src/HOL/Transcendental.thy
 author huffman Mon, 05 Sep 2011 17:00:56 -0700 changeset 44728 86f43cca4886 parent 44727 d45acd50a894 child 44730 11a1290fd0ac permissions -rw-r--r--
convert lemma sin_gt_zero to Isar style; remove duplicate lemma sin_gt_zero1;
```
(*  Title:      HOL/Transcendental.thy
Author:     Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
Author:     Lawrence C Paulson
*)

theory Transcendental
imports Fact Series Deriv NthRoot
begin

subsection {* Properties of Power Series *}

lemma lemma_realpow_diff:
fixes y :: "'a::monoid_mult"
shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
proof -
assume "p \<le> n"
hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
thus ?thesis by (simp add: power_commutes)
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)) =
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)

lemma lemma_realpow_diff_sumr2:
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: right_distrib del: setsum_op_ivl_Suc)
apply (subst mult_left_commute [of "x - y"])
apply (erule subst)
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

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>"}.*}

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"
shows "summable (\<lambda>n. norm (f n * z ^ n))"
proof -
from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
by (rule summable_LIMSEQ_zero)
hence "convergent (\<lambda>n. f n * x ^ n)"
by (rule convergentI)
hence "Cauchy (\<lambda>n. f n * x ^ n)"
by (rule convergent_Cauchy)
hence "Bseq (\<lambda>n. f n * x ^ n)"
by (rule Cauchy_Bseq)
then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
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"
have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
norm (f n * x ^ n) * norm (z ^ n)"
also have "\<dots> \<le> K * norm (z ^ n)"
by (simp only: mult_right_mono 4 norm_ge_zero)
also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
by (simp only: mult_assoc)
finally show "norm (norm (f n * z ^ n)) \<le>
K * norm (z ^ n) * inverse (norm (x ^ n))"
qed
moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
proof -
from 2 have "norm (norm (z * inverse x)) < 1"
using x_neq_0
by (simp add: nonzero_norm_divide divide_inverse [symmetric])
hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
by (rule summable_geometric)
hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
by (rule summable_mult)
thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
using x_neq_0
by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
power_inverse norm_power mult_assoc)
qed
ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
by (rule summable_comparison_test)
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))"
proof (induct n)
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))"
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
finally show ?case .
qed auto

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"
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
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
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]] ..
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]]
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 ..
also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
finally show ?thesis by auto
qed
ultimately have "(norm (?SUM m - x) < r)" by auto
}
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"
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`] .
{
have "?s 0 = 0" by auto
have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto

have "?s sums y" using sums_if'[OF `f sums y`] .
from this[unfolded sums_def, THEN LIMSEQ_Suc]
have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
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 .
show ?thesis unfolding if_sum .
qed

subsection {* Alternating series test / Leibniz formula *}

lemma sums_alternating_upper_lower:
fixes a :: "nat \<Rightarrow> real"
assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
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 -
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
qed
ultimately
show ?thesis by (rule lemma_nest_unique)
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"
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)"
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"
using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast

let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n"
have "?Sa ----> l"
proof (rule LIMSEQ_I)
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
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 .
next
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 .
qed
}
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] .
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
qed

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")
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
from leibniz[OF mono]
show ?thesis using `0 \<le> a 0` by auto
next
let ?a = "\<lambda> n. - a n"
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 ?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

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 ?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
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]
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))"

text{*Lemma about distributing negation over it*}
lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"

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

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
(\<Sum>n. (diffs c)(n) * (x ^ n))"
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))))"

lemma sumr_diff_mult_const2:
"setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"

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 (cases "n", simp)
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 (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])
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"
shows "setsum f {0..<n-k} \<le> of_nat n * K"
apply (rule order_trans [OF setsum_mono])
apply (rule f, simp)
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"
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 -
have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
(z + h) ^ q * z ^ (n - 2 - q)) * norm h"
apply (subst lemma_termdiff2 [OF 1])
apply (subst norm_mult)
apply (rule mult_commute)
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)
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))
\<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
apply (intro
order_trans [OF norm_setsum]
real_setsum_nat_ivl_bounded2
mult_nonneg_nonneg
zero_le_imp_of_nat
zero_le_power K)
apply (rule le_Kn, simp)
done
qed
also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
by (simp only: mult_assoc)
finally show ?thesis .
qed

lemma lemma_termdiff4:
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"
shows "f -- 0 --> 0"
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"
show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
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
thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
next
assume K_neq_zero: "K \<noteq> 0"
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)"
next
fix x::'a
assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
by simp_all
from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
also from x4 K have "K * norm x < K * (r * inverse K / 2)"
by (rule mult_strict_left_mono)
also have "\<dots> = r / 2"
using K_neq_zero by simp
also have "r / 2 < r"
using r by simp
finally show "norm (f x) < r" .
qed
qed
qed

lemma lemma_termdiff5:
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"
hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
by simp
moreover from f have B: "summable (\<lambda>n. f n * norm h)"
by (rule summable_mult2)
ultimately have C: "summable (\<lambda>n. norm (g h n))"
by (rule summable_comparison_test)
hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
by (rule summable_norm)
also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
by (rule summable_le)
also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
by (rule suminf_mult2 [symmetric])
finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
qed

text{* FIXME: Long proofs*}

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"
shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
- of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
proof -
from dense [OF 2]
obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
from norm_ge_zero r1 have r: "0 < r"
by (rule order_le_less_trans)
hence r_neq_0: "r \<noteq> 0" by simp
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)))"
by (rule powser_insidea)
hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
using r
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))"
apply (rule ext)
apply (case_tac n, simp_all add: r_neq_0)
done
finally have "summable
(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
by (rule diffs_equiv [THEN sums_summable])
also have
"(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
r ^ (n - Suc 0)) =
(\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
apply (rule ext)
apply (case_tac "n", simp)
apply (case_tac "nat", simp)
done
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"
assume "norm h < r - norm x"
hence "norm x + norm h < r" by simp
with norm_triangle_ineq have xh: "norm (x + h) < r"
by (rule order_le_less_trans)
show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
\<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
apply (simp only: norm_mult mult_assoc)
apply (rule mult_left_mono [OF _ norm_ge_zero])
apply (simp (no_asm) add: mult_assoc [symmetric])
apply (rule lemma_termdiff3)
apply (rule h)
apply (rule r1 [THEN order_less_imp_le])
apply (rule xh [THEN order_less_imp_le])
done
qed
qed

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"
shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
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"
proof (rule LIM_equal2)
show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
next
fix h :: 'a
assume "h \<noteq> 0"
assume "norm (h - 0) < norm K - norm x"
hence "norm x + norm h < norm K" by simp
hence 5: "norm (x + h) < norm K"
by (rule norm_triangle_ineq [THEN order_le_less_trans])
have A: "summable (\<lambda>n. c n * x ^ n)"
by (rule powser_inside [OF 1 4])
have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
by (rule powser_inside [OF 1 5])
have C: "summable (\<lambda>n. diffs c n * x ^ n)"
by (rule powser_inside [OF 2 4])
show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
- (\<Sum>n. diffs c n * x ^ n) =
(\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
apply (subst sums_unique [OF diffs_equiv [OF C]])
apply (subst suminf_diff [OF B A])
apply (subst suminf_divide [symmetric])
apply (rule summable_diff [OF B A])
apply (subst suminf_diff)
apply (rule summable_divide)
apply (rule summable_diff [OF B A])
apply (rule sums_summable [OF diffs_equiv [OF C]])
apply (rule arg_cong [where f="suminf"], rule ext)
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])
qed
qed

subsection {* Derivability of power series *}

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>"
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

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

obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto

let ?N = "Suc (max N_L N_f')"
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 ?r = "r / (3 * real ?N)"
have "0 < 3 * real ?N" by auto
from divide_pos_pos[OF `0 < r` this]
have "0 < ?r" .

let "?s n" = "SOME s. 0 < s \<and> (\<forall> x. x \<noteq> 0 \<and> \<bar> x \<bar> < s \<longrightarrow> \<bar> ?diff n x - f' x0 n \<bar> < ?r)"
def S' \<equiv> "Min (?s ` { 0 ..< ?N })"

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
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)
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'`
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]
note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
note ign = summable_ignore_initial_segment[where k="?N"]
note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
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
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
} 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>)" ..
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` .
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
thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
qed auto
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
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> = 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
also have "\<dots> < r /3 + r/3 + r/3"
using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
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
qed

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"
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
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
qed
{ 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
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)
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'}"
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
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
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
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
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])
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'}` .
qed
} note for_subinterval = this
let ?R = "(R + \<bar>x0\<bar>) / 2"
have "\<bar>x0\<bar> < ?R" using assms by auto
hence "- ?R < x0"
proof (cases "x0 < 0")
case True
hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
next
case False
have "- ?R < 0" using assms by auto
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
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))"

lemma summable_exp_generic:
fixes x :: "'a::{real_normed_algebra_1,banach}"
defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
shows "summable S"
proof -
have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
unfolding S_def by (simp del: mult_Suc)
obtain r :: real where r0: "0 < r" and r1: "r < 1"
using dense [OF zero_less_one] by fast
obtain N :: nat where N: "norm x < real N * r"
using reals_Archimedean3 [OF r0] by fast
from r1 show ?thesis
proof (rule ratio_test [rule_format])
fix n :: nat
assume n: "N \<le> n"
have "norm x \<le> real N * r"
using N by (rule order_less_imp_le)
also have "real N * r \<le> real (Suc n) * r"
using r0 n by (simp add: mult_right_mono)
finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
using norm_ge_zero by (rule mult_right_mono)
hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
by (rule order_trans [OF norm_mult_ineq])
hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
thus "norm (S (Suc n)) \<le> r * norm (S n)"
qed
qed

lemma summable_norm_exp:
fixes x :: "'a::{real_normed_algebra_1,banach}"
shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
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)"
qed

lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
by (insert summable_exp_generic [where x=x], 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])

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)

lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"

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])+
done

lemma isCont_exp: "isCont exp x"
by (rule DERIV_exp [THEN DERIV_isCont])

lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
by (rule isCont_o2 [OF _ isCont_exp])

lemma tendsto_exp [tendsto_intros]:
"(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
by (rule isCont_tendsto_compose [OF isCont_exp])

subsubsection {* Properties of the Exponential Function *}

lemma powser_zero:
fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
shows "(\<Sum>n. f n * 0 ^ n) = f 0"
proof -
have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
by (rule sums_unique [OF series_zero], simp add: power_0_left)
thus ?thesis unfolding One_nat_def by simp
qed

lemma exp_zero [simp]: "exp 0 = 1"
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)))"
del: setsum_cl_ivl_Suc)

fixes x y :: "'a::{real_field}"
defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
proof (induct n)
case 0
show ?case
unfolding S_def by simp
next
case (Suc n)
have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
unfolding S_def by (simp del: mult_Suc)
hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
by simp

have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
by (simp only: times_S)
also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
by (simp only: Suc)
also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
+ y * (\<Sum>i=0..n. S x i * S y (n-i))"
by (rule left_distrib)
also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
+ (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
by (simp only: setsum_right_distrib mult_ac)
also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
+ (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
by (subst setsum_cl_ivl_Suc2, simp)
also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
(\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
by (subst setsum_cl_ivl_Suc, simp)
also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
(\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
(\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
by (simp only: scaleR_right.setsum)
finally show
"S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
by (simp del: setsum_cl_ivl_Suc)
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)

lemma mult_exp_exp: "exp x * exp y = exp (x + y)"

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)
done

lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
proof
have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
also assume "exp x = 0"
finally show "False" by simp
qed

lemma exp_minus: "exp (- x) = inverse (exp x)"
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

subsubsection {* Properties of the Exponential Function on Reals *}

text {* Comparisons of @{term "exp x"} with zero. *}

text{*Proof: because every exponential can be seen as a square.*}
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
proof -
have "0 \<le> exp (x/2) * exp (x/2)" by simp
qed

lemma exp_gt_zero [simp]: "0 < exp (x::real)"

lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"

lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"

lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
by simp

lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
apply (induct "n")
done

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 (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_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
proof -
assume x: "0 < x"
hence "1 < 1 + x" by simp
also from x have "1 + x \<le> exp x"
finally show ?thesis .
qed

lemma exp_less_mono:
fixes x y :: real
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)
hence "1 < exp y / exp x" by (simp only: exp_diff)
thus "exp x < exp y" by simp
qed

lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
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)

lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
by (auto simp add: linorder_not_less [symmetric])

lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"

text {* Comparisons of @{term "exp x"} with one. *}

lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
using exp_less_cancel_iff [where x=0 and y=x] by simp

lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
using exp_less_cancel_iff [where x=x and y=0] by simp

lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
using exp_le_cancel_iff [where x=0 and y=x] by simp

lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
using exp_le_cancel_iff [where x=x and y=0] by simp

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"
apply (rule IVT)
apply (auto intro: isCont_exp simp add: le_diff_eq)
apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)")
apply simp
done

lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
apply (rule_tac x = 1 and y = y in linorder_cases)
apply (drule order_less_imp_le [THEN lemma_exp_total])
apply (rule_tac [2] x = 0 in exI)
apply (frule_tac [3] one_less_inverse)
apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
apply (rule_tac x = "-x" in exI)
done

subsection {* Natural Logarithm *}

definition ln :: "real \<Rightarrow> real" where
"ln x = (THE u. exp u = x)"

lemma ln_exp [simp]: "ln (exp x) = x"

lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
by (auto dest: exp_total)

lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
by (metis exp_gt_zero exp_ln)

lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
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"

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)

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"

lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"

lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
apply (rule exp_le_cancel_iff [THEN iffD1])
done

lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all

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"
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)"
using ln_le_cancel_iff [of 1 x] by simp

lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x < 0) = (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"
using ln_less_cancel_iff [of 1 x] by simp

lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> (0 < ln x) = (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)"
using ln_inj_iff [of x 1] by simp

lemma ln_less_zero: "\<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> ln x < 0"
by simp

lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
apply (subgoal_tac "isCont ln (exp (ln x))", simp)
apply (rule isCont_inverse_function [where f=exp], simp_all)
done

lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
apply (erule DERIV_cong [OF DERIV_exp exp_ln])
done

lemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x"
by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)

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"

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
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
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)"
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`])
}
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 .
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
thus ?thesis by auto
qed

subsection {* Sine and Cosine *}

definition sin_coeff :: "nat \<Rightarrow> real" where
"sin_coeff = (\<lambda>n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"

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)"

lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
unfolding sin_coeff_def by simp

lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
unfolding cos_coeff_def by simp

lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
unfolding cos_coeff_def sin_coeff_def
by (simp del: mult_Suc)

lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
unfolding cos_coeff_def sin_coeff_def
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

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

lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
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])

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)

lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)

text{*Now at last we can get the derivatives of exp, sin and cos*}

lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
unfolding sin_def cos_def
apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
summable_minus summable_sin summable_cos)
done

lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
unfolding cos_def sin_def
apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus
summable_minus summable_sin summable_cos suminf_minus)
done

lemma isCont_sin: "isCont sin x"
by (rule DERIV_sin [THEN DERIV_isCont])

lemma isCont_cos: "isCont cos x"
by (rule DERIV_cos [THEN DERIV_isCont])

lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
by (rule isCont_o2 [OF _ isCont_sin])

lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
by (rule isCont_o2 [OF _ isCont_cos])

lemma tendsto_sin [tendsto_intros]:
"(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
by (rule isCont_tendsto_compose [OF isCont_sin])

lemma tendsto_cos [tendsto_intros]:
"(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
by (rule isCont_tendsto_compose [OF isCont_cos])

declare
DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]

subsection {* Properties of Sine and Cosine *}

lemma sin_zero [simp]: "sin 0 = 0"
unfolding sin_def sin_coeff_def by (simp add: powser_zero)

lemma cos_zero [simp]: "cos 0 = 1"
unfolding cos_def cos_coeff_def by (simp add: powser_zero)

lemma sin_cos_squared_add [simp]: "(sin x)\<twosuperior> + (cos x)\<twosuperior> = 1"
proof -
have "\<forall>x. DERIV (\<lambda>x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
by (auto intro!: DERIV_intros)
hence "(sin x)\<twosuperior> + (cos x)\<twosuperior> = (sin 0)\<twosuperior> + (cos 0)\<twosuperior>"
by (rule DERIV_isconst_all)
thus "(sin x)\<twosuperior> + (cos x)\<twosuperior> = 1" by simp
qed

lemma sin_cos_squared_add2 [simp]: "(cos x)\<twosuperior> + (sin x)\<twosuperior> = 1"

lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"

lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"

lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"

lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
by (rule power2_le_imp_le, simp_all add: sin_squared_eq)

lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
using abs_sin_le_one [of x] unfolding abs_le_iff by simp

lemma sin_le_one [simp]: "sin x \<le> 1"
using abs_sin_le_one [of x] unfolding abs_le_iff by simp

lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
by (rule power2_le_imp_le, simp_all add: cos_squared_eq)

lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
using abs_cos_le_one [of x] unfolding abs_le_iff by simp

lemma cos_le_one [simp]: "cos x \<le> 1"
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"
by (auto intro!: DERIV_intros)

lemma DERIV_fun_exp:
"DERIV g x :> m ==> DERIV (%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"
by (auto intro!: DERIV_intros)

lemma DERIV_fun_cos:
"DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
by (auto intro!: DERIV_intros)

"(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
(cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
(is "?f x = 0")
proof -
have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
by (auto intro!: DERIV_intros simp add: algebra_simps)
hence "?f x = ?f 0"
by (rule DERIV_isconst_all)
thus ?thesis by simp
qed

lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp

lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp

lemma sin_cos_minus_lemma:
"(sin(-x) + sin(x))\<twosuperior> + (cos(-x) - cos(x))\<twosuperior> = 0" (is "?f x = 0")
proof -
have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
by (auto intro!: DERIV_intros simp add: algebra_simps)
hence "?f x = ?f 0"
by (rule DERIV_isconst_all)
thus ?thesis by simp
qed

lemma sin_minus [simp]: "sin (-x) = -sin(x)"
using sin_cos_minus_lemma [where x=x] by simp

lemma cos_minus [simp]: "cos (-x) = cos(x)"
using sin_cos_minus_lemma [where x=x] by simp

lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"

lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"

lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"

lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"

lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
using sin_add [where x=x and y=x] by simp

lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
using cos_add [where x=x and y=x]

subsection {* The Constant Pi *}

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"
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)
thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)
qed

lemma sin_gt_zero:
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"
proof
fix n :: nat
let ?k2 = "real (Suc (Suc (4 * n)))"
let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
have "x * x < ?k2 * ?k3"
using assms by (intro mult_strict_mono', simp_all)
hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
by (intro mult_strict_right_mono zero_less_power `0 < x`)
thus "0 < ?f n"
by (simp del: mult_Suc,
simp add: less_divide_eq mult_pos_pos field_simps del: mult_Suc)
qed
have sums: "?f sums sin x"
by (rule sin_paired [THEN sums_group], simp)
show "0 < sin x"
unfolding sums_unique [OF sums]
using sums_summable [OF sums] pos
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"
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)
thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)
qed

lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
by simp

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 (rule_tac c=x1 in mult_less_imp_less_right)
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

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)

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
del: fact_Suc)
apply (rule real_mult_inverse_cancel2)
apply (simp del: fact_Suc)
apply (simp del: fact_Suc)
apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
apply (subst fact_lemma)
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"
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")
apply (rule_tac [2] IVT2)
apply (auto intro: DERIV_isCont DERIV_cos)
apply (cut_tac x = xa and y = y in linorder_less_linear)
apply (rule ccontr)
apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")
apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
apply (drule_tac f = cos in Rolle)
apply (drule_tac [5] f = cos in Rolle)
apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
apply (metis order_less_le_trans less_le sin_gt_zero)
apply (metis order_less_le_trans less_le sin_gt_zero)
done

lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"

lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
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

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

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)

lemma pi_ge_zero [simp]: "0 \<le> pi"
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])

lemma pi_not_less_zero [simp]: "\<not> pi < 0"

lemma minus_pi_half_less_zero: "-(pi/2) < 0"
by simp

lemma m2pi_less_pi: "- (2 * pi) < pi"
proof -
have "- (2 * pi) < 0" and "0 < pi" by auto
from order_less_trans[OF this] show ?thesis .
qed

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])
done

lemma cos_pi [simp]: "cos pi = -1"
by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)

lemma sin_pi [simp]: "sin pi = 0"
by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)

lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
declare sin_cos_eq [symmetric, simp]

lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
declare minus_sin_cos_eq [symmetric, simp]

lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
declare cos_sin_eq [symmetric, simp]

lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"

lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"

lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"

lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"

lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"

lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
apply (induct "n")
apply (auto simp add: real_of_nat_Suc left_distrib)
done

lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
proof -
have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
also have "... = -1 ^ n" by (rule cos_npi)
finally show ?thesis .
qed

lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
apply (induct "n")
apply (auto simp add: real_of_nat_Suc left_distrib)
done

lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
by (simp add: mult_commute [of pi])

lemma cos_two_pi [simp]: "cos (2 * pi) = 1"

lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
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

lemma sin_less_zero:
assumes lb: "- 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)

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

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

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

lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
apply (subst sin_cos_eq)
apply (rotate_tac 1)
apply (drule real_sum_of_halves [THEN ssubst])
apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
done

lemma pi_ge_two: "2 \<le> pi"
proof (rule ccontr)
assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
proof (cases "2 < 2 * pi")
case True with dense[OF `pi < 2`] show ?thesis by auto
next
case False have "pi < 2 * pi" by auto
from dense[OF this] and False show ?thesis by auto
qed
then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
hence "0 < sin y" using sin_gt_zero by auto
moreover
have "sin y < 0" using sin_gt_zero_pi[of "y - pi"] `pi < y` and `y < 2 * pi` sin_periodic_pi[of "y - pi"] by auto
ultimately show False by auto
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)

lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
apply (rule_tac [2] IVT2)
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
apply (cut_tac x = xa and y = y in linorder_less_linear)
apply (rule ccontr, auto)
apply (drule_tac f = cos in Rolle)
apply (drule_tac [5] f = cos in Rolle)
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
dest!: DERIV_cos [THEN DERIV_unique]
apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
done

lemma sin_total:
"[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
apply (rule ccontr)
apply (subgoal_tac "\<forall>x. (- (pi/2) \<le> x & x \<le> pi/2 & (sin x = y)) = (0 \<le> (x + pi/2) & (x + pi/2) \<le> pi & (cos (x + pi/2) = -y))")
apply (erule contrapos_np)
apply (simp del: minus_sin_cos_eq [symmetric])
apply (cut_tac y="-y" in cos_total, simp) apply simp
apply (erule ex1E)
apply (rule_tac a = "x - (pi/2)" in ex1I)
apply (rotate_tac 3)
apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all)
done

lemma reals_Archimedean4:
"[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
apply (auto dest!: reals_Archimedean3)
apply (drule_tac x = x in spec, clarify)
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
prefer 2 apply (erule LeastI)
apply (case_tac "LEAST m::nat. x < real m * y", simp)
apply (subgoal_tac "~ x < real nat * y")
prefer 2 apply (rule not_less_Least, simp, force)
done

(* Pre Isabelle99-2 proof was simpler- numerals arithmetic
now causes some unwanted re-arrangements of literals!   *)
lemma cos_zero_lemma:
"[| 0 \<le> x; cos x = 0 |] ==>
\<exists>n::nat. ~even n & x = real n * (pi/2)"
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
apply (subgoal_tac "0 \<le> x - real n * pi &
(x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
apply (auto simp add: algebra_simps real_of_nat_Suc)
prefer 2 apply (simp add: cos_diff)
apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
apply (rule_tac [2] cos_total, safe)
apply (drule_tac x = "x - real n * pi" in spec)
apply (drule_tac x = "pi/2" in spec)
apply (rule_tac x = "Suc (2 * n)" in exI)
apply (simp add: real_of_nat_Suc algebra_simps, auto)
done

lemma sin_zero_lemma:
"[| 0 \<le> x; sin x = 0 |] ==>
\<exists>n::nat. even n & x = real n * (pi/2)"
apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
apply (clarify, rule_tac x = "n - 1" in exI)
apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
apply (rule cos_zero_lemma)
done

lemma cos_zero_iff:
"(cos x = 0) =
((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
(\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
apply (rule iffI)
apply (cut_tac linorder_linear [of 0 x], safe)
apply (drule cos_zero_lemma, assumption+)
apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
apply (force simp add: minus_equation_iff [of x])
apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
done

(* ditto: but to a lesser extent *)
lemma sin_zero_iff:
"(sin x = 0) =
((\<exists>n::nat. even n & (x = real n * (pi/2))) |
(\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
apply (rule iffI)
apply (cut_tac linorder_linear [of 0 x], safe)
apply (drule sin_zero_lemma, assumption+)
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
apply (force simp add: minus_equation_iff [of x])
done

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
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)
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"
proof (cases "y < x")
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
thus ?thesis by auto
qed

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 .
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"
proof (cases "y < x")
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
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"
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
qed

subsection {* Tangent *}

definition tan :: "real \<Rightarrow> real" where
"tan = (\<lambda>x. sin x / cos x)"

lemma tan_zero [simp]: "tan 0 = 0"

lemma tan_pi [simp]: "tan pi = 0"

lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"

lemma tan_minus [simp]: "tan (-x) = - tan x"

lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"

"\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"

"\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"

"[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"

lemma tan_double:
"[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"

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)

lemma tan_less_zero:
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
qed

lemma tan_half: fixes x :: real assumes "- (pi / 2) < x" and "x < pi / 2"
shows "tan x = sin (2 * x) / (cos (2 * x) + 1)"
proof -
from cos_gt_zero_pi[OF `- (pi / 2) < x` `x < pi / 2`]
have "cos x \<noteq> 0" by auto

have minus_cos_2x: "\<And>X. X - cos (2*x) = X - (cos x) ^ 2 + (sin x) ^ 2" unfolding cos_double by algebra

have "tan x = (tan x + tan x) / 2" by auto
also have "\<dots> = sin (x + x) / (cos x * cos x) / 2" unfolding add_tan_eq[OF `cos x \<noteq> 0` `cos x \<noteq> 0`] ..
also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + (cos x) ^ 2 + cos (2*x) - cos (2*x))" unfolding divide_divide_eq_left numeral_2_eq_2 by auto
also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + cos (2*x) + (sin x)^2)" unfolding minus_cos_2x by auto
also have "\<dots> = sin (2 * x) / (cos (2*x) + 1)" by auto
finally show ?thesis .
qed

lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<twosuperior>)"
unfolding tan_def
by (auto intro!: DERIV_intros, simp add: divide_inverse power2_eq_square)

lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
by (rule DERIV_tan [THEN DERIV_isCont])

lemma isCont_tan' [simp]:
"\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
by (rule isCont_o2 [OF _ isCont_tan])

lemma tendsto_tan [tendsto_intros]:
"\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F"
by (rule isCont_tendsto_compose [OF isCont_tan])

lemma LIM_cos_div_sin: "(%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 (rule inverse_less_iff_less [THEN iffD1])
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

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

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
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'^2)"
proof (rule allI, rule impI)
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'^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)\<twosuperior>)" 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)\<twosuperior>)" by auto
have "0 < x - y" using `y < x` by auto
from mult_pos_pos [OF this inv_pos]
have "0 < tan x - tan y" unfolding tan_diff by auto
thus ?thesis by auto
qed

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
next
assume "tan y < tan x"
show "y < x"
proof (rule ccontr)
assume "\<not> y < x" hence "x \<le> y" by auto
hence "tan x \<le> tan y"
proof (cases "x = y")
case True thus ?thesis by auto
next
case False hence "x < y" using `x \<le> y` by auto
from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
qed
thus False using `tan y < tan x` by auto
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_periodic_pi[simp]: "tan (x + 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 (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 left_distrib by auto
show ?case unfolding split_pi_off using Suc by auto
qed auto

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
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)
finally show ?thesis by auto
qed

lemma tan_periodic_n[simp]: "tan (x + number_of n * pi) = tan x"
using tan_periodic_int[of _ "number_of n" ] unfolding real_number_of .

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)"

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])

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)

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

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

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])

lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
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)

lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
by (blast dest: arccos)

lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
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

lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
apply (auto intro!: the1_equality cos_total)
done

lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
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\<twosuperior>)"
apply (subgoal_tac "x\<twosuperior> \<le> 1")
apply (rule power2_eq_imp_eq)
apply (rule cos_ge_zero)
apply (erule (1) arcsin_lbound)
apply (erule (1) arcsin_ubound)
apply simp
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", 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\<twosuperior>)"
apply (subgoal_tac "x\<twosuperior> \<le> 1")
apply (rule power2_eq_imp_eq)
apply (rule sin_ge_zero)
apply (erule (1) arccos_lbound)
apply (erule (1) arccos_ubound)
apply simp
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", 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)

lemma arctan_lbound: "- (pi/2) < arctan y"
by auto

lemma arctan_ubound: "arctan y < pi/2"
by (auto simp only: arctan)

lemma arctan_tan:
"[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
apply (unfold arctan_def)
apply (rule the1_equality)
apply (rule tan_total, auto)
done

lemma arctan_zero_zero [simp]: "arctan 0 = 0"
by (insert arctan_tan [of 0], simp)

lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
by (intro less_imp_neq [symmetric] cos_gt_zero_pi
arctan_lbound arctan_ubound)

lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<twosuperior>)"
proof (rule power2_eq_imp_eq)
show "0 \<le> 1 / sqrt (1 + x\<twosuperior>)" by simp
show "0 \<le> cos (arctan x)"
by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
have "(cos (arctan x))\<twosuperior> * (1 + (tan (arctan x))\<twosuperior>) = 1"
unfolding tan_def by (simp add: right_distrib power_divide)
thus "(cos (arctan x))\<twosuperior> = (1 / sqrt (1 + x\<twosuperior>))\<twosuperior>"
using `0 < 1 + x\<twosuperior>` by (simp add: power_divide eq_divide_eq)
qed

lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<twosuperior>)"
using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
using tan_arctan [of x] unfolding tan_def cos_arctan

lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
apply (rule power_inverse [THEN subst])
apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
apply (auto dest: field_power_not_zero
simp add: power_mult_distrib left_distrib power_divide tan_def
mult_assoc power_inverse [symmetric])
done

lemma isCont_inverse_function2:
fixes f g :: "real \<Rightarrow> real" shows
"\<lbrakk>a < x; x < b;
\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
\<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
\<Longrightarrow> isCont g (f x)"
apply (rule isCont_inverse_function
[where f=f and d="min (x - a) (b - x)"])
done

lemma isCont_arcsin: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arcsin x"
apply (subgoal_tac "isCont arcsin (sin (arcsin x))", simp)
apply (rule isCont_inverse_function2 [where f=sin])
apply (erule (1) arcsin_lt_bounded [THEN conjunct1])
apply (erule (1) arcsin_lt_bounded [THEN conjunct2])
apply (fast intro: arcsin_sin, simp)
done

lemma isCont_arccos: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arccos x"
apply (subgoal_tac "isCont arccos (cos (arccos x))", simp)
apply (rule isCont_inverse_function2 [where f=cos])
apply (erule (1) arccos_lt_bounded [THEN conjunct1])
apply (erule (1) arccos_lt_bounded [THEN conjunct2])
apply (fast intro: arccos_cos, simp)
done

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

lemma DERIV_arcsin:
"\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<twosuperior>))"
apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
apply (rule DERIV_cong [OF DERIV_sin])
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", 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\<twosuperior>))"
apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
apply (rule DERIV_cong [OF DERIV_cos])
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", 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\<twosuperior>)"
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\<twosuperior>", simp)
apply (simp, simp, simp, rule isCont_arctan)
done

declare
DERIV_arcsin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
DERIV_arccos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
DERIV_arctan[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]

subsection {* More Theorems about Sin and Cos *}

lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
proof -
let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
have nonneg: "0 \<le> ?c"
by (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
have "0 = cos (pi / 4 + pi / 4)"
by simp
also have "cos (pi / 4 + pi / 4) = ?c\<twosuperior> - ?s\<twosuperior>"
also have "\<dots> = 2 * ?c\<twosuperior> - 1"
finally have "?c\<twosuperior> = (sqrt 2 / 2)\<twosuperior>"
thus ?thesis
using nonneg by (rule power2_eq_imp_eq) simp
qed

lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"
proof -
let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
have pos_c: "0 < ?c"
by (rule cos_gt_zero, simp, simp)
have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
by simp
also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
also have "\<dots> = ?c * (?c\<twosuperior> - 3 * ?s\<twosuperior>)"
finally have "?c\<twosuperior> = (sqrt 3 / 2)\<twosuperior>"
using pos_c by (simp add: sin_squared_eq power_divide)
thus ?thesis
using pos_c [THEN order_less_imp_le]
by (rule power2_eq_imp_eq) simp
qed

lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
proof -
have "sin (pi / 4) = cos (pi / 2 - pi / 4)" by (rule sin_cos_eq)
also have "pi / 2 - pi / 4 = pi / 4" by simp
also have "cos (pi / 4) = sqrt 2 / 2" by (rule cos_45)
finally show ?thesis .
qed

lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
proof -
have "sin (pi / 3) = cos (pi / 2 - pi / 3)" by (rule sin_cos_eq)
also have "pi / 2 - pi / 3 = pi / 6" by simp
also have "cos (pi / 6) = sqrt 3 / 2" by (rule cos_30)
finally show ?thesis .
qed

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

lemma sin_30: "sin (pi / 6) = 1 / 2"
proof -
have "sin (pi / 6) = cos (pi / 2 - pi / 6)" by (rule sin_cos_eq)
also have "pi / 2 - pi / 6 = pi / 3" by simp
also have "cos (pi / 3) = 1 / 2" by (rule cos_60)
finally show ?thesis .
qed

lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
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)

lemma tan_60: "tan (pi / 3) = sqrt 3"
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 -
have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
thus ?thesis
mult_commute [of pi])
qed

lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"

lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
done

lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"

lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
done

lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"

lemma DERIV_cos_add [simp]: "DERIV (%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)

lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
by (cut_tac x = x in sin_cos_squared_add3, auto)

subsection {* Machins formula *}

lemma tan_total_pi4: assumes "\<bar>x\<bar> < 1"
shows "\<exists> z. - (pi / 4) < z \<and> z < pi / 4 \<and> tan z = x"
proof -
obtain z where "- (pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast
have "tan (pi / 4) = 1" and "tan (- (pi / 4)) = - 1" using tan_45 tan_minus by auto
have "z \<noteq> pi / 4"
proof (rule ccontr)
assume "\<not> (z \<noteq> pi / 4)" hence "z = pi / 4" by auto
have "tan z = 1" unfolding `z = pi / 4` `tan (pi / 4) = 1` ..
thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto
qed
have "z \<noteq> - (pi / 4)"
proof (rule ccontr)
assume "\<not> (z \<noteq> - (pi / 4))" hence "z = - (pi / 4)" by auto
have "tan z = - 1" unfolding `z = - (pi / 4)` `tan (- (pi / 4)) = - 1` ..
thus False unfolding `tan z = x` using `\<bar>x\<bar> < 1` by auto
qed

have "z < pi / 4"
proof (rule ccontr)
assume "\<not> (z < pi / 4)" hence "pi / 4 < z" using `z \<noteq> pi / 4` by auto
have "- (pi / 2) < pi / 4" using m2pi_less_pi by auto
from tan_monotone[OF this `pi / 4 < z` `z < pi / 2`]
have "1 < x" unfolding `tan z = x` `tan (pi / 4) = 1` .
thus False using `\<bar>x\<bar> < 1` by auto
qed
moreover
have "-(pi / 4) < z"
proof (rule ccontr)
assume "\<not> (-(pi / 4) < z)" hence "z < - (pi / 4)" using `z \<noteq> - (pi / 4)` by auto
have "-(pi / 4) < pi / 2" using m2pi_less_pi by auto
from tan_monotone[OF `-(pi / 2) < z` `z < -(pi / 4)` this]
have "x < - 1" unfolding `tan z = x` `tan (-(pi / 4)) = - 1` .
thus False using `\<bar>x\<bar> < 1` by auto
qed
ultimately show ?thesis using `tan z = x` by auto
qed

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 -
obtain y' where "-(pi/4) < y'" and "y' < pi/4" and "tan y' = y" using tan_total_pi4[OF `\<bar>y\<bar> < 1`] by blast

have "pi / 4 < pi / 2" by auto

have "\<exists> x'. -(pi/4) \<le> x' \<and> x' \<le> pi/4 \<and> tan x' = x"
proof (cases "\<bar>x\<bar> < 1")
case True from tan_total_pi4[OF this] obtain x' where "-(pi/4) < x'" and "x' < pi/4" and "tan x' = x" by blast
hence "-(pi/4) \<le> x'" and "x' \<le> pi/4" and "tan x' = x" by auto
thus ?thesis by auto
next
case False
show ?thesis
proof (cases "x = 1")
case True hence "tan (pi/4) = x" using tan_45 by auto
moreover
have "- pi \<le> pi" unfolding minus_le_self_iff by auto
hence "-(pi/4) \<le> pi/4" and "pi/4 \<le> pi/4" by auto
ultimately show ?thesis by blast
next
case False hence "x = -1" using `\<not> \<bar>x\<bar> < 1` and `\<bar>x\<bar> \<le> 1` by auto
hence "tan (-(pi/4)) = x" using tan_45 tan_minus by auto
moreover
have "- pi \<le> pi" unfolding minus_le_self_iff by auto
hence "-(pi/4) \<le> pi/4" and "-(pi/4) \<le> -(pi/4)" by auto
ultimately show ?thesis by blast
qed
qed
then obtain x' where "-(pi/4) \<le> x'" and "x' \<le> pi/4" and "tan x' = x" by blast
hence "-(pi/2) < x'" and "x' < pi/2" using order_le_less_trans[OF `x' \<le> pi/4` `pi / 4 < pi / 2`] by auto

have "cos x' \<noteq> 0" using cos_gt_zero_pi[THEN less_imp_neq] and `-(pi/2) < x'` and `x' < pi/2` by auto
moreover have "cos y' \<noteq> 0" using cos_gt_zero_pi[THEN less_imp_neq] and `-(pi/4) < y'` and `y' < pi/4` by auto
ultimately have "cos x' * cos y' \<noteq> 0" by auto

have divide_nonzero_divide: "\<And> A B C :: real. C \<noteq> 0 \<Longrightarrow> (A / C) / (B / C) = A / B" by auto
have divide_mult_commute: "\<And> A B C D :: real. A * B / (C * D) = (A / C) * (B / D)" by auto

have "tan (x' + y') = sin (x' + y') / (cos x' * cos y' - sin x' * sin y')" unfolding tan_def cos_add ..
also have "\<dots> = (tan x' + tan y') / ((cos x' * cos y' - sin x' * sin y') / (cos x' * cos y'))" unfolding add_tan_eq[OF `cos x' \<noteq> 0` `cos y' \<noteq> 0`] divide_nonzero_divide[OF `cos x' * cos y' \<noteq> 0`] ..
also have "\<dots> = (tan x' + tan y') / (1 - tan x' * tan y')" unfolding tan_def diff_divide_distrib divide_self[OF `cos x' * cos y' \<noteq> 0`] unfolding divide_mult_commute ..
finally have tan_eq: "tan (x' + y') = (x + y) / (1 - x * y)" unfolding `tan y' = y` `tan x' = x` .

have "arctan (tan (x' + y')) = x' + y'" using `-(pi/4) < y'` `-(pi/4) \<le> x'` `y' < pi/4` and `x' \<le> pi/4` by (auto intro!: arctan_tan)
moreover have "arctan (tan (x')) = x'" using `-(pi/2) < x'` and `x' < pi/2` by (auto intro!: arctan_tan)
moreover have "arctan (tan (y')) = y'" using `-(pi/4) < y'` and `y' < pi/4` by (auto intro!: arctan_tan)
ultimately have "arctan x + arctan y = arctan (tan (x' + y'))" unfolding `tan y' = y` [symmetric] `tan x' = x`[symmetric] by auto
thus "arctan x + arctan y = arctan ((x + y) / (1 - x * y))" unfolding tan_eq .
qed

lemma arctan1_eq_pi4: "arctan 1 = pi / 4" unfolding tan_45[symmetric] by (rule arctan_tan, auto simp add: m2pi_less_pi)

theorem machin: "pi / 4 = 4 * arctan (1/5) - arctan (1 / 239)"
proof -
have "\<bar>1 / 5\<bar> < (1 :: real)" by auto
have "2 * arctan (1 / 5) = arctan (5 / 12)" by auto
moreover
have "\<bar>5 / 12\<bar> < (1 :: real)" by auto
have "2 * arctan (5 / 12) = arctan (120 / 119)" by auto
moreover
have "\<bar>1\<bar> \<le> (1::real)" and "\<bar>1 / 239\<bar> < (1::real)" by auto
have "arctan 1 + arctan (1 / 239) = arctan (120 / 119)" by auto
ultimately have "arctan 1 + arctan (1 / 239) = 4 * arctan (1 / 5)" by auto
thus ?thesis unfolding arctan1_eq_pi4 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
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)"
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`)
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)
next
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
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)
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
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
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)")
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^2 < 1"
proof -
from mult_left_mono[OF less_imp_le[OF `\<bar>x\<bar> < 1`] abs_ge_zero[of x]]
have "\<bar> x^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")
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^2 < 1" by (rule less_one_imp_sqr_less_one)
have "summable (\<lambda> n. -1 ^ n * (x^2) ^n)"
by (rule summable_Leibniz(1), auto intro!: LIMSEQ_realpow_zero monoseq_realpow `x^2 < 1` order_less_imp_le[OF `x^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"
proof
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
next
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]
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 =
(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]
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}"
hence "\<bar>x'\<bar> < 1" by auto

let ?S = "\<Sum> n. (-1)^n * x'^(2 * n)"
show "summable (\<lambda> n. ?f n * real (Suc n) * x'^n)" unfolding if_eq
by (rule sums_summable[where l="0 + ?S"], rule sums_if, rule sums_zero, rule summable_sums, rule summable_Integral[OF `\<bar>x'\<bar> < 1`])
}
qed auto
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)")
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"
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))" .
} 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"
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^2) \<bar> < 1" using less_one_imp_sqr_less_one[OF `\<bar>x\<bar> < 1`] by auto
hence "(\<lambda> n. (- (x^2)) ^ n) sums (1 / (1 - (- (x^2))))" unfolding real_norm_def[symmetric] by (rule geometric_sums)
hence "(?c' x) sums (1 / (1 - (- (x^2))))" unfolding power_mult_distrib[symmetric] power_mult nat_mult_commute[of _ 2] by auto
hence suminf_c'_eq_geom: "inverse (1 + x^2) = suminf (?c' x)" using sums_unique unfolding inverse_eq_divide by auto
have "DERIV (\<lambda> x. suminf (?c x)) x :> (inverse (1 + x^2))" unfolding suminf_c'_eq_geom
by (rule DERIV_arctan_suminf[OF `0 < r` `r < 1` `\<bar>x\<bar> < r`])
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
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

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
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
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 "?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
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
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
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
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 .
qed
hence "0 \<le> ?a x n - ?diff x n" by auto
}
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)
hence "?diff 1 n \<le> ?a 1 n" by auto
}
have "?a 1 ----> 0"
unfolding tendsto_rabs_zero_iff power_one divide_inverse One_nat_def
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 }
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]
have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
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

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
finally show ?thesis using `x = -1` by auto
qed
qed
qed

lemma arctan_half: fixes x :: real
shows "arctan x = 2 * arctan (x / (1 + sqrt(1 + x^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

have "0 < cos y" using cos_gt_zero_pi[OF low high] .
hence "cos y \<noteq> 0" and cos_sqrt: "sqrt ((cos y) ^ 2) = cos y" by auto

have "1 + (tan y)^2 = 1 + sin y^2 / cos y^2" unfolding tan_def power_divide ..
also have "\<dots> = cos y^2 / cos y^2 + sin y^2 / cos y^2" using `cos y \<noteq> 0` by auto
also have "\<dots> = 1 / cos y^2" unfolding add_divide_distrib[symmetric] sin_cos_squared_add2 ..
finally have "1 + (tan y)^2 = 1 / cos y^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^2))" unfolding cos_sqrt ..
also have "\<dots> = tan y / (1 + sqrt(1 / cos y^2))" unfolding real_sqrt_divide by auto
finally have eq: "sin y / (cos y + 1) = tan y / (1 + sqrt(1 + (tan y)^2))" unfolding `1 + (tan y)^2 = 1 / cos y^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[OF low2 high2] by auto
finally show ?thesis unfolding eq `tan y = x` .
qed

lemma arctan_monotone: assumes "x < y"
shows "arctan x < arctan y"
proof -
obtain z where "-(pi / 2) < z" and "z < pi / 2" and "tan z = x" using tan_total by blast
obtain w where "-(pi / 2) < w" and "w < pi / 2" and "tan w = y" using tan_total by blast
have "z < w" unfolding tan_monotone'[OF `-(pi / 2) < z` `z < pi / 2` `-(pi / 2) < w` `w < pi / 2`] `tan z = x` `tan w = y` using `x < y` .
thus ?thesis
unfolding `tan z = x`[symmetric] arctan_tan[OF `-(pi / 2) < z` `z < pi / 2`]
unfolding `tan w = y`[symmetric] arctan_tan[OF `-(pi / 2) < w` `w < pi / 2`] .
qed

lemma arctan_monotone': assumes "x \<le> y" shows "arctan x \<le> arctan y"
proof (cases "x = y")
case False hence "x < y" using `x \<le> y` by auto from arctan_monotone[OF this] show ?thesis by auto
qed auto

lemma arctan_minus: "arctan (- x) = - arctan x"
proof -
obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast
thus ?thesis unfolding `tan y = x`[symmetric] tan_minus[symmetric] using arctan_tan[of y] arctan_tan[of "-y"] by auto
qed

lemma arctan_inverse: assumes "x \<noteq> 0" shows "arctan (1 / x) = sgn x * pi / 2 - arctan x"
proof -
obtain y where "- (pi / 2) < y" and "y < pi / 2" and "tan y = x" using tan_total by blast
hence "y = arctan x" unfolding `tan y = x`[symmetric] using arctan_tan by auto

{ fix y x :: real assume "0 < y" and "y < pi /2" and "y = arctan x" and "tan y = x" hence "- (pi / 2) < y" by auto
have "tan y > 0" using tan_monotone'[OF _ _ `- (pi / 2) < y` `y < pi / 2`, of 0] tan_zero `0 < y` by auto
hence "x > 0" using `tan y = x` by auto

have "- (pi / 2) < pi / 2 - y" using `y > 0` `y < pi / 2` by auto
moreover have "pi / 2 - y < pi / 2" using `y > 0` `y < pi / 2` by auto
ultimately have "arctan (1 / x) = pi / 2 - y" unfolding `tan y = x`[symmetric] tan_inverse using arctan_tan by auto
hence "arctan (1 / x) = sgn x * pi / 2 - arctan x" unfolding `y = arctan x` real_sgn_pos[OF `x > 0`] by auto
} note pos_y = this

show ?thesis
proof (cases "y > 0")
case True from pos_y[OF this `y < pi / 2` `y = arctan x` `tan y = x`] show ?thesis .
next
case False hence "y \<le> 0" by auto
moreover have "y \<noteq> 0"
proof (rule ccontr)
assume "\<not> y \<noteq> 0" hence "y = 0" by auto
have "x = 0" unfolding `tan y = x`[symmetric] `y = 0` tan_zero ..
thus False using `x \<noteq> 0` by auto
qed
ultimately have "y < 0" by auto
hence "0 < - y" and "-y < pi / 2" using `- (pi / 2) < y` by auto
moreover have "-y = arctan (-x)" unfolding arctan_minus `y = arctan x` ..
moreover have "tan (-y) = -x" unfolding tan_minus `tan y = x` ..
ultimately have "arctan (1 / -x) = sgn (-x) * pi / 2 - arctan (-x)" using pos_y by blast
hence "arctan (- (1 / x)) = - (sgn x * pi / 2 - arctan x)" unfolding arctan_minus[of x] divide_minus_right sgn_minus by auto
thus ?thesis unfolding arctan_minus neg_equal_iff_equal .
qed
qed

theorem pi_series: "pi / 4 = (\<Sum> k. (-1)^k * 1 / real (k*2+1))" (is "_ = ?SUM")
proof -
have "pi / 4 = arctan 1" using arctan1_eq_pi4 by auto
also have "\<dots> = ?SUM" using arctan_series[of 1] by auto
finally show ?thesis by auto
qed

subsection {* Existence of Polar Coordinates *}

lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<twosuperior> + y\<twosuperior>)\<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

lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"

lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<twosuperior>)"

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\<twosuperior> + y\<twosuperior>)" in exI)
apply (rule_tac x = "arccos (x / sqrt (x\<twosuperior> + y\<twosuperior>))" in exI)
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_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

end
```