src/HOL/Transcendental.thy
author hoelzl
Fri Mar 22 10:41:43 2013 +0100 (2013-03-22)
changeset 51481 ef949192e5d6
parent 51478 270b21f3ae0a
child 51482 80efd8c49f52
permissions -rw-r--r--
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
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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: distrib_left 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 nested_sequence_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
wenzelm@32960
   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@47489
   424
         of_nat_0_le_iff
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
hoelzl@51478
   884
lemma continuous_exp [continuous_intros]: "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
hoelzl@51478
   885
  unfolding continuous_def by (rule tendsto_exp)
hoelzl@51478
   886
hoelzl@51478
   887
lemma continuous_on_exp [continuous_on_intros]: "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
hoelzl@51478
   888
  unfolding continuous_on_def by (auto intro: tendsto_exp)
hoelzl@51478
   889
huffman@29167
   890
subsubsection {* Properties of the Exponential Function *}
paulson@15077
   891
huffman@23278
   892
lemma powser_zero:
haftmann@31017
   893
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
huffman@23278
   894
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
paulson@15077
   895
proof -
huffman@23278
   896
  have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
huffman@23115
   897
    by (rule sums_unique [OF series_zero], simp add: power_0_left)
huffman@30082
   898
  thus ?thesis unfolding One_nat_def by simp
paulson@15077
   899
qed
paulson@15077
   900
huffman@23278
   901
lemma exp_zero [simp]: "exp 0 = 1"
huffman@23278
   902
unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
huffman@23278
   903
huffman@23115
   904
lemma setsum_cl_ivl_Suc2:
huffman@23115
   905
  "(\<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
   906
by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
huffman@23115
   907
         del: setsum_cl_ivl_Suc)
huffman@23115
   908
huffman@23115
   909
lemma exp_series_add:
haftmann@31017
   910
  fixes x y :: "'a::{real_field}"
haftmann@25062
   911
  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
huffman@23115
   912
  shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
huffman@23115
   913
proof (induct n)
huffman@23115
   914
  case 0
huffman@23115
   915
  show ?case
huffman@23115
   916
    unfolding S_def by simp
huffman@23115
   917
next
huffman@23115
   918
  case (Suc n)
haftmann@25062
   919
  have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
huffman@30273
   920
    unfolding S_def by (simp del: mult_Suc)
haftmann@25062
   921
  hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
huffman@23115
   922
    by simp
huffman@23115
   923
haftmann@25062
   924
  have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
huffman@23115
   925
    by (simp only: times_S)
huffman@23115
   926
  also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
huffman@23115
   927
    by (simp only: Suc)
huffman@23115
   928
  also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
huffman@23115
   929
                + y * (\<Sum>i=0..n. S x i * S y (n-i))"
webertj@49962
   930
    by (rule distrib_right)
huffman@23115
   931
  also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
huffman@23115
   932
                + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
huffman@23115
   933
    by (simp only: setsum_right_distrib mult_ac)
haftmann@25062
   934
  also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
haftmann@25062
   935
                + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
   936
    by (simp add: times_S Suc_diff_le)
haftmann@25062
   937
  also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
haftmann@25062
   938
             (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
   939
    by (subst setsum_cl_ivl_Suc2, simp)
haftmann@25062
   940
  also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
haftmann@25062
   941
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
   942
    by (subst setsum_cl_ivl_Suc, simp)
haftmann@25062
   943
  also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
haftmann@25062
   944
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
haftmann@25062
   945
             (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
   946
    by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
huffman@23115
   947
              real_of_nat_add [symmetric], simp)
haftmann@25062
   948
  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
huffman@23127
   949
    by (simp only: scaleR_right.setsum)
huffman@23115
   950
  finally show
huffman@23115
   951
    "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
huffman@35216
   952
    by (simp del: setsum_cl_ivl_Suc)
huffman@23115
   953
qed
huffman@23115
   954
huffman@23115
   955
lemma exp_add: "exp (x + y) = exp x * exp y"
huffman@23115
   956
unfolding exp_def
huffman@23115
   957
by (simp only: Cauchy_product summable_norm_exp exp_series_add)
huffman@23115
   958
huffman@29170
   959
lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
huffman@29170
   960
by (rule exp_add [symmetric])
huffman@29170
   961
huffman@23241
   962
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
huffman@23241
   963
unfolding exp_def
huffman@44282
   964
apply (subst suminf_of_real)
huffman@23241
   965
apply (rule summable_exp_generic)
huffman@23241
   966
apply (simp add: scaleR_conv_of_real)
huffman@23241
   967
done
huffman@23241
   968
huffman@29170
   969
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
huffman@29170
   970
proof
huffman@29170
   971
  have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
huffman@29170
   972
  also assume "exp x = 0"
huffman@29170
   973
  finally show "False" by simp
paulson@15077
   974
qed
paulson@15077
   975
huffman@29170
   976
lemma exp_minus: "exp (- x) = inverse (exp x)"
huffman@29170
   977
by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
paulson@15077
   978
huffman@29170
   979
lemma exp_diff: "exp (x - y) = exp x / exp y"
huffman@29170
   980
  unfolding diff_minus divide_inverse
huffman@29170
   981
  by (simp add: exp_add exp_minus)
paulson@15077
   982
huffman@29167
   983
huffman@29167
   984
subsubsection {* Properties of the Exponential Function on Reals *}
huffman@29167
   985
huffman@29170
   986
text {* Comparisons of @{term "exp x"} with zero. *}
huffman@29167
   987
huffman@29167
   988
text{*Proof: because every exponential can be seen as a square.*}
huffman@29167
   989
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
huffman@29167
   990
proof -
huffman@29167
   991
  have "0 \<le> exp (x/2) * exp (x/2)" by simp
huffman@29167
   992
  thus ?thesis by (simp add: exp_add [symmetric])
huffman@29167
   993
qed
huffman@29167
   994
huffman@23115
   995
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
paulson@15077
   996
by (simp add: order_less_le)
paulson@15077
   997
huffman@29170
   998
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
huffman@29170
   999
by (simp add: not_less)
huffman@29170
  1000
huffman@29170
  1001
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
huffman@29170
  1002
by (simp add: not_le)
paulson@15077
  1003
huffman@23115
  1004
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
huffman@29165
  1005
by simp
paulson@15077
  1006
paulson@15077
  1007
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
paulson@15251
  1008
apply (induct "n")
webertj@49962
  1009
apply (auto simp add: real_of_nat_Suc distrib_left exp_add mult_commute)
paulson@15077
  1010
done
paulson@15077
  1011
huffman@29170
  1012
text {* Strict monotonicity of exponential. *}
huffman@29170
  1013
huffman@29170
  1014
lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
huffman@29170
  1015
apply (drule order_le_imp_less_or_eq, auto)
huffman@29170
  1016
apply (simp add: exp_def)
huffman@36777
  1017
apply (rule order_trans)
huffman@29170
  1018
apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
huffman@29170
  1019
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
huffman@29170
  1020
done
huffman@29170
  1021
huffman@29170
  1022
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
huffman@29170
  1023
proof -
huffman@29170
  1024
  assume x: "0 < x"
huffman@29170
  1025
  hence "1 < 1 + x" by simp
huffman@29170
  1026
  also from x have "1 + x \<le> exp x"
huffman@29170
  1027
    by (simp add: exp_ge_add_one_self_aux)
huffman@29170
  1028
  finally show ?thesis .
huffman@29170
  1029
qed
huffman@29170
  1030
paulson@15077
  1031
lemma exp_less_mono:
huffman@23115
  1032
  fixes x y :: real
huffman@29165
  1033
  assumes "x < y" shows "exp x < exp y"
paulson@15077
  1034
proof -
huffman@29165
  1035
  from `x < y` have "0 < y - x" by simp
huffman@29165
  1036
  hence "1 < exp (y - x)" by (rule exp_gt_one)
huffman@29165
  1037
  hence "1 < exp y / exp x" by (simp only: exp_diff)
huffman@29165
  1038
  thus "exp x < exp y" by simp
paulson@15077
  1039
qed
paulson@15077
  1040
huffman@23115
  1041
lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
huffman@29170
  1042
apply (simp add: linorder_not_le [symmetric])
huffman@29170
  1043
apply (auto simp add: order_le_less exp_less_mono)
paulson@15077
  1044
done
paulson@15077
  1045
huffman@29170
  1046
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
paulson@15077
  1047
by (auto intro: exp_less_mono exp_less_cancel)
paulson@15077
  1048
huffman@29170
  1049
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
paulson@15077
  1050
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
  1051
huffman@29170
  1052
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
paulson@15077
  1053
by (simp add: order_eq_iff)
paulson@15077
  1054
huffman@29170
  1055
text {* Comparisons of @{term "exp x"} with one. *}
huffman@29170
  1056
huffman@29170
  1057
lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
huffman@29170
  1058
  using exp_less_cancel_iff [where x=0 and y=x] by simp
huffman@29170
  1059
huffman@29170
  1060
lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
huffman@29170
  1061
  using exp_less_cancel_iff [where x=x and y=0] by simp
huffman@29170
  1062
huffman@29170
  1063
lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
huffman@29170
  1064
  using exp_le_cancel_iff [where x=0 and y=x] by simp
huffman@29170
  1065
huffman@29170
  1066
lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
huffman@29170
  1067
  using exp_le_cancel_iff [where x=x and y=0] by simp
huffman@29170
  1068
huffman@29170
  1069
lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
huffman@29170
  1070
  using exp_inj_iff [where x=x and y=0] by simp
huffman@29170
  1071
huffman@23115
  1072
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
huffman@44755
  1073
proof (rule IVT)
huffman@44755
  1074
  assume "1 \<le> y"
huffman@44755
  1075
  hence "0 \<le> y - 1" by simp
huffman@44755
  1076
  hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
huffman@44755
  1077
  thus "y \<le> exp (y - 1)" by simp
huffman@44755
  1078
qed (simp_all add: le_diff_eq)
paulson@15077
  1079
huffman@23115
  1080
lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
huffman@44755
  1081
proof (rule linorder_le_cases [of 1 y])
huffman@44755
  1082
  assume "1 \<le> y" thus "\<exists>x. exp x = y"
huffman@44755
  1083
    by (fast dest: lemma_exp_total)
huffman@44755
  1084
next
huffman@44755
  1085
  assume "0 < y" and "y \<le> 1"
huffman@44755
  1086
  hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
huffman@44755
  1087
  then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
huffman@44755
  1088
  hence "exp (- x) = y" by (simp add: exp_minus)
huffman@44755
  1089
  thus "\<exists>x. exp x = y" ..
huffman@44755
  1090
qed
paulson@15077
  1091
paulson@15077
  1092
huffman@29164
  1093
subsection {* Natural Logarithm *}
paulson@15077
  1094
huffman@44308
  1095
definition ln :: "real \<Rightarrow> real" where
huffman@23043
  1096
  "ln x = (THE u. exp u = x)"
huffman@23043
  1097
huffman@23043
  1098
lemma ln_exp [simp]: "ln (exp x) = x"
huffman@44308
  1099
  by (simp add: ln_def)
paulson@15077
  1100
huffman@22654
  1101
lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
huffman@44308
  1102
  by (auto dest: exp_total)
huffman@22654
  1103
huffman@29171
  1104
lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
huffman@44308
  1105
  by (metis exp_gt_zero exp_ln)
paulson@15077
  1106
huffman@29171
  1107
lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
huffman@44308
  1108
  by (erule subst, rule ln_exp)
huffman@29171
  1109
huffman@29171
  1110
lemma ln_one [simp]: "ln 1 = 0"
huffman@44308
  1111
  by (rule ln_unique, simp)
huffman@29171
  1112
huffman@29171
  1113
lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"
huffman@44308
  1114
  by (rule ln_unique, simp add: exp_add)
huffman@29171
  1115
huffman@29171
  1116
lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
huffman@44308
  1117
  by (rule ln_unique, simp add: exp_minus)
huffman@29171
  1118
huffman@29171
  1119
lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"
huffman@44308
  1120
  by (rule ln_unique, simp add: exp_diff)
paulson@15077
  1121
huffman@29171
  1122
lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
huffman@44308
  1123
  by (rule ln_unique, simp add: exp_real_of_nat_mult)
huffman@29171
  1124
huffman@29171
  1125
lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
huffman@44308
  1126
  by (subst exp_less_cancel_iff [symmetric], simp)
huffman@29171
  1127
huffman@29171
  1128
lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
huffman@44308
  1129
  by (simp add: linorder_not_less [symmetric])
huffman@29171
  1130
huffman@29171
  1131
lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
huffman@44308
  1132
  by (simp add: order_eq_iff)
huffman@29171
  1133
huffman@29171
  1134
lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
huffman@44308
  1135
  apply (rule exp_le_cancel_iff [THEN iffD1])
huffman@44308
  1136
  apply (simp add: exp_ge_add_one_self_aux)
huffman@44308
  1137
  done
paulson@15077
  1138
huffman@29171
  1139
lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
huffman@44308
  1140
  by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
huffman@44308
  1141
huffman@44308
  1142
lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
huffman@44308
  1143
  using ln_le_cancel_iff [of 1 x] by simp
huffman@44308
  1144
huffman@44308
  1145
lemma ln_ge_zero_imp_ge_one: "\<lbrakk>0 \<le> ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 \<le> x"
huffman@44308
  1146
  using ln_le_cancel_iff [of 1 x] by simp
huffman@44308
  1147
huffman@44308
  1148
lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> (0 \<le> ln x) = (1 \<le> x)"
huffman@44308
  1149
  using ln_le_cancel_iff [of 1 x] by simp
huffman@44308
  1150
huffman@44308
  1151
lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x < 0) = (x < 1)"
huffman@44308
  1152
  using ln_less_cancel_iff [of x 1] by simp
huffman@44308
  1153
huffman@44308
  1154
lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
huffman@44308
  1155
  using ln_less_cancel_iff [of 1 x] by simp
huffman@44308
  1156
huffman@44308
  1157
lemma ln_gt_zero_imp_gt_one: "\<lbrakk>0 < ln x; 0 < x\<rbrakk> \<Longrightarrow> 1 < x"
huffman@44308
  1158
  using ln_less_cancel_iff [of 1 x] by simp
huffman@44308
  1159
huffman@44308
  1160
lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> (0 < ln x) = (1 < x)"
huffman@44308
  1161
  using ln_less_cancel_iff [of 1 x] by simp
huffman@44308
  1162
huffman@44308
  1163
lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> (ln x = 0) = (x = 1)"
huffman@44308
  1164
  using ln_inj_iff [of x 1] by simp
huffman@44308
  1165
huffman@44308
  1166
lemma ln_less_zero: "\<lbrakk>0 < x; x < 1\<rbrakk> \<Longrightarrow> ln x < 0"
huffman@44308
  1167
  by simp
paulson@15077
  1168
huffman@23045
  1169
lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
huffman@44308
  1170
  apply (subgoal_tac "isCont ln (exp (ln x))", simp)
huffman@44308
  1171
  apply (rule isCont_inverse_function [where f=exp], simp_all)
huffman@44308
  1172
  done
huffman@23045
  1173
huffman@45915
  1174
lemma tendsto_ln [tendsto_intros]:
huffman@45915
  1175
  "\<lbrakk>(f ---> a) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
huffman@45915
  1176
  by (rule isCont_tendsto_compose [OF isCont_ln])
huffman@45915
  1177
hoelzl@51478
  1178
lemma continuous_ln:
hoelzl@51478
  1179
  "continuous F f \<Longrightarrow> 0 < f (Lim F (\<lambda>x. x)) \<Longrightarrow> continuous F (\<lambda>x. ln (f x))"
hoelzl@51478
  1180
  unfolding continuous_def by (rule tendsto_ln)
hoelzl@51478
  1181
hoelzl@51478
  1182
lemma isCont_ln' [continuous_intros]:
hoelzl@51478
  1183
  "continuous (at x) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x))"
hoelzl@51478
  1184
  unfolding continuous_at by (rule tendsto_ln)
hoelzl@51478
  1185
hoelzl@51478
  1186
lemma continuous_within_ln [continuous_intros]:
hoelzl@51478
  1187
  "continuous (at x within s) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x))"
hoelzl@51478
  1188
  unfolding continuous_within by (rule tendsto_ln)
hoelzl@51478
  1189
hoelzl@51478
  1190
lemma continuous_on_ln [continuous_on_intros]:
hoelzl@51478
  1191
  "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. 0 < f x) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x))"
hoelzl@51478
  1192
  unfolding continuous_on_def by (auto intro: tendsto_ln)
hoelzl@51478
  1193
huffman@23045
  1194
lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
huffman@44308
  1195
  apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
huffman@44317
  1196
  apply (erule DERIV_cong [OF DERIV_exp exp_ln])
huffman@44308
  1197
  apply (simp_all add: abs_if isCont_ln)
huffman@44308
  1198
  done
huffman@23045
  1199
paulson@33667
  1200
lemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x"
paulson@33667
  1201
  by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
paulson@33667
  1202
hoelzl@29803
  1203
lemma ln_series: assumes "0 < x" and "x < 2"
hoelzl@29803
  1204
  shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))")
hoelzl@29803
  1205
proof -
hoelzl@29803
  1206
  let "?f' x n" = "(-1)^n * (x - 1)^n"
hoelzl@29803
  1207
hoelzl@29803
  1208
  have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
hoelzl@29803
  1209
  proof (rule DERIV_isconst3[where x=x])
hoelzl@29803
  1210
    fix x :: real assume "x \<in> {0 <..< 2}" hence "0 < x" and "x < 2" by auto
hoelzl@29803
  1211
    have "norm (1 - x) < 1" using `0 < x` and `x < 2` by auto
hoelzl@29803
  1212
    have "1 / x = 1 / (1 - (1 - x))" by auto
hoelzl@29803
  1213
    also have "\<dots> = (\<Sum> n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
hoelzl@29803
  1214
    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
  1215
    finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
hoelzl@29803
  1216
    moreover
hoelzl@29803
  1217
    have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
hoelzl@29803
  1218
    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
  1219
    proof (rule DERIV_power_series')
hoelzl@29803
  1220
      show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto
hoelzl@29803
  1221
      { fix x :: real assume "x \<in> {- 1<..<1}" hence "norm (-x) < 1" by auto
wenzelm@32960
  1222
        show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
huffman@30082
  1223
          unfolding One_nat_def
huffman@35216
  1224
          by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
hoelzl@29803
  1225
      }
hoelzl@29803
  1226
    qed
huffman@30082
  1227
    hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" unfolding One_nat_def by auto
hoelzl@29803
  1228
    hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos .
hoelzl@29803
  1229
    ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
hoelzl@29803
  1230
      by (rule DERIV_diff)
hoelzl@29803
  1231
    thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
hoelzl@29803
  1232
  qed (auto simp add: assms)
huffman@44289
  1233
  thus ?thesis by auto
hoelzl@29803
  1234
qed
paulson@15077
  1235
hoelzl@50326
  1236
lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) * (x ^ (n+2)))"
hoelzl@50326
  1237
proof -
hoelzl@50326
  1238
  have "exp x = suminf (%n. inverse(fact n) * (x ^ n))"
hoelzl@50326
  1239
    by (simp add: exp_def)
hoelzl@50326
  1240
  also from summable_exp have "... = (\<Sum> n::nat = 0 ..< 2. inverse(fact n) * (x ^ n)) +
hoelzl@50326
  1241
      (\<Sum> n. inverse(fact(n+2)) * (x ^ (n+2)))" (is "_ = ?a + _")
hoelzl@50326
  1242
    by (rule suminf_split_initial_segment)
hoelzl@50326
  1243
  also have "?a = 1 + x"
hoelzl@50326
  1244
    by (simp add: numeral_2_eq_2)
hoelzl@50326
  1245
  finally show ?thesis .
hoelzl@50326
  1246
qed
hoelzl@50326
  1247
hoelzl@50326
  1248
lemma exp_bound: "0 <= (x::real) ==> x <= 1 ==> exp x <= 1 + x + x^2"
hoelzl@50326
  1249
proof -
hoelzl@50326
  1250
  assume a: "0 <= x"
hoelzl@50326
  1251
  assume b: "x <= 1"
hoelzl@50326
  1252
  { fix n :: nat
hoelzl@50326
  1253
    have "2 * 2 ^ n \<le> fact (n + 2)"
hoelzl@50326
  1254
      by (induct n, simp, simp)
hoelzl@50326
  1255
    hence "real ((2::nat) * 2 ^ n) \<le> real (fact (n + 2))"
hoelzl@50326
  1256
      by (simp only: real_of_nat_le_iff)
hoelzl@50326
  1257
    hence "2 * 2 ^ n \<le> real (fact (n + 2))"
hoelzl@50326
  1258
      by simp
hoelzl@50326
  1259
    hence "inverse (fact (n + 2)) \<le> inverse (2 * 2 ^ n)"
hoelzl@50326
  1260
      by (rule le_imp_inverse_le) simp
hoelzl@50326
  1261
    hence "inverse (fact (n + 2)) \<le> 1/2 * (1/2)^n"
hoelzl@50326
  1262
      by (simp add: inverse_mult_distrib power_inverse)
hoelzl@50326
  1263
    hence "inverse (fact (n + 2)) * (x^n * x\<twosuperior>) \<le> 1/2 * (1/2)^n * (1 * x\<twosuperior>)"
hoelzl@50326
  1264
      by (rule mult_mono)
hoelzl@50326
  1265
        (rule mult_mono, simp_all add: power_le_one a b mult_nonneg_nonneg)
hoelzl@50326
  1266
    hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<twosuperior>/2) * ((1/2)^n)"
hoelzl@50326
  1267
      unfolding power_add by (simp add: mult_ac del: fact_Suc) }
hoelzl@50326
  1268
  note aux1 = this
hoelzl@50326
  1269
  have "(\<lambda>n. x\<twosuperior> / 2 * (1 / 2) ^ n) sums (x\<twosuperior> / 2 * (1 / (1 - 1 / 2)))"
hoelzl@50326
  1270
    by (intro sums_mult geometric_sums, simp)
hoelzl@50326
  1271
  hence aux2: "(\<lambda>n. (x::real) ^ 2 / 2 * (1 / 2) ^ n) sums x^2"
hoelzl@50326
  1272
    by simp
hoelzl@50326
  1273
  have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <= x^2"
hoelzl@50326
  1274
  proof -
hoelzl@50326
  1275
    have "suminf (%n. inverse(fact (n+2)) * (x ^ (n+2))) <=
hoelzl@50326
  1276
        suminf (%n. (x^2/2) * ((1/2)^n))"
hoelzl@50326
  1277
      apply (rule summable_le)
hoelzl@50326
  1278
      apply (rule allI, rule aux1)
hoelzl@50326
  1279
      apply (rule summable_exp [THEN summable_ignore_initial_segment])
hoelzl@50326
  1280
      by (rule sums_summable, rule aux2)
hoelzl@50326
  1281
    also have "... = x^2"
hoelzl@50326
  1282
      by (rule sums_unique [THEN sym], rule aux2)
hoelzl@50326
  1283
    finally show ?thesis .
hoelzl@50326
  1284
  qed
hoelzl@50326
  1285
  thus ?thesis unfolding exp_first_two_terms by auto
hoelzl@50326
  1286
qed
hoelzl@50326
  1287
hoelzl@50326
  1288
lemma ln_one_minus_pos_upper_bound: "0 <= x ==> x < 1 ==> ln (1 - x) <= - x"
hoelzl@50326
  1289
proof -
hoelzl@50326
  1290
  assume a: "0 <= (x::real)" and b: "x < 1"
hoelzl@50326
  1291
  have "(1 - x) * (1 + x + x^2) = (1 - x^3)"
hoelzl@50326
  1292
    by (simp add: algebra_simps power2_eq_square power3_eq_cube)
hoelzl@50326
  1293
  also have "... <= 1"
hoelzl@50326
  1294
    by (auto simp add: a)
hoelzl@50326
  1295
  finally have "(1 - x) * (1 + x + x ^ 2) <= 1" .
hoelzl@50326
  1296
  moreover have c: "0 < 1 + x + x\<twosuperior>"
hoelzl@50326
  1297
    by (simp add: add_pos_nonneg a)
hoelzl@50326
  1298
  ultimately have "1 - x <= 1 / (1 + x + x^2)"
hoelzl@50326
  1299
    by (elim mult_imp_le_div_pos)
hoelzl@50326
  1300
  also have "... <= 1 / exp x"
hoelzl@50326
  1301
    apply (rule divide_left_mono)
hoelzl@50326
  1302
    apply (rule exp_bound, rule a)
hoelzl@50326
  1303
    apply (rule b [THEN less_imp_le])
hoelzl@50326
  1304
    apply simp
hoelzl@50326
  1305
    apply (rule mult_pos_pos)
hoelzl@50326
  1306
    apply (rule c)
hoelzl@50326
  1307
    apply simp
hoelzl@50326
  1308
    done
hoelzl@50326
  1309
  also have "... = exp (-x)"
hoelzl@50326
  1310
    by (auto simp add: exp_minus divide_inverse)
hoelzl@50326
  1311
  finally have "1 - x <= exp (- x)" .
hoelzl@50326
  1312
  also have "1 - x = exp (ln (1 - x))"
hoelzl@50326
  1313
  proof -
hoelzl@50326
  1314
    have "0 < 1 - x"
hoelzl@50326
  1315
      by (insert b, auto)
hoelzl@50326
  1316
    thus ?thesis
hoelzl@50326
  1317
      by (auto simp only: exp_ln_iff [THEN sym])
hoelzl@50326
  1318
  qed
hoelzl@50326
  1319
  finally have "exp (ln (1 - x)) <= exp (- x)" .
hoelzl@50326
  1320
  thus ?thesis by (auto simp only: exp_le_cancel_iff)
hoelzl@50326
  1321
qed
hoelzl@50326
  1322
hoelzl@50326
  1323
lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
hoelzl@50326
  1324
  apply (case_tac "0 <= x")
hoelzl@50326
  1325
  apply (erule exp_ge_add_one_self_aux)
hoelzl@50326
  1326
  apply (case_tac "x <= -1")
hoelzl@50326
  1327
  apply (subgoal_tac "1 + x <= 0")
hoelzl@50326
  1328
  apply (erule order_trans)
hoelzl@50326
  1329
  apply simp
hoelzl@50326
  1330
  apply simp
hoelzl@50326
  1331
  apply (subgoal_tac "1 + x = exp(ln (1 + x))")
hoelzl@50326
  1332
  apply (erule ssubst)
hoelzl@50326
  1333
  apply (subst exp_le_cancel_iff)
hoelzl@50326
  1334
  apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
hoelzl@50326
  1335
  apply simp
hoelzl@50326
  1336
  apply (rule ln_one_minus_pos_upper_bound)
hoelzl@50326
  1337
  apply auto
hoelzl@50326
  1338
done
hoelzl@50326
  1339
hoelzl@50326
  1340
lemma exp_at_bot: "(exp ---> (0::real)) at_bot"
hoelzl@50326
  1341
  unfolding tendsto_Zfun_iff
hoelzl@50326
  1342
proof (rule ZfunI, simp add: eventually_at_bot_dense)
hoelzl@50326
  1343
  fix r :: real assume "0 < r"
hoelzl@50326
  1344
  { fix x assume "x < ln r"
hoelzl@50326
  1345
    then have "exp x < exp (ln r)"
hoelzl@50326
  1346
      by simp
hoelzl@50326
  1347
    with `0 < r` have "exp x < r"
hoelzl@50326
  1348
      by simp }
hoelzl@50326
  1349
  then show "\<exists>k. \<forall>n<k. exp n < r" by auto
hoelzl@50326
  1350
qed
hoelzl@50326
  1351
hoelzl@50326
  1352
lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
hoelzl@50346
  1353
  by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"])
hoelzl@50346
  1354
     (auto intro: eventually_gt_at_top)
hoelzl@50326
  1355
hoelzl@50326
  1356
lemma ln_at_0: "LIM x at_right 0. ln x :> at_bot"
hoelzl@50346
  1357
  by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
hoelzl@50346
  1358
     (auto simp: eventually_within)
hoelzl@50326
  1359
hoelzl@50326
  1360
lemma ln_at_top: "LIM x at_top. ln x :> at_top"
hoelzl@50346
  1361
  by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
hoelzl@50346
  1362
     (auto intro: eventually_gt_at_top)
hoelzl@50326
  1363
hoelzl@50347
  1364
lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) ---> (0::real)) at_top"
hoelzl@50347
  1365
proof (induct k)
hoelzl@50347
  1366
  show "((\<lambda>x. x ^ 0 / exp x) ---> (0::real)) at_top"
hoelzl@50347
  1367
    by (simp add: inverse_eq_divide[symmetric])
hoelzl@50347
  1368
       (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
hoelzl@50347
  1369
              at_top_le_at_infinity order_refl)
hoelzl@50347
  1370
next
hoelzl@50347
  1371
  case (Suc k)
hoelzl@50347
  1372
  show ?case
hoelzl@50347
  1373
  proof (rule lhospital_at_top_at_top)
hoelzl@50347
  1374
    show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"
hoelzl@50347
  1375
      by eventually_elim (intro DERIV_intros, simp, simp)
hoelzl@50347
  1376
    show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top"
hoelzl@50347
  1377
      by eventually_elim (auto intro!: DERIV_intros)
hoelzl@50347
  1378
    show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
hoelzl@50347
  1379
      by auto
hoelzl@50347
  1380
    from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
hoelzl@50347
  1381
    show "((\<lambda>x. real (Suc k) * x ^ k / exp x) ---> 0) at_top"
hoelzl@50347
  1382
      by simp
hoelzl@50347
  1383
  qed (rule exp_at_top)
hoelzl@50347
  1384
qed
hoelzl@50347
  1385
huffman@29164
  1386
subsection {* Sine and Cosine *}
huffman@29164
  1387
huffman@44308
  1388
definition sin_coeff :: "nat \<Rightarrow> real" where
huffman@31271
  1389
  "sin_coeff = (\<lambda>n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"
huffman@31271
  1390
huffman@44308
  1391
definition cos_coeff :: "nat \<Rightarrow> real" where
huffman@31271
  1392
  "cos_coeff = (\<lambda>n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"
huffman@31271
  1393
huffman@44308
  1394
definition sin :: "real \<Rightarrow> real" where
huffman@44308
  1395
  "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
huffman@44308
  1396
huffman@44308
  1397
definition cos :: "real \<Rightarrow> real" where
huffman@44308
  1398
  "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
huffman@31271
  1399
huffman@44319
  1400
lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
huffman@44319
  1401
  unfolding sin_coeff_def by simp
huffman@44319
  1402
huffman@44319
  1403
lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
huffman@44319
  1404
  unfolding cos_coeff_def by simp
huffman@44319
  1405
huffman@44319
  1406
lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
huffman@44319
  1407
  unfolding cos_coeff_def sin_coeff_def
huffman@44319
  1408
  by (simp del: mult_Suc)
huffman@44319
  1409
huffman@44319
  1410
lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
huffman@44319
  1411
  unfolding cos_coeff_def sin_coeff_def
huffman@44319
  1412
  by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)
huffman@44319
  1413
huffman@31271
  1414
lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
huffman@31271
  1415
unfolding sin_coeff_def
huffman@44308
  1416
apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
huffman@29164
  1417
apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
huffman@29164
  1418
done
huffman@29164
  1419
huffman@31271
  1420
lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"
huffman@31271
  1421
unfolding cos_coeff_def
huffman@44308
  1422
apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
huffman@29164
  1423
apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
huffman@29164
  1424
done
huffman@29164
  1425
huffman@31271
  1426
lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
huffman@29164
  1427
unfolding sin_def by (rule summable_sin [THEN summable_sums])
huffman@29164
  1428
huffman@31271
  1429
lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
huffman@29164
  1430
unfolding cos_def by (rule summable_cos [THEN summable_sums])
huffman@29164
  1431
huffman@44319
  1432
lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
huffman@44319
  1433
  by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
huffman@44319
  1434
huffman@44319
  1435
lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
huffman@44319
  1436
  by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
huffman@29164
  1437
huffman@29164
  1438
text{*Now at last we can get the derivatives of exp, sin and cos*}
huffman@29164
  1439
huffman@29164
  1440
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
huffman@44319
  1441
  unfolding sin_def cos_def
huffman@44319
  1442
  apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
huffman@44319
  1443
  apply (simp_all add: diffs_sin_coeff diffs_cos_coeff
huffman@44319
  1444
    summable_minus summable_sin summable_cos)
huffman@44319
  1445
  done
huffman@29164
  1446
huffman@29164
  1447
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
huffman@44319
  1448
  unfolding cos_def sin_def
huffman@44319
  1449
  apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
huffman@44319
  1450
  apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus
huffman@44319
  1451
    summable_minus summable_sin summable_cos suminf_minus)
huffman@44319
  1452
  done
huffman@29164
  1453
huffman@44311
  1454
lemma isCont_sin: "isCont sin x"
huffman@44311
  1455
  by (rule DERIV_sin [THEN DERIV_isCont])
huffman@44311
  1456
huffman@44311
  1457
lemma isCont_cos: "isCont cos x"
huffman@44311
  1458
  by (rule DERIV_cos [THEN DERIV_isCont])
huffman@44311
  1459
huffman@44311
  1460
lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
huffman@44311
  1461
  by (rule isCont_o2 [OF _ isCont_sin])
huffman@44311
  1462
huffman@44311
  1463
lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
huffman@44311
  1464
  by (rule isCont_o2 [OF _ isCont_cos])
huffman@44311
  1465
huffman@44311
  1466
lemma tendsto_sin [tendsto_intros]:
huffman@44311
  1467
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
huffman@44311
  1468
  by (rule isCont_tendsto_compose [OF isCont_sin])
huffman@44311
  1469
huffman@44311
  1470
lemma tendsto_cos [tendsto_intros]:
huffman@44311
  1471
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
huffman@44311
  1472
  by (rule isCont_tendsto_compose [OF isCont_cos])
huffman@29164
  1473
hoelzl@51478
  1474
lemma continuous_sin [continuous_intros]:
hoelzl@51478
  1475
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
hoelzl@51478
  1476
  unfolding continuous_def by (rule tendsto_sin)
hoelzl@51478
  1477
hoelzl@51478
  1478
lemma continuous_on_sin [continuous_on_intros]:
hoelzl@51478
  1479
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
hoelzl@51478
  1480
  unfolding continuous_on_def by (auto intro: tendsto_sin)
hoelzl@51478
  1481
hoelzl@51478
  1482
lemma continuous_cos [continuous_intros]:
hoelzl@51478
  1483
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
hoelzl@51478
  1484
  unfolding continuous_def by (rule tendsto_cos)
hoelzl@51478
  1485
hoelzl@51478
  1486
lemma continuous_on_cos [continuous_on_intros]:
hoelzl@51478
  1487
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
hoelzl@51478
  1488
  unfolding continuous_on_def by (auto intro: tendsto_cos)
hoelzl@51478
  1489
hoelzl@31880
  1490
declare
hoelzl@31880
  1491
  DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@43335
  1492
  DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  1493
  DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  1494
  DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  1495
huffman@29164
  1496
subsection {* Properties of Sine and Cosine *}
paulson@15077
  1497
paulson@15077
  1498
lemma sin_zero [simp]: "sin 0 = 0"
huffman@44311
  1499
  unfolding sin_def sin_coeff_def by (simp add: powser_zero)
paulson@15077
  1500
paulson@15077
  1501
lemma cos_zero [simp]: "cos 0 = 1"
huffman@44311
  1502
  unfolding cos_def cos_coeff_def by (simp add: powser_zero)
paulson@15077
  1503
huffman@44308
  1504
lemma sin_cos_squared_add [simp]: "(sin x)\<twosuperior> + (cos x)\<twosuperior> = 1"
huffman@44308
  1505
proof -
huffman@44308
  1506
  have "\<forall>x. DERIV (\<lambda>x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
huffman@44308
  1507
    by (auto intro!: DERIV_intros)
huffman@44308
  1508
  hence "(sin x)\<twosuperior> + (cos x)\<twosuperior> = (sin 0)\<twosuperior> + (cos 0)\<twosuperior>"
huffman@44308
  1509
    by (rule DERIV_isconst_all)
huffman@44308
  1510
  thus "(sin x)\<twosuperior> + (cos x)\<twosuperior> = 1" by simp
huffman@44308
  1511
qed
huffman@44308
  1512
huffman@44308
  1513
lemma sin_cos_squared_add2 [simp]: "(cos x)\<twosuperior> + (sin x)\<twosuperior> = 1"
huffman@44308
  1514
  by (subst add_commute, rule sin_cos_squared_add)
paulson@15077
  1515
paulson@15077
  1516
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
huffman@44308
  1517
  using sin_cos_squared_add2 [unfolded power2_eq_square] .
paulson@15077
  1518
paulson@15077
  1519
lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
huffman@44308
  1520
  unfolding eq_diff_eq by (rule sin_cos_squared_add)
paulson@15077
  1521
paulson@15077
  1522
lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
huffman@44308
  1523
  unfolding eq_diff_eq by (rule sin_cos_squared_add2)
paulson@15077
  1524
paulson@15081
  1525
lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
huffman@44308
  1526
  by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
paulson@15077
  1527
paulson@15077
  1528
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
huffman@44308
  1529
  using abs_sin_le_one [of x] unfolding abs_le_iff by simp
paulson@15077
  1530
paulson@15077
  1531
lemma sin_le_one [simp]: "sin x \<le> 1"
huffman@44308
  1532
  using abs_sin_le_one [of x] unfolding abs_le_iff by simp
paulson@15077
  1533
paulson@15081
  1534
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
huffman@44308
  1535
  by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
paulson@15077
  1536
paulson@15077
  1537
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
huffman@44308
  1538
  using abs_cos_le_one [of x] unfolding abs_le_iff by simp
paulson@15077
  1539
paulson@15077
  1540
lemma cos_le_one [simp]: "cos x \<le> 1"
huffman@44308
  1541
  using abs_cos_le_one [of x] unfolding abs_le_iff by simp
paulson@15077
  1542
hoelzl@41970
  1543
lemma DERIV_fun_pow: "DERIV g x :> m ==>
paulson@15077
  1544
      DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
huffman@44311
  1545
  by (auto intro!: DERIV_intros)
paulson@15077
  1546
paulson@15229
  1547
lemma DERIV_fun_exp:
paulson@15229
  1548
     "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
huffman@44311
  1549
  by (auto intro!: DERIV_intros)
paulson@15077
  1550
paulson@15229
  1551
lemma DERIV_fun_sin:
paulson@15229
  1552
     "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
huffman@44311
  1553
  by (auto intro!: DERIV_intros)
paulson@15077
  1554
paulson@15229
  1555
lemma DERIV_fun_cos:
paulson@15229
  1556
     "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
huffman@44311
  1557
  by (auto intro!: DERIV_intros)
paulson@15077
  1558
huffman@44308
  1559
lemma sin_cos_add_lemma:
hoelzl@41970
  1560
     "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
paulson@15077
  1561
      (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
huffman@44308
  1562
  (is "?f x = 0")
huffman@44308
  1563
proof -
huffman@44308
  1564
  have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
huffman@44308
  1565
    by (auto intro!: DERIV_intros simp add: algebra_simps)
huffman@44308
  1566
  hence "?f x = ?f 0"
huffman@44308
  1567
    by (rule DERIV_isconst_all)
huffman@44308
  1568
  thus ?thesis by simp
huffman@44308
  1569
qed
paulson@15077
  1570
paulson@15077
  1571
lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
huffman@44308
  1572
  using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
paulson@15077
  1573
paulson@15077
  1574
lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
huffman@44308
  1575
  using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
huffman@44308
  1576
huffman@44308
  1577
lemma sin_cos_minus_lemma:
huffman@44308
  1578
  "(sin(-x) + sin(x))\<twosuperior> + (cos(-x) - cos(x))\<twosuperior> = 0" (is "?f x = 0")
huffman@44308
  1579
proof -
huffman@44308
  1580
  have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
huffman@44308
  1581
    by (auto intro!: DERIV_intros simp add: algebra_simps)
huffman@44308
  1582
  hence "?f x = ?f 0"
huffman@44308
  1583
    by (rule DERIV_isconst_all)
huffman@44308
  1584
  thus ?thesis by simp
huffman@44308
  1585
qed
paulson@15077
  1586
paulson@15077
  1587
lemma sin_minus [simp]: "sin (-x) = -sin(x)"
huffman@44308
  1588
  using sin_cos_minus_lemma [where x=x] by simp
paulson@15077
  1589
paulson@15077
  1590
lemma cos_minus [simp]: "cos (-x) = cos(x)"
huffman@44308
  1591
  using sin_cos_minus_lemma [where x=x] by simp
paulson@15077
  1592
paulson@15077
  1593
lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
huffman@44308
  1594
  by (simp add: diff_minus sin_add)
paulson@15077
  1595
paulson@15077
  1596
lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
huffman@44308
  1597
  by (simp add: sin_diff mult_commute)
paulson@15077
  1598
paulson@15077
  1599
lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
huffman@44308
  1600
  by (simp add: diff_minus cos_add)
paulson@15077
  1601
paulson@15077
  1602
lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
huffman@44308
  1603
  by (simp add: cos_diff mult_commute)
paulson@15077
  1604
paulson@15077
  1605
lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
huffman@29165
  1606
  using sin_add [where x=x and y=x] by simp
paulson@15077
  1607
paulson@15077
  1608
lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
huffman@29165
  1609
  using cos_add [where x=x and y=x]
huffman@29165
  1610
  by (simp add: power2_eq_square)
paulson@15077
  1611
paulson@15077
  1612
huffman@29164
  1613
subsection {* The Constant Pi *}
paulson@15077
  1614
huffman@44308
  1615
definition pi :: "real" where
huffman@23053
  1616
  "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
huffman@23043
  1617
hoelzl@41970
  1618
text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
paulson@15077
  1619
   hence define pi.*}
paulson@15077
  1620
paulson@15077
  1621
lemma sin_paired:
hoelzl@41970
  1622
     "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1))
paulson@15077
  1623
      sums  sin x"
paulson@15077
  1624
proof -
huffman@31271
  1625
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
huffman@44727
  1626
    by (rule sin_converges [THEN sums_group], simp)
huffman@31271
  1627
  thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)
paulson@15077
  1628
qed
paulson@15077
  1629
huffman@44728
  1630
lemma sin_gt_zero:
huffman@44728
  1631
  assumes "0 < x" and "x < 2" shows "0 < sin x"
huffman@44728
  1632
proof -
huffman@44728
  1633
  let ?f = "\<lambda>n. \<Sum>k = n*2..<n*2+2. -1 ^ k / real (fact (2*k+1)) * x^(2*k+1)"
huffman@44728
  1634
  have pos: "\<forall>n. 0 < ?f n"
huffman@44728
  1635
  proof
huffman@44728
  1636
    fix n :: nat
huffman@44728
  1637
    let ?k2 = "real (Suc (Suc (4 * n)))"
huffman@44728
  1638
    let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
huffman@44728
  1639
    have "x * x < ?k2 * ?k3"
huffman@44728
  1640
      using assms by (intro mult_strict_mono', simp_all)
huffman@44728
  1641
    hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
huffman@44728
  1642
      by (intro mult_strict_right_mono zero_less_power `0 < x`)
huffman@44728
  1643
    thus "0 < ?f n"
huffman@44728
  1644
      by (simp del: mult_Suc,
huffman@44728
  1645
        simp add: less_divide_eq mult_pos_pos field_simps del: mult_Suc)
huffman@44728
  1646
  qed
huffman@44728
  1647
  have sums: "?f sums sin x"
huffman@44728
  1648
    by (rule sin_paired [THEN sums_group], simp)
huffman@44728
  1649
  show "0 < sin x"
huffman@44728
  1650
    unfolding sums_unique [OF sums]
huffman@44728
  1651
    using sums_summable [OF sums] pos
huffman@44728
  1652
    by (rule suminf_gt_zero)
huffman@44728
  1653
qed
paulson@15077
  1654
paulson@15077
  1655
lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
huffman@44311
  1656
apply (cut_tac x = x in sin_gt_zero)
paulson@15077
  1657
apply (auto simp add: cos_squared_eq cos_double)
paulson@15077
  1658
done
paulson@15077
  1659
paulson@15077
  1660
lemma cos_paired:
huffman@23177
  1661
     "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
paulson@15077
  1662
proof -
huffman@31271
  1663
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
huffman@44727
  1664
    by (rule cos_converges [THEN sums_group], simp)
huffman@31271
  1665
  thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)
paulson@15077
  1666
qed
paulson@15077
  1667
huffman@36824
  1668
lemma real_mult_inverse_cancel:
hoelzl@41970
  1669
     "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
huffman@36824
  1670
      ==> inverse x * y < inverse x1 * u"
hoelzl@41970
  1671
apply (rule_tac c=x in mult_less_imp_less_left)
huffman@36824
  1672
apply (auto simp add: mult_assoc [symmetric])
huffman@36824
  1673
apply (simp (no_asm) add: mult_ac)
hoelzl@41970
  1674
apply (rule_tac c=x1 in mult_less_imp_less_right)
huffman@36824
  1675
apply (auto simp add: mult_ac)
huffman@36824
  1676
done
huffman@36824
  1677
huffman@36824
  1678
lemma real_mult_inverse_cancel2:
huffman@36824
  1679
     "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
huffman@36824
  1680
apply (auto dest: real_mult_inverse_cancel simp add: mult_ac)
huffman@36824
  1681
done
huffman@36824
  1682
huffman@36824
  1683
lemma realpow_num_eq_if:
huffman@36824
  1684
  fixes m :: "'a::power"
huffman@36824
  1685
  shows "m ^ n = (if n=0 then 1 else m * m ^ (n - 1))"
huffman@36824
  1686
by (cases n, auto)
huffman@36824
  1687
huffman@23053
  1688
lemma cos_two_less_zero [simp]: "cos (2) < 0"
paulson@15077
  1689
apply (cut_tac x = 2 in cos_paired)
paulson@15077
  1690
apply (drule sums_minus)
hoelzl@41970
  1691
apply (rule neg_less_iff_less [THEN iffD1])
nipkow@15539
  1692
apply (frule sums_unique, auto)
nipkow@15539
  1693
apply (rule_tac y =
huffman@23177
  1694
 "\<Sum>n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
paulson@15481
  1695
       in order_less_trans)
avigad@32047
  1696
apply (simp (no_asm) add: fact_num_eq_if_nat realpow_num_eq_if del: fact_Suc)
nipkow@15561
  1697
apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
paulson@15077
  1698
apply (rule sumr_pos_lt_pair)
paulson@15077
  1699
apply (erule sums_summable, safe)
huffman@30082
  1700
unfolding One_nat_def
hoelzl@41970
  1701
apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
avigad@32047
  1702
            del: fact_Suc)
huffman@46240
  1703
apply (simp add: inverse_eq_divide less_divide_eq del: fact_Suc)
avigad@32047
  1704
apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
paulson@15481
  1705
apply (simp only: real_of_nat_mult)
huffman@23007
  1706
apply (rule mult_strict_mono, force)
huffman@27483
  1707
  apply (rule_tac [3] real_of_nat_ge_zero)
paulson@15481
  1708
 prefer 2 apply force
paulson@15077
  1709
apply (rule real_of_nat_less_iff [THEN iffD2])
avigad@32036
  1710
apply (rule fact_less_mono_nat, auto)
paulson@15077
  1711
done
huffman@23053
  1712
huffman@23053
  1713
lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
huffman@23053
  1714
lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
paulson@15077
  1715
paulson@15077
  1716
lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
huffman@44730
  1717
proof (rule ex_ex1I)
huffman@44730
  1718
  show "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0"
huffman@44730
  1719
    by (rule IVT2, simp_all)
huffman@44730
  1720
next
huffman@44730
  1721
  fix x y
huffman@44730
  1722
  assume x: "0 \<le> x \<and> x \<le> 2 \<and> cos x = 0"
huffman@44730
  1723
  assume y: "0 \<le> y \<and> y \<le> 2 \<and> cos y = 0"
huffman@44730
  1724
  have [simp]: "\<forall>x. cos differentiable x"
huffman@44730
  1725
    unfolding differentiable_def by (auto intro: DERIV_cos)
huffman@44730
  1726
  from x y show "x = y"
huffman@44730
  1727
    apply (cut_tac less_linear [of x y], auto)
huffman@44730
  1728
    apply (drule_tac f = cos in Rolle)
huffman@44730
  1729
    apply (drule_tac [5] f = cos in Rolle)
huffman@44730
  1730
    apply (auto dest!: DERIV_cos [THEN DERIV_unique])
huffman@44730
  1731
    apply (metis order_less_le_trans less_le sin_gt_zero)
huffman@44730
  1732
    apply (metis order_less_le_trans less_le sin_gt_zero)
huffman@44730
  1733
    done
huffman@44730
  1734
qed
hoelzl@31880
  1735
huffman@23053
  1736
lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
paulson@15077
  1737
by (simp add: pi_def)
paulson@15077
  1738
paulson@15077
  1739
lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
huffman@23053
  1740
by (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1741
huffman@23053
  1742
lemma pi_half_gt_zero [simp]: "0 < pi / 2"
huffman@23053
  1743
apply (rule order_le_neq_trans)
huffman@23053
  1744
apply (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1745
apply (rule notI, drule arg_cong [where f=cos], simp)
paulson@15077
  1746
done
paulson@15077
  1747
huffman@23053
  1748
lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
huffman@23053
  1749
lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
paulson@15077
  1750
huffman@23053
  1751
lemma pi_half_less_two [simp]: "pi / 2 < 2"
huffman@23053
  1752
apply (rule order_le_neq_trans)
huffman@23053
  1753
apply (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1754
apply (rule notI, drule arg_cong [where f=cos], simp)
paulson@15077
  1755
done
huffman@23053
  1756
huffman@23053
  1757
lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
huffman@23053
  1758
lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
paulson@15077
  1759
paulson@15077
  1760
lemma pi_gt_zero [simp]: "0 < pi"
huffman@23053
  1761
by (insert pi_half_gt_zero, simp)
huffman@23053
  1762
huffman@23053
  1763
lemma pi_ge_zero [simp]: "0 \<le> pi"
huffman@23053
  1764
by (rule pi_gt_zero [THEN order_less_imp_le])
paulson@15077
  1765
paulson@15077
  1766
lemma pi_neq_zero [simp]: "pi \<noteq> 0"
huffman@22998
  1767
by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym])
paulson@15077
  1768
huffman@23053
  1769
lemma pi_not_less_zero [simp]: "\<not> pi < 0"
huffman@23053
  1770
by (simp add: linorder_not_less)
paulson@15077
  1771
huffman@29165
  1772
lemma minus_pi_half_less_zero: "-(pi/2) < 0"
huffman@29165
  1773
by simp
paulson@15077
  1774
hoelzl@29803
  1775
lemma m2pi_less_pi: "- (2 * pi) < pi"
huffman@45308
  1776
by simp
hoelzl@29803
  1777
paulson@15077
  1778
lemma sin_pi_half [simp]: "sin(pi/2) = 1"
paulson@15077
  1779
apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
paulson@15077
  1780
apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
huffman@36970
  1781
apply (simp add: power2_eq_1_iff)
paulson@15077
  1782
done
paulson@15077
  1783
paulson@15077
  1784
lemma cos_pi [simp]: "cos pi = -1"
nipkow@15539
  1785
by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
paulson@15077
  1786
paulson@15077
  1787
lemma sin_pi [simp]: "sin pi = 0"
nipkow@15539
  1788
by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
paulson@15077
  1789
paulson@15077
  1790
lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
huffman@45309
  1791
by (simp add: cos_diff)
paulson@15077
  1792
paulson@15077
  1793
lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
paulson@15229
  1794
by (simp add: cos_add)
paulson@15077
  1795
paulson@15077
  1796
lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
huffman@45309
  1797
by (simp add: sin_diff)
paulson@15077
  1798
paulson@15077
  1799
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
paulson@15229
  1800
by (simp add: sin_add)
paulson@15077
  1801
paulson@15077
  1802
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
paulson@15229
  1803
by (simp add: sin_add)
paulson@15077
  1804
paulson@15077
  1805
lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
paulson@15229
  1806
by (simp add: cos_add)
paulson@15077
  1807
paulson@15077
  1808
lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
paulson@15077
  1809
by (simp add: sin_add cos_double)
paulson@15077
  1810
paulson@15077
  1811
lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
paulson@15077
  1812
by (simp add: cos_add cos_double)
paulson@15077
  1813
paulson@15077
  1814
lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
paulson@15251
  1815
apply (induct "n")
webertj@49962
  1816
apply (auto simp add: real_of_nat_Suc distrib_right)
paulson@15077
  1817
done
paulson@15077
  1818
paulson@15383
  1819
lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
paulson@15383
  1820
proof -
paulson@15383
  1821
  have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
hoelzl@41970
  1822
  also have "... = -1 ^ n" by (rule cos_npi)
paulson@15383
  1823
  finally show ?thesis .
paulson@15383
  1824
qed
paulson@15383
  1825
paulson@15077
  1826
lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
paulson@15251
  1827
apply (induct "n")
webertj@49962
  1828
apply (auto simp add: real_of_nat_Suc distrib_right)
paulson@15077
  1829
done
paulson@15077
  1830
paulson@15077
  1831
lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
hoelzl@41970
  1832
by (simp add: mult_commute [of pi])
paulson@15077
  1833
paulson@15077
  1834
lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
paulson@15077
  1835
by (simp add: cos_double)
paulson@15077
  1836
paulson@15077
  1837
lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
paulson@15229
  1838
by simp
paulson@15077
  1839
paulson@15077
  1840
lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
paulson@15077
  1841
apply (rule sin_gt_zero, assumption)
paulson@15077
  1842
apply (rule order_less_trans, assumption)
paulson@15077
  1843
apply (rule pi_half_less_two)
paulson@15077
  1844
done
paulson@15077
  1845
hoelzl@41970
  1846
lemma sin_less_zero:
paulson@15077
  1847
  assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
paulson@15077
  1848
proof -
hoelzl@41970
  1849
  have "0 < sin (- x)" using assms by (simp only: sin_gt_zero2)
paulson@15077
  1850
  thus ?thesis by simp
paulson@15077
  1851
qed
paulson@15077
  1852
paulson@15077
  1853
lemma pi_less_4: "pi < 4"
paulson@15077
  1854
by (cut_tac pi_half_less_two, auto)
paulson@15077
  1855
paulson@15077
  1856
lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1857
apply (cut_tac pi_less_4)
paulson@15077
  1858
apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
paulson@15077
  1859
apply (cut_tac cos_is_zero, safe)
paulson@15077
  1860
apply (rename_tac y z)
paulson@15077
  1861
apply (drule_tac x = y in spec)
hoelzl@41970
  1862
apply (drule_tac x = "pi/2" in spec, simp)
paulson@15077
  1863
done
paulson@15077
  1864
paulson@15077
  1865
lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1866
apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15077
  1867
apply (rule cos_minus [THEN subst])
paulson@15077
  1868
apply (rule cos_gt_zero)
paulson@15077
  1869
apply (auto intro: cos_gt_zero)
paulson@15077
  1870
done
hoelzl@41970
  1871
paulson@15077
  1872
lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
paulson@15077
  1873
apply (auto simp add: order_le_less cos_gt_zero_pi)
hoelzl@41970
  1874
apply (subgoal_tac "x = pi/2", auto)
paulson@15077
  1875
done
paulson@15077
  1876
paulson@15077
  1877
lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
huffman@45309
  1878
by (simp add: sin_cos_eq cos_gt_zero_pi)
hoelzl@29803
  1879
hoelzl@29803
  1880
lemma pi_ge_two: "2 \<le> pi"
hoelzl@29803
  1881
proof (rule ccontr)
hoelzl@29803
  1882
  assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
hoelzl@29803
  1883
  have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
hoelzl@29803
  1884
  proof (cases "2 < 2 * pi")
hoelzl@29803
  1885
    case True with dense[OF `pi < 2`] show ?thesis by auto
hoelzl@29803
  1886
  next
hoelzl@29803
  1887
    case False have "pi < 2 * pi" by auto
hoelzl@29803
  1888
    from dense[OF this] and False show ?thesis by auto
hoelzl@29803
  1889
  qed
hoelzl@29803
  1890
  then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
hoelzl@29803
  1891
  hence "0 < sin y" using sin_gt_zero by auto
hoelzl@41970
  1892
  moreover
hoelzl@29803
  1893
  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
  1894
  ultimately show False by auto
hoelzl@29803
  1895
qed
hoelzl@29803
  1896
paulson@15077
  1897
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
paulson@15077
  1898
by (auto simp add: order_le_less sin_gt_zero_pi)
paulson@15077
  1899
huffman@44745
  1900
text {* FIXME: This proof is almost identical to lemma @{text cos_is_zero}.
huffman@44745
  1901
  It should be possible to factor out some of the common parts. *}
huffman@44745
  1902
paulson@15077
  1903
lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
huffman@44745
  1904
proof (rule ex_ex1I)
huffman@44745
  1905
  assume y: "-1 \<le> y" "y \<le> 1"
huffman@44745
  1906
  show "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y"
huffman@44745
  1907
    by (rule IVT2, simp_all add: y)
huffman@44745
  1908
next
huffman@44745
  1909
  fix a b
huffman@44745
  1910
  assume a: "0 \<le> a \<and> a \<le> pi \<and> cos a = y"
huffman@44745
  1911
  assume b: "0 \<le> b \<and> b \<le> pi \<and> cos b = y"
huffman@44745
  1912
  have [simp]: "\<forall>x. cos differentiable x"
huffman@44745
  1913
    unfolding differentiable_def by (auto intro: DERIV_cos)
huffman@44745
  1914
  from a b show "a = b"
huffman@44745
  1915
    apply (cut_tac less_linear [of a b], auto)
huffman@44745
  1916
    apply (drule_tac f = cos in Rolle)
huffman@44745
  1917
    apply (drule_tac [5] f = cos in Rolle)
huffman@44745
  1918
    apply (auto dest!: DERIV_cos [THEN DERIV_unique])
huffman@44745
  1919
    apply (metis order_less_le_trans less_le sin_gt_zero_pi)
huffman@44745
  1920
    apply (metis order_less_le_trans less_le sin_gt_zero_pi)
huffman@44745
  1921
    done
huffman@44745
  1922
qed
paulson@15077
  1923
paulson@15077
  1924
lemma sin_total:
paulson@15077
  1925
     "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
paulson@15077
  1926
apply (rule ccontr)
paulson@15077
  1927
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
  1928
apply (erule contrapos_np)
huffman@45309
  1929
apply simp
hoelzl@41970
  1930
apply (cut_tac y="-y" in cos_total, simp) apply simp
paulson@15077
  1931
apply (erule ex1E)
paulson@15229
  1932
apply (rule_tac a = "x - (pi/2)" in ex1I)
huffman@23286
  1933
apply (simp (no_asm) add: add_assoc)
paulson@15077
  1934
apply (rotate_tac 3)
huffman@45309
  1935
apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all add: cos_add)
paulson@15077
  1936
done
paulson@15077
  1937
paulson@15077
  1938
lemma reals_Archimedean4:
paulson@15077
  1939
     "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
paulson@15077
  1940
apply (auto dest!: reals_Archimedean3)
hoelzl@41970
  1941
apply (drule_tac x = x in spec, clarify)
paulson@15077
  1942
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
hoelzl@41970
  1943
 prefer 2 apply (erule LeastI)
hoelzl@41970
  1944
apply (case_tac "LEAST m::nat. x < real m * y", simp)
paulson@15077
  1945
apply (subgoal_tac "~ x < real nat * y")
hoelzl@41970
  1946
 prefer 2 apply (rule not_less_Least, simp, force)
paulson@15077
  1947
done
paulson@15077
  1948
hoelzl@41970
  1949
(* Pre Isabelle99-2 proof was simpler- numerals arithmetic
paulson@15077
  1950
   now causes some unwanted re-arrangements of literals!   *)
paulson@15229
  1951
lemma cos_zero_lemma:
hoelzl@41970
  1952
     "[| 0 \<le> x; cos x = 0 |] ==>
paulson@15077
  1953
      \<exists>n::nat. ~even n & x = real n * (pi/2)"
paulson@15077
  1954
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
hoelzl@41970
  1955
apply (subgoal_tac "0 \<le> x - real n * pi &
paulson@15086
  1956
                    (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
nipkow@29667
  1957
apply (auto simp add: algebra_simps real_of_nat_Suc)
nipkow@29667
  1958
 prefer 2 apply (simp add: cos_diff)
paulson@15077
  1959
apply (simp add: cos_diff)
paulson@15077
  1960
apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
paulson@15077
  1961
apply (rule_tac [2] cos_total, safe)
paulson@15077
  1962
apply (drule_tac x = "x - real n * pi" in spec)
paulson@15077
  1963
apply (drule_tac x = "pi/2" in spec)
paulson@15077
  1964
apply (simp add: cos_diff)
paulson@15229
  1965
apply (rule_tac x = "Suc (2 * n)" in exI)
nipkow@29667
  1966
apply (simp add: real_of_nat_Suc algebra_simps, auto)
paulson@15077
  1967
done
paulson@15077
  1968
paulson@15229
  1969
lemma sin_zero_lemma:
hoelzl@41970
  1970
     "[| 0 \<le> x; sin x = 0 |] ==>
paulson@15077
  1971
      \<exists>n::nat. even n & x = real n * (pi/2)"
paulson@15077
  1972
apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
paulson@15077
  1973
 apply (clarify, rule_tac x = "n - 1" in exI)
webertj@49962
  1974
 apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc distrib_right)
paulson@15085
  1975
apply (rule cos_zero_lemma)
huffman@45309
  1976
apply (simp_all add: cos_add)
paulson@15077
  1977
done
paulson@15077
  1978
paulson@15077
  1979
paulson@15229
  1980
lemma cos_zero_iff:
hoelzl@41970
  1981
     "(cos x = 0) =
hoelzl@41970
  1982
      ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
paulson@15077
  1983
       (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1984
apply (rule iffI)
paulson@15077
  1985
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1986
apply (drule cos_zero_lemma, assumption+)
hoelzl@41970
  1987
apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
hoelzl@41970
  1988
apply (force simp add: minus_equation_iff [of x])
webertj@49962
  1989
apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc distrib_right)
nipkow@15539
  1990
apply (auto simp add: cos_add)
paulson@15077
  1991
done
paulson@15077
  1992
paulson@15077
  1993
(* ditto: but to a lesser extent *)
paulson@15229
  1994
lemma sin_zero_iff:
hoelzl@41970
  1995
     "(sin x = 0) =
hoelzl@41970
  1996
      ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
paulson@15077
  1997
       (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1998
apply (rule iffI)
paulson@15077
  1999
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  2000
apply (drule sin_zero_lemma, assumption+)
paulson@15077
  2001
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
hoelzl@41970
  2002
apply (force simp add: minus_equation_iff [of x])
nipkow@15539
  2003
apply (auto simp add: even_mult_two_ex)
paulson@15077
  2004
done
paulson@15077
  2005
hoelzl@29803
  2006
lemma cos_monotone_0_pi: assumes "0 \<le> y" and "y < x" and "x \<le> pi"
hoelzl@29803
  2007
  shows "cos x < cos y"
hoelzl@29803
  2008
proof -
wenzelm@33549
  2009
  have "- (x - y) < 0" using assms by auto
hoelzl@29803
  2010
hoelzl@29803
  2011
  from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
hoelzl@29803
  2012
  obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" by auto
wenzelm@33549
  2013
  hence "0 < z" and "z < pi" using assms by auto
hoelzl@29803
  2014
  hence "0 < sin z" using sin_gt_zero_pi by auto
hoelzl@29803
  2015
  hence "cos x - cos y < 0" unfolding cos_diff minus_mult_commute[symmetric] using `- (x - y) < 0` by (rule mult_pos_neg2)
hoelzl@29803
  2016
  thus ?thesis by auto
hoelzl@29803
  2017
qed
hoelzl@29803
  2018
hoelzl@29803
  2019
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
  2020
proof (cases "y < x")
hoelzl@29803
  2021
  case True show ?thesis using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
hoelzl@29803
  2022
next
hoelzl@29803
  2023
  case False hence "y = x" using `y \<le> x` by auto
hoelzl@29803
  2024
  thus ?thesis by auto
hoelzl@29803
  2025
qed
hoelzl@29803
  2026
hoelzl@29803
  2027
lemma cos_monotone_minus_pi_0: assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
hoelzl@29803
  2028
  shows "cos y < cos x"
hoelzl@29803
  2029
proof -
wenzelm@33549
  2030
  have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" using assms by auto
hoelzl@29803
  2031
  from cos_monotone_0_pi[OF this]
hoelzl@29803
  2032
  show ?thesis unfolding cos_minus .
hoelzl@29803
  2033
qed
hoelzl@29803
  2034
hoelzl@29803
  2035
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
  2036
proof (cases "y < x")
hoelzl@29803
  2037
  case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] by auto
hoelzl@29803
  2038
next
hoelzl@29803
  2039
  case False hence "y = x" using `y \<le> x` by auto
hoelzl@29803
  2040
  thus ?thesis by auto
hoelzl@29803
  2041
qed
hoelzl@29803
  2042
hoelzl@29803
  2043
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
  2044
proof -
wenzelm@33549
  2045
  have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi"
wenzelm@33549
  2046
    using pi_ge_two and assms by auto
hoelzl@29803
  2047
  from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] by auto
hoelzl@29803
  2048
qed
paulson@15077
  2049
huffman@29164
  2050
subsection {* Tangent *}
paulson@15077
  2051
huffman@44311
  2052
definition tan :: "real \<Rightarrow> real" where
huffman@44311
  2053
  "tan = (\<lambda>x. sin x / cos x)"
huffman@23043
  2054
paulson@15077
  2055
lemma tan_zero [simp]: "tan 0 = 0"
huffman@44311
  2056
  by (simp add: tan_def)
paulson@15077
  2057
paulson@15077
  2058
lemma tan_pi [simp]: "tan pi = 0"
huffman@44311
  2059
  by (simp add: tan_def)
paulson@15077
  2060
paulson@15077
  2061
lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
huffman@44311
  2062
  by (simp add: tan_def)
paulson@15077
  2063
paulson@15077
  2064
lemma tan_minus [simp]: "tan (-x) = - tan x"
huffman@44311
  2065
  by (simp add: tan_def)
paulson@15077
  2066
paulson@15077
  2067
lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
huffman@44311
  2068
  by (simp add: tan_def)
paulson@15077
  2069
hoelzl@41970
  2070
lemma lemma_tan_add1:
huffman@44311
  2071
  "\<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
  2072
  by (simp add: tan_def cos_add field_simps)
paulson@15077
  2073
hoelzl@41970
  2074
lemma add_tan_eq:
huffman@44311
  2075
  "\<lbrakk>cos x \<noteq> 0; cos y \<noteq> 0\<rbrakk> \<Longrightarrow> tan x + tan y = sin(x + y)/(cos x * cos y)"
huffman@44311
  2076
  by (simp add: tan_def sin_add field_simps)
paulson@15077
  2077
paulson@15229
  2078
lemma tan_add:
hoelzl@41970
  2079
     "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
paulson@15077
  2080
      ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
huffman@44311
  2081
  by (simp add: add_tan_eq lemma_tan_add1, simp add: tan_def)
paulson@15077
  2082
paulson@15229
  2083
lemma tan_double:
hoelzl@41970
  2084
     "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
paulson@15077
  2085
      ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
huffman@44311
  2086
  using tan_add [of x x] by (simp add: power2_eq_square)
paulson@15077
  2087
paulson@15077
  2088
lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
hoelzl@41970
  2089
by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi)
hoelzl@41970
  2090
hoelzl@41970
  2091
lemma tan_less_zero:
paulson@15077
  2092
  assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
paulson@15077
  2093
proof -
hoelzl@41970
  2094
  have "0 < tan (- x)" using assms by (simp only: tan_gt_zero)
paulson@15077
  2095
  thus ?thesis by simp
paulson@15077
  2096
qed
paulson@15077
  2097
huffman@44756
  2098
lemma tan_half: "tan x = sin (2 * x) / (cos (2 * x) + 1)"
huffman@44756
  2099
  unfolding tan_def sin_double cos_double sin_squared_eq
huffman@44756
  2100
  by (simp add: power2_eq_square)
hoelzl@29803
  2101
huffman@44311
  2102
lemma DERIV_tan [simp]: "cos x \<noteq> 0 \<Longrightarrow> DERIV tan x :> inverse ((cos x)\<twosuperior>)"
huffman@44311
  2103
  unfolding tan_def
huffman@44311
  2104
  by (auto intro!: DERIV_intros, simp add: divide_inverse power2_eq_square)
huffman@44311
  2105
huffman@44311
  2106
lemma isCont_tan: "cos x \<noteq> 0 \<Longrightarrow> isCont tan x"
huffman@44311
  2107
  by (rule DERIV_tan [THEN DERIV_isCont])
huffman@44311
  2108
huffman@44311
  2109
lemma isCont_tan' [simp]:
huffman@44311
  2110
  "\<lbrakk>isCont f a; cos (f a) \<noteq> 0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. tan (f x)) a"
huffman@44311
  2111
  by (rule isCont_o2 [OF _ isCont_tan])
huffman@44311
  2112
huffman@44311
  2113
lemma tendsto_tan [tendsto_intros]:
huffman@44311
  2114
  "\<lbrakk>(f ---> a) F; cos a \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. tan (f x)) ---> tan a) F"
huffman@44311
  2115
  by (rule isCont_tendsto_compose [OF isCont_tan])
huffman@44311
  2116
hoelzl@51478
  2117
lemma continuous_tan:
hoelzl@51478
  2118
  "continuous F f \<Longrightarrow> cos (f (Lim F (\<lambda>x. x))) \<noteq> 0 \<Longrightarrow> continuous F (\<lambda>x. tan (f x))"
hoelzl@51478
  2119
  unfolding continuous_def by (rule tendsto_tan)
hoelzl@51478
  2120
hoelzl@51478
  2121
lemma isCont_tan'' [continuous_intros]:
hoelzl@51478
  2122
  "continuous (at x) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x) (\<lambda>x. tan (f x))"
hoelzl@51478
  2123
  unfolding continuous_at by (rule tendsto_tan)
hoelzl@51478
  2124
hoelzl@51478
  2125
lemma continuous_within_tan [continuous_intros]:
hoelzl@51478
  2126
  "continuous (at x within s) f \<Longrightarrow> cos (f x) \<noteq> 0 \<Longrightarrow> continuous (at x within s) (\<lambda>x. tan (f x))"
hoelzl@51478
  2127
  unfolding continuous_within by (rule tendsto_tan)
hoelzl@51478
  2128
hoelzl@51478
  2129
lemma continuous_on_tan [continuous_on_intros]:
hoelzl@51478
  2130
  "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. cos (f x) \<noteq> 0) \<Longrightarrow> continuous_on s (\<lambda>x. tan (f x))"
hoelzl@51478
  2131
  unfolding continuous_on_def by (auto intro: tendsto_tan)
hoelzl@51478
  2132
huffman@44311
  2133
lemma LIM_cos_div_sin: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
huffman@44311
  2134
  by (rule LIM_cong_limit, (rule tendsto_intros)+, simp_all)
paulson@15077
  2135
paulson@15077
  2136
lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
paulson@15077
  2137
apply (cut_tac LIM_cos_div_sin)
huffman@31338
  2138
apply (simp only: LIM_eq)
paulson@15077
  2139
apply (drule_tac x = "inverse y" in spec, safe, force)
paulson@15077
  2140
apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
paulson@15229
  2141
apply (rule_tac x = "(pi/2) - e" in exI)
paulson@15077
  2142
apply (simp (no_asm_simp))
paulson@15229
  2143
apply (drule_tac x = "(pi/2) - e" in spec)
huffman@45309
  2144
apply (auto simp add: tan_def sin_diff cos_diff)
paulson@15077
  2145
apply (rule inverse_less_iff_less [THEN iffD1])
paulson@15079
  2146
apply (auto simp add: divide_inverse)
huffman@36777
  2147
apply (rule mult_pos_pos)
paulson@15229
  2148
apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
huffman@36777
  2149
apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute)
paulson@15077
  2150
done
paulson@15077
  2151
paulson@15077
  2152
lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
huffman@22998
  2153
apply (frule order_le_imp_less_or_eq, safe)
paulson@15077
  2154
 prefer 2 apply force
paulson@15077
  2155
apply (drule lemma_tan_total, safe)
paulson@15077
  2156
apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
paulson@15077
  2157
apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
paulson@15077
  2158
apply (drule_tac y = xa in order_le_imp_less_or_eq)
paulson@15077
  2159
apply (auto dest: cos_gt_zero)
paulson@15077
  2160
done
paulson@15077
  2161
paulson@15077
  2162
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  2163
apply (cut_tac linorder_linear [of 0 y], safe)
paulson@15077
  2164
apply (drule tan_total_pos)
paulson@15077
  2165
apply (cut_tac [2] y="-y" in tan_total_pos, safe)
paulson@15077
  2166
apply (rule_tac [3] x = "-x" in exI)
huffman@44710
  2167
apply (auto del: exI intro!: exI)
paulson@15077
  2168
done
paulson@15077
  2169
paulson@15077
  2170
lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  2171
apply (cut_tac y = y in lemma_tan_total1, auto)
paulson@15077
  2172
apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
paulson@15077
  2173
apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
paulson@15077
  2174
apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
paulson@15077
  2175
apply (rule_tac [4] Rolle)
paulson@15077
  2176
apply (rule_tac [2] Rolle)
huffman@44710
  2177
apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
paulson@15077
  2178
            simp add: differentiable_def)
paulson@15077
  2179
txt{*Now, simulate TRYALL*}
paulson@15077
  2180
apply (rule_tac [!] DERIV_tan asm_rl)
paulson@15077
  2181
apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
hoelzl@41970
  2182
            simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym])
paulson@15077
  2183
done
paulson@15077
  2184
hoelzl@29803
  2185
lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
hoelzl@29803
  2186
  shows "tan y < tan x"
hoelzl@29803
  2187
proof -
hoelzl@29803
  2188
  have "\<forall> x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse (cos x'^2)"
hoelzl@29803
  2189
  proof (rule allI, rule impI)
hoelzl@29803
  2190
    fix x' :: real assume "y \<le> x' \<and> x' \<le> x"
wenzelm@33549
  2191
    hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
hoelzl@29803
  2192
    from cos_gt_zero_pi[OF this]
hoelzl@29803
  2193
    have "cos x' \<noteq> 0" by auto
hoelzl@29803
  2194
    thus "DERIV tan x' :> inverse (cos x'^2)" by (rule DERIV_tan)
hoelzl@29803
  2195
  qed
hoelzl@41970
  2196
  from MVT2[OF `y < x` this]
hoelzl@29803
  2197
  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
  2198
  hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
hoelzl@29803
  2199
  hence "0 < cos z" using cos_gt_zero_pi by auto
hoelzl@29803
  2200
  hence inv_pos: "0 < inverse ((cos z)\<twosuperior>)" by auto
hoelzl@29803
  2201
  have "0 < x - y" using `y < x` by auto
huffman@36777
  2202
  from mult_pos_pos [OF this inv_pos]
hoelzl@29803
  2203
  have "0 < tan x - tan y" unfolding tan_diff by auto
hoelzl@29803
  2204
  thus ?thesis by auto
hoelzl@29803
  2205
qed
hoelzl@29803
  2206
hoelzl@29803
  2207
lemma tan_monotone': assumes "- (pi / 2) < y" and "y < pi / 2" and "- (pi / 2) < x" and "x < pi / 2"
hoelzl@29803
  2208
  shows "(y < x) = (tan y < tan x)"
hoelzl@29803
  2209
proof
hoelzl@29803
  2210
  assume "y < x" thus "tan y < tan x" using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
hoelzl@29803
  2211
next
hoelzl@29803
  2212
  assume "tan y < tan x"
hoelzl@29803
  2213
  show "y < x"
hoelzl@29803
  2214
  proof (rule ccontr)
hoelzl@29803
  2215
    assume "\<not> y < x" hence "x \<le> y" by auto
hoelzl@41970
  2216
    hence "tan x \<le> tan y"
hoelzl@29803
  2217
    proof (cases "x = y")
hoelzl@29803
  2218
      case True thus ?thesis by auto
hoelzl@29803
  2219
    next
hoelzl@29803
  2220
      case False hence "x < y" using `x \<le> y` by auto
hoelzl@29803
  2221
      from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
hoelzl@29803
  2222
    qed
hoelzl@29803
  2223
    thus False using `tan y < tan x` by auto
hoelzl@29803
  2224
  qed
hoelzl@29803
  2225
qed
hoelzl@29803
  2226
hoelzl@29803
  2227
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
  2228
hoelzl@41970
  2229
lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x"
hoelzl@29803
  2230
  by (simp add: tan_def)
hoelzl@29803
  2231
hoelzl@41970
  2232
lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x"
hoelzl@29803
  2233
proof (induct n arbitrary: x)
hoelzl@29803
  2234
  case (Suc n)
webertj@49962
  2235
  have split_pi_off: "x + real (Suc n) * pi = (x + real n * pi) + pi" unfolding Suc_eq_plus1 real_of_nat_add real_of_one distrib_right by auto
hoelzl@29803
  2236
  show ?case unfolding split_pi_off using Suc by auto
hoelzl@29803
  2237
qed auto
hoelzl@29803
  2238
hoelzl@29803
  2239
lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
hoelzl@29803
  2240
proof (cases "0 \<le> i")
hoelzl@29803
  2241
  case True hence i_nat: "real i = real (nat i)" by auto
hoelzl@29803
  2242
  show ?thesis unfolding i_nat by auto
hoelzl@29803
  2243
next
hoelzl@29803
  2244
  case False hence i_nat: "real i = - real (nat (-i))" by auto
hoelzl@29803
  2245
  have "tan x = tan (x + real i * pi - real i * pi)" by auto
hoelzl@29803
  2246
  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
  2247
  finally show ?thesis by auto
hoelzl@29803
  2248
qed
hoelzl@29803
  2249
huffman@47108
  2250
lemma tan_periodic_n[simp]: "tan (x + numeral n * pi) = tan x"
huffman@47108
  2251
  using tan_periodic_int[of _ "numeral n" ] unfolding real_numeral .
huffman@23043
  2252
huffman@23043
  2253
subsection {* Inverse Trigonometric Functions *}
huffman@23043
  2254
huffman@23043
  2255
definition
huffman@23043
  2256
  arcsin :: "real => real" where
huffman@23043
  2257
  "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
huffman@23043
  2258
huffman@23043
  2259
definition
huffman@23043
  2260
  arccos :: "real => real" where
huffman@23043
  2261
  "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
huffman@23043
  2262
hoelzl@41970
  2263
definition
huffman@23043
  2264
  arctan :: "real => real" where
huffman@23043
  2265
  "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
huffman@23043
  2266
paulson@15229
  2267
lemma arcsin:
hoelzl@41970
  2268
     "[| -1 \<le> y; y \<le> 1 |]
hoelzl@41970
  2269
      ==> -(pi/2) \<le> arcsin y &
paulson@15077
  2270
           arcsin y \<le> pi/2 & sin(arcsin y) = y"
huffman@23011
  2271
unfolding arcsin_def by (rule theI' [OF sin_total])
huffman@23011
  2272
huffman@23011
  2273
lemma arcsin_pi:
hoelzl@41970
  2274
     "[| -1 \<le> y; y \<le> 1 |]
huffman@23011
  2275
      ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
huffman@23011
  2276
apply (drule (1) arcsin)
huffman@23011
  2277
apply (force intro: order_trans)
paulson@15077
  2278
done
paulson@15077
  2279
paulson@15077
  2280
lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
paulson@15077
  2281
by (blast dest: arcsin)
hoelzl@41970
  2282
paulson@15077
  2283
lemma arcsin_bounded:
paulson@15077
  2284
     "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
paulson@15077
  2285
by (blast dest: arcsin)
paulson@15077
  2286
paulson@15077
  2287
lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
paulson@15077
  2288
by (blast dest: arcsin)
paulson@15077
  2289
paulson@15077
  2290
lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
paulson@15077
  2291
by (blast dest: arcsin)
paulson@15077
  2292
paulson@15077
  2293
lemma arcsin_lt_bounded:
paulson@15077
  2294
     "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
paulson@15077
  2295
apply (frule order_less_imp_le)
paulson@15077
  2296
apply (frule_tac y = y in order_less_imp_le)
paulson@15077
  2297
apply (frule arcsin_bounded)
paulson@15077
  2298
apply (safe, simp)
paulson@15077
  2299
apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
paulson@15077
  2300
apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
paulson@15077
  2301
apply (drule_tac [!] f = sin in arg_cong, auto)
paulson@15077
  2302
done
paulson@15077
  2303
paulson@15077
  2304
lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
paulson@15077
  2305
apply (unfold arcsin_def)
huffman@23011
  2306
apply (rule the1_equality)
paulson@15077
  2307
apply (rule sin_total, auto)
paulson@15077
  2308
done
paulson@15077
  2309
huffman@22975
  2310
lemma arccos:
hoelzl@41970
  2311
     "[| -1 \<le> y; y \<le> 1 |]
huffman@22975
  2312
      ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
huffman@23011
  2313
unfolding arccos_def by (rule theI' [OF cos_total])
paulson@15077
  2314
huffman@22975
  2315
lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
huffman@22975
  2316
by (blast dest: arccos)
hoelzl@41970
  2317
huffman@22975
  2318
lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
huffman@22975
  2319
by (blast dest: arccos)
paulson@15077
  2320
huffman@22975
  2321
lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
huffman@22975
  2322
by (blast dest: arccos)
paulson@15077
  2323
huffman@22975
  2324
lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
huffman@22975
  2325
by (blast dest: arccos)
paulson@15077
  2326
huffman@22975
  2327
lemma arccos_lt_bounded:
hoelzl@41970
  2328
     "[| -1 < y; y < 1 |]
huffman@22975
  2329
      ==> 0 < arccos y & arccos y < pi"
paulson@15077
  2330
apply (frule order_less_imp_le)
paulson@15077
  2331
apply (frule_tac y = y in order_less_imp_le)
huffman@22975
  2332
apply (frule arccos_bounded, auto)
huffman@22975
  2333
apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
paulson@15077
  2334
apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
paulson@15077
  2335
apply (drule_tac [!] f = cos in arg_cong, auto)
paulson@15077
  2336
done
paulson@15077
  2337
huffman@22975
  2338
lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
huffman@22975
  2339
apply (simp add: arccos_def)
huffman@23011
  2340
apply (auto intro!: the1_equality cos_total)
paulson@15077
  2341
done
paulson@15077
  2342
huffman@22975
  2343
lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
huffman@22975
  2344
apply (simp add: arccos_def)
huffman@23011
  2345
apply (auto intro!: the1_equality cos_total)
paulson@15077
  2346
done
paulson@15077
  2347
huffman@23045
  2348
lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<twosuperior>)"
huffman@23045
  2349
apply (subgoal_tac "x\<twosuperior> \<le> 1")
huffman@23052
  2350
apply (rule power2_eq_imp_eq)
huffman@23045
  2351
apply (simp add: cos_squared_eq)
huffman@23045
  2352
apply (rule cos_ge_zero)
huffman@23045
  2353
apply (erule (1) arcsin_lbound)
huffman@23045
  2354
apply (erule (1) arcsin_ubound)
huffman@23045
  2355
apply simp
huffman@23045
  2356
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
huffman@23045
  2357
apply (rule power_mono, simp, simp)
huffman@23045
  2358
done
huffman@23045
  2359
huffman@23045
  2360
lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<twosuperior>)"
huffman@23045
  2361
apply (subgoal_tac "x\<twosuperior> \<le> 1")
huffman@23052
  2362
apply (rule power2_eq_imp_eq)
huffman@23045
  2363
apply (simp add: sin_squared_eq)
huffman@23045
  2364
apply (rule sin_ge_zero)
huffman@23045
  2365
apply (erule (1) arccos_lbound)
huffman@23045
  2366
apply (erule (1) arccos_ubound)
huffman@23045
  2367
apply simp
huffman@23045
  2368
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
huffman@23045
  2369
apply (rule power_mono, simp, simp)
huffman@23045
  2370
done