src/HOL/Transcendental.thy
author nipkow
Mon Jan 30 21:49:41 2012 +0100 (2012-01-30)
changeset 46372 6fa9cdb8b850
parent 46240 933f35c4e126
child 47108 2a1953f0d20d
permissions -rw-r--r--
added "'a rel"
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(*  Title:      HOL/Transcendental.thy
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    Author:     Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
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    Author:     Lawrence C Paulson
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*)
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header{*Power Series, Transcendental Functions etc.*}
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theory Transcendental
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imports Fact Series Deriv NthRoot
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begin
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subsection {* Properties of Power Series *}
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lemma lemma_realpow_diff:
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  fixes y :: "'a::monoid_mult"
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  shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
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proof -
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  assume "p \<le> n"
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  hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
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  thus ?thesis by (simp add: power_commutes)
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qed
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lemma lemma_realpow_diff_sumr:
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  fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows
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     "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =
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      y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
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by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac
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         del: setsum_op_ivl_Suc)
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lemma lemma_realpow_diff_sumr2:
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  fixes y :: "'a::{comm_ring,monoid_mult}" shows
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     "x ^ (Suc n) - y ^ (Suc n) =
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      (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
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apply (induct n, simp)
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apply (simp del: setsum_op_ivl_Suc)
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apply (subst setsum_op_ivl_Suc)
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apply (subst lemma_realpow_diff_sumr)
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apply (simp add: right_distrib del: setsum_op_ivl_Suc)
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apply (subst mult_left_commute [of "x - y"])
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apply (erule subst)
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apply (simp add: algebra_simps)
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done
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lemma lemma_realpow_rev_sumr:
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     "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =
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      (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
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apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
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apply (rule inj_onI, simp)
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apply auto
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apply (rule_tac x="n - x" in image_eqI, simp, simp)
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done
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text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
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x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
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lemma powser_insidea:
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  fixes x z :: "'a::real_normed_field"
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  assumes 1: "summable (\<lambda>n. f n * x ^ n)"
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  assumes 2: "norm z < norm x"
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  shows "summable (\<lambda>n. norm (f n * z ^ n))"
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proof -
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  from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
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  from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
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    by (rule summable_LIMSEQ_zero)
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  hence "convergent (\<lambda>n. f n * x ^ n)"
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    by (rule convergentI)
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  hence "Cauchy (\<lambda>n. f n * x ^ n)"
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    by (rule convergent_Cauchy)
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  hence "Bseq (\<lambda>n. f n * x ^ n)"
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    by (rule Cauchy_Bseq)
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  then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
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    by (simp add: Bseq_def, safe)
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  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
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                   K * norm (z ^ n) * inverse (norm (x ^ n))"
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  proof (intro exI allI impI)
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    fix n::nat assume "0 \<le> n"
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    have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
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          norm (f n * x ^ n) * norm (z ^ n)"
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      by (simp add: norm_mult abs_mult)
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    also have "\<dots> \<le> K * norm (z ^ n)"
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      by (simp only: mult_right_mono 4 norm_ge_zero)
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    also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
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      by (simp add: x_neq_0)
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    also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
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      by (simp only: mult_assoc)
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    finally show "norm (norm (f n * z ^ n)) \<le>
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                  K * norm (z ^ n) * inverse (norm (x ^ n))"
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      by (simp add: mult_le_cancel_right x_neq_0)
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  qed
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  moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
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  proof -
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    from 2 have "norm (norm (z * inverse x)) < 1"
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      using x_neq_0
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      by (simp add: nonzero_norm_divide divide_inverse [symmetric])
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    hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
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      by (rule summable_geometric)
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    hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
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      by (rule summable_mult)
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    thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
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      using x_neq_0
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      by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
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                    power_inverse norm_power mult_assoc)
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  qed
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  ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
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    by (rule summable_comparison_test)
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qed
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lemma powser_inside:
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  fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach}" shows
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     "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]
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      ==> summable (%n. f(n) * (z ^ n))"
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by (rule powser_insidea [THEN summable_norm_cancel])
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lemma sum_split_even_odd: fixes f :: "nat \<Rightarrow> real" shows
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  "(\<Sum> i = 0 ..< 2 * n. if even i then f i else g i) =
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   (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1))"
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proof (induct n)
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  case (Suc n)
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  have "(\<Sum> i = 0 ..< 2 * Suc n. if even i then f i else g i) =
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        (\<Sum> i = 0 ..< n. f (2 * i)) + (\<Sum> i = 0 ..< n. g (2 * i + 1)) + (f (2 * n) + g (2 * n + 1))"
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    using Suc.hyps unfolding One_nat_def by auto
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  also have "\<dots> = (\<Sum> i = 0 ..< Suc n. f (2 * i)) + (\<Sum> i = 0 ..< Suc n. g (2 * i + 1))" by auto
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  finally show ?case .
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qed auto
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lemma sums_if': fixes g :: "nat \<Rightarrow> real" assumes "g sums x"
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  shows "(\<lambda> n. if even n then 0 else g ((n - 1) div 2)) sums x"
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  unfolding sums_def
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proof (rule LIMSEQ_I)
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  fix r :: real assume "0 < r"
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  from `g sums x`[unfolded sums_def, THEN LIMSEQ_D, OF this]
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  obtain no where no_eq: "\<And> n. n \<ge> no \<Longrightarrow> (norm (setsum g { 0..<n } - x) < r)" by blast
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  let ?SUM = "\<lambda> m. \<Sum> i = 0 ..< m. if even i then 0 else g ((i - 1) div 2)"
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  { fix m assume "m \<ge> 2 * no" hence "m div 2 \<ge> no" by auto
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    have sum_eq: "?SUM (2 * (m div 2)) = setsum g { 0 ..< m div 2 }"
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      using sum_split_even_odd by auto
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    hence "(norm (?SUM (2 * (m div 2)) - x) < r)" using no_eq unfolding sum_eq using `m div 2 \<ge> no` by auto
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    moreover
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    have "?SUM (2 * (m div 2)) = ?SUM m"
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    proof (cases "even m")
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      case True show ?thesis unfolding even_nat_div_two_times_two[OF True, unfolded numeral_2_eq_2[symmetric]] ..
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    next
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      case False hence "even (Suc m)" by auto
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      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]]
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      have eq: "Suc (2 * (m div 2)) = m" by auto
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      hence "even (2 * (m div 2))" using `odd m` by auto
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      have "?SUM m = ?SUM (Suc (2 * (m div 2)))" unfolding eq ..
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      also have "\<dots> = ?SUM (2 * (m div 2))" using `even (2 * (m div 2))` by auto
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      finally show ?thesis by auto
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    qed
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    ultimately have "(norm (?SUM m - x) < r)" by auto
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  }
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  thus "\<exists> no. \<forall> m \<ge> no. norm (?SUM m - x) < r" by blast
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qed
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lemma sums_if: fixes g :: "nat \<Rightarrow> real" assumes "g sums x" and "f sums y"
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  shows "(\<lambda> n. if even n then f (n div 2) else g ((n - 1) div 2)) sums (x + y)"
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proof -
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  let ?s = "\<lambda> n. if even n then 0 else f ((n - 1) div 2)"
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  { fix B T E have "(if B then (0 :: real) else E) + (if B then T else 0) = (if B then T else E)"
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      by (cases B) auto } note if_sum = this
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  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`] .
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  {
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    have "?s 0 = 0" by auto
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    have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
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    have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
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    have "?s sums y" using sums_if'[OF `f sums y`] .
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    from this[unfolded sums_def, THEN LIMSEQ_Suc]
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    have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
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      unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
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                image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
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                even_Suc Suc_m1 if_eq .
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  } from sums_add[OF g_sums this]
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  show ?thesis unfolding if_sum .
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qed
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subsection {* Alternating series test / Leibniz formula *}
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lemma sums_alternating_upper_lower:
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  fixes a :: "nat \<Rightarrow> real"
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  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
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  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>
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             ((\<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)"
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  (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
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proof -
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  have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
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  have "\<forall> n. ?f n \<le> ?f (Suc n)"
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  proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed
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  moreover
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  have "\<forall> n. ?g (Suc n) \<le> ?g n"
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  proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
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    unfolding One_nat_def by auto qed
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  moreover
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  have "\<forall> n. ?f n \<le> ?g n"
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  proof fix n show "?f n \<le> ?g n" using fg_diff a_pos
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    unfolding One_nat_def by auto qed
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  moreover
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  have "(\<lambda> n. ?f n - ?g n) ----> 0" unfolding fg_diff
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  proof (rule LIMSEQ_I)
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    fix r :: real assume "0 < r"
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    with `a ----> 0`[THEN LIMSEQ_D]
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    obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" by auto
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    hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
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    thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
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  qed
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  ultimately
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  show ?thesis by (rule lemma_nest_unique)
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qed
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lemma summable_Leibniz': fixes a :: "nat \<Rightarrow> real"
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  assumes a_zero: "a ----> 0" and a_pos: "\<And> n. 0 \<le> a n"
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  and a_monotone: "\<And> n. a (Suc n) \<le> a n"
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  shows summable: "summable (\<lambda> n. (-1)^n * a n)"
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  and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
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  and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
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  and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
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  and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
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proof -
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  let "?S n" = "(-1)^n * a n"
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  let "?P n" = "\<Sum>i=0..<n. ?S i"
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  let "?f n" = "?P (2 * n)"
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  let "?g n" = "?P (2 * n + 1)"
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  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"
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    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
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  let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n"
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  have "?Sa ----> l"
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  proof (rule LIMSEQ_I)
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    fix r :: real assume "0 < r"
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    with `?f ----> l`[THEN LIMSEQ_D]
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    obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
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    from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
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    obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
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    { fix n :: nat
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      assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
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      have "norm (?Sa n - l) < r"
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      proof (cases "even n")
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        case True from even_nat_div_two_times_two[OF this]
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        have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto
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        with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" by auto
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        from f[OF this]
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        show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
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      next
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        case False hence "even (n - 1)" by simp
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        from even_nat_div_two_times_two[OF this]
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        have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto
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   253
        hence range_eq: "n - 1 + 1 = n" using odd_pos[OF False] by auto
wenzelm@32960
   254
wenzelm@32960
   255
        from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no" by auto
wenzelm@32960
   256
        from g[OF this]
wenzelm@32960
   257
        show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
hoelzl@29803
   258
      qed
hoelzl@29803
   259
    }
hoelzl@29803
   260
    thus "\<exists> no. \<forall> n \<ge> no. norm (?Sa n - l) < r" by blast
hoelzl@29803
   261
  qed
hoelzl@29803
   262
  hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l" unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
hoelzl@29803
   263
  thus "summable ?S" using summable_def by auto
hoelzl@29803
   264
hoelzl@29803
   265
  have "l = suminf ?S" using sums_unique[OF sums_l] .
hoelzl@29803
   266
hoelzl@29803
   267
  { fix n show "suminf ?S \<le> ?g n" unfolding sums_unique[OF sums_l, symmetric] using above_l by auto }
hoelzl@29803
   268
  { fix n show "?f n \<le> suminf ?S" unfolding sums_unique[OF sums_l, symmetric] using below_l by auto }
hoelzl@29803
   269
  show "?g ----> suminf ?S" using `?g ----> l` `l = suminf ?S` by auto
hoelzl@29803
   270
  show "?f ----> suminf ?S" using `?f ----> l` `l = suminf ?S` by auto
hoelzl@29803
   271
qed
hoelzl@29803
   272
hoelzl@29803
   273
theorem summable_Leibniz: fixes a :: "nat \<Rightarrow> real"
hoelzl@29803
   274
  assumes a_zero: "a ----> 0" and "monoseq a"
hoelzl@29803
   275
  shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
hoelzl@29803
   276
  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")
hoelzl@29803
   277
  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")
hoelzl@29803
   278
  and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
hoelzl@29803
   279
  and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
hoelzl@29803
   280
proof -
hoelzl@29803
   281
  have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
hoelzl@29803
   282
  proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
hoelzl@29803
   283
    case True
hoelzl@29803
   284
    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n" by auto
hoelzl@29803
   285
    { fix n have "a (Suc n) \<le> a n" using ord[where n="Suc n" and m=n] by auto }
hoelzl@29803
   286
    note leibniz = summable_Leibniz'[OF `a ----> 0` ge0] and mono = this
hoelzl@29803
   287
    from leibniz[OF mono]
hoelzl@29803
   288
    show ?thesis using `0 \<le> a 0` by auto
hoelzl@29803
   289
  next
hoelzl@29803
   290
    let ?a = "\<lambda> n. - a n"
hoelzl@29803
   291
    case False
hoelzl@29803
   292
    with monoseq_le[OF `monoseq a` `a ----> 0`]
hoelzl@29803
   293
    have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
hoelzl@29803
   294
    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n" by auto
hoelzl@29803
   295
    { fix n have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n] by auto }
hoelzl@29803
   296
    note monotone = this
huffman@44568
   297
    note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
hoelzl@29803
   298
    have "summable (\<lambda> n. (-1)^n * ?a n)" using leibniz(1) by auto
hoelzl@29803
   299
    then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l" unfolding summable_def by auto
hoelzl@29803
   300
    from this[THEN sums_minus]
hoelzl@29803
   301
    have "(\<lambda> n. (-1)^n * a n) sums -l" by auto
hoelzl@29803
   302
    hence ?summable unfolding summable_def by auto
hoelzl@29803
   303
    moreover
hoelzl@29803
   304
    have "\<And> a b :: real. \<bar> - a - - b \<bar> = \<bar>a - b\<bar>" unfolding minus_diff_minus by auto
hoelzl@41970
   305
hoelzl@29803
   306
    from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
hoelzl@29803
   307
    have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)" by auto
hoelzl@29803
   308
hoelzl@29803
   309
    have ?pos using `0 \<le> ?a 0` by auto
hoelzl@29803
   310
    moreover have ?neg using leibniz(2,4) unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le by auto
huffman@44568
   311
    moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel] by auto
hoelzl@29803
   312
    ultimately show ?thesis by auto
hoelzl@29803
   313
  qed
hoelzl@29803
   314
  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]
hoelzl@29803
   315
       this[THEN conjunct2, THEN conjunct2, THEN conjunct2, THEN conjunct2]
hoelzl@29803
   316
  show ?summable and ?pos and ?neg and ?f and ?g .
hoelzl@29803
   317
qed
paulson@15077
   318
huffman@29164
   319
subsection {* Term-by-Term Differentiability of Power Series *}
huffman@23043
   320
huffman@23043
   321
definition
huffman@23082
   322
  diffs :: "(nat => 'a::ring_1) => nat => 'a" where
huffman@23082
   323
  "diffs c = (%n. of_nat (Suc n) * c(Suc n))"
paulson@15077
   324
paulson@15077
   325
text{*Lemma about distributing negation over it*}
paulson@15077
   326
lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
paulson@15077
   327
by (simp add: diffs_def)
paulson@15077
   328
huffman@29163
   329
lemma sums_Suc_imp:
huffman@29163
   330
  assumes f: "f 0 = 0"
huffman@29163
   331
  shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
huffman@29163
   332
unfolding sums_def
huffman@29163
   333
apply (rule LIMSEQ_imp_Suc)
huffman@29163
   334
apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
huffman@29163
   335
apply (simp only: setsum_shift_bounds_Suc_ivl)
paulson@15077
   336
done
paulson@15077
   337
paulson@15229
   338
lemma diffs_equiv:
hoelzl@41970
   339
  fixes x :: "'a::{real_normed_vector, ring_1}"
hoelzl@41970
   340
  shows "summable (%n. (diffs c)(n) * (x ^ n)) ==>
hoelzl@41970
   341
      (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
nipkow@15546
   342
         (\<Sum>n. (diffs c)(n) * (x ^ n))"
huffman@29163
   343
unfolding diffs_def
huffman@29163
   344
apply (drule summable_sums)
huffman@29163
   345
apply (rule sums_Suc_imp, simp_all)
paulson@15077
   346
done
paulson@15077
   347
paulson@15077
   348
lemma lemma_termdiff1:
haftmann@31017
   349
  fixes z :: "'a :: {monoid_mult,comm_ring}" shows
hoelzl@41970
   350
  "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
huffman@23082
   351
   (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
wenzelm@41550
   352
by(auto simp add: algebra_simps power_add [symmetric])
paulson@15077
   353
huffman@23082
   354
lemma sumr_diff_mult_const2:
huffman@23082
   355
  "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
huffman@23082
   356
by (simp add: setsum_subtractf)
huffman@23082
   357
paulson@15229
   358
lemma lemma_termdiff2:
haftmann@31017
   359
  fixes h :: "'a :: {field}"
huffman@20860
   360
  assumes h: "h \<noteq> 0" shows
huffman@23082
   361
  "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
huffman@20860
   362
   h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
huffman@23082
   363
        (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
huffman@23082
   364
apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
huffman@20860
   365
apply (simp add: right_diff_distrib diff_divide_distrib h)
paulson@15077
   366
apply (simp add: mult_assoc [symmetric])
huffman@20860
   367
apply (cases "n", simp)
huffman@20860
   368
apply (simp add: lemma_realpow_diff_sumr2 h
huffman@20860
   369
                 right_diff_distrib [symmetric] mult_assoc
huffman@30273
   370
            del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
huffman@20860
   371
apply (subst lemma_realpow_rev_sumr)
huffman@23082
   372
apply (subst sumr_diff_mult_const2)
huffman@20860
   373
apply simp
huffman@20860
   374
apply (simp only: lemma_termdiff1 setsum_right_distrib)
huffman@20860
   375
apply (rule setsum_cong [OF refl])
nipkow@15539
   376
apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
huffman@20860
   377
apply (clarify)
huffman@20860
   378
apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
huffman@30273
   379
            del: setsum_op_ivl_Suc power_Suc)
huffman@20860
   380
apply (subst mult_assoc [symmetric], subst power_add [symmetric])
huffman@20860
   381
apply (simp add: mult_ac)
huffman@20860
   382
done
huffman@20860
   383
huffman@20860
   384
lemma real_setsum_nat_ivl_bounded2:
haftmann@35028
   385
  fixes K :: "'a::linordered_semidom"
huffman@23082
   386
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
huffman@23082
   387
  assumes K: "0 \<le> K"
huffman@23082
   388
  shows "setsum f {0..<n-k} \<le> of_nat n * K"
huffman@23082
   389
apply (rule order_trans [OF setsum_mono])
huffman@23082
   390
apply (rule f, simp)
huffman@23082
   391
apply (simp add: mult_right_mono K)
paulson@15077
   392
done
paulson@15077
   393
paulson@15229
   394
lemma lemma_termdiff3:
haftmann@31017
   395
  fixes h z :: "'a::{real_normed_field}"
huffman@20860
   396
  assumes 1: "h \<noteq> 0"
huffman@23082
   397
  assumes 2: "norm z \<le> K"
huffman@23082
   398
  assumes 3: "norm (z + h) \<le> K"
huffman@23082
   399
  shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
huffman@23082
   400
          \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
huffman@20860
   401
proof -
huffman@23082
   402
  have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
huffman@23082
   403
        norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
huffman@23082
   404
          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
huffman@20860
   405
    apply (subst lemma_termdiff2 [OF 1])
huffman@23082
   406
    apply (subst norm_mult)
huffman@20860
   407
    apply (rule mult_commute)
huffman@20860
   408
    done
huffman@23082
   409
  also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
huffman@23082
   410
  proof (rule mult_right_mono [OF _ norm_ge_zero])
huffman@23082
   411
    from norm_ge_zero 2 have K: "0 \<le> K" by (rule order_trans)
huffman@23082
   412
    have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
huffman@20860
   413
      apply (erule subst)
huffman@23082
   414
      apply (simp only: norm_mult norm_power power_add)
huffman@23082
   415
      apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
huffman@20860
   416
      done
huffman@23082
   417
    show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
huffman@23082
   418
              (z + h) ^ q * z ^ (n - 2 - q))
huffman@23082
   419
          \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
huffman@20860
   420
      apply (intro
huffman@23082
   421
         order_trans [OF norm_setsum]
huffman@20860
   422
         real_setsum_nat_ivl_bounded2
huffman@20860
   423
         mult_nonneg_nonneg
huffman@23082
   424
         zero_le_imp_of_nat
huffman@20860
   425
         zero_le_power K)
huffman@20860
   426
      apply (rule le_Kn, simp)
huffman@20860
   427
      done
huffman@20860
   428
  qed
huffman@23082
   429
  also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
huffman@20860
   430
    by (simp only: mult_assoc)
huffman@20860
   431
  finally show ?thesis .
huffman@20860
   432
qed
paulson@15077
   433
huffman@20860
   434
lemma lemma_termdiff4:
haftmann@31017
   435
  fixes f :: "'a::{real_normed_field} \<Rightarrow>
huffman@23082
   436
              'b::real_normed_vector"
huffman@20860
   437
  assumes k: "0 < (k::real)"
huffman@23082
   438
  assumes le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
huffman@20860
   439
  shows "f -- 0 --> 0"
huffman@31338
   440
unfolding LIM_eq diff_0_right
huffman@29163
   441
proof (safe)
huffman@29163
   442
  let ?h = "of_real (k / 2)::'a"
huffman@29163
   443
  have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
huffman@29163
   444
  hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
huffman@29163
   445
  hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
huffman@29163
   446
  hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
huffman@29163
   447
huffman@20860
   448
  fix r::real assume r: "0 < r"
huffman@23082
   449
  show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
huffman@20860
   450
  proof (cases)
huffman@20860
   451
    assume "K = 0"
huffman@23082
   452
    with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
huffman@20860
   453
      by simp
huffman@23082
   454
    thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
huffman@20860
   455
  next
huffman@20860
   456
    assume K_neq_zero: "K \<noteq> 0"
huffman@20860
   457
    with zero_le_K have K: "0 < K" by simp
huffman@23082
   458
    show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
huffman@20860
   459
    proof (rule exI, safe)
huffman@20860
   460
      from k r K show "0 < min k (r * inverse K / 2)"
huffman@20860
   461
        by (simp add: mult_pos_pos positive_imp_inverse_positive)
huffman@20860
   462
    next
huffman@23082
   463
      fix x::'a
huffman@23082
   464
      assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
huffman@23082
   465
      from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
huffman@20860
   466
        by simp_all
huffman@23082
   467
      from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
huffman@23082
   468
      also from x4 K have "K * norm x < K * (r * inverse K / 2)"
huffman@20860
   469
        by (rule mult_strict_left_mono)
huffman@20860
   470
      also have "\<dots> = r / 2"
huffman@20860
   471
        using K_neq_zero by simp
huffman@20860
   472
      also have "r / 2 < r"
huffman@20860
   473
        using r by simp
huffman@23082
   474
      finally show "norm (f x) < r" .
huffman@20860
   475
    qed
huffman@20860
   476
  qed
huffman@20860
   477
qed
paulson@15077
   478
paulson@15229
   479
lemma lemma_termdiff5:
haftmann@31017
   480
  fixes g :: "'a::{real_normed_field} \<Rightarrow>
huffman@23082
   481
              nat \<Rightarrow> 'b::banach"
huffman@20860
   482
  assumes k: "0 < (k::real)"
huffman@20860
   483
  assumes f: "summable f"
huffman@23082
   484
  assumes le: "\<And>h n. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (g h n) \<le> f n * norm h"
huffman@20860
   485
  shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
huffman@20860
   486
proof (rule lemma_termdiff4 [OF k])
huffman@23082
   487
  fix h::'a assume "h \<noteq> 0" and "norm h < k"
huffman@23082
   488
  hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
huffman@20860
   489
    by (simp add: le)
huffman@23082
   490
  hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
huffman@20860
   491
    by simp
huffman@23082
   492
  moreover from f have B: "summable (\<lambda>n. f n * norm h)"
huffman@20860
   493
    by (rule summable_mult2)
huffman@23082
   494
  ultimately have C: "summable (\<lambda>n. norm (g h n))"
huffman@20860
   495
    by (rule summable_comparison_test)
huffman@23082
   496
  hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
huffman@23082
   497
    by (rule summable_norm)
huffman@23082
   498
  also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
huffman@20860
   499
    by (rule summable_le)
huffman@23082
   500
  also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
huffman@20860
   501
    by (rule suminf_mult2 [symmetric])
huffman@23082
   502
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
huffman@20860
   503
qed
paulson@15077
   504
paulson@15077
   505
paulson@15077
   506
text{* FIXME: Long proofs*}
paulson@15077
   507
paulson@15077
   508
lemma termdiffs_aux:
haftmann@31017
   509
  fixes x :: "'a::{real_normed_field,banach}"
huffman@20849
   510
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
huffman@23082
   511
  assumes 2: "norm x < norm K"
huffman@20860
   512
  shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
huffman@23082
   513
             - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
huffman@20849
   514
proof -
huffman@20860
   515
  from dense [OF 2]
huffman@23082
   516
  obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
huffman@23082
   517
  from norm_ge_zero r1 have r: "0 < r"
huffman@20860
   518
    by (rule order_le_less_trans)
huffman@20860
   519
  hence r_neq_0: "r \<noteq> 0" by simp
huffman@20860
   520
  show ?thesis
huffman@20849
   521
  proof (rule lemma_termdiff5)
huffman@23082
   522
    show "0 < r - norm x" using r1 by simp
huffman@20849
   523
  next
huffman@23082
   524
    from r r2 have "norm (of_real r::'a) < norm K"
huffman@23082
   525
      by simp
huffman@23082
   526
    with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
huffman@20860
   527
      by (rule powser_insidea)
huffman@23082
   528
    hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
huffman@23082
   529
      using r
huffman@23082
   530
      by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
huffman@23082
   531
    hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
huffman@20860
   532
      by (rule diffs_equiv [THEN sums_summable])
huffman@23082
   533
    also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))
huffman@23082
   534
      = (\<lambda>n. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
huffman@20849
   535
      apply (rule ext)
huffman@20849
   536
      apply (simp add: diffs_def)
huffman@20849
   537
      apply (case_tac n, simp_all add: r_neq_0)
huffman@20849
   538
      done
hoelzl@41970
   539
    finally have "summable
huffman@23082
   540
      (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
huffman@20860
   541
      by (rule diffs_equiv [THEN sums_summable])
huffman@20860
   542
    also have
huffman@23082
   543
      "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
huffman@20860
   544
           r ^ (n - Suc 0)) =
huffman@23082
   545
       (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
huffman@20849
   546
      apply (rule ext)
huffman@20849
   547
      apply (case_tac "n", simp)
huffman@20849
   548
      apply (case_tac "nat", simp)
huffman@20849
   549
      apply (simp add: r_neq_0)
huffman@20849
   550
      done
huffman@20860
   551
    finally show
huffman@23082
   552
      "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
huffman@20849
   553
  next
huffman@23082
   554
    fix h::'a and n::nat
huffman@20860
   555
    assume h: "h \<noteq> 0"
huffman@23082
   556
    assume "norm h < r - norm x"
huffman@23082
   557
    hence "norm x + norm h < r" by simp
huffman@23082
   558
    with norm_triangle_ineq have xh: "norm (x + h) < r"
huffman@20860
   559
      by (rule order_le_less_trans)
huffman@23082
   560
    show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
huffman@23082
   561
          \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
huffman@23082
   562
      apply (simp only: norm_mult mult_assoc)
huffman@23082
   563
      apply (rule mult_left_mono [OF _ norm_ge_zero])
huffman@20860
   564
      apply (simp (no_asm) add: mult_assoc [symmetric])
huffman@20860
   565
      apply (rule lemma_termdiff3)
huffman@20860
   566
      apply (rule h)
huffman@20860
   567
      apply (rule r1 [THEN order_less_imp_le])
huffman@20860
   568
      apply (rule xh [THEN order_less_imp_le])
huffman@20860
   569
      done
huffman@20849
   570
  qed
huffman@20849
   571
qed
webertj@20217
   572
huffman@20860
   573
lemma termdiffs:
haftmann@31017
   574
  fixes K x :: "'a::{real_normed_field,banach}"
huffman@20860
   575
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
huffman@20860
   576
  assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
huffman@20860
   577
  assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
huffman@23082
   578
  assumes 4: "norm x < norm K"
huffman@20860
   579
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
huffman@29163
   580
unfolding deriv_def
huffman@29163
   581
proof (rule LIM_zero_cancel)
huffman@20860
   582
  show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
huffman@20860
   583
            - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
huffman@20860
   584
  proof (rule LIM_equal2)
huffman@29163
   585
    show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
huffman@20860
   586
  next
huffman@23082
   587
    fix h :: 'a
huffman@20860
   588
    assume "h \<noteq> 0"
huffman@23082
   589
    assume "norm (h - 0) < norm K - norm x"
huffman@23082
   590
    hence "norm x + norm h < norm K" by simp
huffman@23082
   591
    hence 5: "norm (x + h) < norm K"
huffman@23082
   592
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
huffman@20860
   593
    have A: "summable (\<lambda>n. c n * x ^ n)"
huffman@20860
   594
      by (rule powser_inside [OF 1 4])
huffman@20860
   595
    have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
huffman@20860
   596
      by (rule powser_inside [OF 1 5])
huffman@20860
   597
    have C: "summable (\<lambda>n. diffs c n * x ^ n)"
huffman@20860
   598
      by (rule powser_inside [OF 2 4])
huffman@20860
   599
    show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
hoelzl@41970
   600
             - (\<Sum>n. diffs c n * x ^ n) =
huffman@23082
   601
          (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
huffman@20860
   602
      apply (subst sums_unique [OF diffs_equiv [OF C]])
huffman@20860
   603
      apply (subst suminf_diff [OF B A])
huffman@20860
   604
      apply (subst suminf_divide [symmetric])
huffman@20860
   605
      apply (rule summable_diff [OF B A])
huffman@20860
   606
      apply (subst suminf_diff)
huffman@20860
   607
      apply (rule summable_divide)
huffman@20860
   608
      apply (rule summable_diff [OF B A])
huffman@20860
   609
      apply (rule sums_summable [OF diffs_equiv [OF C]])
huffman@29163
   610
      apply (rule arg_cong [where f="suminf"], rule ext)
nipkow@29667
   611
      apply (simp add: algebra_simps)
huffman@20860
   612
      done
huffman@20860
   613
  next
huffman@20860
   614
    show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h -
huffman@23082
   615
               of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
huffman@20860
   616
        by (rule termdiffs_aux [OF 3 4])
huffman@20860
   617
  qed
huffman@20860
   618
qed
huffman@20860
   619
paulson@15077
   620
hoelzl@29803
   621
subsection {* Derivability of power series *}
hoelzl@29803
   622
hoelzl@29803
   623
lemma DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
hoelzl@29803
   624
  assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
hoelzl@29803
   625
  and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
hoelzl@29803
   626
  and "summable (f' x0)"
hoelzl@29803
   627
  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>"
hoelzl@29803
   628
  shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
hoelzl@29803
   629
  unfolding deriv_def
hoelzl@29803
   630
proof (rule LIM_I)
hoelzl@29803
   631
  fix r :: real assume "0 < r" hence "0 < r/3" by auto
hoelzl@29803
   632
hoelzl@41970
   633
  obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
hoelzl@29803
   634
    using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
hoelzl@29803
   635
hoelzl@41970
   636
  obtain N_f' where N_f': "\<And> n. N_f' \<le> n \<Longrightarrow> \<bar> \<Sum> i. f' x0 (i + n) \<bar> < r/3"
hoelzl@29803
   637
    using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
hoelzl@29803
   638
hoelzl@29803
   639
  let ?N = "Suc (max N_L N_f')"
hoelzl@29803
   640
  have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
hoelzl@29803
   641
    L_estimate: "\<bar> \<Sum> i. L (i + ?N) \<bar> < r/3" using N_L[of "?N"] and N_f' [of "?N"] by auto
hoelzl@29803
   642
hoelzl@29803
   643
  let "?diff i x" = "(f (x0 + x) i - f x0 i) / x"
hoelzl@29803
   644
hoelzl@29803
   645
  let ?r = "r / (3 * real ?N)"
hoelzl@29803
   646
  have "0 < 3 * real ?N" by auto
hoelzl@29803
   647
  from divide_pos_pos[OF `0 < r` this]
hoelzl@29803
   648
  have "0 < ?r" .
hoelzl@29803
   649
hoelzl@29803
   650
  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)"
hoelzl@29803
   651
  def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
hoelzl@29803
   652
hoelzl@29803
   653
  have "0 < S'" unfolding S'_def
hoelzl@29803
   654
  proof (rule iffD2[OF Min_gr_iff])
hoelzl@29803
   655
    show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x"
hoelzl@29803
   656
    proof (rule ballI)
hoelzl@29803
   657
      fix x assume "x \<in> ?s ` {0..<?N}"
hoelzl@29803
   658
      then obtain n where "x = ?s n" and "n \<in> {0..<?N}" using image_iff[THEN iffD1] by blast
hoelzl@41970
   659
      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
hoelzl@29803
   660
      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
hoelzl@29803
   661
      have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)
hoelzl@29803
   662
      thus "0 < x" unfolding `x = ?s n` .
hoelzl@29803
   663
    qed
hoelzl@29803
   664
  qed auto
hoelzl@29803
   665
hoelzl@29803
   666
  def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
hoelzl@29803
   667
  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'`
hoelzl@29803
   668
    by auto
hoelzl@29803
   669
hoelzl@29803
   670
  { fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
hoelzl@29803
   671
    hence x_in_I: "x0 + x \<in> { a <..< b }" using S_a S_b by auto
hoelzl@41970
   672
hoelzl@29803
   673
    note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
hoelzl@29803
   674
    note div_smbl = summable_divide[OF diff_smbl]
hoelzl@29803
   675
    note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
hoelzl@29803
   676
    note ign = summable_ignore_initial_segment[where k="?N"]
hoelzl@29803
   677
    note diff_shft_smbl = summable_diff[OF ign[OF allf_summable[OF x_in_I]] ign[OF allf_summable[OF x0_in_I]]]
hoelzl@29803
   678
    note div_shft_smbl = summable_divide[OF diff_shft_smbl]
hoelzl@29803
   679
    note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
hoelzl@29803
   680
hoelzl@29803
   681
    { fix n
hoelzl@41970
   682
      have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
wenzelm@32960
   683
        using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .
hoelzl@29803
   684
      hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 0` by auto
hoelzl@29803
   685
    } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
hoelzl@29803
   686
    from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
hoelzl@29803
   687
    have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
hoelzl@29803
   688
    hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3") using L_estimate by auto
hoelzl@29803
   689
hoelzl@29803
   690
    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>)" ..
hoelzl@29803
   691
    also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
hoelzl@29803
   692
    proof (rule setsum_strict_mono)
hoelzl@29803
   693
      fix n assume "n \<in> { 0 ..< ?N}"
hoelzl@29803
   694
      have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .
hoelzl@29803
   695
      also have "S \<le> S'" using `S \<le> S'` .
hoelzl@41970
   696
      also have "S' \<le> ?s n" unfolding S'_def
hoelzl@29803
   697
      proof (rule Min_le_iff[THEN iffD2])
wenzelm@32960
   698
        have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto
wenzelm@32960
   699
        thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
hoelzl@29803
   700
      qed auto
hoelzl@29803
   701
      finally have "\<bar> x \<bar> < ?s n" .
hoelzl@29803
   702
hoelzl@29803
   703
      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]
hoelzl@29803
   704
      have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
hoelzl@29803
   705
      with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n`
hoelzl@29803
   706
      show "\<bar>?diff n x - f' x0 n\<bar> < ?r" by blast
hoelzl@29803
   707
    qed auto
hoelzl@29803
   708
    also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)
hoelzl@29803
   709
    also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto
hoelzl@29803
   710
    also have "\<dots> = r/3" by auto
hoelzl@29803
   711
    finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
hoelzl@29803
   712
hoelzl@29803
   713
    from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
hoelzl@41970
   714
    have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> =
hoelzl@29803
   715
                    \<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
hoelzl@29803
   716
    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)
hoelzl@29803
   717
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto
hoelzl@41970
   718
    also have "\<dots> < r /3 + r/3 + r/3"
huffman@36842
   719
      using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
huffman@36842
   720
      by (rule add_strict_mono [OF add_less_le_mono])
hoelzl@29803
   721
    finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"
hoelzl@29803
   722
      by auto
hoelzl@41970
   723
  } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
hoelzl@29803
   724
      norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`
hoelzl@29803
   725
    unfolding real_norm_def diff_0_right by blast
hoelzl@29803
   726
qed
hoelzl@29803
   727
hoelzl@29803
   728
lemma DERIV_power_series': fixes f :: "nat \<Rightarrow> real"
hoelzl@29803
   729
  assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
hoelzl@29803
   730
  and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
hoelzl@29803
   731
  shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
hoelzl@29803
   732
  (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
hoelzl@29803
   733
proof -
hoelzl@29803
   734
  { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
hoelzl@29803
   735
    hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" by auto
hoelzl@29803
   736
    have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
hoelzl@29803
   737
    proof (rule DERIV_series')
hoelzl@29803
   738
      show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
hoelzl@29803
   739
      proof -
wenzelm@32960
   740
        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto
wenzelm@32960
   741
        hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" using `R' < R` by auto
wenzelm@32960
   742
        have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto
wenzelm@32960
   743
        from powser_insidea[OF converges[OF in_Rball] this] show ?thesis by auto
hoelzl@29803
   744
      qed
hoelzl@29803
   745
      { fix n x y assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
wenzelm@32960
   746
        show "\<bar>?f x n - ?f y n\<bar> \<le> \<bar>f n * real (Suc n) * R'^n\<bar> * \<bar>x-y\<bar>"
wenzelm@32960
   747
        proof -
hoelzl@41970
   748
          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>"
wenzelm@32960
   749
            unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto
hoelzl@41970
   750
          also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
wenzelm@32960
   751
          proof (rule mult_left_mono)
wenzelm@32960
   752
            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)
wenzelm@32960
   753
            also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
wenzelm@32960
   754
            proof (rule setsum_mono)
wenzelm@32960
   755
              fix p assume "p \<in> {0..<Suc n}" hence "p \<le> n" by auto
wenzelm@32960
   756
              { fix n fix x :: real assume "x \<in> {-R'<..<R'}"
wenzelm@32960
   757
                hence "\<bar>x\<bar> \<le> R'"  by auto
wenzelm@32960
   758
                hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto)
wenzelm@32960
   759
              } from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
wenzelm@32960
   760
              have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" unfolding abs_mult by auto
wenzelm@32960
   761
              thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> n` by auto
wenzelm@32960
   762
            qed
wenzelm@32960
   763
            also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
wenzelm@32960
   764
            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'`]]] .
wenzelm@32960
   765
            show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult[symmetric] by auto
wenzelm@32960
   766
          qed
huffman@36777
   767
          also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult mult_assoc[symmetric] by algebra
wenzelm@32960
   768
          finally show ?thesis .
wenzelm@32960
   769
        qed }
hoelzl@31881
   770
      { fix n show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
wenzelm@32960
   771
          by (auto intro!: DERIV_intros simp del: power_Suc) }
hoelzl@29803
   772
      { fix x assume "x \<in> {-R' <..< R'}" hence "R' \<in> {-R <..< R}" and "norm x < norm R'" using assms `R' < R` by auto
wenzelm@32960
   773
        have "summable (\<lambda> n. f n * x^n)"
wenzelm@32960
   774
        proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
wenzelm@32960
   775
          fix n
wenzelm@32960
   776
          have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto)
wenzelm@32960
   777
          show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult
wenzelm@32960
   778
            by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right])
wenzelm@32960
   779
        qed
huffman@36777
   780
        from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
wenzelm@32960
   781
        show "summable (?f x)" by auto }
hoelzl@29803
   782
      show "summable (?f' x0)" using converges[OF `x0 \<in> {-R <..< R}`] .
hoelzl@29803
   783
      show "x0 \<in> {-R' <..< R'}" using `x0 \<in> {-R' <..< R'}` .
hoelzl@29803
   784
    qed
hoelzl@29803
   785
  } note for_subinterval = this
hoelzl@29803
   786
  let ?R = "(R + \<bar>x0\<bar>) / 2"
hoelzl@29803
   787
  have "\<bar>x0\<bar> < ?R" using assms by auto
hoelzl@29803
   788
  hence "- ?R < x0"
hoelzl@29803
   789
  proof (cases "x0 < 0")
hoelzl@29803
   790
    case True
hoelzl@29803
   791
    hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
hoelzl@29803
   792
    thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
hoelzl@29803
   793
  next
hoelzl@29803
   794
    case False
hoelzl@29803
   795
    have "- ?R < 0" using assms by auto
hoelzl@41970
   796
    also have "\<dots> \<le> x0" using False by auto
hoelzl@29803
   797
    finally show ?thesis .
hoelzl@29803
   798
  qed
hoelzl@29803
   799
  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto
hoelzl@29803
   800
  from for_subinterval[OF this]
hoelzl@29803
   801
  show ?thesis .
hoelzl@29803
   802
qed
chaieb@29695
   803
huffman@29164
   804
subsection {* Exponential Function *}
huffman@23043
   805
huffman@44308
   806
definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" where
huffman@44308
   807
  "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
huffman@23043
   808
huffman@23115
   809
lemma summable_exp_generic:
haftmann@31017
   810
  fixes x :: "'a::{real_normed_algebra_1,banach}"
haftmann@25062
   811
  defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
huffman@23115
   812
  shows "summable S"
huffman@23115
   813
proof -
haftmann@25062
   814
  have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
huffman@30273
   815
    unfolding S_def by (simp del: mult_Suc)
huffman@23115
   816
  obtain r :: real where r0: "0 < r" and r1: "r < 1"
huffman@23115
   817
    using dense [OF zero_less_one] by fast
huffman@23115
   818
  obtain N :: nat where N: "norm x < real N * r"
huffman@23115
   819
    using reals_Archimedean3 [OF r0] by fast
huffman@23115
   820
  from r1 show ?thesis
huffman@23115
   821
  proof (rule ratio_test [rule_format])
huffman@23115
   822
    fix n :: nat
huffman@23115
   823
    assume n: "N \<le> n"
huffman@23115
   824
    have "norm x \<le> real N * r"
huffman@23115
   825
      using N by (rule order_less_imp_le)
huffman@23115
   826
    also have "real N * r \<le> real (Suc n) * r"
huffman@23115
   827
      using r0 n by (simp add: mult_right_mono)
huffman@23115
   828
    finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
huffman@23115
   829
      using norm_ge_zero by (rule mult_right_mono)
huffman@23115
   830
    hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
huffman@23115
   831
      by (rule order_trans [OF norm_mult_ineq])
huffman@23115
   832
    hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
huffman@23115
   833
      by (simp add: pos_divide_le_eq mult_ac)
huffman@23115
   834
    thus "norm (S (Suc n)) \<le> r * norm (S n)"
huffman@35216
   835
      by (simp add: S_Suc inverse_eq_divide)
huffman@23115
   836
  qed
huffman@23115
   837
qed
huffman@23115
   838
huffman@23115
   839
lemma summable_norm_exp:
haftmann@31017
   840
  fixes x :: "'a::{real_normed_algebra_1,banach}"
haftmann@25062
   841
  shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
huffman@23115
   842
proof (rule summable_norm_comparison_test [OF exI, rule_format])
haftmann@25062
   843
  show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
huffman@23115
   844
    by (rule summable_exp_generic)
huffman@23115
   845
next
haftmann@25062
   846
  fix n show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
huffman@35216
   847
    by (simp add: norm_power_ineq)
huffman@23115
   848
qed
huffman@23115
   849
huffman@23043
   850
lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
huffman@23115
   851
by (insert summable_exp_generic [where x=x], simp)
huffman@23043
   852
haftmann@25062
   853
lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
huffman@23115
   854
unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
huffman@23043
   855
huffman@23043
   856
hoelzl@41970
   857
lemma exp_fdiffs:
paulson@15077
   858
      "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
huffman@23431
   859
by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
huffman@23082
   860
         del: mult_Suc of_nat_Suc)
paulson@15077
   861
huffman@23115
   862
lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
huffman@23115
   863
by (simp add: diffs_def)
huffman@23115
   864
paulson@15077
   865
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
huffman@44308
   866
unfolding exp_def scaleR_conv_of_real
huffman@44308
   867
apply (rule DERIV_cong)
huffman@44308
   868
apply (rule termdiffs [where K="of_real (1 + norm x)"])
huffman@23115
   869
apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
huffman@23115
   870
apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
huffman@23115
   871
apply (simp del: of_real_add)
paulson@15077
   872
done
paulson@15077
   873
huffman@44311
   874
lemma isCont_exp: "isCont exp x"
huffman@44311
   875
  by (rule DERIV_exp [THEN DERIV_isCont])
huffman@44311
   876
huffman@44311
   877
lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
huffman@44311
   878
  by (rule isCont_o2 [OF _ isCont_exp])
huffman@44311
   879
huffman@44311
   880
lemma tendsto_exp [tendsto_intros]:
huffman@44311
   881
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
huffman@44311
   882
  by (rule isCont_tendsto_compose [OF isCont_exp])
huffman@23045
   883
huffman@23045
   884
huffman@29167
   885
subsubsection {* Properties of the Exponential Function *}
paulson@15077
   886
huffman@23278
   887
lemma powser_zero:
haftmann@31017
   888
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
huffman@23278
   889
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
paulson@15077
   890
proof -
huffman@23278
   891
  have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
huffman@23115
   892
    by (rule sums_unique [OF series_zero], simp add: power_0_left)
huffman@30082
   893
  thus ?thesis unfolding One_nat_def by simp
paulson@15077
   894
qed
paulson@15077
   895
huffman@23278
   896
lemma exp_zero [simp]: "exp 0 = 1"
huffman@23278
   897
unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
huffman@23278
   898
huffman@23115
   899
lemma setsum_cl_ivl_Suc2:
huffman@23115
   900
  "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
nipkow@28069
   901
by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
huffman@23115
   902
         del: setsum_cl_ivl_Suc)
huffman@23115
   903
huffman@23115
   904
lemma exp_series_add:
haftmann@31017
   905
  fixes x y :: "'a::{real_field}"
haftmann@25062
   906
  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
huffman@23115
   907
  shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
huffman@23115
   908
proof (induct n)
huffman@23115
   909
  case 0
huffman@23115
   910
  show ?case
huffman@23115
   911
    unfolding S_def by simp
huffman@23115
   912
next
huffman@23115
   913
  case (Suc n)
haftmann@25062
   914
  have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
huffman@30273
   915
    unfolding S_def by (simp del: mult_Suc)
haftmann@25062
   916
  hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
huffman@23115
   917
    by simp
huffman@23115
   918
haftmann@25062
   919
  have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
huffman@23115
   920
    by (simp only: times_S)
huffman@23115
   921
  also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
huffman@23115
   922
    by (simp only: Suc)
huffman@23115
   923
  also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
huffman@23115
   924
                + y * (\<Sum>i=0..n. S x i * S y (n-i))"
huffman@23115
   925
    by (rule left_distrib)
huffman@23115
   926
  also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
huffman@23115
   927
                + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
huffman@23115
   928
    by (simp only: setsum_right_distrib mult_ac)
haftmann@25062
   929
  also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
haftmann@25062
   930
                + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
   931
    by (simp add: times_S Suc_diff_le)
haftmann@25062
   932
  also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
haftmann@25062
   933
             (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
   934
    by (subst setsum_cl_ivl_Suc2, simp)
haftmann@25062
   935
  also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
haftmann@25062
   936
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
   937
    by (subst setsum_cl_ivl_Suc, simp)
haftmann@25062
   938
  also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
haftmann@25062
   939
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
haftmann@25062
   940
             (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
   941
    by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
huffman@23115
   942
              real_of_nat_add [symmetric], simp)
haftmann@25062
   943
  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
huffman@23127
   944
    by (simp only: scaleR_right.setsum)
huffman@23115
   945
  finally show
huffman@23115
   946
    "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
huffman@35216
   947
    by (simp del: setsum_cl_ivl_Suc)
huffman@23115
   948
qed
huffman@23115
   949
huffman@23115
   950
lemma exp_add: "exp (x + y) = exp x * exp y"
huffman@23115
   951
unfolding exp_def
huffman@23115
   952
by (simp only: Cauchy_product summable_norm_exp exp_series_add)
huffman@23115
   953
huffman@29170
   954
lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
huffman@29170
   955
by (rule exp_add [symmetric])
huffman@29170
   956
huffman@23241
   957
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
huffman@23241
   958
unfolding exp_def
huffman@44282
   959
apply (subst suminf_of_real)
huffman@23241
   960
apply (rule summable_exp_generic)
huffman@23241
   961
apply (simp add: scaleR_conv_of_real)
huffman@23241
   962
done
huffman@23241
   963
huffman@29170
   964
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
huffman@29170
   965
proof
huffman@29170
   966
  have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
huffman@29170
   967
  also assume "exp x = 0"
huffman@29170
   968
  finally show "False" by simp
paulson@15077
   969
qed
paulson@15077
   970
huffman@29170
   971
lemma exp_minus: "exp (- x) = inverse (exp x)"
huffman@29170
   972
by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
paulson@15077
   973
huffman@29170
   974
lemma exp_diff: "exp (x - y) = exp x / exp y"
huffman@29170
   975
  unfolding diff_minus divide_inverse
huffman@29170
   976
  by (simp add: exp_add exp_minus)
paulson@15077
   977
huffman@29167
   978
huffman@29167
   979
subsubsection {* Properties of the Exponential Function on Reals *}
huffman@29167
   980
huffman@29170
   981
text {* Comparisons of @{term "exp x"} with zero. *}
huffman@29167
   982
huffman@29167
   983
text{*Proof: because every exponential can be seen as a square.*}
huffman@29167
   984
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
huffman@29167
   985
proof -
huffman@29167
   986
  have "0 \<le> exp (x/2) * exp (x/2)" by simp
huffman@29167
   987
  thus ?thesis by (simp add: exp_add [symmetric])
huffman@29167
   988
qed
huffman@29167
   989
huffman@23115
   990
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
paulson@15077
   991
by (simp add: order_less_le)
paulson@15077
   992
huffman@29170
   993
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
huffman@29170
   994
by (simp add: not_less)
huffman@29170
   995
huffman@29170
   996
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
huffman@29170
   997
by (simp add: not_le)
paulson@15077
   998
huffman@23115
   999
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
huffman@29165
  1000
by simp
paulson@15077
  1001
paulson@15077
  1002
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
paulson@15251
  1003
apply (induct "n")
paulson@15077
  1004
apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
paulson@15077
  1005
done
paulson@15077
  1006
huffman@29170
  1007
text {* Strict monotonicity of exponential. *}
huffman@29170
  1008
huffman@29170
  1009
lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
huffman@29170
  1010
apply (drule order_le_imp_less_or_eq, auto)
huffman@29170
  1011
apply (simp add: exp_def)
huffman@36777
  1012
apply (rule order_trans)
huffman@29170
  1013
apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
huffman@29170
  1014
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
huffman@29170
  1015
done
huffman@29170
  1016
huffman@29170
  1017
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
huffman@29170
  1018
proof -
huffman@29170
  1019
  assume x: "0 < x"
huffman@29170
  1020
  hence "1 < 1 + x" by simp
huffman@29170
  1021
  also from x have "1 + x \<le> exp x"
huffman@29170
  1022
    by (simp add: exp_ge_add_one_self_aux)
huffman@29170
  1023
  finally show ?thesis .
huffman@29170
  1024
qed
huffman@29170
  1025
paulson@15077
  1026
lemma exp_less_mono:
huffman@23115
  1027
  fixes x y :: real
huffman@29165
  1028
  assumes "x < y" shows "exp x < exp y"
paulson@15077
  1029
proof -
huffman@29165
  1030
  from `x < y` have "0 < y - x" by simp
huffman@29165
  1031
  hence "1 < exp (y - x)" by (rule exp_gt_one)
huffman@29165
  1032
  hence "1 < exp y / exp x" by (simp only: exp_diff)
huffman@29165
  1033
  thus "exp x < exp y" by simp
paulson@15077
  1034
qed
paulson@15077
  1035
huffman@23115
  1036
lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
huffman@29170
  1037
apply (simp add: linorder_not_le [symmetric])
huffman@29170
  1038
apply (auto simp add: order_le_less exp_less_mono)
paulson@15077
  1039
done
paulson@15077
  1040
huffman@29170
  1041
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
paulson@15077
  1042
by (auto intro: exp_less_mono exp_less_cancel)
paulson@15077
  1043
huffman@29170
  1044
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
paulson@15077
  1045
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
  1046
huffman@29170
  1047
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
paulson@15077
  1048
by (simp add: order_eq_iff)
paulson@15077
  1049
huffman@29170
  1050
text {* Comparisons of @{term "exp x"} with one. *}
huffman@29170
  1051
huffman@29170
  1052
lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
huffman@29170
  1053
  using exp_less_cancel_iff [where x=0 and y=x] by simp
huffman@29170
  1054
huffman@29170
  1055
lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
huffman@29170
  1056
  using exp_less_cancel_iff [where x=x and y=0] by simp
huffman@29170
  1057
huffman@29170
  1058
lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
huffman@29170
  1059
  using exp_le_cancel_iff [where x=0 and y=x] by simp
huffman@29170
  1060
huffman@29170
  1061
lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
huffman@29170
  1062
  using exp_le_cancel_iff [where x=x and y=0] by simp
huffman@29170
  1063
huffman@29170
  1064
lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
huffman@29170
  1065
  using exp_inj_iff [where x=x and y=0] by simp
huffman@29170
  1066
huffman@23115
  1067
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
huffman@44755
  1068
proof (rule IVT)
huffman@44755
  1069
  assume "1 \<le> y"
huffman@44755
  1070
  hence "0 \<le> y - 1" by simp
huffman@44755
  1071
  hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
huffman@44755
  1072
  thus "y \<le> exp (y - 1)" by simp
huffman@44755
  1073
qed (simp_all add: le_diff_eq)
paulson@15077
  1074
huffman@23115
  1075
lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
huffman@44755
  1076
proof (rule linorder_le_cases [of 1 y])
huffman@44755
  1077
  assume "1 \<le> y" thus "\<exists>x. exp x = y"
huffman@44755
  1078
    by (fast dest: lemma_exp_total)
huffman@44755
  1079
next
huffman@44755
  1080
  assume "0 < y" and "y \<le> 1"
huffman@44755
  1081
  hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
huffman@44755
  1082
  then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
huffman@44755
  1083
  hence "exp (- x) = y" by (simp add: exp_minus)
huffman@44755
  1084
  thus "\<exists>x. exp x = y" ..
huffman@44755
  1085
qed
paulson@15077
  1086
paulson@15077
  1087
huffman@29164
  1088
subsection {* Natural Logarithm *}
paulson@15077
  1089
huffman@44308
  1090
definition ln :: "real \<Rightarrow> real" where
huffman@23043
  1091
  "ln x = (THE u. exp u = x)"
huffman@23043
  1092
huffman@23043
  1093
lemma ln_exp [simp]: "ln (exp x) = x"
huffman@44308
  1094
  by (simp add: ln_def)
paulson@15077
  1095
huffman@22654
  1096
lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
huffman@44308
  1097
  by (auto dest: exp_total)
huffman@22654
  1098
huffman@29171
  1099
lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
huffman@44308
  1100
  by (metis exp_gt_zero exp_ln)
paulson@15077
  1101
huffman@29171
  1102
lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
huffman@44308
  1103
  by (erule subst, rule ln_exp)
huffman@29171
  1104
huffman@29171
  1105
lemma ln_one [simp]: "ln 1 = 0"
huffman@44308
  1106
  by (rule ln_unique, simp)
huffman@29171
  1107
huffman@29171
  1108
lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"
huffman@44308
  1109
  by (rule ln_unique, simp add: exp_add)
huffman@29171
  1110
huffman@29171
  1111
lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
huffman@44308
  1112
  by (rule ln_unique, simp add: exp_minus)
huffman@29171
  1113
huffman@29171
  1114
lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"
huffman@44308
  1115
  by (rule ln_unique, simp add: exp_diff)
paulson@15077
  1116
huffman@29171
  1117
lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
huffman@44308
  1118
  by (rule ln_unique, simp add: exp_real_of_nat_mult)
huffman@29171
  1119
huffman@29171
  1120
lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
huffman@44308
  1121
  by (subst exp_less_cancel_iff [symmetric], simp)
huffman@29171
  1122
huffman@29171
  1123
lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
huffman@44308
  1124
  by (simp add: linorder_not_less [symmetric])
huffman@29171
  1125
huffman@29171
  1126
lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
huffman@44308
  1127
  by (simp add: order_eq_iff)
huffman@29171
  1128
huffman@29171
  1129
lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
huffman@44308
  1130
  apply (rule exp_le_cancel_iff [THEN iffD1])
huffman@44308
  1131
  apply (simp add: exp_ge_add_one_self_aux)
huffman@44308
  1132
  done
paulson@15077
  1133
huffman@29171
  1134
lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
huffman@44308
  1135
  by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
huffman@44308
  1136
huffman@44308
  1137
lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
huffman@44308
  1138
  using ln_le_cancel_iff [of 1 x] by simp
huffman@44308
  1139
huffman@44308
  1140
lemma ln_ge_zero_imp_ge_one: "\<lbrakk>0 \<le> ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 \<le> x"
huffman@44308
  1141
  using ln_le_cancel_iff [of 1 x] by simp
huffman@44308
  1142
huffman@44308
  1143
lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> (0 \<le> ln x) = (1 \<le> x)"
huffman@44308
  1144
  using ln_le_cancel_iff [of 1 x] by simp
huffman@44308
  1145
huffman@44308
  1146
lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x < 0) = (x < 1)"
huffman@44308
  1147
  using ln_less_cancel_iff [of x 1] by simp
huffman@44308
  1148
huffman@44308
  1149
lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
huffman@44308
  1150
  using ln_less_cancel_iff [of 1 x] by simp
huffman@44308
  1151
huffman@44308
  1152
lemma ln_gt_zero_imp_gt_one: "\<lbrakk>0 < ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 < x"
huffman@44308
  1153
  using ln_less_cancel_iff [of 1 x] by simp
huffman@44308
  1154
huffman@44308
  1155
lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> (0 < ln x) = (1 < x)"
huffman@44308
  1156
  using ln_less_cancel_iff [of 1 x] by simp
huffman@44308
  1157
huffman@44308
  1158
lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x = 0) = (x = 1)"
huffman@44308
  1159
  using ln_inj_iff [of x 1] by simp
huffman@44308
  1160
huffman@44308
  1161
lemma ln_less_zero: "\<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> ln x < 0"
huffman@44308
  1162
  by simp
paulson@15077
  1163
huffman@23045
  1164
lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
huffman@44308
  1165
  apply (subgoal_tac "isCont ln (exp (ln x))", simp)
huffman@44308
  1166
  apply (rule isCont_inverse_function [where f=exp], simp_all)
huffman@44308
  1167
  done
huffman@23045
  1168
huffman@45915
  1169
lemma tendsto_ln [tendsto_intros]:
huffman@45915
  1170
  "\<lbrakk>(f ---> a) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
huffman@45915
  1171
  by (rule isCont_tendsto_compose [OF isCont_ln])
huffman@45915
  1172
huffman@23045
  1173
lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
huffman@44308
  1174
  apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
huffman@44317
  1175
  apply (erule DERIV_cong [OF DERIV_exp exp_ln])
huffman@44308
  1176
  apply (simp_all add: abs_if isCont_ln)
huffman@44308
  1177
  done
huffman@23045
  1178
paulson@33667
  1179
lemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x"
paulson@33667
  1180
  by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
paulson@33667
  1181
hoelzl@29803
  1182
lemma ln_series: assumes "0 < x" and "x < 2"
hoelzl@29803
  1183
  shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))")
hoelzl@29803
  1184
proof -
hoelzl@29803
  1185
  let "?f' x n" = "(-1)^n * (x - 1)^n"
hoelzl@29803
  1186
hoelzl@29803
  1187
  have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
hoelzl@29803
  1188
  proof (rule DERIV_isconst3[where x=x])
hoelzl@29803
  1189
    fix x :: real assume "x \<in> {0 <..< 2}" hence "0 < x" and "x < 2" by auto
hoelzl@29803
  1190
    have "norm (1 - x) < 1" using `0 < x` and `x < 2` by auto
hoelzl@29803
  1191
    have "1 / x = 1 / (1 - (1 - x))" by auto
hoelzl@29803
  1192
    also have "\<dots> = (\<Sum> n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
hoelzl@29803
  1193
    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)
huffman@36777
  1194
    finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
hoelzl@29803
  1195
    moreover
hoelzl@29803
  1196
    have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
hoelzl@29803
  1197
    have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
hoelzl@29803
  1198
    proof (rule DERIV_power_series')
hoelzl@29803
  1199
      show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto
hoelzl@29803
  1200
      { fix x :: real assume "x \<in> {- 1<..<1}" hence "norm (-x) < 1" by auto
wenzelm@32960
  1201
        show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
huffman@30082
  1202
          unfolding One_nat_def
huffman@35216
  1203
          by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
hoelzl@29803
  1204
      }
hoelzl@29803
  1205
    qed
huffman@30082
  1206
    hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" unfolding One_nat_def by auto
hoelzl@29803
  1207
    hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos .
hoelzl@29803
  1208
    ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
hoelzl@29803
  1209
      by (rule DERIV_diff)
hoelzl@29803
  1210
    thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
hoelzl@29803
  1211
  qed (auto simp add: assms)
huffman@44289
  1212
  thus ?thesis by auto
hoelzl@29803
  1213
qed
paulson@15077
  1214
huffman@29164
  1215
subsection {* Sine and Cosine *}
huffman@29164
  1216
huffman@44308
  1217
definition sin_coeff :: "nat \<Rightarrow> real" where
huffman@31271
  1218
  "sin_coeff = (\<lambda>n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"
huffman@31271
  1219
huffman@44308
  1220
definition cos_coeff :: "nat \<Rightarrow> real" where
huffman@31271
  1221
  "cos_coeff = (\<lambda>n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"
huffman@31271
  1222
huffman@44308
  1223
definition sin :: "real \<Rightarrow> real" where
huffman@44308
  1224
  "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
huffman@44308
  1225
huffman@44308
  1226
definition cos :: "real \<Rightarrow> real" where
huffman@44308
  1227
  "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
huffman@31271
  1228
huffman@44319
  1229
lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
huffman@44319
  1230
  unfolding sin_coeff_def by simp
huffman@44319
  1231
huffman@44319
  1232
lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
huffman@44319
  1233
  unfolding cos_coeff_def by simp
huffman@44319
  1234
huffman@44319
  1235
lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
huffman@44319
  1236
  unfolding cos_coeff_def sin_coeff_def
huffman@44319
  1237
  by (simp del: mult_Suc)
huffman@44319
  1238
huffman@44319
  1239
lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
huffman@44319
  1240
  unfolding cos_coeff_def sin_coeff_def
huffman@44319
  1241
  by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)
huffman@44319
  1242
huffman@31271
  1243
lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
huffman@31271
  1244
unfolding sin_coeff_def
huffman@44308
  1245
apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
huffman@29164
  1246
apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
huffman@29164
  1247
done
huffman@29164
  1248
huffman@31271
  1249
lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"
huffman@31271
  1250
unfolding cos_coeff_def
huffman@44308
  1251
apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
huffman@29164
  1252
apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
huffman@29164
  1253
done
huffman@29164
  1254
huffman@31271
  1255
lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
huffman@29164
  1256
unfolding sin_def by (rule summable_sin [THEN summable_sums])
huffman@29164
  1257
huffman@31271
  1258
lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
huffman@29164
  1259
unfolding cos_def by (rule summable_cos [THEN summable_sums])
huffman@29164
  1260
huffman@44319
  1261
lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
huffman@44319
  1262
  by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
huffman@44319
  1263
huffman@44319
  1264
lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
huffman@44319
  1265
  by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
huffman@29164
  1266
huffman@29164
  1267
text{*Now at last we can get the derivatives of exp, sin and cos*}
huffman@29164
  1268
huffman@29164
  1269
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
huffman@44319
  1270
  unfolding sin_def cos_def
huffman@44319
  1271
  apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
huffman@44319
  1272
  apply (simp_all add: diffs_sin_coeff diffs_cos_coeff
huffman@44319
  1273
    summable_minus summable_sin summable_cos)
huffman@44319
  1274
  done
huffman@29164
  1275
huffman@29164
  1276
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
huffman@44319
  1277
  unfolding cos_def sin_def
huffman@44319
  1278
  apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
huffman@44319
  1279
  apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus
huffman@44319
  1280
    summable_minus summable_sin summable_cos suminf_minus)
huffman@44319
  1281
  done
huffman@29164
  1282
huffman@44311
  1283
lemma isCont_sin: "isCont sin x"
huffman@44311
  1284
  by (rule DERIV_sin [THEN DERIV_isCont])
huffman@44311
  1285
huffman@44311
  1286
lemma isCont_cos: "isCont cos x"
huffman@44311
  1287
  by (rule DERIV_cos [THEN DERIV_isCont])
huffman@44311
  1288
huffman@44311
  1289
lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
huffman@44311
  1290
  by (rule isCont_o2 [OF _ isCont_sin])
huffman@44311
  1291
huffman@44311
  1292
lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
huffman@44311
  1293
  by (rule isCont_o2 [OF _ isCont_cos])
huffman@44311
  1294
huffman@44311
  1295
lemma tendsto_sin [tendsto_intros]:
huffman@44311
  1296
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
huffman@44311
  1297
  by (rule isCont_tendsto_compose [OF isCont_sin])
huffman@44311
  1298
huffman@44311
  1299
lemma tendsto_cos [tendsto_intros]:
huffman@44311
  1300
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
huffman@44311
  1301
  by (rule isCont_tendsto_compose [OF isCont_cos])
huffman@29164
  1302
hoelzl@31880
  1303
declare
hoelzl@31880
  1304
  DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@43335
  1305
  DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  1306
  DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  1307
  DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  1308
huffman@29164
  1309
subsection {* Properties of Sine and Cosine *}
paulson@15077
  1310
paulson@15077
  1311
lemma sin_zero [simp]: "sin 0 = 0"
huffman@44311
  1312
  unfolding sin_def sin_coeff_def by (simp add: powser_zero)
paulson@15077
  1313
paulson@15077
  1314
lemma cos_zero [simp]: "cos 0 = 1"
huffman@44311
  1315
  unfolding cos_def cos_coeff_def by (simp add: powser_zero)
paulson@15077
  1316
huffman@44308
  1317
lemma sin_cos_squared_add [simp]: "(sin x)\<twosuperior> + (cos x)\<twosuperior> = 1"
huffman@44308
  1318
proof -
huffman@44308
  1319
  have "\<forall>x. DERIV (\<lambda>x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
huffman@44308
  1320
    by (auto intro!: DERIV_intros)
huffman@44308
  1321
  hence "(sin x)\<twosuperior> + (cos x)\<twosuperior> = (sin 0)\<twosuperior> + (cos 0)\<twosuperior>"
huffman@44308
  1322
    by (rule DERIV_isconst_all)
huffman@44308
  1323
  thus "(sin x)\<twosuperior> + (cos x)\<twosuperior> = 1" by simp
huffman@44308
  1324
qed
huffman@44308
  1325
huffman@44308
  1326
lemma sin_cos_squared_add2 [simp]: "(cos x)\<twosuperior> + (sin x)\<twosuperior> = 1"
huffman@44308
  1327
  by (subst add_commute, rule sin_cos_squared_add)
paulson@15077
  1328
paulson@15077
  1329
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
huffman@44308
  1330
  using sin_cos_squared_add2 [unfolded power2_eq_square] .
paulson@15077
  1331
paulson@15077
  1332
lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
huffman@44308
  1333
  unfolding eq_diff_eq by (rule sin_cos_squared_add)
paulson@15077
  1334
paulson@15077
  1335
lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
huffman@44308
  1336
  unfolding eq_diff_eq by (rule sin_cos_squared_add2)
paulson@15077
  1337
paulson@15081
  1338
lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
huffman@44308
  1339
  by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
paulson@15077
  1340
paulson@15077
  1341
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
huffman@44308
  1342
  using abs_sin_le_one [of x] unfolding abs_le_iff by simp
paulson@15077
  1343
paulson@15077
  1344
lemma sin_le_one [simp]: "sin x \<le> 1"
huffman@44308
  1345
  using abs_sin_le_one [of x] unfolding abs_le_iff by simp
paulson@15077
  1346
paulson@15081
  1347
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
huffman@44308
  1348
  by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
paulson@15077
  1349
paulson@15077
  1350
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
huffman@44308
  1351
  using abs_cos_le_one [of x] unfolding abs_le_iff by simp
paulson@15077
  1352
paulson@15077
  1353
lemma cos_le_one [simp]: "cos x \<le> 1"
huffman@44308
  1354
  using abs_cos_le_one [of x] unfolding abs_le_iff by simp
paulson@15077
  1355
hoelzl@41970
  1356
lemma DERIV_fun_pow: "DERIV g x :> m ==>
paulson@15077
  1357
      DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
huffman@44311
  1358
  by (auto intro!: DERIV_intros)
paulson@15077
  1359
paulson@15229
  1360
lemma DERIV_fun_exp:
paulson@15229
  1361
     "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
huffman@44311
  1362
  by (auto intro!: DERIV_intros)
paulson@15077
  1363
paulson@15229
  1364
lemma DERIV_fun_sin:
paulson@15229
  1365
     "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
huffman@44311
  1366
  by (auto intro!: DERIV_intros)
paulson@15077
  1367
paulson@15229
  1368
lemma DERIV_fun_cos:
paulson@15229
  1369
     "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
huffman@44311
  1370
  by (auto intro!: DERIV_intros)
paulson@15077
  1371
huffman@44308
  1372
lemma sin_cos_add_lemma:
hoelzl@41970
  1373
     "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
paulson@15077
  1374
      (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
huffman@44308
  1375
  (is "?f x = 0")
huffman@44308
  1376
proof -
huffman@44308
  1377
  have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
huffman@44308
  1378
    by (auto intro!: DERIV_intros simp add: algebra_simps)
huffman@44308
  1379
  hence "?f x = ?f 0"
huffman@44308
  1380
    by (rule DERIV_isconst_all)
huffman@44308
  1381
  thus ?thesis by simp
huffman@44308
  1382
qed
paulson@15077
  1383
paulson@15077
  1384
lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
huffman@44308
  1385
  using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
paulson@15077
  1386
paulson@15077
  1387
lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
huffman@44308
  1388
  using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
huffman@44308
  1389
huffman@44308
  1390
lemma sin_cos_minus_lemma:
huffman@44308
  1391
  "(sin(-x) + sin(x))\<twosuperior> + (cos(-x) - cos(x))\<twosuperior> = 0" (is "?f x = 0")
huffman@44308
  1392
proof -
huffman@44308
  1393
  have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
huffman@44308
  1394
    by (auto intro!: DERIV_intros simp add: algebra_simps)
huffman@44308
  1395
  hence "?f x = ?f 0"
huffman@44308
  1396
    by (rule DERIV_isconst_all)
huffman@44308
  1397
  thus ?thesis by simp
huffman@44308
  1398
qed
paulson@15077
  1399
paulson@15077
  1400
lemma sin_minus [simp]: "sin (-x) = -sin(x)"
huffman@44308
  1401
  using sin_cos_minus_lemma [where x=x] by simp
paulson@15077
  1402
paulson@15077
  1403
lemma cos_minus [simp]: "cos (-x) = cos(x)"
huffman@44308
  1404
  using sin_cos_minus_lemma [where x=x] by simp
paulson@15077
  1405
paulson@15077
  1406
lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
huffman@44308
  1407
  by (simp add: diff_minus sin_add)
paulson@15077
  1408
paulson@15077
  1409
lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
huffman@44308
  1410
  by (simp add: sin_diff mult_commute)
paulson@15077
  1411
paulson@15077
  1412
lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
huffman@44308
  1413
  by (simp add: diff_minus cos_add)
paulson@15077
  1414
paulson@15077
  1415
lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
huffman@44308
  1416
  by (simp add: cos_diff mult_commute)
paulson@15077
  1417
paulson@15077
  1418
lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
huffman@29165
  1419
  using sin_add [where x=x and y=x] by simp
paulson@15077
  1420
paulson@15077
  1421
lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
huffman@29165
  1422
  using cos_add [where x=x and y=x]
huffman@29165
  1423
  by (simp add: power2_eq_square)
paulson@15077
  1424
paulson@15077
  1425
huffman@29164
  1426
subsection {* The Constant Pi *}
paulson@15077
  1427
huffman@44308
  1428
definition pi :: "real" where
huffman@23053
  1429
  "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
huffman@23043
  1430
hoelzl@41970
  1431
text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
paulson@15077
  1432
   hence define pi.*}
paulson@15077
  1433
paulson@15077
  1434
lemma sin_paired:
hoelzl@41970
  1435
     "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1))
paulson@15077
  1436
      sums  sin x"
paulson@15077
  1437
proof -
huffman@31271
  1438
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
huffman@44727
  1439
    by (rule sin_converges [THEN sums_group], simp)
huffman@31271
  1440
  thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)
paulson@15077
  1441
qed
paulson@15077
  1442
huffman@44728
  1443
lemma sin_gt_zero:
huffman@44728
  1444
  assumes "0 < x" and "x < 2" shows "0 < sin x"
huffman@44728
  1445
proof -
huffman@44728
  1446
  let ?f = "\<lambda>n. \<Sum>k = n*2..<n*2+2. -1 ^ k / real (fact (2*k+1)) * x^(2*k+1)"
huffman@44728
  1447
  have pos: "\<forall>n. 0 < ?f n"
huffman@44728
  1448
  proof
huffman@44728
  1449
    fix n :: nat
huffman@44728
  1450
    let ?k2 = "real (Suc (Suc (4 * n)))"
huffman@44728
  1451
    let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
huffman@44728
  1452
    have "x * x < ?k2 * ?k3"
huffman@44728
  1453
      using assms by (intro mult_strict_mono', simp_all)
huffman@44728
  1454
    hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
huffman@44728
  1455
      by (intro mult_strict_right_mono zero_less_power `0 < x`)
huffman@44728
  1456
    thus "0 < ?f n"
huffman@44728
  1457
      by (simp del: mult_Suc,
huffman@44728
  1458
        simp add: less_divide_eq mult_pos_pos field_simps del: mult_Suc)
huffman@44728
  1459
  qed
huffman@44728
  1460
  have sums: "?f sums sin x"
huffman@44728
  1461
    by (rule sin_paired [THEN sums_group], simp)
huffman@44728
  1462
  show "0 < sin x"
huffman@44728
  1463
    unfolding sums_unique [OF sums]
huffman@44728
  1464
    using sums_summable [OF sums] pos
huffman@44728
  1465
    by (rule suminf_gt_zero)
huffman@44728
  1466
qed
paulson@15077
  1467
paulson@15077
  1468
lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
huffman@44311
  1469
apply (cut_tac x = x in sin_gt_zero)
paulson@15077
  1470
apply (auto simp add: cos_squared_eq cos_double)
paulson@15077
  1471
done
paulson@15077
  1472
paulson@15077
  1473
lemma cos_paired:
huffman@23177
  1474
     "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
paulson@15077
  1475
proof -
huffman@31271
  1476
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
huffman@44727
  1477
    by (rule cos_converges [THEN sums_group], simp)
huffman@31271
  1478
  thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)
paulson@15077
  1479
qed
paulson@15077
  1480
huffman@36824
  1481
lemma real_mult_inverse_cancel:
hoelzl@41970
  1482
     "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
huffman@36824
  1483
      ==> inverse x * y < inverse x1 * u"
hoelzl@41970
  1484
apply (rule_tac c=x in mult_less_imp_less_left)
huffman@36824
  1485
apply (auto simp add: mult_assoc [symmetric])
huffman@36824
  1486
apply (simp (no_asm) add: mult_ac)
hoelzl@41970
  1487
apply (rule_tac c=x1 in mult_less_imp_less_right)
huffman@36824
  1488
apply (auto simp add: mult_ac)
huffman@36824
  1489
done
huffman@36824
  1490
huffman@36824
  1491
lemma real_mult_inverse_cancel2:
huffman@36824
  1492
     "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
huffman@36824
  1493
apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)
huffman@36824
  1494
done
huffman@36824
  1495
huffman@36824
  1496
lemma realpow_num_eq_if:
huffman@36824
  1497
  fixes m :: "'a::power"
huffman@36824
  1498
  shows "m ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
huffman@36824
  1499
by (cases n, auto)
huffman@36824
  1500
huffman@23053
  1501
lemma cos_two_less_zero [simp]: "cos (2) < 0"
paulson@15077
  1502
apply (cut_tac x = 2 in cos_paired)
paulson@15077
  1503
apply (drule sums_minus)
hoelzl@41970
  1504
apply (rule neg_less_iff_less [THEN iffD1])
nipkow@15539
  1505
apply (frule sums_unique, auto)
nipkow@15539
  1506
apply (rule_tac y =
huffman@23177
  1507
 "\<Sum>n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
paulson@15481
  1508
       in order_less_trans)
avigad@32047
  1509
apply (simp (no_asm) add: fact_num_eq_if_nat realpow_num_eq_if del: fact_Suc)
nipkow@15561
  1510
apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
paulson@15077
  1511
apply (rule sumr_pos_lt_pair)
paulson@15077
  1512
apply (erule sums_summable, safe)
huffman@30082
  1513
unfolding One_nat_def
hoelzl@41970
  1514
apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
avigad@32047
  1515
            del: fact_Suc)
huffman@46240
  1516
apply (simp add: inverse_eq_divide less_divide_eq del: fact_Suc)
avigad@32047
  1517
apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
paulson@15481
  1518
apply (simp only: real_of_nat_mult)
huffman@23007
  1519
apply (rule mult_strict_mono, force)
huffman@27483
  1520
  apply (rule_tac [3] real_of_nat_ge_zero)
paulson@15481
  1521
 prefer 2 apply force
paulson@15077
  1522
apply (rule real_of_nat_less_iff [THEN iffD2])
avigad@32036
  1523
apply (rule fact_less_mono_nat, auto)
paulson@15077
  1524
done
huffman@23053
  1525
huffman@23053
  1526
lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
huffman@23053
  1527
lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
paulson@15077
  1528
paulson@15077
  1529
lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
huffman@44730
  1530
proof (rule ex_ex1I)
huffman@44730
  1531
  show "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0"
huffman@44730
  1532
    by (rule IVT2, simp_all)
huffman@44730
  1533
next
huffman@44730
  1534
  fix x y
huffman@44730
  1535
  assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
huffman@44730
  1536
  assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
huffman@44730
  1537
  have [simp]: "\<forall>x. cos differentiable x"
huffman@44730
  1538
    unfolding differentiable_def by (auto intro: DERIV_cos)
huffman@44730
  1539
  from x y show "x = y"
huffman@44730
  1540
    apply (cut_tac less_linear [of x y], auto)
huffman@44730
  1541
    apply (drule_tac f = cos in Rolle)
huffman@44730
  1542
    apply (drule_tac [5] f = cos in Rolle)
huffman@44730
  1543
    apply (auto dest!: DERIV_cos [THEN DERIV_unique])
huffman@44730
  1544
    apply (metis order_less_le_trans less_le sin_gt_zero)
huffman@44730
  1545
    apply (metis order_less_le_trans less_le sin_gt_zero)
huffman@44730
  1546
    done
huffman@44730
  1547
qed
hoelzl@31880
  1548
huffman@23053
  1549
lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
paulson@15077
  1550
by (simp add: pi_def)
paulson@15077
  1551
paulson@15077
  1552
lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
huffman@23053
  1553
by (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1554
huffman@23053
  1555
lemma pi_half_gt_zero [simp]: "0 < pi / 2"
huffman@23053
  1556
apply (rule order_le_neq_trans)
huffman@23053
  1557
apply (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1558
apply (rule notI, drule arg_cong [where f=cos], simp)
paulson@15077
  1559
done
paulson@15077
  1560
huffman@23053
  1561
lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
huffman@23053
  1562
lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
paulson@15077
  1563
huffman@23053
  1564
lemma pi_half_less_two [simp]: "pi / 2 < 2"
huffman@23053
  1565
apply (rule order_le_neq_trans)
huffman@23053
  1566
apply (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1567
apply (rule notI, drule arg_cong [where f=cos], simp)
paulson@15077
  1568
done
huffman@23053
  1569
huffman@23053
  1570
lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
huffman@23053
  1571
lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
paulson@15077
  1572
paulson@15077
  1573
lemma pi_gt_zero [simp]: "0 < pi"
huffman@23053
  1574
by (insert pi_half_gt_zero, simp)
huffman@23053
  1575
huffman@23053
  1576
lemma pi_ge_zero [simp]: "0 \<le> pi"
huffman@23053
  1577
by (rule pi_gt_zero [THEN order_less_imp_le])
paulson@15077
  1578
paulson@15077
  1579
lemma pi_neq_zero [simp]: "pi \<noteq> 0"
huffman@22998
  1580
by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym])
paulson@15077
  1581
huffman@23053
  1582
lemma pi_not_less_zero [simp]: "\<not> pi < 0"
huffman@23053
  1583
by (simp add: linorder_not_less)
paulson@15077
  1584
huffman@29165
  1585
lemma minus_pi_half_less_zero: "-(pi/2) < 0"
huffman@29165
  1586
by simp
paulson@15077
  1587
hoelzl@29803
  1588
lemma m2pi_less_pi: "- (2 * pi) < pi"
huffman@45308
  1589
by simp
hoelzl@29803
  1590
paulson@15077
  1591
lemma sin_pi_half [simp]: "sin(pi/2) = 1"
paulson@15077
  1592
apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
paulson@15077
  1593
apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
huffman@36970
  1594
apply (simp add: power2_eq_1_iff)
paulson@15077
  1595
done
paulson@15077
  1596
paulson@15077
  1597
lemma cos_pi [simp]: "cos pi = -1"
nipkow@15539
  1598
by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
paulson@15077
  1599
paulson@15077
  1600
lemma sin_pi [simp]: "sin pi = 0"
nipkow@15539
  1601
by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
paulson@15077
  1602
paulson@15077
  1603
lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
huffman@45309
  1604
by (simp add: cos_diff)
paulson@15077
  1605
paulson@15077
  1606
lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
paulson@15229
  1607
by (simp add: cos_add)
paulson@15077
  1608
paulson@15077
  1609
lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
huffman@45309
  1610
by (simp add: sin_diff)
paulson@15077
  1611
paulson@15077
  1612
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
paulson@15229
  1613
by (simp add: sin_add)
paulson@15077
  1614
paulson@15077
  1615
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
paulson@15229
  1616
by (simp add: sin_add)
paulson@15077
  1617
paulson@15077
  1618
lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
paulson@15229
  1619
by (simp add: cos_add)
paulson@15077
  1620
paulson@15077
  1621
lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
paulson@15077
  1622
by (simp add: sin_add cos_double)
paulson@15077
  1623
paulson@15077
  1624
lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
paulson@15077
  1625
by (simp add: cos_add cos_double)
paulson@15077
  1626
paulson@15077
  1627
lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
paulson@15251
  1628
apply (induct "n")
paulson@15077
  1629
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1630
done
paulson@15077
  1631
paulson@15383
  1632
lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
paulson@15383
  1633
proof -
paulson@15383
  1634
  have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
hoelzl@41970
  1635
  also have "... = -1 ^ n" by (rule cos_npi)
paulson@15383
  1636
  finally show ?thesis .
paulson@15383
  1637
qed
paulson@15383
  1638
paulson@15077
  1639
lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
paulson@15251
  1640
apply (induct "n")
paulson@15077
  1641
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1642
done
paulson@15077
  1643
paulson@15077
  1644
lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
hoelzl@41970
  1645
by (simp add: mult_commute [of pi])
paulson@15077
  1646
paulson@15077
  1647
lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
paulson@15077
  1648
by (simp add: cos_double)
paulson@15077
  1649
paulson@15077
  1650
lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
paulson@15229
  1651
by simp
paulson@15077
  1652
paulson@15077
  1653
lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
paulson@15077
  1654
apply (rule sin_gt_zero, assumption)
paulson@15077
  1655
apply (rule order_less_trans, assumption)
paulson@15077
  1656
apply (rule pi_half_less_two)
paulson@15077
  1657
done
paulson@15077
  1658
hoelzl@41970
  1659
lemma sin_less_zero:
paulson@15077
  1660
  assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
paulson@15077
  1661
proof -
hoelzl@41970
  1662
  have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
paulson@15077
  1663
  thus ?thesis by simp
paulson@15077
  1664
qed
paulson@15077
  1665
paulson@15077
  1666
lemma pi_less_4: "pi < 4"
paulson@15077
  1667
by (cut_tac pi_half_less_two, auto)
paulson@15077
  1668
paulson@15077
  1669
lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1670
apply (cut_tac pi_less_4)
paulson@15077
  1671
apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
paulson@15077
  1672
apply (cut_tac cos_is_zero, safe)
paulson@15077
  1673
apply (rename_tac y z)
paulson@15077
  1674
apply (drule_tac x = y in spec)
hoelzl@41970
  1675
apply (drule_tac x = "pi/2" in spec, simp)
paulson@15077
  1676
done
paulson@15077
  1677
paulson@15077
  1678
lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1679
apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15077
  1680
apply (rule cos_minus [THEN subst])
paulson@15077
  1681
apply (rule cos_gt_zero)
paulson@15077
  1682
apply (auto intro: cos_gt_zero)
paulson@15077
  1683
done
hoelzl@41970
  1684
paulson@15077
  1685
lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
paulson@15077
  1686
apply (auto simp add: order_le_less cos_gt_zero_pi)
hoelzl@41970
  1687
apply (subgoal_tac "x = pi/2", auto)
paulson@15077
  1688
done
paulson@15077
  1689
paulson@15077
  1690
lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
huffman@45309
  1691
by (simp add: sin_cos_eq cos_gt_zero_pi)
hoelzl@29803
  1692
hoelzl@29803
  1693
lemma pi_ge_two: "2 \<le> pi"
hoelzl@29803
  1694
proof (rule ccontr)
hoelzl@29803
  1695
  assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
hoelzl@29803
  1696
  have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
hoelzl@29803
  1697
  proof (cases "2 < 2 * pi")
hoelzl@29803
  1698
    case True with dense[OF `pi < 2`] show ?thesis by auto
hoelzl@29803
  1699
  next
hoelzl@29803
  1700
    case False have "pi < 2 * pi" by auto
hoelzl@29803
  1701
    from dense[OF this] and False show ?thesis by auto
hoelzl@29803
  1702
  qed
hoelzl@29803
  1703
  then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
hoelzl@29803
  1704
  hence "0 < sin y" using sin_gt_zero by auto
hoelzl@41970
  1705
  moreover
hoelzl@29803
  1706
  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
hoelzl@29803
  1707
  ultimately show False by auto
hoelzl@29803
  1708
qed
hoelzl@29803
  1709
paulson@15077
  1710
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
paulson@15077
  1711
by (auto simp add: order_le_less sin_gt_zero_pi)
paulson@15077
  1712
huffman@44745
  1713
text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}.
huffman@44745
  1714
  It should be possible to factor out some of the common parts. *}
huffman@44745
  1715
paulson@15077
  1716
lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
huffman@44745
  1717
proof (rule ex_ex1I)
huffman@44745
  1718
  assume y: "-1 \<le> y" "y \<le> 1"
huffman@44745
  1719
  show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
huffman@44745
  1720
    by (rule IVT2, simp_all add: y)
huffman@44745
  1721
next
huffman@44745
  1722
  fix a b
huffman@44745
  1723
  assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
huffman@44745
  1724
  assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
huffman@44745
  1725
  have [simp]: "\<forall>x. cos differentiable x"
huffman@44745
  1726
    unfolding differentiable_def by (auto intro: DERIV_cos)
huffman@44745
  1727
  from a b show "a = b"
huffman@44745
  1728
    apply (cut_tac less_linear [of a b], auto)
huffman@44745
  1729
    apply (drule_tac f = cos in Rolle)
huffman@44745
  1730
    apply (drule_tac [5] f = cos in Rolle)
huffman@44745
  1731
    apply (auto dest!: DERIV_cos [THEN DERIV_unique])
huffman@44745
  1732
    apply (metis order_less_le_trans less_le sin_gt_zero_pi)
huffman@44745
  1733
    apply (metis order_less_le_trans less_le sin_gt_zero_pi)
huffman@44745
  1734
    done
huffman@44745
  1735
qed
paulson@15077
  1736
paulson@15077
  1737
lemma sin_total:
paulson@15077
  1738
     "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
paulson@15077
  1739
apply (rule ccontr)
paulson@15077
  1740
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))")
wenzelm@18585
  1741
apply (erule contrapos_np)
huffman@45309
  1742
apply simp
hoelzl@41970
  1743
apply (cut_tac y="-y" in cos_total, simp) apply simp
paulson@15077
  1744
apply (erule ex1E)
paulson@15229
  1745
apply (rule_tac a = "x - (pi/2)" in ex1I)
huffman@23286
  1746
apply (simp (no_asm) add: add_assoc)
paulson@15077
  1747
apply (rotate_tac 3)
huffman@45309
  1748
apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all add: cos_add)
paulson@15077
  1749
done
paulson@15077
  1750
paulson@15077
  1751
lemma reals_Archimedean4:
paulson@15077
  1752
     "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
paulson@15077
  1753
apply (auto dest!: reals_Archimedean3)
hoelzl@41970
  1754
apply (drule_tac x = x in spec, clarify)
paulson@15077
  1755
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
hoelzl@41970
  1756
 prefer 2 apply (erule LeastI)
hoelzl@41970
  1757
apply (case_tac "LEAST m::nat. x < real m * y", simp)
paulson@15077
  1758
apply (subgoal_tac "~ x < real nat * y")
hoelzl@41970
  1759
 prefer 2 apply (rule not_less_Least, simp, force)
paulson@15077
  1760
done
paulson@15077
  1761
hoelzl@41970
  1762
(* Pre Isabelle99-2 proof was simpler- numerals arithmetic
paulson@15077
  1763
   now causes some unwanted re-arrangements of literals!   *)
paulson@15229
  1764
lemma cos_zero_lemma:
hoelzl@41970
  1765
     "[| 0 \<le> x; cos x = 0 |] ==>
paulson@15077
  1766
      \<exists>n::nat. ~even n & x = real n * (pi/2)"
paulson@15077
  1767
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
hoelzl@41970
  1768
apply (subgoal_tac "0 \<le> x - real n * pi &
paulson@15086
  1769
                    (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
nipkow@29667
  1770
apply (auto simp add: algebra_simps real_of_nat_Suc)
nipkow@29667
  1771
 prefer 2 apply (simp add: cos_diff)
paulson@15077
  1772
apply (simp add: cos_diff)
paulson@15077
  1773
apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
paulson@15077
  1774
apply (rule_tac [2] cos_total, safe)
paulson@15077
  1775
apply (drule_tac x = "x - real n * pi" in spec)
paulson@15077
  1776
apply (drule_tac x = "pi/2" in spec)
paulson@15077
  1777
apply (simp add: cos_diff)
paulson@15229
  1778
apply (rule_tac x = "Suc (2 * n)" in exI)
nipkow@29667
  1779
apply (simp add: real_of_nat_Suc algebra_simps, auto)
paulson@15077
  1780
done
paulson@15077
  1781
paulson@15229
  1782
lemma sin_zero_lemma:
hoelzl@41970
  1783
     "[| 0 \<le> x; sin x = 0 |] ==>
paulson@15077
  1784
      \<exists>n::nat. even n & x = real n * (pi/2)"
paulson@15077
  1785
apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
paulson@15077
  1786
 apply (clarify, rule_tac x = "n - 1" in exI)
paulson@15077
  1787
 apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
paulson@15085
  1788
apply (rule cos_zero_lemma)
huffman@45309
  1789
apply (simp_all add: cos_add)
paulson@15077
  1790
done
paulson@15077
  1791
paulson@15077
  1792
paulson@15229
  1793
lemma cos_zero_iff:
hoelzl@41970
  1794
     "(cos x = 0) =
hoelzl@41970
  1795
      ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
paulson@15077
  1796
       (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1797
apply (rule iffI)
paulson@15077
  1798
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1799
apply (drule cos_zero_lemma, assumption+)
hoelzl@41970
  1800
apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
hoelzl@41970
  1801
apply (force simp add: minus_equation_iff [of x])
hoelzl@41970
  1802
apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
nipkow@15539
  1803
apply (auto simp add: cos_add)
paulson@15077
  1804
done
paulson@15077
  1805
paulson@15077
  1806
(* ditto: but to a lesser extent *)
paulson@15229
  1807
lemma sin_zero_iff:
hoelzl@41970
  1808
     "(sin x = 0) =
hoelzl@41970
  1809
      ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
paulson@15077
  1810
       (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1811
apply (rule iffI)
paulson@15077
  1812
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1813
apply (drule sin_zero_lemma, assumption+)
paulson@15077
  1814
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
hoelzl@41970
  1815
apply (force simp add: minus_equation_iff [of x])
nipkow@15539
  1816
apply (auto simp add: even_mult_two_ex)
paulson@15077
  1817
done
paulson@15077
  1818
hoelzl@29803
  1819
lemma cos_monotone_0_pi: assumes "0 \<le> y" and "y < x" and "x \<le> pi"
hoelzl@29803
  1820
  shows "cos x < cos y"
hoelzl@29803
  1821
proof -
wenzelm@33549
  1822
  have "- (x - y) < 0" using assms by auto
hoelzl@29803
  1823
hoelzl@29803
  1824
  from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
hoelzl@29803
  1825
  obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" by auto
wenzelm@33549
  1826
  hence "0 < z" and "z < pi" using assms by auto
hoelzl@29803
  1827
  hence "0 < sin z" using sin_gt_zero_pi by auto
hoelzl@29803
  1828
  hence "cos x - cos y < 0" unfolding cos_diff minus_mult_commute[symmetric] using `- (x - y) < 0` by (rule mult_pos_neg2)
hoelzl@29803
  1829
  thus ?thesis by auto
hoelzl@29803
  1830
qed
hoelzl@29803
  1831
hoelzl@29803
  1832
lemma cos_monotone_0_pi': assumes "0 \<le> y" and "y \<le> x" and "x \<le> pi" shows "cos x \<le> cos y"
hoelzl@29803
  1833
proof (cases "y < x")
hoelzl@29803
  1834
  case True show ?thesis using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
hoelzl@29803
  1835
next
hoelzl@29803
  1836
  case False hence "y = x" using `y \<le> x` by auto
hoelzl@29803
  1837
  thus ?thesis by auto
hoelzl@29803
  1838
qed
hoelzl@29803
  1839
hoelzl@29803
  1840
lemma cos_monotone_minus_pi_0: assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
hoelzl@29803
  1841
  shows "cos y < cos x"
hoelzl@29803
  1842
proof -
wenzelm@33549
  1843
  have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" using assms by auto
hoelzl@29803
  1844
  from cos_monotone_0_pi[OF this]
hoelzl@29803
  1845
  show ?thesis unfolding cos_minus .
hoelzl@29803
  1846
qed
hoelzl@29803
  1847
hoelzl@29803
  1848
lemma cos_monotone_minus_pi_0': assumes "-pi \<le> y" and "y \<le> x" and "x \<le> 0" shows "cos y \<le> cos x"
hoelzl@29803
  1849
proof (cases "y < x")
hoelzl@29803
  1850
  case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] by auto
hoelzl@29803
  1851
next
hoelzl@29803
  1852
  case False hence "y = x" using `y \<le> x` by auto
hoelzl@29803
  1853
  thus ?thesis by auto
hoelzl@29803
  1854
qed
hoelzl@29803
  1855
hoelzl@29803
  1856
lemma sin_monotone_2pi': assumes "- (pi / 2) \<le> y" and "y \<le> x" and "x \<le> pi / 2" shows "sin y \<le> sin x"
hoelzl@29803
  1857
proof -
wenzelm@33549
  1858
  have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi"
wenzelm@33549
  1859
    using pi_ge_two and assms by auto
hoelzl@29803
  1860
  from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] by auto
hoelzl@29803
  1861
qed
paulson@15077
  1862
huffman@29164
  1863
subsection {* Tangent *}
paulson@15077
  1864
huffman@44311
  1865
definition tan :: "real \<Rightarrow> real" where
huffman@44311
  1866
  "tan = (\<lambda>x. sin x / cos x)"
huffman@23043
  1867
paulson@15077
  1868
lemma tan_zero [simp]: "tan 0 = 0"
huffman@44311
  1869
  by (simp add: tan_def)
paulson@15077
  1870
paulson@15077
  1871
lemma tan_pi [simp]: "tan pi = 0"
huffman@44311
  1872
  by (simp add: tan_def)
paulson@15077
  1873
paulson@15077
  1874
lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
huffman@44311
  1875
  by (simp add: tan_def)
paulson@15077
  1876
paulson@15077
  1877
lemma tan_minus [simp]: "tan (-x) = - tan x"
huffman@44311
  1878
  by (simp add: tan_def)
paulson@15077
  1879
paulson@15077
  1880
lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
huffman@44311
  1881
  by (simp add: tan_def)
paulson@15077
  1882
hoelzl@41970
  1883
lemma lemma_tan_add1:
huffman@44311
  1884
  "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> 1 - tan x * tan y = cos (x + y)/(cos x * cos y)"
huffman@44311
  1885
  by (simp add: tan_def cos_add field_simps)
paulson@15077
  1886
hoelzl@41970
  1887
lemma add_tan_eq:
huffman@44311
  1888
  "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
huffman@44311
  1889
  by (simp add: tan_def sin_add field_simps)
paulson@15077
  1890
paulson@15229
  1891
lemma tan_add:
hoelzl@41970
  1892
     "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
paulson@15077
  1893
      ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
huffman@44311
  1894
  by (simp add: add_tan_eq lemma_tan_add1, simp add: tan_def)
paulson@15077
  1895
paulson@15229
  1896
lemma tan_double:
hoelzl@41970
  1897
     "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
paulson@15077
  1898
      ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
huffman@44311
  1899
  using tan_add [of x x] by (simp add: power2_eq_square)
paulson@15077
  1900
paulson@15077
  1901
lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
hoelzl@41970
  1902
by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
hoelzl@41970
  1903
hoelzl@41970
  1904
lemma tan_less_zero:
paulson@15077
  1905
  assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
paulson@15077
  1906
proof -
hoelzl@41970
  1907
  have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
paulson@15077
  1908
  thus ?thesis by simp
paulson@15077
  1909
qed
paulson@15077
  1910
huffman@44756
  1911
lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"
huffman@44756
  1912
  unfolding tan_def sin_double cos_double sin_squared_eq
huffman@44756
  1913
  by (simp add: power2_eq_square)
hoelzl@29803
  1914
huffman@44311
  1915
lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<twosuperior>)"
huffman@44311
  1916
  unfolding tan_def
huffman@44311
  1917
  by (auto intro!: DERIV_intros, simp add: divide_inverse power2_eq_square)
huffman@44311
  1918
huffman@44311
  1919
lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
huffman@44311
  1920
  by (rule DERIV_tan [THEN DERIV_isCont])
huffman@44311
  1921
huffman@44311
  1922
lemma isCont_tan' [simp]:
huffman@44311
  1923
  "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
huffman@44311
  1924
  by (rule isCont_o2 [OF _ isCont_tan])
huffman@44311
  1925
huffman@44311
  1926
lemma tendsto_tan [tendsto_intros]:
huffman@44311
  1927
  "\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F"
huffman@44311
  1928
  by (rule isCont_tendsto_compose [OF isCont_tan])
huffman@44311
  1929
huffman@44311
  1930
lemma LIM_cos_div_sin: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
huffman@44311
  1931
  by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
paulson@15077
  1932
paulson@15077
  1933
lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
paulson@15077
  1934
apply (cut_tac LIM_cos_div_sin)
huffman@31338
  1935
apply (simp only: LIM_eq)
paulson@15077
  1936
apply (drule_tac x = "inverse y" in spec, safe, force)
paulson@15077
  1937
apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
paulson@15229
  1938
apply (rule_tac x = "(pi/2) - e" in exI)
paulson@15077
  1939
apply (simp (no_asm_simp))
paulson@15229
  1940
apply (drule_tac x = "(pi/2) - e" in spec)
huffman@45309
  1941
apply (auto simp add: tan_def sin_diff cos_diff)
paulson@15077
  1942
apply (rule inverse_less_iff_less [THEN iffD1])
paulson@15079
  1943
apply (auto simp add: divide_inverse)
huffman@36777
  1944
apply (rule mult_pos_pos)
paulson@15229
  1945
apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
huffman@36777
  1946
apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
paulson@15077
  1947
done
paulson@15077
  1948
paulson@15077
  1949
lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
huffman@22998
  1950
apply (frule order_le_imp_less_or_eq, safe)
paulson@15077
  1951
 prefer 2 apply force
paulson@15077
  1952
apply (drule lemma_tan_total, safe)
paulson@15077
  1953
apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
paulson@15077
  1954
apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
paulson@15077
  1955
apply (drule_tac y = xa in order_le_imp_less_or_eq)
paulson@15077
  1956
apply (auto dest: cos_gt_zero)
paulson@15077
  1957
done
paulson@15077
  1958
paulson@15077
  1959
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  1960
apply (cut_tac linorder_linear [of 0 y], safe)
paulson@15077
  1961
apply (drule tan_total_pos)
paulson@15077
  1962
apply (cut_tac [2] y="-y" in tan_total_pos, safe)
paulson@15077
  1963
apply (rule_tac [3] x = "-x" in exI)
huffman@44710
  1964
apply (auto del: exI intro!: exI)
paulson@15077
  1965
done
paulson@15077
  1966
paulson@15077
  1967
lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  1968
apply (cut_tac y = y in lemma_tan_total1, auto)
paulson@15077
  1969
apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
paulson@15077
  1970
apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
paulson@15077
  1971
apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
paulson@15077
  1972
apply (rule_tac [4] Rolle)
paulson@15077
  1973
apply (rule_tac [2] Rolle)
huffman@44710
  1974
apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
paulson@15077
  1975
            simp add: differentiable_def)
paulson@15077
  1976
txt{*Now, simulate TRYALL*}
paulson@15077
  1977
apply (rule_tac [!] DERIV_tan asm_rl)
paulson@15077
  1978
apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
hoelzl@41970
  1979
            simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
paulson@15077
  1980
done
paulson@15077
  1981
hoelzl@29803
  1982
lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
hoelzl@29803
  1983
  shows "tan y < tan x"
hoelzl@29803
  1984
proof -
hoelzl@29803
  1985
  have "\<forall> x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse (cos x'^2)"
hoelzl@29803
  1986
  proof (rule allI, rule impI)
hoelzl@29803
  1987
    fix x' :: real assume "y \<le> x' \<and> x' \<le> x"
wenzelm@33549
  1988
    hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
hoelzl@29803
  1989
    from cos_gt_zero_pi[OF this]
hoelzl@29803
  1990
    have "cos x' \<noteq> 0" by auto
hoelzl@29803
  1991
    thus "DERIV tan x' :> inverse (cos x'^2)" by (rule DERIV_tan)
hoelzl@29803
  1992
  qed
hoelzl@41970
  1993
  from MVT2[OF `y < x` this]
hoelzl@29803
  1994
  obtain z where "y < z" and "z < x" and tan_diff: "tan x - tan y = (x - y) * inverse ((cos z)\<twosuperior>)" by auto
wenzelm@33549
  1995
  hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
hoelzl@29803
  1996
  hence "0 < cos z" using cos_gt_zero_pi by auto
hoelzl@29803
  1997
  hence inv_pos: "0 < inverse ((cos z)\<twosuperior>)" by auto
hoelzl@29803
  1998
  have "0 < x - y" using `y < x` by auto
huffman@36777
  1999
  from mult_pos_pos [OF this inv_pos]
hoelzl@29803
  2000
  have "0 < tan x - tan y" unfolding tan_diff by auto
hoelzl@29803
  2001
  thus ?thesis by auto
hoelzl@29803
  2002
qed
hoelzl@29803
  2003
hoelzl@29803
  2004
lemma tan_monotone': assumes "- (pi / 2) < y" and "y < pi / 2" and "- (pi / 2) < x" and "x < pi / 2"
hoelzl@29803
  2005
  shows "(y < x) = (tan y < tan x)"
hoelzl@29803
  2006
proof
hoelzl@29803
  2007
  assume "y < x" thus "tan y < tan x" using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
hoelzl@29803
  2008
next
hoelzl@29803
  2009
  assume "tan y < tan x"
hoelzl@29803
  2010
  show "y < x"
hoelzl@29803
  2011
  proof (rule ccontr)
hoelzl@29803
  2012
    assume "\<not> y < x" hence "x \<le> y" by auto
hoelzl@41970
  2013
    hence "tan x \<le> tan y"
hoelzl@29803
  2014
    proof (cases "x = y")
hoelzl@29803
  2015
      case True thus ?thesis by auto
hoelzl@29803
  2016
    next
hoelzl@29803
  2017
      case False hence "x < y" using `x \<le> y` by auto
hoelzl@29803
  2018
      from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
hoelzl@29803
  2019
    qed
hoelzl@29803
  2020
    thus False using `tan y < tan x` by auto
hoelzl@29803
  2021
  qed
hoelzl@29803
  2022
qed
hoelzl@29803
  2023
hoelzl@29803
  2024
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
hoelzl@29803
  2025
hoelzl@41970
  2026
lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
hoelzl@29803
  2027
  by (simp add: tan_def)
hoelzl@29803
  2028
hoelzl@41970
  2029
lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x"
hoelzl@29803
  2030
proof (induct n arbitrary: x)
hoelzl@29803
  2031
  case (Suc n)
huffman@36777
  2032
  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
hoelzl@29803
  2033
  show ?case unfolding split_pi_off using Suc by auto
hoelzl@29803
  2034
qed auto
hoelzl@29803
  2035
hoelzl@29803
  2036
lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
hoelzl@29803
  2037
proof (cases "0 \<le> i")
hoelzl@29803
  2038
  case True hence i_nat: "real i = real (nat i)" by auto
hoelzl@29803
  2039
  show ?thesis unfolding i_nat by auto
hoelzl@29803
  2040
next
hoelzl@29803
  2041
  case False hence i_nat: "real i = - real (nat (-i))" by auto
hoelzl@29803
  2042
  have "tan x = tan (x + real i * pi - real i * pi)" by auto
hoelzl@29803
  2043
  also have "\<dots> = tan (x + real i * pi)" unfolding i_nat mult_minus_left diff_minus_eq_add by (rule tan_periodic_nat)
hoelzl@29803
  2044
  finally show ?thesis by auto
hoelzl@29803
  2045
qed
hoelzl@29803
  2046
hoelzl@29803
  2047
lemma tan_periodic_n[simp]: "tan (x + number_of n * pi) = tan x"
hoelzl@29803
  2048
  using tan_periodic_int[of _ "number_of n" ] unfolding real_number_of .
huffman@23043
  2049
huffman@23043
  2050
subsection {* Inverse Trigonometric Functions *}
huffman@23043
  2051
huffman@23043
  2052
definition
huffman@23043
  2053
  arcsin :: "real => real" where
huffman@23043
  2054
  "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
huffman@23043
  2055
huffman@23043
  2056
definition
huffman@23043
  2057
  arccos :: "real => real" where
huffman@23043
  2058
  "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
huffman@23043
  2059
hoelzl@41970
  2060
definition
huffman@23043
  2061
  arctan :: "real => real" where
huffman@23043
  2062
  "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
huffman@23043
  2063
paulson@15229
  2064
lemma arcsin:
hoelzl@41970
  2065
     "[| -1 \<le> y; y \<le> 1 |]
hoelzl@41970
  2066
      ==> -(pi/2) \<le> arcsin y &
paulson@15077
  2067
           arcsin y \<le> pi/2 & sin(arcsin y) = y"
huffman@23011
  2068
unfolding arcsin_def by (rule theI' [OF sin_total])
huffman@23011
  2069
huffman@23011
  2070
lemma arcsin_pi:
hoelzl@41970
  2071
     "[| -1 \<le> y; y \<le> 1 |]
huffman@23011
  2072
      ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
huffman@23011
  2073
apply (drule (1) arcsin)
huffman@23011
  2074
apply (force intro: order_trans)
paulson@15077
  2075
done
paulson@15077
  2076
paulson@15077
  2077
lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
paulson@15077
  2078
by (blast dest: arcsin)
hoelzl@41970
  2079
paulson@15077
  2080
lemma arcsin_bounded:
paulson@15077
  2081
     "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
paulson@15077
  2082
by (blast dest: arcsin)
paulson@15077
  2083
paulson@15077
  2084
lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
paulson@15077
  2085
by (blast dest: arcsin)
paulson@15077
  2086
paulson@15077
  2087
lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
paulson@15077
  2088
by (blast dest: arcsin)
paulson@15077
  2089
paulson@15077
  2090
lemma arcsin_lt_bounded:
paulson@15077
  2091
     "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
paulson@15077
  2092
apply (frule order_less_imp_le)
paulson@15077
  2093
apply (frule_tac y = y in order_less_imp_le)
paulson@15077
  2094
apply (frule arcsin_bounded)
paulson@15077
  2095
apply (safe, simp)
paulson@15077
  2096
apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
paulson@15077
  2097
apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
paulson@15077
  2098
apply (drule_tac [!] f = sin in arg_cong, auto)
paulson@15077
  2099
done
paulson@15077
  2100
paulson@15077
  2101
lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
paulson@15077
  2102
apply (unfold arcsin_def)
huffman@23011
  2103
apply (rule the1_equality)
paulson@15077
  2104
apply (rule sin_total, auto)
paulson@15077
  2105
done
paulson@15077
  2106
huffman@22975
  2107
lemma arccos:
hoelzl@41970
  2108
     "[| -1 \<le> y; y \<le> 1 |]
huffman@22975
  2109
      ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
huffman@23011
  2110
unfolding arccos_def by (rule theI' [OF cos_total])
paulson@15077
  2111
huffman@22975
  2112
lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
huffman@22975
  2113
by (blast dest: arccos)
hoelzl@41970
  2114
huffman@22975
  2115
lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
huffman@22975
  2116
by (blast dest: arccos)
paulson@15077
  2117
huffman@22975
  2118
lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
huffman@22975
  2119
by (blast dest: arccos)
paulson@15077
  2120
huffman@22975
  2121
lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
huffman@22975
  2122
by (blast dest: arccos)
paulson@15077
  2123
huffman@22975
  2124
lemma arccos_lt_bounded:
hoelzl@41970
  2125
     "[| -1 < y; y < 1 |]
huffman@22975
  2126
      ==> 0 < arccos y & arccos y < pi"
paulson@15077
  2127
apply (frule order_less_imp_le)
paulson@15077
  2128
apply (frule_tac y = y in order_less_imp_le)
huffman@22975
  2129
apply (frule arccos_bounded, auto)
huffman@22975
  2130
apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
paulson@15077
  2131
apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
paulson@15077
  2132
apply (drule_tac [!] f = cos in arg_cong, auto)
paulson@15077
  2133
done
paulson@15077
  2134
huffman@22975
  2135
lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
huffman@22975
  2136
apply (simp add: arccos_def)
huffman@23011
  2137
apply (auto intro!: the1_equality cos_total)
paulson@15077
  2138
done
paulson@15077
  2139
huffman@22975
  2140
lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
huffman@22975
  2141
apply (simp add: arccos_def)
huffman@23011
  2142
apply (auto intro!: the1_equality cos_total)
paulson@15077
  2143
done
paulson@15077
  2144
huffman@23045
  2145
lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<twosuperior>)"
huffman@23045
  2146
apply (subgoal_tac "x\<twosuperior> \<le> 1")
huffman@23052
  2147
apply (rule power2_eq_imp_eq)
huffman@23045
  2148
apply (simp add: cos_squared_eq)
huffman@23045
  2149
apply (rule cos_ge_zero)
huffman@23045
  2150
apply (erule (1) arcsin_lbound)
huffman@23045
  2151
apply (erule (1) arcsin_ubound)
huffman@23045
  2152
apply simp
huffman@23045
  2153
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
huffman@23045
  2154
apply (rule power_mono, simp, simp)
huffman@23045
  2155
done
huffman@23045
  2156
huffman@23045
  2157
lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<twosuperior>)"
huffman@23045
  2158
apply (subgoal_tac "x\<twosuperior> \<le> 1")
huffman@23052
  2159
apply (rule power2_eq_imp_eq)
huffman@23045
  2160
apply (simp add: sin_squared_eq)
huffman@23045
  2161
apply (rule sin_ge_zero)
huffman@23045
  2162
apply (erule (1) arccos_lbound)
huffman@23045
  2163
apply (erule (1) arccos_ubound)
huffman@23045
  2164
apply simp
huffman@23045
  2165
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
huffman@23045
  2166
apply (rule power_mono, simp, simp)
huffman@23045
  2167
done
huffman@23045
  2168
paulson@15077
  2169
lemma arctan [simp]:
paulson@15077
  2170
     "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
huffman@23011
  2171
unfolding arctan_def by (rule theI' [OF tan_total])
paulson@15077
  2172
paulson@15077
  2173
lemma tan_arctan: "tan(arctan y) = y"
paulson@15077
  2174
by auto
paulson@15077
  2175
paulson@15077
  2176
lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
paulson@15077
  2177
by (auto simp only: arctan)
paulson@15077
  2178
paulson@15077
  2179
lemma arctan_lbound: "- (pi/2) < arctan y"
paulson@15077
  2180
by auto
paulson@15077
  2181
paulson@15077
  2182
lemma arctan_ubound: "arctan y < pi/2"
paulson@15077
  2183
by (auto simp only: arctan)
paulson@15077
  2184
huffman@44746
  2185
lemma arctan_unique:
huffman@44746
  2186
  assumes "-(pi/2) < x" and "x < pi/2" and "tan x = y"
huffman@44746
  2187
  shows "arctan y = x"
huffman@44746
  2188
  using assms arctan [of y] tan_total [of y] by (fast elim: ex1E)
huffman@44746
  2189
hoelzl@41970
  2190
lemma arctan_tan:
paulson@15077
  2191
      "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
huffman@44746
  2192
  by (rule arctan_unique, simp_all)
paulson@15077
  2193
paulson@15077
  2194
lemma arctan_zero_zero [simp]: "arctan 0 = 0"
huffman@44746
  2195
  by (rule arctan_unique, simp_all)
huffman@44746
  2196
huffman@44746
  2197
lemma arctan_minus: "arctan (- x) = - arctan x"
huffman@44746
  2198
  apply (rule arctan_unique)
huffman@44746
  2199
  apply (simp only: neg_less_iff_less arctan_ubound)
huffman@44746
  2200
  apply (metis minus_less_iff arctan_lbound)
huffman@44746
  2201
  apply simp
huffman@44746
  2202
  done
paulson@15077
  2203
huffman@44725
  2204
lemma cos_arctan_not_zero [simp]: "cos (arctan x) \<noteq> 0"
huffman@44725
  2205
  by (intro less_imp_neq [symmetric] cos_gt_zero_pi
huffman@44725
  2206
    arctan_lbound arctan_ubound)
huffman@44725
  2207
huffman@44725
  2208
lemma cos_arctan: "cos (arctan x) = 1 / sqrt (1 + x\<twosuperior>)"
huffman@44725
  2209
proof (rule power2_eq_imp_eq)
huffman@44725
  2210
  have "0 < 1 + x\<twosuperior>" by (simp add: add_pos_nonneg)
huffman@44725
  2211
  show "0 \<le> 1 / sqrt (1 + x\<twosuperior>)" by simp
huffman@44725
  2212
  show "0 \<le> cos (arctan x)"
huffman@44725
  2213
    by (intro less_imp_le cos_gt_zero_pi arctan_lbound arctan_ubound)
huffman@44725
  2214
  have "(cos (arctan x))\<twosuperior> * (1 + (tan (arctan x))\<twosuperior>) = 1"
huffman@44725
  2215
    unfolding tan_def by (simp add: right_distrib power_divide)
huffman@44725
  2216
  thus "(cos (arctan x))\<twosuperior> = (1 / sqrt (1 + x\<twosuperior>))\<twosuperior>"
huffman@44725
  2217
    using `0 < 1 + x\<twosuperior>` by (simp add: power_divide eq_divide_eq)
huffman@44725
  2218
qed
huffman@44725
  2219
huffman@44725
  2220
lemma sin_arctan: "sin (arctan x) = x / sqrt (1 + x\<twosuperior>)"
huffman@44725
  2221
  using add_pos_nonneg [OF zero_less_one zero_le_power2 [of x]]
huffman@44725
  2222
  using tan_arctan [of x] unfolding tan_def cos_arctan
huffman@44725
  2223
  by (simp add: eq_divide_eq)
paulson@15077
  2224
paulson@15077
  2225
lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
paulson@15077
  2226
apply (rule power_inverse [THEN subst])
paulson@15077
  2227
apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
huffman@22960
  2228
apply (auto dest: field_power_not_zero
hoelzl@41970
  2229
        simp add: power_mult_distrib left_distrib power_divide tan_def
huffman@30273
  2230
                  mult_assoc power_inverse [symmetric])
paulson@15077
  2231
done
paulson@15077
  2232
huffman@44746
  2233
lemma arctan_less_iff: "arctan x < arctan y \<longleftrightarrow> x < y"
huffman@44746
  2234
  by (metis tan_monotone' arctan_lbound arctan_ubound tan_arctan)
huffman@44746
  2235
huffman@44746
  2236
lemma arctan_le_iff: "arctan x \<le> arctan y \<longleftrightarrow> x \<le> y"
huffman@44746
  2237
  by (simp only: not_less [symmetric] arctan_less_iff)
huffman@44746
  2238
huffman@44746
  2239
lemma arctan_eq_iff: "arctan x = arctan y \<longleftrightarrow> x = y"
huffman@44746
  2240
  by (simp only: eq_iff [where 'a=real] arctan_le_iff)
huffman@44746
  2241
huffman@44746
  2242
lemma zero_less_arctan_iff [simp]: "0 < arctan x \<longleftrightarrow> 0 < x"
huffman@44746
  2243
  using arctan_less_iff [of 0 x] by simp
huffman@44746
  2244
huffman@44746
  2245
lemma arctan_less_zero_iff [simp]: "arctan x < 0 \<longleftrightarrow> x < 0"
huffman@44746
  2246
  using arctan_less_iff [of x 0] by simp
huffman@44746
  2247
huffman@44746
  2248
lemma zero_le_arctan_iff [simp]: "0 \<le> arctan x \<longleftrightarrow> 0 \<le> x"
huffman@44746
  2249
  using arctan_le_iff [of 0 x] by simp
huffman@44746
  2250
huffman@44746
  2251
lemma arctan_le_zero_iff [simp]: "arctan x \<le> 0 \<longleftrightarrow> x \<le> 0"
huffman@44746
  2252
  using arctan_le_iff [of x 0] by simp
huffman@44746
  2253
huffman@44746
  2254
lemma arctan_eq_zero_iff [simp]: "arctan x = 0 \<longleftrightarrow> x = 0"
huffman@44746
  2255
  using arctan_eq_iff [of x 0] by simp
huffman@44746
  2256
huffman@23045
  2257
lemma isCont_inverse_function2:
huffman@23045
  2258
  fixes f g :: "real \<Rightarrow> real" shows
huffman@23045
  2259
  "\<lbrakk>a < x; x < b;
huffman@23045
  2260
    \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
huffman@23045
  2261
    \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
huffman@23045
  2262
   \<Longrightarrow> isCont g (f x)"
huffman@23045
  2263
apply (rule isCont_inverse_function
huffman@23045
  2264
       [where f=f and d="min (x - a) (b - x)"])
huffman@23045
  2265
apply (simp_all add: abs_le_iff)
huffman@23045
  2266
done
huffman@23045
  2267
huffman@23045
  2268
lemma isCont_arcsin: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arcsin x"
huffman@23045
  2269
apply (subgoal_tac "isCont arcsin (sin (arcsin x))", simp)
huffman@23045
  2270
apply (rule isCont_inverse_function2 [where f=sin])
huffman@23045
  2271
apply (erule (1) arcsin_lt_bounded [THEN conjunct1])
huffman@23045
  2272
apply (erule (1) arcsin_lt_bounded [THEN conjunct2])
huffman@23045
  2273
apply (fast intro: arcsin_sin, simp)
huffman@23045
  2274
done
huffman@23045
  2275
huffman@23045
  2276
lemma isCont_arccos: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arccos x"
huffman@23045
  2277
apply (subgoal_tac "isCont arccos (cos (arccos x))", simp)
huffman@23045
  2278
apply (rule isCont_inverse_function2 [where f=cos])
huffman@23045
  2279
apply (erule (1) arccos_lt_bounded [THEN conjunct1])
huffman@23045
  2280
apply (erule (1) arccos_lt_bounded [THEN conjunct2])
huffman@23045
  2281
apply (fast intro: arccos_cos, simp)
huffman@23045
  2282
done
huffman@23045
  2283
huffman@23045
  2284
lemma isCont_arctan: "isCont arctan x"
huffman@23045
  2285
apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
huffman@23045
  2286
apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
huffman@23045
  2287
apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
huffman@23045
  2288
apply (erule (1) isCont_inverse_function2 [where f=tan])
paulson@33667
  2289
apply (metis arctan_tan order_le_less_trans order_less_le_trans)
huffman@36777
  2290
apply (metis cos_gt_zero_pi isCont_tan order_less_le_trans less_le)
huffman@23045
  2291
done
huffman@23045
  2292
huffman@23045
  2293
lemma DERIV_arcsin:
huffman@23045
  2294
  "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<twosuperior>))"
huffman@23045
  2295
apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
huffman@44308
  2296
apply (rule DERIV_cong [OF DERIV_sin])
huffman@23045
  2297
apply (simp add: cos_arcsin)
huffman@23045
  2298
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
huffman@23045
  2299
apply (rule power_strict_mono, simp, simp, simp)
huffman@23045
  2300
apply assumption
huffman@23045
  2301
apply assumption
huffman@23045
  2302
apply simp
huffman@23045
  2303
apply (erule (1) isCont_arcsin)
huffman@23045
  2304
done
huffman@23045
  2305
huffman@23045
  2306
lemma DERIV_arccos:
huffman@23045
  2307
  "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<twosuperior>))"
huffman@23045
  2308
apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
huffman@44308
  2309
apply (rule DERIV_cong [OF DERIV_cos])
huffman@23045
  2310
apply (simp add: sin_arccos)
huffman@23045
  2311
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
huffman@23045
  2312
apply (rule power_strict_mono, simp, simp, simp)
huffman@23045
  2313
apply assumption
huffman@23045
  2314
apply assumption
huffman@23045
  2315
apply simp
huffman@23045
  2316
apply (erule (1) isCont_arccos)
huffman@23045
  2317
done
huffman@23045
  2318
huffman@23045
  2319
lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<twosuperior>)"
huffman@23045
  2320
apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
huffman@44308
  2321
apply (rule DERIV_cong [OF DERIV_tan])
huffman@23045
  2322
apply (rule cos_arctan_not_zero)
huffman@23045
  2323
apply (simp add: power_inverse tan_sec [symmetric])
huffman@23045
  2324
apply (subgoal_tac "0 < 1 + x\<twosuperior>", simp)
huffman@23045
  2325
apply (simp add: add_pos_nonneg)
huffman@23045
  2326
apply (simp, simp, simp, rule isCont_arctan)
huffman@23045
  2327
done
huffman@23045
  2328
hoelzl@31880
  2329
declare
hoelzl@31880
  2330
  DERIV_arcsin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  2331
  DERIV_arccos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  2332
  DERIV_arctan[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  2333
huffman@23043
  2334
subsection {* More Theorems about Sin and Cos *}
huffman@23043
  2335
huffman@23052
  2336
lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
huffman@23052
  2337
proof -
huffman@23052
  2338
  let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
huffman@23052
  2339
  have nonneg: "0 \<le> ?c"
huffman@45308
  2340
    by (simp add: cos_ge_zero)
huffman@23052
  2341
  have "0 = cos (pi / 4 + pi / 4)"
huffman@23052
  2342
    by simp
huffman@23052
  2343
  also have "cos (pi / 4 + pi / 4) = ?c\<twosuperior> - ?s\<twosuperior>"
huffman@23052
  2344
    by (simp only: cos_add power2_eq_square)
huffman@23052
  2345
  also have "\<dots> = 2 * ?c\<twosuperior> - 1"
huffman@23052
  2346
    by (simp add: sin_squared_eq)
huffman@23052
  2347
  finally have "?c\<twosuperior> = (sqrt 2 / 2)\<twosuperior>"
huffman@23052
  2348
    by (simp add: power_divide)
huffman@23052
  2349
  thus ?thesis
huffman@23052
  2350
    using nonneg by (rule power2_eq_imp_eq) simp
huffman@23052
  2351
qed
huffman@23052
  2352
huffman@23052
  2353
lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"
huffman@23052
  2354
proof -
huffman@23052
  2355
  let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
huffman@23052
  2356
  have pos_c: "0 < ?c"
huffman@23052
  2357
    by (rule cos_gt_zero, simp, simp)
huffman@23052
  2358
  have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
huffman@23066
  2359
    by simp
huffman@23052
  2360
  also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
huffman@23052
  2361
    by (simp only: cos_add sin_add)
huffman@23052
  2362
  also have "\<dots> = ?c * (?c\<twosuperior> - 3 * ?s\<twosuperior>)"
nipkow@29667
  2363
    by (simp add: algebra_simps power2_eq_square)
huffman@23052
  2364
  finally have "?c\<twosuperior> = (sqrt 3 / 2)\<twosuperior>"
huffman@23052
  2365
    using pos_c by (simp add: sin_squared_eq power_divide)
huffman@23052
  2366
  thus ?thesis
huffman@23052
  2367
    using pos_c [THEN order_less_imp_le]
huffman@23052
  2368
    by (rule power2_eq_imp_eq) simp
huffman@23052
  2369
qed
huffman@23052
  2370
huffman@23052
  2371
lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
huffman@45309
  2372
by (simp add: sin_cos_eq cos_45)
huffman@23052
  2373
huffman@23052
  2374
lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
huffman@45309
  2375
by (simp add: sin_cos_eq cos_30)
huffman@23052
  2376
huffman@23052
  2377
lemma cos_60: "cos (pi / 3) = 1 / 2"