+ by (rule frac_le) simp_all
+ show "0 \<le> 1 / real (Suc (n * 2))"
+ by auto
+ show "x ^ Suc (Suc n * 2) \<le> x ^ Suc (n * 2)"
+ by (rule power_decreasing) (simp_all add: `0 \<le> x` `x \<le> 1`)
+ show "0 \<le> x ^ Suc (Suc n * 2)"
+ by (rule zero_le_power) (simp add: `0 \<le> x`)
qed
} note mono = this
show ?thesis
proof (cases "0 \<le> x")
case True from mono[OF this `x \<le> 1`, THEN allI]
- show ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI2)
+ show ?thesis unfolding Suc_eq_plus1[symmetric]
+ by (rule mono_SucI2)
next
- case False hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto
+ case False
+ hence "0 \<le> -x" and "-x \<le> 1" using `-1 \<le> x` by auto
from mono[OF this]
- have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge> 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using `0 \<le> -x` by auto
+ have "\<And>n. 1 / real (Suc (Suc n * 2)) * x ^ Suc (Suc n * 2) \<ge>
+ 1 / real (Suc (n * 2)) * x ^ Suc (n * 2)" using `0 \<le> -x` by auto
thus ?thesis unfolding Suc_eq_plus1[symmetric] by (rule mono_SucI1[OF allI])
qed
qed
qed
-lemma zeroseq_arctan_series: fixes x :: real
- assumes "\<bar>x\<bar> \<le> 1" shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
-proof (cases "x = 0") case True thus ?thesis unfolding One_nat_def by (auto simp add: tendsto_const)
+lemma zeroseq_arctan_series:
+ fixes x :: real
+ assumes "\<bar>x\<bar> \<le> 1"
+ shows "(\<lambda> n. 1 / real (n*2+1) * x^(n*2+1)) ----> 0" (is "?a ----> 0")
+proof (cases "x = 0")
+ case True
+ thus ?thesis
+ unfolding One_nat_def by (auto simp add: tendsto_const)
next
case False
have "norm x \<le> 1" and "x \<le> 1" and "-1 \<le> x" using assms by auto
show "?a ----> 0"
proof (cases "\<bar>x\<bar> < 1")
- case True hence "norm x < 1" by auto
+ case True
+ hence "norm x < 1" by auto
from tendsto_mult[OF LIMSEQ_inverse_real_of_nat LIMSEQ_power_zero[OF `norm x < 1`, THEN LIMSEQ_Suc]]
have "(\<lambda>n. 1 / real (n + 1) * x ^ (n + 1)) ----> 0"
unfolding inverse_eq_divide Suc_eq_plus1 by simp
then show ?thesis using pos2 by (rule LIMSEQ_linear)
next
- case False hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto
- hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x" unfolding One_nat_def by auto
+ case False
+ hence "x = -1 \<or> x = 1" using `\<bar>x\<bar> \<le> 1` by auto
+ hence n_eq: "\<And> n. x ^ (n * 2 + 1) = x"
+ unfolding One_nat_def by auto
from tendsto_mult[OF LIMSEQ_inverse_real_of_nat[THEN LIMSEQ_linear, OF pos2, unfolded inverse_eq_divide] tendsto_const[of x]]
show ?thesis unfolding n_eq Suc_eq_plus1 by auto
qed
qed
-lemma summable_arctan_series: fixes x :: real and n :: nat
- assumes "\<bar>x\<bar> \<le> 1" shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "summable (?c x)")
+lemma summable_arctan_series:
+ fixes x :: real and n :: nat
+ assumes "\<bar>x\<bar> \<le> 1"
+ shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
+ (is "summable (?c x)")
by (rule summable_Leibniz(1), rule zeroseq_arctan_series[OF assms], rule monoseq_arctan_series[OF assms])
-lemma less_one_imp_sqr_less_one: fixes x :: real assumes "\<bar>x\<bar> < 1" shows "x\<^sup>2 < 1"
+lemma less_one_imp_sqr_less_one:
+ fixes x :: real
+ assumes "\<bar>x\<bar> < 1"
+ shows "x\<^sup>2 < 1"
proof -
from mult_left_mono[OF less_imp_le[OF `\<bar>x\<bar> < 1`] abs_ge_zero[of x]]
have "\<bar>x\<^sup>2\<bar> < 1" using `\<bar>x\<bar> < 1` unfolding numeral_2_eq_2 power_Suc2 by auto
thus ?thesis using zero_le_power2 by auto
qed
-lemma DERIV_arctan_series: assumes "\<bar> x \<bar> < 1"
- shows "DERIV (\<lambda> x'. \<Sum> k. (-1)^k * (1 / real (k*2+1) * x' ^ (k*2+1))) x :> (\<Sum> k. (-1)^k * x^(k*2))" (is "DERIV ?arctan _ :> ?Int")
+lemma DERIV_arctan_series:
+ assumes "\<bar> x \<bar> < 1"
+ shows "DERIV (\<lambda> x'. \<Sum> k. (-1)^k * (1 / real (k*2+1) * x' ^ (k*2+1))) x :> (\<Sum> k. (-1)^k * x^(k*2))"
+ (is "DERIV ?arctan _ :> ?Int")
proof -
- let "?f n" = "if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
-
- { fix n :: nat assume "even n" hence "2 * (n div 2) = n" by presburger } note n_even=this
- have if_eq: "\<And> n x'. ?f n * real (Suc n) * x'^n = (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)" using n_even by auto
-
- { fix x :: real assume "\<bar>x\<bar> < 1" hence "x\<^sup>2 < 1" by (rule less_one_imp_sqr_less_one)
+ let ?f = "\<lambda>n. if even n then (-1)^(n div 2) * 1 / real (Suc n) else 0"
+
+ have n_even: "\<And>n :: nat. even n \<Longrightarrow> 2 * (n div 2) = n"
+ by presburger
+ then have if_eq: "\<And>n x'. ?f n * real (Suc n) * x'^n =
+ (if even n then (-1)^(n div 2) * x'^(2 * (n div 2)) else 0)"
+ by auto
+
+ {
+ fix x :: real
+ assume "\<bar>x\<bar> < 1"
+ hence "x\<^sup>2 < 1" by (rule less_one_imp_sqr_less_one)
have "summable (\<lambda> n. -1 ^ n * (x\<^sup>2) ^n)"
by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x\<^sup>2 < 1` order_less_imp_le[OF `x\<^sup>2 < 1`])
hence "summable (\<lambda> n. -1 ^ n * x^(2*n))" unfolding power_mult .
} note summable_Integral = this
- { fix f :: "nat \<Rightarrow> real"
- have "\<And> x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
+ {
+ fix f :: "nat \<Rightarrow> real"
+ have "\<And>x. f sums x = (\<lambda> n. if even n then f (n div 2) else 0) sums x"
proof
- fix x :: real assume "f sums x"
+ fix x :: real
+ assume "f sums x"
from sums_if[OF sums_zero this]
- show "(\<lambda> n. if even n then f (n div 2) else 0) sums x" by auto
+ show "(\<lambda>n. if even n then f (n div 2) else 0) sums x"
+ by auto
next
- fix x :: real assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
+ fix x :: real
+ assume "(\<lambda> n. if even n then f (n div 2) else 0) sums x"
from LIMSEQ_linear[OF this[unfolded sums_def] pos2, unfolded sum_split_even_odd[unfolded mult_commute]]
show "f sums x" unfolding sums_def by auto
qed
hence "op sums f = op sums (\<lambda> n. if even n then f (n div 2) else 0)" ..
} note sums_even = this
- have Int_eq: "(\<Sum> n. ?f n * real (Suc n) * x^n) = ?Int" unfolding if_eq mult_commute[of _ 2] suminf_def sums_even[of "\<lambda> n. -1 ^ n * x ^ (2 * n)", symmetric]
+ have Int_eq: "(\<Sum>n. ?f n * real (Suc n) * x^n) = ?Int"
+ unfolding if_eq mult_commute[of _ 2] suminf_def sums_even[of "\<lambda> n. -1 ^ n * x ^ (2 * n)", symmetric]
by auto
- { fix x :: real
- have if_eq': "\<And> n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
+ {
+ fix x :: real
+ have if_eq': "\<And>n. (if even n then -1 ^ (n div 2) * 1 / real (Suc n) else 0) * x ^ Suc n =
(if even n then -1 ^ (n div 2) * (1 / real (Suc (2 * (n div 2))) * x ^ Suc (2 * (n div 2))) else 0)"
using n_even by auto
- have idx_eq: "\<And> n. n * 2 + 1 = Suc (2 * n)" by auto
- have "(\<Sum> n. ?f n * x^(Suc n)) = ?arctan x" unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. -1 ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
+ have idx_eq: "\<And>n. n * 2 + 1 = Suc (2 * n)" by auto
+ have "(\<Sum>n. ?f n * x^(Suc n)) = ?arctan x"
+ unfolding if_eq' idx_eq suminf_def sums_even[of "\<lambda> n. -1 ^ n * (1 / real (Suc (2 * n)) * x ^ Suc (2 * n))", symmetric]
by auto
} note arctan_eq = this
have "DERIV (\<lambda> x. \<Sum> n. ?f n * x^(Suc n)) x :> (\<Sum> n. ?f n * real (Suc n) * x^n)"
proof (rule DERIV_power_series')
show "x \<in> {- 1 <..< 1}" using `\<bar> x \<bar> < 1` by auto
- { fix x' :: real assume x'_bounds: "x' \<in> {- 1 <..< 1}"
+ {
+ fix x' :: real
+ assume x'_bounds: "x' \<in> {- 1 <..< 1}"
hence "\<bar>x'\<bar> < 1" by auto
let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
@@ -3488,101 +3763,145 @@
thus ?thesis unfolding Int_eq arctan_eq .
qed
-lemma arctan_series: assumes "\<bar> x \<bar> \<le> 1"
- shows "arctan x = (\<Sum> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))" (is "_ = suminf (\<lambda> n. ?c x n)")
+lemma arctan_series:
+ assumes "\<bar> x \<bar> \<le> 1"
+ shows "arctan x = (\<Sum>k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
+ (is "_ = suminf (\<lambda> n. ?c x n)")
proof -
- let "?c' x n" = "(-1)^n * x^(n*2)"
-
- { fix r x :: real assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
+ let ?c' = "\<lambda>x n. (-1)^n * x^(n*2)"
+
+ {
+ fix r x :: real
+ assume "0 < r" and "r < 1" and "\<bar> x \<bar> < r"
have "\<bar>x\<bar> < 1" using `r < 1` and `\<bar>x\<bar> < r` by auto
- from DERIV_arctan_series[OF this]
- have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
+ from DERIV_arctan_series[OF this] have "DERIV (\<lambda> x. suminf (?c x)) x :> (suminf (?c' x))" .
} note DERIV_arctan_suminf = this
- { fix x :: real assume "\<bar>x\<bar> \<le> 1" note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]] }
- note arctan_series_borders = this
-
- { fix x :: real assume "\<bar>x\<bar> < 1" have "arctan x = (\<Sum> k. ?c x k)"
- proof -
- obtain r where "\<bar>x\<bar> < r" and "r < 1" using dense[OF `\<bar>x\<bar> < 1`] by blast
- hence "0 < r" and "-r < x" and "x < r" by auto
-
- have suminf_eq_arctan_bounded: "\<And> x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow> suminf (?c x) - arctan x = suminf (?c a) - arctan a"
+ {
+ fix x :: real
+ assume "\<bar>x\<bar> \<le> 1"
+ note summable_Leibniz[OF zeroseq_arctan_series[OF this] monoseq_arctan_series[OF this]]
+ } note arctan_series_borders = this
+
+ {
+ fix x :: real
+ assume "\<bar>x\<bar> < 1"
+ have "arctan x = (\<Sum>k. ?c x k)"
proof -
- fix x a b assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
- hence "\<bar>x\<bar> < r" by auto
- show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
- proof (rule DERIV_isconst2[of "a" "b"])
- show "a < b" and "a \<le> x" and "x \<le> b" using `a < b` `a \<le> x` `x \<le> b` by auto
- have "\<forall> x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
- proof (rule allI, rule impI)
- fix x assume "-r < x \<and> x < r" hence "\<bar>x\<bar> < r" by auto
- hence "\<bar>x\<bar> < 1" using `r < 1` by auto
- have "\<bar> - (x\<^sup>2) \<bar> < 1" using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
- hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))" unfolding real_norm_def[symmetric] by (rule geometric_sums)
- hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))" unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto
- hence suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)" using sums_unique unfolding inverse_eq_divide by auto
- have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))" unfolding suminf_c'_eq_geom
- by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
- from DERIV_add_minus[OF this DERIV_arctan]
- show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0" unfolding diff_minus by auto
+ obtain r where "\<bar>x\<bar> < r" and "r < 1"
+ using dense[OF `\<bar>x\<bar> < 1`] by blast
+ hence "0 < r" and "-r < x" and "x < r" by auto
+
+ have suminf_eq_arctan_bounded: "\<And>x a b. \<lbrakk> -r < a ; b < r ; a < b ; a \<le> x ; x \<le> b \<rbrakk> \<Longrightarrow>
+ suminf (?c x) - arctan x = suminf (?c a) - arctan a"
+ proof -
+ fix x a b
+ assume "-r < a" and "b < r" and "a < b" and "a \<le> x" and "x \<le> b"
+ hence "\<bar>x\<bar> < r" by auto
+ show "suminf (?c x) - arctan x = suminf (?c a) - arctan a"
+ proof (rule DERIV_isconst2[of "a" "b"])
+ show "a < b" and "a \<le> x" and "x \<le> b"
+ using `a < b` `a \<le> x` `x \<le> b` by auto
+ have "\<forall>x. -r < x \<and> x < r \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
+ proof (rule allI, rule impI)
+ fix x
+ assume "-r < x \<and> x < r"
+ hence "\<bar>x\<bar> < r" by auto
+ hence "\<bar>x\<bar> < 1" using `r < 1` by auto
+ have "\<bar> - (x\<^sup>2) \<bar> < 1"
+ using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
+ hence "(\<lambda> n. (- (x\<^sup>2)) ^ n) sums (1 / (1 - (- (x\<^sup>2))))"
+ unfolding real_norm_def[symmetric] by (rule geometric_sums)
+ hence "(?c' x) sums (1 / (1 - (- (x\<^sup>2))))"
+ unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto
+ hence suminf_c'_eq_geom: "inverse (1 + x\<^sup>2) = suminf (?c' x)"
+ using sums_unique unfolding inverse_eq_divide by auto
+ have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x\<^sup>2))"
+ unfolding suminf_c'_eq_geom
+ by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
+ from DERIV_add_minus[OF this DERIV_arctan]
+ show "DERIV (\<lambda> x. suminf (?c x) - arctan x) x :> 0"
+ unfolding diff_minus by auto
+ qed
+ hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
+ using `-r < a` `b < r` by auto
+ thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0"
+ using `\<bar>x\<bar> < r` by auto
+ show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y"
+ using DERIV_in_rball DERIV_isCont by auto
qed
- hence DERIV_in_rball: "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0" using `-r < a` `b < r` by auto
- thus "\<forall> y. a < y \<and> y < b \<longrightarrow> DERIV (\<lambda> x. suminf (?c x) - arctan x) y :> 0" using `\<bar>x\<bar> < r` by auto
- show "\<forall> y. a \<le> y \<and> y \<le> b \<longrightarrow> isCont (\<lambda> x. suminf (?c x) - arctan x) y" using DERIV_in_rball DERIV_isCont by auto
qed
+
+ have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
+ unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero
+ by auto
+
+ have "suminf (?c x) - arctan x = 0"
+ proof (cases "x = 0")
+ case True
+ thus ?thesis using suminf_arctan_zero by auto
+ next
+ case False
+ hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
+ have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
+ by (rule suminf_eq_arctan_bounded[where x="0" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>", symmetric])
+ (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
+ moreover
+ have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
+ by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"])
+ (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
+ ultimately
+ show ?thesis using suminf_arctan_zero by auto
+ qed
+ thus ?thesis by auto
qed
-
- have suminf_arctan_zero: "suminf (?c 0) - arctan 0 = 0"
- unfolding Suc_eq_plus1[symmetric] power_Suc2 mult_zero_right arctan_zero_zero suminf_zero by auto
-
- have "suminf (?c x) - arctan x = 0"
- proof (cases "x = 0")
- case True thus ?thesis using suminf_arctan_zero by auto
- next
- case False hence "0 < \<bar>x\<bar>" and "- \<bar>x\<bar> < \<bar>x\<bar>" by auto
- have "suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>) = suminf (?c 0) - arctan 0"
- by (rule suminf_eq_arctan_bounded[where x="0" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>", symmetric])
- (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
- moreover
- have "suminf (?c x) - arctan x = suminf (?c (-\<bar>x\<bar>)) - arctan (-\<bar>x\<bar>)"
- by (rule suminf_eq_arctan_bounded[where x="x" and a="-\<bar>x\<bar>" and b="\<bar>x\<bar>"])
- (simp_all only: `\<bar>x\<bar> < r` `-\<bar>x\<bar> < \<bar>x\<bar>` neg_less_iff_less)
- ultimately
- show ?thesis using suminf_arctan_zero by auto
- qed
- thus ?thesis by auto
- qed } note when_less_one = this
+ } note when_less_one = this
show "arctan x = suminf (\<lambda> n. ?c x n)"
proof (cases "\<bar>x\<bar> < 1")
- case True thus ?thesis by (rule when_less_one)
- next case False hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto
- let "?a x n" = "\<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
- let "?diff x n" = "\<bar> arctan x - (\<Sum> i = 0..<n. ?c x i)\<bar>"
- { fix n :: nat
+ case True
+ thus ?thesis by (rule when_less_one)
+ next
+ case False
+ hence "\<bar>x\<bar> = 1" using `\<bar>x\<bar> \<le> 1` by auto
+ let ?a = "\<lambda>x n. \<bar>1 / real (n*2+1) * x^(n*2+1)\<bar>"
+ let ?diff = "\<lambda> x n. \<bar> arctan x - (\<Sum> i = 0..<n. ?c x i)\<bar>"
+ {
+ fix n :: nat
have "0 < (1 :: real)" by auto
moreover
- { fix x :: real assume "0 < x" and "x < 1" hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
- from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)" by auto
+ {
+ fix x :: real
+ assume "0 < x" and "x < 1"
+ hence "\<bar>x\<bar> \<le> 1" and "\<bar>x\<bar> < 1" by auto
+ from `0 < x` have "0 < 1 / real (0 * 2 + (1::nat)) * x ^ (0 * 2 + 1)"
+ by auto
note bounds = mp[OF arctan_series_borders(2)[OF `\<bar>x\<bar> \<le> 1`] this, unfolded when_less_one[OF `\<bar>x\<bar> < 1`, symmetric], THEN spec]
- have "0 < 1 / real (n*2+1) * x^(n*2+1)" by (rule mult_pos_pos, auto simp only: zero_less_power[OF `0 < x`], auto)
- hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)" by (rule abs_of_pos)
+ have "0 < 1 / real (n*2+1) * x^(n*2+1)"
+ by (rule mult_pos_pos, auto simp only: zero_less_power[OF `0 < x`], auto)
+ hence a_pos: "?a x n = 1 / real (n*2+1) * x^(n*2+1)"
+ by (rule abs_of_pos)
have "?diff x n \<le> ?a x n"
proof (cases "even n")
- case True hence sgn_pos: "(-1)^n = (1::real)" by auto
- from `even n` obtain m where "2 * m = n" unfolding even_mult_two_ex by auto
+ case True
+ hence sgn_pos: "(-1)^n = (1::real)" by auto
+ from `even n` obtain m where "2 * m = n"
+ unfolding even_mult_two_ex by auto
from bounds[of m, unfolded this atLeastAtMost_iff]
- have "\<bar>arctan x - (\<Sum>i = 0..<n. (?c x i))\<bar> \<le> (\<Sum>i = 0..<n + 1. (?c x i)) - (\<Sum>i = 0..<n. (?c x i))" by auto
+ have "\<bar>arctan x - (\<Sum>i = 0..<n. (?c x i))\<bar> \<le> (\<Sum>i = 0..<n + 1. (?c x i)) - (\<Sum>i = 0..<n. (?c x i))"
+ by auto
also have "\<dots> = ?c x n" unfolding One_nat_def by auto
also have "\<dots> = ?a x n" unfolding sgn_pos a_pos by auto
finally show ?thesis .
next
- case False hence sgn_neg: "(-1)^n = (-1::real)" by auto
- from `odd n` obtain m where m_def: "2 * m + 1 = n" unfolding odd_Suc_mult_two_ex by auto
+ case False
+ hence sgn_neg: "(-1)^n = (-1::real)" by auto
+ from `odd n` obtain m where m_def: "2 * m + 1 = n"
+ unfolding odd_Suc_mult_two_ex by auto
hence m_plus: "2 * (m + 1) = n + 1" by auto
from bounds[of "m + 1", unfolded this atLeastAtMost_iff, THEN conjunct1] bounds[of m, unfolded m_def atLeastAtMost_iff, THEN conjunct2]
- have "\<bar>arctan x - (\<Sum>i = 0..<n. (?c x i))\<bar> \<le> (\<Sum>i = 0..<n. (?c x i)) - (\<Sum>i = 0..<n+1. (?c x i))" by auto
+ have "\<bar>arctan x - (\<Sum>i = 0..<n. (?c x i))\<bar> \<le> (\<Sum>i = 0..<n. (?c x i)) - (\<Sum>i = 0..<n+1. (?c x i))"
+ by auto
also have "\<dots> = - ?c x n" unfolding One_nat_def by auto
also have "\<dots> = ?a x n" unfolding sgn_neg a_pos by auto
finally show ?thesis .
@@ -3592,8 +3911,10 @@
hence "\<forall> x \<in> { 0 <..< 1 }. 0 \<le> ?a x n - ?diff x n" by auto
moreover have "\<And>x. isCont (\<lambda> x. ?a x n - ?diff x n) x"
unfolding diff_minus divide_inverse
- by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum)
- ultimately have "0 \<le> ?a 1 n - ?diff 1 n" by (rule LIM_less_bound)
+ by (auto intro!: isCont_add isCont_rabs isCont_ident isCont_minus isCont_arctan
+ isCont_inverse isCont_mult isCont_power isCont_const isCont_setsum)
+ ultimately have "0 \<le> ?a 1 n - ?diff 1 n"
+ by (rule LIM_less_bound)
hence "?diff 1 n \<le> ?a 1 n" by auto
}
have "?a 1 ----> 0"
@@ -3601,10 +3922,15 @@
by (auto intro!: tendsto_mult LIMSEQ_linear LIMSEQ_inverse_real_of_nat)
have "?diff 1 ----> 0"
proof (rule LIMSEQ_I)
- fix r :: real assume "0 < r"
- obtain N :: nat where N_I: "\<And> n. N \<le> n \<Longrightarrow> ?a 1 n < r" using LIMSEQ_D[OF `?a 1 ----> 0` `0 < r`] by auto
- { fix n assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this]
- have "norm (?diff 1 n - 0) < r" by auto }
+ fix r :: real
+ assume "0 < r"
+ obtain N :: nat where N_I: "\<And>n. N \<le> n \<Longrightarrow> ?a 1 n < r"
+ using LIMSEQ_D[OF `?a 1 ----> 0` `0 < r`] by auto
+ {
+ fix n
+ assume "N \<le> n" from `?diff 1 n \<le> ?a 1 n` N_I[OF this]
+ have "norm (?diff 1 n - 0) < r" by auto
+ }
thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
qed
from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
@@ -3612,62 +3938,91 @@
hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
show ?thesis
- proof (cases "x = 1", simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
- assume "x \<noteq> 1" hence "x = -1" using `\<bar>x\<bar> = 1` by auto
+ proof (cases "x = 1")
+ case True
+ then show ?thesis by (simp add: `arctan 1 = (\<Sum> i. ?c 1 i)`)
+ next
+ case False
+ hence "x = -1" using `\<bar>x\<bar> = 1` by auto
have "- (pi / 2) < 0" using pi_gt_zero by auto
have "- (2 * pi) < 0" using pi_gt_zero by auto
- have c_minus_minus: "\<And> i. ?c (- 1) i = - ?c 1 i" unfolding One_nat_def by auto
-
- have "arctan (- 1) = arctan (tan (-(pi / 4)))" unfolding tan_45 tan_minus ..
- also have "\<dots> = - (pi / 4)" by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
- also have "\<dots> = - (arctan (tan (pi / 4)))" unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero])
- also have "\<dots> = - (arctan 1)" unfolding tan_45 ..
- also have "\<dots> = - (\<Sum> i. ?c 1 i)" using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto
- also have "\<dots> = (\<Sum> i. ?c (- 1) i)" using suminf_minus[OF sums_summable[OF `(?c 1) sums (arctan 1)`]] unfolding c_minus_minus by auto
+ have c_minus_minus: "\<And>i. ?c (- 1) i = - ?c 1 i"
+ unfolding One_nat_def by auto
+
+ have "arctan (- 1) = arctan (tan (-(pi / 4)))"
+ unfolding tan_45 tan_minus ..
+ also have "\<dots> = - (pi / 4)"
+ by (rule arctan_tan, auto simp add: order_less_trans[OF `- (pi / 2) < 0` pi_gt_zero])
+ also have "\<dots> = - (arctan (tan (pi / 4)))"
+ unfolding neg_equal_iff_equal by (rule arctan_tan[symmetric], auto simp add: order_less_trans[OF `- (2 * pi) < 0` pi_gt_zero])
+ also have "\<dots> = - (arctan 1)"
+ unfolding tan_45 ..
+ also have "\<dots> = - (\<Sum> i. ?c 1 i)"
+ using `arctan 1 = (\<Sum> i. ?c 1 i)` by auto
+ also have "\<dots> = (\<Sum> i. ?c (- 1) i)"
+ using suminf_minus[OF sums_summable[OF `(?c 1) sums (arctan 1)`]]
+ unfolding c_minus_minus by auto
finally show ?thesis using `x = -1` by auto
qed
qed
qed
-lemma arctan_half: fixes x :: real
+lemma arctan_half:
+ fixes x :: real
shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x\<^sup>2)))"
proof -
- obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x" using tan_total by blast
- hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2" by auto
-
- have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> A / B = (A / C) / (B / C)" by auto
+ obtain y where low: "- (pi / 2) < y" and high: "y < pi / 2" and y_eq: "tan y = x"
+ using tan_total by blast
+ hence low2: "- (pi / 2) < y / 2" and high2: "y / 2 < pi / 2"
+ by auto
+
+ have divide_nonzero_divide: "\<And>A B C :: real. C \<noteq> 0 \<Longrightarrow> A / B = (A / C) / (B / C)"
+ by auto
have "0 < cos y" using cos_gt_zero_pi[OF low high] .
- hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y" by auto
-
- have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2" unfolding tan_def power_divide ..
- also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2" using `cos y \<noteq> 0` by auto
- also have "\<dots> = 1 / (cos y)\<^sup>2" unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
+ hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y)\<^sup>2) = cos y"
+ by auto
+
+ have "1 + (tan y)\<^sup>2 = 1 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
+ unfolding tan_def power_divide ..
+ also have "\<dots> = (cos y)\<^sup>2 / (cos y)\<^sup>2 + (sin y)\<^sup>2 / (cos y)\<^sup>2"
+ using `cos y \<noteq> 0` by auto
+ also have "\<dots> = 1 / (cos y)\<^sup>2"
+ unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
finally have "1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2" .
- have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)" unfolding tan_def divide_nonzero_divide[OF `cos y \<noteq> 0`, symmetric] ..
- also have "\<dots> = tan y / (1 + 1 / cos y)" using `cos y \<noteq> 0` unfolding add_divide_distrib by auto
- also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))" unfolding cos_sqrt ..
- also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))" unfolding real_sqrt_divide by auto
- finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))" unfolding `1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2` .
-
- have "arctan x = y" using arctan_tan low high y_eq by auto
- also have "\<dots> = 2 * (arctan (tan (y/2)))" using arctan_tan[OF low2 high2] by auto
- also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))" unfolding tan_half by auto
- finally show ?thesis unfolding eq `tan y = x` .
+ have "sin y / (cos y + 1) = tan y / ((cos y + 1) / cos y)"
+ unfolding tan_def divide_nonzero_divide[OF `cos y \<noteq> 0`, symmetric] ..
+ also have "\<dots> = tan y / (1 + 1 / cos y)"
+ using `cos y \<noteq> 0` unfolding add_divide_distrib by auto
+ also have "\<dots> = tan y / (1 + 1 / sqrt ((cos y)\<^sup>2))"
+ unfolding cos_sqrt ..
+ also have "\<dots> = tan y / (1 + sqrt (1 / (cos y)\<^sup>2))"
+ unfolding real_sqrt_divide by auto
+ finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)\<^sup>2))"
+ unfolding `1 + (tan y)\<^sup>2 = 1 / (cos y)\<^sup>2` .
+
+ have "arctan x = y"
+ using arctan_tan low high y_eq by auto
+ also have "\<dots> = 2 * (arctan (tan (y/2)))"
+ using arctan_tan[OF low2 high2] by auto
+ also have "\<dots> = 2 * (arctan (sin y / (cos y + 1)))"
+ unfolding tan_half by auto
+ finally show ?thesis
+ unfolding eq `tan y = x` .
qed
-lemma arctan_monotone: assumes "x < y"
- shows "arctan x < arctan y"
- using assms by (simp only: arctan_less_iff)
-
-lemma arctan_monotone': assumes "x \<le> y" shows "arctan x \<le> arctan y"
- using assms by (simp only: arctan_le_iff)
+lemma arctan_monotone: "x < y \<Longrightarrow> arctan x < arctan y"
+ by (simp only: arctan_less_iff)
+
+lemma arctan_monotone': "x \<le> y \<Longrightarrow> arctan x \<le> arctan y"
+ by (simp only: arctan_le_iff)
lemma arctan_inverse:
- assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
+ assumes "x \<noteq> 0"
+ shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
proof (rule arctan_unique)
show "- (pi / 2) < sgn x * pi / 2 - arctan x"
using arctan_bounded [of x] assms
@@ -3693,45 +4048,46 @@
finally show ?thesis by auto
qed
+
subsection {* Existence of Polar Coordinates *}
lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<^sup>2 + y\<^sup>2)\<bar> \<le> 1"
-apply (rule power2_le_imp_le [OF _ zero_le_one])
-apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
-done
+ apply (rule power2_le_imp_le [OF _ zero_le_one])
+ apply (simp add: power_divide divide_le_eq not_sum_power2_lt_zero)
+ done
lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
-by (simp add: abs_le_iff)
+ by (simp add: abs_le_iff)
lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<^sup>2)"
-by (simp add: sin_arccos abs_le_iff)
+ by (simp add: sin_arccos abs_le_iff)
lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
-lemma polar_ex1:
- "0 < y ==> \<exists>r a. x = r * cos a & y = r * sin a"
-apply (rule_tac x = "sqrt (x\<^sup>2 + y\<^sup>2)" in exI)
-apply (rule_tac x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))" in exI)
-apply (simp add: cos_arccos_lemma1)
-apply (simp add: sin_arccos_lemma1)
-apply (simp add: power_divide)
-apply (simp add: real_sqrt_mult [symmetric])
-apply (simp add: right_diff_distrib)
-done
-
-lemma polar_ex2:
- "y < 0 ==> \<exists>r a. x = r * cos a & y = r * sin a"
-apply (insert polar_ex1 [where x=x and y="-y"], simp, clarify)
-apply (metis cos_minus minus_minus minus_mult_right sin_minus)
-done
+lemma polar_ex1: "0 < y \<Longrightarrow> \<exists>r a. x = r * cos a & y = r * sin a"
+ apply (rule_tac x = "sqrt (x\<^sup>2 + y\<^sup>2)" in exI)
+ apply (rule_tac x = "arccos (x / sqrt (x\<^sup>2 + y\<^sup>2))" in exI)
+ apply (simp add: cos_arccos_lemma1)
+ apply (simp add: sin_arccos_lemma1)
+ apply (simp add: power_divide)
+ apply (simp add: real_sqrt_mult [symmetric])
+ apply (simp add: right_diff_distrib)
+ done
+
+lemma polar_ex2: "y < 0 ==> \<exists>r a. x = r * cos a & y = r * sin a"
+ using polar_ex1 [where x=x and y="-y"]
+ apply simp
+ apply clarify
+ apply (metis cos_minus minus_minus minus_mult_right sin_minus)
+ done
lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
-apply (rule_tac x=0 and y=y in linorder_cases)
-apply (erule polar_ex1)
-apply (rule_tac x=x in exI, rule_tac x=0 in exI, simp)
-apply (erule polar_ex2)
-done
+ apply (rule_tac x=0 and y=y in linorder_cases)
+ apply (erule polar_ex1)
+ apply (rule_tac x=x in exI, rule_tac x=0 in exI, simp)
+ apply (erule polar_ex2)
+ done
end