src/HOL/Transcendental.thy
author huffman
Thu Feb 18 14:21:44 2010 -0800 (2010-02-18)
changeset 35216 7641e8d831d2
parent 35213 b9866ad4e3be
child 36776 c137ae7673d3
permissions -rw-r--r--
get rid of many duplicate simp rule warnings
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(*  Title:      HOL/Transcendental.thy
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    Author:     Jacques D. Fleuriot, University of Cambridge, University of Edinburgh
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    Author:     Lawrence C Paulson
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*)
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header{*Power Series, Transcendental Functions etc.*}
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theory Transcendental
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imports Fact Series Deriv NthRoot
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begin
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subsection {* Properties of Power Series *}
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lemma lemma_realpow_diff:
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  fixes y :: "'a::monoid_mult"
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  shows "p \<le> n \<Longrightarrow> y ^ (Suc n - p) = (y ^ (n - p)) * y"
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proof -
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  assume "p \<le> n"
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  hence "Suc n - p = Suc (n - p)" by (rule Suc_diff_le)
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  thus ?thesis by (simp add: power_commutes)
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qed
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lemma lemma_realpow_diff_sumr:
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  fixes y :: "'a::{comm_semiring_0,monoid_mult}" shows
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     "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ (Suc n - p)) =  
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      y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
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by (simp add: setsum_right_distrib lemma_realpow_diff mult_ac
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         del: setsum_op_ivl_Suc)
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lemma lemma_realpow_diff_sumr2:
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  fixes y :: "'a::{comm_ring,monoid_mult}" shows
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     "x ^ (Suc n) - y ^ (Suc n) =  
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      (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
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apply (induct n, simp)
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apply (simp del: setsum_op_ivl_Suc)
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apply (subst setsum_op_ivl_Suc)
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apply (subst lemma_realpow_diff_sumr)
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apply (simp add: right_distrib del: setsum_op_ivl_Suc)
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apply (subst mult_left_commute [of "x - y"])
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apply (erule subst)
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apply (simp add: algebra_simps)
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done
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lemma lemma_realpow_rev_sumr:
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     "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =  
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      (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
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apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
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apply (rule inj_onI, simp)
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apply auto
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apply (rule_tac x="n - x" in image_eqI, simp, simp)
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done
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text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
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x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
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lemma powser_insidea:
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  fixes x z :: "'a::{real_normed_field,banach}"
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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 (simp add: Cauchy_convergent_iff)
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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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    { fix B T E have "(if \<not> B then T else E) = (if B then E else T)" by auto } note if_eq = this
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    have "?s sums y" using sums_if'[OF `f sums y`] .
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    from this[unfolded sums_def, THEN LIMSEQ_Suc] 
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    have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
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      unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
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                image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
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                even_Suc Suc_m1 if_eq .
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  } from sums_add[OF g_sums this]
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  show ?thesis unfolding if_sum .
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qed
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subsection {* Alternating series test / Leibniz formula *}
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lemma sums_alternating_upper_lower:
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  fixes a :: "nat \<Rightarrow> real"
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  assumes mono: "\<And>n. a (Suc n) \<le> a n" and a_pos: "\<And>n. 0 \<le> a n" and "a ----> 0"
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  shows "\<exists>l. ((\<forall>n. (\<Sum>i=0..<2*n. -1^i*a i) \<le> l) \<and> (\<lambda> n. \<Sum>i=0..<2*n. -1^i*a i) ----> l) \<and> 
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             ((\<forall>n. l \<le> (\<Sum>i=0..<2*n + 1. -1^i*a i)) \<and> (\<lambda> n. \<Sum>i=0..<2*n + 1. -1^i*a i) ----> l)"
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  (is "\<exists>l. ((\<forall>n. ?f n \<le> l) \<and> _) \<and> ((\<forall>n. l \<le> ?g n) \<and> _)")
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proof -
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  have fg_diff: "\<And>n. ?f n - ?g n = - a (2 * n)" unfolding One_nat_def by auto
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  have "\<forall> n. ?f n \<le> ?f (Suc n)"
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  proof fix n show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto qed
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  moreover
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  have "\<forall> n. ?g (Suc n) \<le> ?g n"
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  proof fix n show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
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    unfolding One_nat_def by auto qed
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  moreover
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  have "\<forall> n. ?f n \<le> ?g n" 
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  proof fix n show "?f n \<le> ?g n" using fg_diff a_pos
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    unfolding One_nat_def by auto qed
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  moreover
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  have "(\<lambda> n. ?f n - ?g n) ----> 0" unfolding fg_diff
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  proof (rule LIMSEQ_I)
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    fix r :: real assume "0 < r"
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    with `a ----> 0`[THEN LIMSEQ_D] 
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    obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r" by auto
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    hence "\<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
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    thus "\<exists> N. \<forall> n \<ge> N. norm (- a (2 * n) - 0) < r" by auto
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  qed
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  ultimately
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  show ?thesis by (rule lemma_nest_unique)
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qed 
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lemma summable_Leibniz': fixes a :: "nat \<Rightarrow> real"
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  assumes a_zero: "a ----> 0" and a_pos: "\<And> n. 0 \<le> a n"
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  and a_monotone: "\<And> n. a (Suc n) \<le> a n"
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  shows summable: "summable (\<lambda> n. (-1)^n * a n)"
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  and "\<And>n. (\<Sum>i=0..<2*n. (-1)^i*a i) \<le> (\<Sum>i. (-1)^i*a i)"
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  and "(\<lambda>n. \<Sum>i=0..<2*n. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
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  and "\<And>n. (\<Sum>i. (-1)^i*a i) \<le> (\<Sum>i=0..<2*n+1. (-1)^i*a i)"
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  and "(\<lambda>n. \<Sum>i=0..<2*n+1. (-1)^i*a i) ----> (\<Sum>i. (-1)^i*a i)"
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proof -
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  let "?S n" = "(-1)^n * a n"
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  let "?P n" = "\<Sum>i=0..<n. ?S i"
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  let "?f n" = "?P (2 * n)"
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  let "?g n" = "?P (2 * n + 1)"
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  obtain l :: real where below_l: "\<forall> n. ?f n \<le> l" and "?f ----> l" and above_l: "\<forall> n. l \<le> ?g n" and "?g ----> l"
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    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
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  let ?Sa = "\<lambda> m. \<Sum> n = 0..<m. ?S n"
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  have "?Sa ----> l"
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  proof (rule LIMSEQ_I)
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    fix r :: real assume "0 < r"
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    with `?f ----> l`[THEN LIMSEQ_D] 
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    obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
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    from `0 < r` `?g ----> l`[THEN LIMSEQ_D] 
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    obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
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    { fix n :: nat
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      assume "n \<ge> (max (2 * f_no) (2 * g_no))" hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
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      have "norm (?Sa n - l) < r"
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      proof (cases "even n")
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        case True from even_nat_div_two_times_two[OF this]
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        have n_eq: "2 * (n div 2) = n" unfolding numeral_2_eq_2[symmetric] by auto
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        with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no" by auto
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        from f[OF this]
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        show ?thesis unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
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      next
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        case False hence "even (n - 1)" by simp
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        from even_nat_div_two_times_two[OF this]
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        have n_eq: "2 * ((n - 1) div 2) = n - 1" unfolding numeral_2_eq_2[symmetric] by auto
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
hoelzl@29803
   297
    note leibniz = summable_Leibniz'[OF _ ge0, of "\<lambda>x. x", OF LIMSEQ_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@29803
   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
hoelzl@29803
   311
    moreover have ?f and ?g using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN LIMSEQ_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:
paulson@15229
   339
     "summable (%n. (diffs c)(n) * (x ^ n)) ==>  
huffman@23082
   340
      (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums  
nipkow@15546
   341
         (\<Sum>n. (diffs c)(n) * (x ^ n))"
huffman@29163
   342
unfolding diffs_def
huffman@29163
   343
apply (drule summable_sums)
huffman@29163
   344
apply (rule sums_Suc_imp, simp_all)
paulson@15077
   345
done
paulson@15077
   346
paulson@15077
   347
lemma lemma_termdiff1:
haftmann@31017
   348
  fixes z :: "'a :: {monoid_mult,comm_ring}" shows
nipkow@15539
   349
  "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =  
huffman@23082
   350
   (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
nipkow@29667
   351
by(auto simp add: algebra_simps power_add [symmetric] cong: strong_setsum_cong)
paulson@15077
   352
huffman@23082
   353
lemma sumr_diff_mult_const2:
huffman@23082
   354
  "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
huffman@23082
   355
by (simp add: setsum_subtractf)
huffman@23082
   356
paulson@15229
   357
lemma lemma_termdiff2:
haftmann@31017
   358
  fixes h :: "'a :: {field}"
huffman@20860
   359
  assumes h: "h \<noteq> 0" shows
huffman@23082
   360
  "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
huffman@20860
   361
   h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
huffman@23082
   362
        (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
huffman@23082
   363
apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
huffman@20860
   364
apply (simp add: right_diff_distrib diff_divide_distrib h)
paulson@15077
   365
apply (simp add: mult_assoc [symmetric])
huffman@20860
   366
apply (cases "n", simp)
huffman@20860
   367
apply (simp add: lemma_realpow_diff_sumr2 h
huffman@20860
   368
                 right_diff_distrib [symmetric] mult_assoc
huffman@30273
   369
            del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
huffman@20860
   370
apply (subst lemma_realpow_rev_sumr)
huffman@23082
   371
apply (subst sumr_diff_mult_const2)
huffman@20860
   372
apply simp
huffman@20860
   373
apply (simp only: lemma_termdiff1 setsum_right_distrib)
huffman@20860
   374
apply (rule setsum_cong [OF refl])
nipkow@15539
   375
apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
huffman@20860
   376
apply (clarify)
huffman@20860
   377
apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
huffman@30273
   378
            del: setsum_op_ivl_Suc power_Suc)
huffman@20860
   379
apply (subst mult_assoc [symmetric], subst power_add [symmetric])
huffman@20860
   380
apply (simp add: mult_ac)
huffman@20860
   381
done
huffman@20860
   382
huffman@20860
   383
lemma real_setsum_nat_ivl_bounded2:
haftmann@35028
   384
  fixes K :: "'a::linordered_semidom"
huffman@23082
   385
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
huffman@23082
   386
  assumes K: "0 \<le> K"
huffman@23082
   387
  shows "setsum f {0..<n-k} \<le> of_nat n * K"
huffman@23082
   388
apply (rule order_trans [OF setsum_mono])
huffman@23082
   389
apply (rule f, simp)
huffman@23082
   390
apply (simp add: mult_right_mono K)
paulson@15077
   391
done
paulson@15077
   392
paulson@15229
   393
lemma lemma_termdiff3:
haftmann@31017
   394
  fixes h z :: "'a::{real_normed_field}"
huffman@20860
   395
  assumes 1: "h \<noteq> 0"
huffman@23082
   396
  assumes 2: "norm z \<le> K"
huffman@23082
   397
  assumes 3: "norm (z + h) \<le> K"
huffman@23082
   398
  shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
huffman@23082
   399
          \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
huffman@20860
   400
proof -
huffman@23082
   401
  have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
huffman@23082
   402
        norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
huffman@23082
   403
          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
huffman@20860
   404
    apply (subst lemma_termdiff2 [OF 1])
huffman@23082
   405
    apply (subst norm_mult)
huffman@20860
   406
    apply (rule mult_commute)
huffman@20860
   407
    done
huffman@23082
   408
  also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
huffman@23082
   409
  proof (rule mult_right_mono [OF _ norm_ge_zero])
huffman@23082
   410
    from norm_ge_zero 2 have K: "0 \<le> K" by (rule order_trans)
huffman@23082
   411
    have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
huffman@20860
   412
      apply (erule subst)
huffman@23082
   413
      apply (simp only: norm_mult norm_power power_add)
huffman@23082
   414
      apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
huffman@20860
   415
      done
huffman@23082
   416
    show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
huffman@23082
   417
              (z + h) ^ q * z ^ (n - 2 - q))
huffman@23082
   418
          \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
huffman@20860
   419
      apply (intro
huffman@23082
   420
         order_trans [OF norm_setsum]
huffman@20860
   421
         real_setsum_nat_ivl_bounded2
huffman@20860
   422
         mult_nonneg_nonneg
huffman@23082
   423
         zero_le_imp_of_nat
huffman@20860
   424
         zero_le_power K)
huffman@20860
   425
      apply (rule le_Kn, simp)
huffman@20860
   426
      done
huffman@20860
   427
  qed
huffman@23082
   428
  also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
huffman@20860
   429
    by (simp only: mult_assoc)
huffman@20860
   430
  finally show ?thesis .
huffman@20860
   431
qed
paulson@15077
   432
huffman@20860
   433
lemma lemma_termdiff4:
haftmann@31017
   434
  fixes f :: "'a::{real_normed_field} \<Rightarrow>
huffman@23082
   435
              'b::real_normed_vector"
huffman@20860
   436
  assumes k: "0 < (k::real)"
huffman@23082
   437
  assumes le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
huffman@20860
   438
  shows "f -- 0 --> 0"
huffman@31338
   439
unfolding LIM_eq diff_0_right
huffman@29163
   440
proof (safe)
huffman@29163
   441
  let ?h = "of_real (k / 2)::'a"
huffman@29163
   442
  have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
huffman@29163
   443
  hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
huffman@29163
   444
  hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
huffman@29163
   445
  hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
huffman@29163
   446
huffman@20860
   447
  fix r::real assume r: "0 < r"
huffman@23082
   448
  show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
huffman@20860
   449
  proof (cases)
huffman@20860
   450
    assume "K = 0"
huffman@23082
   451
    with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
huffman@20860
   452
      by simp
huffman@23082
   453
    thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
huffman@20860
   454
  next
huffman@20860
   455
    assume K_neq_zero: "K \<noteq> 0"
huffman@20860
   456
    with zero_le_K have K: "0 < K" by simp
huffman@23082
   457
    show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
huffman@20860
   458
    proof (rule exI, safe)
huffman@20860
   459
      from k r K show "0 < min k (r * inverse K / 2)"
huffman@20860
   460
        by (simp add: mult_pos_pos positive_imp_inverse_positive)
huffman@20860
   461
    next
huffman@23082
   462
      fix x::'a
huffman@23082
   463
      assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
huffman@23082
   464
      from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
huffman@20860
   465
        by simp_all
huffman@23082
   466
      from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
huffman@23082
   467
      also from x4 K have "K * norm x < K * (r * inverse K / 2)"
huffman@20860
   468
        by (rule mult_strict_left_mono)
huffman@20860
   469
      also have "\<dots> = r / 2"
huffman@20860
   470
        using K_neq_zero by simp
huffman@20860
   471
      also have "r / 2 < r"
huffman@20860
   472
        using r by simp
huffman@23082
   473
      finally show "norm (f x) < r" .
huffman@20860
   474
    qed
huffman@20860
   475
  qed
huffman@20860
   476
qed
paulson@15077
   477
paulson@15229
   478
lemma lemma_termdiff5:
haftmann@31017
   479
  fixes g :: "'a::{real_normed_field} \<Rightarrow>
huffman@23082
   480
              nat \<Rightarrow> 'b::banach"
huffman@20860
   481
  assumes k: "0 < (k::real)"
huffman@20860
   482
  assumes f: "summable f"
huffman@23082
   483
  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
   484
  shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
huffman@20860
   485
proof (rule lemma_termdiff4 [OF k])
huffman@23082
   486
  fix h::'a assume "h \<noteq> 0" and "norm h < k"
huffman@23082
   487
  hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
huffman@20860
   488
    by (simp add: le)
huffman@23082
   489
  hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
huffman@20860
   490
    by simp
huffman@23082
   491
  moreover from f have B: "summable (\<lambda>n. f n * norm h)"
huffman@20860
   492
    by (rule summable_mult2)
huffman@23082
   493
  ultimately have C: "summable (\<lambda>n. norm (g h n))"
huffman@20860
   494
    by (rule summable_comparison_test)
huffman@23082
   495
  hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
huffman@23082
   496
    by (rule summable_norm)
huffman@23082
   497
  also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
huffman@20860
   498
    by (rule summable_le)
huffman@23082
   499
  also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
huffman@20860
   500
    by (rule suminf_mult2 [symmetric])
huffman@23082
   501
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
huffman@20860
   502
qed
paulson@15077
   503
paulson@15077
   504
paulson@15077
   505
text{* FIXME: Long proofs*}
paulson@15077
   506
paulson@15077
   507
lemma termdiffs_aux:
haftmann@31017
   508
  fixes x :: "'a::{real_normed_field,banach}"
huffman@20849
   509
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
huffman@23082
   510
  assumes 2: "norm x < norm K"
huffman@20860
   511
  shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
huffman@23082
   512
             - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
huffman@20849
   513
proof -
huffman@20860
   514
  from dense [OF 2]
huffman@23082
   515
  obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
huffman@23082
   516
  from norm_ge_zero r1 have r: "0 < r"
huffman@20860
   517
    by (rule order_le_less_trans)
huffman@20860
   518
  hence r_neq_0: "r \<noteq> 0" by simp
huffman@20860
   519
  show ?thesis
huffman@20849
   520
  proof (rule lemma_termdiff5)
huffman@23082
   521
    show "0 < r - norm x" using r1 by simp
huffman@20849
   522
  next
huffman@23082
   523
    from r r2 have "norm (of_real r::'a) < norm K"
huffman@23082
   524
      by simp
huffman@23082
   525
    with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
huffman@20860
   526
      by (rule powser_insidea)
huffman@23082
   527
    hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
huffman@23082
   528
      using r
huffman@23082
   529
      by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
huffman@23082
   530
    hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
huffman@20860
   531
      by (rule diffs_equiv [THEN sums_summable])
huffman@23082
   532
    also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))
huffman@23082
   533
      = (\<lambda>n. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
huffman@20849
   534
      apply (rule ext)
huffman@20849
   535
      apply (simp add: diffs_def)
huffman@20849
   536
      apply (case_tac n, simp_all add: r_neq_0)
huffman@20849
   537
      done
huffman@20860
   538
    finally have "summable 
huffman@23082
   539
      (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
huffman@20860
   540
      by (rule diffs_equiv [THEN sums_summable])
huffman@20860
   541
    also have
huffman@23082
   542
      "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
huffman@20860
   543
           r ^ (n - Suc 0)) =
huffman@23082
   544
       (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
huffman@20849
   545
      apply (rule ext)
huffman@20849
   546
      apply (case_tac "n", simp)
huffman@20849
   547
      apply (case_tac "nat", simp)
huffman@20849
   548
      apply (simp add: r_neq_0)
huffman@20849
   549
      done
huffman@20860
   550
    finally show
huffman@23082
   551
      "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
huffman@20849
   552
  next
huffman@23082
   553
    fix h::'a and n::nat
huffman@20860
   554
    assume h: "h \<noteq> 0"
huffman@23082
   555
    assume "norm h < r - norm x"
huffman@23082
   556
    hence "norm x + norm h < r" by simp
huffman@23082
   557
    with norm_triangle_ineq have xh: "norm (x + h) < r"
huffman@20860
   558
      by (rule order_le_less_trans)
huffman@23082
   559
    show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
huffman@23082
   560
          \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
huffman@23082
   561
      apply (simp only: norm_mult mult_assoc)
huffman@23082
   562
      apply (rule mult_left_mono [OF _ norm_ge_zero])
huffman@20860
   563
      apply (simp (no_asm) add: mult_assoc [symmetric])
huffman@20860
   564
      apply (rule lemma_termdiff3)
huffman@20860
   565
      apply (rule h)
huffman@20860
   566
      apply (rule r1 [THEN order_less_imp_le])
huffman@20860
   567
      apply (rule xh [THEN order_less_imp_le])
huffman@20860
   568
      done
huffman@20849
   569
  qed
huffman@20849
   570
qed
webertj@20217
   571
huffman@20860
   572
lemma termdiffs:
haftmann@31017
   573
  fixes K x :: "'a::{real_normed_field,banach}"
huffman@20860
   574
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
huffman@20860
   575
  assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
huffman@20860
   576
  assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
huffman@23082
   577
  assumes 4: "norm x < norm K"
huffman@20860
   578
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
huffman@29163
   579
unfolding deriv_def
huffman@29163
   580
proof (rule LIM_zero_cancel)
huffman@20860
   581
  show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
huffman@20860
   582
            - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
huffman@20860
   583
  proof (rule LIM_equal2)
huffman@29163
   584
    show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
huffman@20860
   585
  next
huffman@23082
   586
    fix h :: 'a
huffman@20860
   587
    assume "h \<noteq> 0"
huffman@23082
   588
    assume "norm (h - 0) < norm K - norm x"
huffman@23082
   589
    hence "norm x + norm h < norm K" by simp
huffman@23082
   590
    hence 5: "norm (x + h) < norm K"
huffman@23082
   591
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
huffman@20860
   592
    have A: "summable (\<lambda>n. c n * x ^ n)"
huffman@20860
   593
      by (rule powser_inside [OF 1 4])
huffman@20860
   594
    have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
huffman@20860
   595
      by (rule powser_inside [OF 1 5])
huffman@20860
   596
    have C: "summable (\<lambda>n. diffs c n * x ^ n)"
huffman@20860
   597
      by (rule powser_inside [OF 2 4])
huffman@20860
   598
    show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
huffman@20860
   599
             - (\<Sum>n. diffs c n * x ^ n) = 
huffman@23082
   600
          (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
huffman@20860
   601
      apply (subst sums_unique [OF diffs_equiv [OF C]])
huffman@20860
   602
      apply (subst suminf_diff [OF B A])
huffman@20860
   603
      apply (subst suminf_divide [symmetric])
huffman@20860
   604
      apply (rule summable_diff [OF B A])
huffman@20860
   605
      apply (subst suminf_diff)
huffman@20860
   606
      apply (rule summable_divide)
huffman@20860
   607
      apply (rule summable_diff [OF B A])
huffman@20860
   608
      apply (rule sums_summable [OF diffs_equiv [OF C]])
huffman@29163
   609
      apply (rule arg_cong [where f="suminf"], rule ext)
nipkow@29667
   610
      apply (simp add: algebra_simps)
huffman@20860
   611
      done
huffman@20860
   612
  next
huffman@20860
   613
    show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h -
huffman@23082
   614
               of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
huffman@20860
   615
        by (rule termdiffs_aux [OF 3 4])
huffman@20860
   616
  qed
huffman@20860
   617
qed
huffman@20860
   618
paulson@15077
   619
chaieb@29695
   620
subsection{* Some properties of factorials *}
chaieb@29695
   621
avigad@32036
   622
lemma real_of_nat_fact_not_zero [simp]: "real (fact (n::nat)) \<noteq> 0"
chaieb@29695
   623
by auto
chaieb@29695
   624
avigad@32036
   625
lemma real_of_nat_fact_gt_zero [simp]: "0 < real(fact (n::nat))"
chaieb@29695
   626
by auto
chaieb@29695
   627
avigad@32036
   628
lemma real_of_nat_fact_ge_zero [simp]: "0 \<le> real(fact (n::nat))"
chaieb@29695
   629
by simp
chaieb@29695
   630
avigad@32036
   631
lemma inv_real_of_nat_fact_gt_zero [simp]: "0 < inverse (real (fact (n::nat)))"
chaieb@29695
   632
by (auto simp add: positive_imp_inverse_positive)
chaieb@29695
   633
avigad@32036
   634
lemma inv_real_of_nat_fact_ge_zero [simp]: "0 \<le> inverse (real (fact (n::nat)))"
chaieb@29695
   635
by (auto intro: order_less_imp_le)
chaieb@29695
   636
hoelzl@29803
   637
subsection {* Derivability of power series *}
hoelzl@29803
   638
hoelzl@29803
   639
lemma DERIV_series': fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
hoelzl@29803
   640
  assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
hoelzl@29803
   641
  and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
hoelzl@29803
   642
  and "summable (f' x0)"
hoelzl@29803
   643
  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
   644
  shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
hoelzl@29803
   645
  unfolding deriv_def
hoelzl@29803
   646
proof (rule LIM_I)
hoelzl@29803
   647
  fix r :: real assume "0 < r" hence "0 < r/3" by auto
hoelzl@29803
   648
hoelzl@29803
   649
  obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3" 
hoelzl@29803
   650
    using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
hoelzl@29803
   651
hoelzl@29803
   652
  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
   653
    using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
hoelzl@29803
   654
hoelzl@29803
   655
  let ?N = "Suc (max N_L N_f')"
hoelzl@29803
   656
  have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
hoelzl@29803
   657
    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
   658
hoelzl@29803
   659
  let "?diff i x" = "(f (x0 + x) i - f x0 i) / x"
hoelzl@29803
   660
hoelzl@29803
   661
  let ?r = "r / (3 * real ?N)"
hoelzl@29803
   662
  have "0 < 3 * real ?N" by auto
hoelzl@29803
   663
  from divide_pos_pos[OF `0 < r` this]
hoelzl@29803
   664
  have "0 < ?r" .
hoelzl@29803
   665
hoelzl@29803
   666
  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
   667
  def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
hoelzl@29803
   668
hoelzl@29803
   669
  have "0 < S'" unfolding S'_def
hoelzl@29803
   670
  proof (rule iffD2[OF Min_gr_iff])
hoelzl@29803
   671
    show "\<forall> x \<in> (?s ` { 0 ..< ?N }). 0 < x"
hoelzl@29803
   672
    proof (rule ballI)
hoelzl@29803
   673
      fix x assume "x \<in> ?s ` {0..<?N}"
hoelzl@29803
   674
      then obtain n where "x = ?s n" and "n \<in> {0..<?N}" using image_iff[THEN iffD1] by blast
hoelzl@29803
   675
      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def] 
hoelzl@29803
   676
      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
   677
      have "0 < ?s n" by (rule someI2[where a=s], auto simp add: s_bound)
hoelzl@29803
   678
      thus "0 < x" unfolding `x = ?s n` .
hoelzl@29803
   679
    qed
hoelzl@29803
   680
  qed auto
hoelzl@29803
   681
hoelzl@29803
   682
  def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
hoelzl@29803
   683
  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
   684
    by auto
hoelzl@29803
   685
hoelzl@29803
   686
  { fix x assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
hoelzl@29803
   687
    hence x_in_I: "x0 + x \<in> { a <..< b }" using S_a S_b by auto
hoelzl@29803
   688
    
hoelzl@29803
   689
    note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
hoelzl@29803
   690
    note div_smbl = summable_divide[OF diff_smbl]
hoelzl@29803
   691
    note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
hoelzl@29803
   692
    note ign = summable_ignore_initial_segment[where k="?N"]
hoelzl@29803
   693
    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
   694
    note div_shft_smbl = summable_divide[OF diff_shft_smbl]
hoelzl@29803
   695
    note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
hoelzl@29803
   696
hoelzl@29803
   697
    { fix n
hoelzl@29803
   698
      have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>" 
wenzelm@32960
   699
        using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero] unfolding abs_divide .
hoelzl@29803
   700
      hence "\<bar> ( \<bar> ?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)" using `x \<noteq> 0` by auto
hoelzl@29803
   701
    } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
hoelzl@29803
   702
    from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
hoelzl@29803
   703
    have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
hoelzl@29803
   704
    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
   705
hoelzl@29803
   706
    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
   707
    also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
hoelzl@29803
   708
    proof (rule setsum_strict_mono)
hoelzl@29803
   709
      fix n assume "n \<in> { 0 ..< ?N}"
hoelzl@29803
   710
      have "\<bar> x \<bar> < S" using `\<bar> x \<bar> < S` .
hoelzl@29803
   711
      also have "S \<le> S'" using `S \<le> S'` .
hoelzl@29803
   712
      also have "S' \<le> ?s n" unfolding S'_def 
hoelzl@29803
   713
      proof (rule Min_le_iff[THEN iffD2])
wenzelm@32960
   714
        have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n" using `n \<in> { 0 ..< ?N}` by auto
wenzelm@32960
   715
        thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
hoelzl@29803
   716
      qed auto
hoelzl@29803
   717
      finally have "\<bar> x \<bar> < ?s n" .
hoelzl@29803
   718
hoelzl@29803
   719
      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
   720
      have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
hoelzl@29803
   721
      with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n`
hoelzl@29803
   722
      show "\<bar>?diff n x - f' x0 n\<bar> < ?r" by blast
hoelzl@29803
   723
    qed auto
hoelzl@29803
   724
    also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r" by (rule setsum_constant)
hoelzl@29803
   725
    also have "\<dots> = real ?N * ?r" unfolding real_eq_of_nat by auto
hoelzl@29803
   726
    also have "\<dots> = r/3" by auto
hoelzl@29803
   727
    finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
hoelzl@29803
   728
hoelzl@29803
   729
    from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
hoelzl@29803
   730
    have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> = 
hoelzl@29803
   731
                    \<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
   732
    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
   733
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part" using abs_triangle_ineq4 by auto
hoelzl@29803
   734
    also have "\<dots> < r /3 + r/3 + r/3" 
hoelzl@29803
   735
      using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3` by auto
hoelzl@29803
   736
    finally have "\<bar> (suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0) \<bar> < r"
hoelzl@29803
   737
      by auto
hoelzl@29803
   738
  } thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow> 
hoelzl@29803
   739
      norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r" using `0 < S`
hoelzl@29803
   740
    unfolding real_norm_def diff_0_right by blast
hoelzl@29803
   741
qed
hoelzl@29803
   742
hoelzl@29803
   743
lemma DERIV_power_series': fixes f :: "nat \<Rightarrow> real"
hoelzl@29803
   744
  assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
hoelzl@29803
   745
  and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
hoelzl@29803
   746
  shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
hoelzl@29803
   747
  (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
hoelzl@29803
   748
proof -
hoelzl@29803
   749
  { fix R' assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
hoelzl@29803
   750
    hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}" by auto
hoelzl@29803
   751
    have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
hoelzl@29803
   752
    proof (rule DERIV_series')
hoelzl@29803
   753
      show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
hoelzl@29803
   754
      proof -
wenzelm@32960
   755
        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2" using `0 < R'` `0 < R` `R' < R` by auto
wenzelm@32960
   756
        hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}" using `R' < R` by auto
wenzelm@32960
   757
        have "norm R' < norm ((R' + R) / 2)" using `0 < R'` `0 < R` `R' < R` by auto
wenzelm@32960
   758
        from powser_insidea[OF converges[OF in_Rball] this] show ?thesis by auto
hoelzl@29803
   759
      qed
hoelzl@29803
   760
      { fix n x y assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
wenzelm@32960
   761
        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
   762
        proof -
wenzelm@32960
   763
          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
   764
            unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult by auto
wenzelm@32960
   765
          also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)" 
wenzelm@32960
   766
          proof (rule mult_left_mono)
wenzelm@32960
   767
            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
   768
            also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
wenzelm@32960
   769
            proof (rule setsum_mono)
wenzelm@32960
   770
              fix p assume "p \<in> {0..<Suc n}" hence "p \<le> n" by auto
wenzelm@32960
   771
              { fix n fix x :: real assume "x \<in> {-R'<..<R'}"
wenzelm@32960
   772
                hence "\<bar>x\<bar> \<le> R'"  by auto
wenzelm@32960
   773
                hence "\<bar>x^n\<bar> \<le> R'^n" unfolding power_abs by (rule power_mono, auto)
wenzelm@32960
   774
              } 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
   775
              have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)" unfolding abs_mult by auto
wenzelm@32960
   776
              thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n" unfolding power_add[symmetric] using `p \<le> n` by auto
wenzelm@32960
   777
            qed
wenzelm@32960
   778
            also have "\<dots> = real (Suc n) * R' ^ n" unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
wenzelm@32960
   779
            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
   780
            show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult[symmetric] by auto
wenzelm@32960
   781
          qed
wenzelm@32960
   782
          also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>" unfolding abs_mult real_mult_assoc[symmetric] by algebra
wenzelm@32960
   783
          finally show ?thesis .
wenzelm@32960
   784
        qed }
hoelzl@31881
   785
      { fix n show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
wenzelm@32960
   786
          by (auto intro!: DERIV_intros simp del: power_Suc) }
hoelzl@29803
   787
      { 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
   788
        have "summable (\<lambda> n. f n * x^n)"
wenzelm@32960
   789
        proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
wenzelm@32960
   790
          fix n
wenzelm@32960
   791
          have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)" by (rule mult_left_mono, auto)
wenzelm@32960
   792
          show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)" unfolding real_norm_def abs_mult
wenzelm@32960
   793
            by (rule mult_right_mono, auto simp add: le[unfolded mult_1_right])
wenzelm@32960
   794
        qed
wenzelm@32960
   795
        from this[THEN summable_mult2[where c=x], unfolded real_mult_assoc, unfolded real_mult_commute]
wenzelm@32960
   796
        show "summable (?f x)" by auto }
hoelzl@29803
   797
      show "summable (?f' x0)" using converges[OF `x0 \<in> {-R <..< R}`] .
hoelzl@29803
   798
      show "x0 \<in> {-R' <..< R'}" using `x0 \<in> {-R' <..< R'}` .
hoelzl@29803
   799
    qed
hoelzl@29803
   800
  } note for_subinterval = this
hoelzl@29803
   801
  let ?R = "(R + \<bar>x0\<bar>) / 2"
hoelzl@29803
   802
  have "\<bar>x0\<bar> < ?R" using assms by auto
hoelzl@29803
   803
  hence "- ?R < x0"
hoelzl@29803
   804
  proof (cases "x0 < 0")
hoelzl@29803
   805
    case True
hoelzl@29803
   806
    hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
hoelzl@29803
   807
    thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
hoelzl@29803
   808
  next
hoelzl@29803
   809
    case False
hoelzl@29803
   810
    have "- ?R < 0" using assms by auto
hoelzl@29803
   811
    also have "\<dots> \<le> x0" using False by auto 
hoelzl@29803
   812
    finally show ?thesis .
hoelzl@29803
   813
  qed
hoelzl@29803
   814
  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R" using assms by auto
hoelzl@29803
   815
  from for_subinterval[OF this]
hoelzl@29803
   816
  show ?thesis .
hoelzl@29803
   817
qed
chaieb@29695
   818
huffman@29164
   819
subsection {* Exponential Function *}
huffman@23043
   820
huffman@23043
   821
definition
haftmann@31017
   822
  exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}" where
haftmann@25062
   823
  "exp x = (\<Sum>n. x ^ n /\<^sub>R real (fact n))"
huffman@23043
   824
huffman@23115
   825
lemma summable_exp_generic:
haftmann@31017
   826
  fixes x :: "'a::{real_normed_algebra_1,banach}"
haftmann@25062
   827
  defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
huffman@23115
   828
  shows "summable S"
huffman@23115
   829
proof -
haftmann@25062
   830
  have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
huffman@30273
   831
    unfolding S_def by (simp del: mult_Suc)
huffman@23115
   832
  obtain r :: real where r0: "0 < r" and r1: "r < 1"
huffman@23115
   833
    using dense [OF zero_less_one] by fast
huffman@23115
   834
  obtain N :: nat where N: "norm x < real N * r"
huffman@23115
   835
    using reals_Archimedean3 [OF r0] by fast
huffman@23115
   836
  from r1 show ?thesis
huffman@23115
   837
  proof (rule ratio_test [rule_format])
huffman@23115
   838
    fix n :: nat
huffman@23115
   839
    assume n: "N \<le> n"
huffman@23115
   840
    have "norm x \<le> real N * r"
huffman@23115
   841
      using N by (rule order_less_imp_le)
huffman@23115
   842
    also have "real N * r \<le> real (Suc n) * r"
huffman@23115
   843
      using r0 n by (simp add: mult_right_mono)
huffman@23115
   844
    finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
huffman@23115
   845
      using norm_ge_zero by (rule mult_right_mono)
huffman@23115
   846
    hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
huffman@23115
   847
      by (rule order_trans [OF norm_mult_ineq])
huffman@23115
   848
    hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
huffman@23115
   849
      by (simp add: pos_divide_le_eq mult_ac)
huffman@23115
   850
    thus "norm (S (Suc n)) \<le> r * norm (S n)"
huffman@35216
   851
      by (simp add: S_Suc inverse_eq_divide)
huffman@23115
   852
  qed
huffman@23115
   853
qed
huffman@23115
   854
huffman@23115
   855
lemma summable_norm_exp:
haftmann@31017
   856
  fixes x :: "'a::{real_normed_algebra_1,banach}"
haftmann@25062
   857
  shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
huffman@23115
   858
proof (rule summable_norm_comparison_test [OF exI, rule_format])
haftmann@25062
   859
  show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
huffman@23115
   860
    by (rule summable_exp_generic)
huffman@23115
   861
next
haftmann@25062
   862
  fix n show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
huffman@35216
   863
    by (simp add: norm_power_ineq)
huffman@23115
   864
qed
huffman@23115
   865
huffman@23043
   866
lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
huffman@23115
   867
by (insert summable_exp_generic [where x=x], simp)
huffman@23043
   868
haftmann@25062
   869
lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
huffman@23115
   870
unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
huffman@23043
   871
huffman@23043
   872
paulson@15077
   873
lemma exp_fdiffs: 
paulson@15077
   874
      "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
huffman@23431
   875
by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
huffman@23082
   876
         del: mult_Suc of_nat_Suc)
paulson@15077
   877
huffman@23115
   878
lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
huffman@23115
   879
by (simp add: diffs_def)
huffman@23115
   880
haftmann@25062
   881
lemma lemma_exp_ext: "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
paulson@15077
   882
by (auto intro!: ext simp add: exp_def)
paulson@15077
   883
paulson@15077
   884
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
paulson@15229
   885
apply (simp add: exp_def)
paulson@15077
   886
apply (subst lemma_exp_ext)
huffman@23115
   887
apply (subgoal_tac "DERIV (\<lambda>u. \<Sum>n. of_real (inverse (real (fact n))) * u ^ n) x :> (\<Sum>n. diffs (\<lambda>n. of_real (inverse (real (fact n)))) n * x ^ n)")
huffman@23115
   888
apply (rule_tac [2] K = "of_real (1 + norm x)" in termdiffs)
huffman@23115
   889
apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
huffman@23115
   890
apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
huffman@23115
   891
apply (simp del: of_real_add)
paulson@15077
   892
done
paulson@15077
   893
huffman@23045
   894
lemma isCont_exp [simp]: "isCont exp x"
huffman@23045
   895
by (rule DERIV_exp [THEN DERIV_isCont])
huffman@23045
   896
huffman@23045
   897
huffman@29167
   898
subsubsection {* Properties of the Exponential Function *}
paulson@15077
   899
huffman@23278
   900
lemma powser_zero:
haftmann@31017
   901
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
huffman@23278
   902
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
paulson@15077
   903
proof -
huffman@23278
   904
  have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
huffman@23115
   905
    by (rule sums_unique [OF series_zero], simp add: power_0_left)
huffman@30082
   906
  thus ?thesis unfolding One_nat_def by simp
paulson@15077
   907
qed
paulson@15077
   908
huffman@23278
   909
lemma exp_zero [simp]: "exp 0 = 1"
huffman@23278
   910
unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
huffman@23278
   911
huffman@23115
   912
lemma setsum_cl_ivl_Suc2:
huffman@23115
   913
  "(\<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
   914
by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
huffman@23115
   915
         del: setsum_cl_ivl_Suc)
huffman@23115
   916
huffman@23115
   917
lemma exp_series_add:
haftmann@31017
   918
  fixes x y :: "'a::{real_field}"
haftmann@25062
   919
  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
huffman@23115
   920
  shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
huffman@23115
   921
proof (induct n)
huffman@23115
   922
  case 0
huffman@23115
   923
  show ?case
huffman@23115
   924
    unfolding S_def by simp
huffman@23115
   925
next
huffman@23115
   926
  case (Suc n)
haftmann@25062
   927
  have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
huffman@30273
   928
    unfolding S_def by (simp del: mult_Suc)
haftmann@25062
   929
  hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
huffman@23115
   930
    by simp
huffman@23115
   931
haftmann@25062
   932
  have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
huffman@23115
   933
    by (simp only: times_S)
huffman@23115
   934
  also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
huffman@23115
   935
    by (simp only: Suc)
huffman@23115
   936
  also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
huffman@23115
   937
                + y * (\<Sum>i=0..n. S x i * S y (n-i))"
huffman@23115
   938
    by (rule left_distrib)
huffman@23115
   939
  also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
huffman@23115
   940
                + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
huffman@23115
   941
    by (simp only: setsum_right_distrib mult_ac)
haftmann@25062
   942
  also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
haftmann@25062
   943
                + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
   944
    by (simp add: times_S Suc_diff_le)
haftmann@25062
   945
  also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
haftmann@25062
   946
             (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
   947
    by (subst setsum_cl_ivl_Suc2, simp)
haftmann@25062
   948
  also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
haftmann@25062
   949
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
   950
    by (subst setsum_cl_ivl_Suc, simp)
haftmann@25062
   951
  also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
haftmann@25062
   952
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
haftmann@25062
   953
             (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
   954
    by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
huffman@23115
   955
              real_of_nat_add [symmetric], simp)
haftmann@25062
   956
  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
huffman@23127
   957
    by (simp only: scaleR_right.setsum)
huffman@23115
   958
  finally show
huffman@23115
   959
    "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
huffman@35216
   960
    by (simp del: setsum_cl_ivl_Suc)
huffman@23115
   961
qed
huffman@23115
   962
huffman@23115
   963
lemma exp_add: "exp (x + y) = exp x * exp y"
huffman@23115
   964
unfolding exp_def
huffman@23115
   965
by (simp only: Cauchy_product summable_norm_exp exp_series_add)
huffman@23115
   966
huffman@29170
   967
lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
huffman@29170
   968
by (rule exp_add [symmetric])
huffman@29170
   969
huffman@23241
   970
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
huffman@23241
   971
unfolding exp_def
huffman@23241
   972
apply (subst of_real.suminf)
huffman@23241
   973
apply (rule summable_exp_generic)
huffman@23241
   974
apply (simp add: scaleR_conv_of_real)
huffman@23241
   975
done
huffman@23241
   976
huffman@29170
   977
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
huffman@29170
   978
proof
huffman@29170
   979
  have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
huffman@29170
   980
  also assume "exp x = 0"
huffman@29170
   981
  finally show "False" by simp
paulson@15077
   982
qed
paulson@15077
   983
huffman@29170
   984
lemma exp_minus: "exp (- x) = inverse (exp x)"
huffman@29170
   985
by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
paulson@15077
   986
huffman@29170
   987
lemma exp_diff: "exp (x - y) = exp x / exp y"
huffman@29170
   988
  unfolding diff_minus divide_inverse
huffman@29170
   989
  by (simp add: exp_add exp_minus)
paulson@15077
   990
huffman@29167
   991
huffman@29167
   992
subsubsection {* Properties of the Exponential Function on Reals *}
huffman@29167
   993
huffman@29170
   994
text {* Comparisons of @{term "exp x"} with zero. *}
huffman@29167
   995
huffman@29167
   996
text{*Proof: because every exponential can be seen as a square.*}
huffman@29167
   997
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
huffman@29167
   998
proof -
huffman@29167
   999
  have "0 \<le> exp (x/2) * exp (x/2)" by simp
huffman@29167
  1000
  thus ?thesis by (simp add: exp_add [symmetric])
huffman@29167
  1001
qed
huffman@29167
  1002
huffman@23115
  1003
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
paulson@15077
  1004
by (simp add: order_less_le)
paulson@15077
  1005
huffman@29170
  1006
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
huffman@29170
  1007
by (simp add: not_less)
huffman@29170
  1008
huffman@29170
  1009
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
huffman@29170
  1010
by (simp add: not_le)
paulson@15077
  1011
huffman@23115
  1012
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
huffman@29165
  1013
by simp
paulson@15077
  1014
paulson@15077
  1015
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
paulson@15251
  1016
apply (induct "n")
paulson@15077
  1017
apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
paulson@15077
  1018
done
paulson@15077
  1019
huffman@29170
  1020
text {* Strict monotonicity of exponential. *}
huffman@29170
  1021
huffman@29170
  1022
lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
huffman@29170
  1023
apply (drule order_le_imp_less_or_eq, auto)
huffman@29170
  1024
apply (simp add: exp_def)
huffman@29170
  1025
apply (rule real_le_trans)
huffman@29170
  1026
apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
huffman@29170
  1027
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_mult_iff)
huffman@29170
  1028
done
huffman@29170
  1029
huffman@29170
  1030
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
huffman@29170
  1031
proof -
huffman@29170
  1032
  assume x: "0 < x"
huffman@29170
  1033
  hence "1 < 1 + x" by simp
huffman@29170
  1034
  also from x have "1 + x \<le> exp x"
huffman@29170
  1035
    by (simp add: exp_ge_add_one_self_aux)
huffman@29170
  1036
  finally show ?thesis .
huffman@29170
  1037
qed
huffman@29170
  1038
paulson@15077
  1039
lemma exp_less_mono:
huffman@23115
  1040
  fixes x y :: real
huffman@29165
  1041
  assumes "x < y" shows "exp x < exp y"
paulson@15077
  1042
proof -
huffman@29165
  1043
  from `x < y` have "0 < y - x" by simp
huffman@29165
  1044
  hence "1 < exp (y - x)" by (rule exp_gt_one)
huffman@29165
  1045
  hence "1 < exp y / exp x" by (simp only: exp_diff)
huffman@29165
  1046
  thus "exp x < exp y" by simp
paulson@15077
  1047
qed
paulson@15077
  1048
huffman@23115
  1049
lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
huffman@29170
  1050
apply (simp add: linorder_not_le [symmetric])
huffman@29170
  1051
apply (auto simp add: order_le_less exp_less_mono)
paulson@15077
  1052
done
paulson@15077
  1053
huffman@29170
  1054
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
paulson@15077
  1055
by (auto intro: exp_less_mono exp_less_cancel)
paulson@15077
  1056
huffman@29170
  1057
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
paulson@15077
  1058
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
  1059
huffman@29170
  1060
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
paulson@15077
  1061
by (simp add: order_eq_iff)
paulson@15077
  1062
huffman@29170
  1063
text {* Comparisons of @{term "exp x"} with one. *}
huffman@29170
  1064
huffman@29170
  1065
lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
huffman@29170
  1066
  using exp_less_cancel_iff [where x=0 and y=x] by simp
huffman@29170
  1067
huffman@29170
  1068
lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
huffman@29170
  1069
  using exp_less_cancel_iff [where x=x and y=0] by simp
huffman@29170
  1070
huffman@29170
  1071
lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
huffman@29170
  1072
  using exp_le_cancel_iff [where x=0 and y=x] by simp
huffman@29170
  1073
huffman@29170
  1074
lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
huffman@29170
  1075
  using exp_le_cancel_iff [where x=x and y=0] by simp
huffman@29170
  1076
huffman@29170
  1077
lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
huffman@29170
  1078
  using exp_inj_iff [where x=x and y=0] by simp
huffman@29170
  1079
huffman@23115
  1080
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
paulson@15077
  1081
apply (rule IVT)
huffman@23045
  1082
apply (auto intro: isCont_exp simp add: le_diff_eq)
paulson@15077
  1083
apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)") 
huffman@29165
  1084
apply simp
avigad@17014
  1085
apply (rule exp_ge_add_one_self_aux, simp)
paulson@15077
  1086
done
paulson@15077
  1087
huffman@23115
  1088
lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
paulson@15077
  1089
apply (rule_tac x = 1 and y = y in linorder_cases)
paulson@15077
  1090
apply (drule order_less_imp_le [THEN lemma_exp_total])
paulson@15077
  1091
apply (rule_tac [2] x = 0 in exI)
paulson@15077
  1092
apply (frule_tac [3] real_inverse_gt_one)
paulson@15077
  1093
apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
paulson@15077
  1094
apply (rule_tac x = "-x" in exI)
paulson@15077
  1095
apply (simp add: exp_minus)
paulson@15077
  1096
done
paulson@15077
  1097
paulson@15077
  1098
huffman@29164
  1099
subsection {* Natural Logarithm *}
paulson@15077
  1100
huffman@23043
  1101
definition
huffman@23043
  1102
  ln :: "real => real" where
huffman@23043
  1103
  "ln x = (THE u. exp u = x)"
huffman@23043
  1104
huffman@23043
  1105
lemma ln_exp [simp]: "ln (exp x) = x"
paulson@15077
  1106
by (simp add: ln_def)
paulson@15077
  1107
huffman@22654
  1108
lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
huffman@22654
  1109
by (auto dest: exp_total)
huffman@22654
  1110
huffman@29171
  1111
lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
huffman@29171
  1112
apply (rule iffI)
huffman@29171
  1113
apply (erule subst, rule exp_gt_zero)
huffman@29171
  1114
apply (erule exp_ln)
paulson@15077
  1115
done
paulson@15077
  1116
huffman@29171
  1117
lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
huffman@29171
  1118
by (erule subst, rule ln_exp)
huffman@29171
  1119
huffman@29171
  1120
lemma ln_one [simp]: "ln 1 = 0"
huffman@29171
  1121
by (rule ln_unique, simp)
huffman@29171
  1122
huffman@29171
  1123
lemma ln_mult: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x * y) = ln x + ln y"
huffman@29171
  1124
by (rule ln_unique, simp add: exp_add)
huffman@29171
  1125
huffman@29171
  1126
lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
huffman@29171
  1127
by (rule ln_unique, simp add: exp_minus)
huffman@29171
  1128
huffman@29171
  1129
lemma ln_div: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln (x / y) = ln x - ln y"
huffman@29171
  1130
by (rule ln_unique, simp add: exp_diff)
paulson@15077
  1131
huffman@29171
  1132
lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
huffman@29171
  1133
by (rule ln_unique, simp add: exp_real_of_nat_mult)
huffman@29171
  1134
huffman@29171
  1135
lemma ln_less_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
huffman@29171
  1136
by (subst exp_less_cancel_iff [symmetric], simp)
huffman@29171
  1137
huffman@29171
  1138
lemma ln_le_cancel_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
huffman@29171
  1139
by (simp add: linorder_not_less [symmetric])
huffman@29171
  1140
huffman@29171
  1141
lemma ln_inj_iff [simp]: "\<lbrakk>0 < x; 0 < y\<rbrakk> \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
huffman@29171
  1142
by (simp add: order_eq_iff)
huffman@29171
  1143
huffman@29171
  1144
lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
huffman@29171
  1145
apply (rule exp_le_cancel_iff [THEN iffD1])
huffman@29171
  1146
apply (simp add: exp_ge_add_one_self_aux)
paulson@15077
  1147
done
paulson@15077
  1148
huffman@29171
  1149
lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
huffman@29171
  1150
by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
paulson@15077
  1151
paulson@15234
  1152
lemma ln_ge_zero [simp]:
paulson@15077
  1153
  assumes x: "1 \<le> x" shows "0 \<le> ln x"
paulson@15077
  1154
proof -
paulson@15077
  1155
  have "0 < x" using x by arith
paulson@15077
  1156
  hence "exp 0 \<le> exp (ln x)"
huffman@22915
  1157
    by (simp add: x)
paulson@15077
  1158
  thus ?thesis by (simp only: exp_le_cancel_iff)
paulson@15077
  1159
qed
paulson@15077
  1160
paulson@15077
  1161
lemma ln_ge_zero_imp_ge_one:
paulson@15077
  1162
  assumes ln: "0 \<le> ln x" 
paulson@15077
  1163
      and x:  "0 < x"
paulson@15077
  1164
  shows "1 \<le> x"
paulson@15077
  1165
proof -
paulson@15077
  1166
  from ln have "ln 1 \<le> ln x" by simp
paulson@15077
  1167
  thus ?thesis by (simp add: x del: ln_one) 
paulson@15077
  1168
qed
paulson@15077
  1169
paulson@15077
  1170
lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
paulson@15077
  1171
by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
paulson@15077
  1172
paulson@15234
  1173
lemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)"
paulson@15234
  1174
by (insert ln_ge_zero_iff [of x], arith)
paulson@15234
  1175
paulson@15077
  1176
lemma ln_gt_zero:
paulson@15077
  1177
  assumes x: "1 < x" shows "0 < ln x"
paulson@15077
  1178
proof -
paulson@15077
  1179
  have "0 < x" using x by arith
huffman@22915
  1180
  hence "exp 0 < exp (ln x)" by (simp add: x)
paulson@15077
  1181
  thus ?thesis  by (simp only: exp_less_cancel_iff)
paulson@15077
  1182
qed
paulson@15077
  1183
paulson@15077
  1184
lemma ln_gt_zero_imp_gt_one:
paulson@15077
  1185
  assumes ln: "0 < ln x" 
paulson@15077
  1186
      and x:  "0 < x"
paulson@15077
  1187
  shows "1 < x"
paulson@15077
  1188
proof -
paulson@15077
  1189
  from ln have "ln 1 < ln x" by simp
paulson@15077
  1190
  thus ?thesis by (simp add: x del: ln_one) 
paulson@15077
  1191
qed
paulson@15077
  1192
paulson@15077
  1193
lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
paulson@15077
  1194
by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
paulson@15077
  1195
paulson@15234
  1196
lemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)"
paulson@15234
  1197
by (insert ln_less_zero_iff [of x] ln_gt_zero_iff [of x], arith)
paulson@15077
  1198
paulson@15077
  1199
lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
paulson@15234
  1200
by simp
paulson@15077
  1201
paulson@15077
  1202
lemma exp_ln_eq: "exp u = x ==> ln x = u"
paulson@15077
  1203
by auto
paulson@15077
  1204
huffman@23045
  1205
lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
huffman@23045
  1206
apply (subgoal_tac "isCont ln (exp (ln x))", simp)
huffman@23045
  1207
apply (rule isCont_inverse_function [where f=exp], simp_all)
huffman@23045
  1208
done
huffman@23045
  1209
huffman@23045
  1210
lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
huffman@23045
  1211
apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
huffman@23045
  1212
apply (erule lemma_DERIV_subst [OF DERIV_exp exp_ln])
huffman@23045
  1213
apply (simp_all add: abs_if isCont_ln)
huffman@23045
  1214
done
huffman@23045
  1215
paulson@33667
  1216
lemma DERIV_ln_divide: "0 < x ==> DERIV ln x :> 1 / x"
paulson@33667
  1217
  by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
paulson@33667
  1218
hoelzl@29803
  1219
lemma ln_series: assumes "0 < x" and "x < 2"
hoelzl@29803
  1220
  shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))" (is "ln x = suminf (?f (x - 1))")
hoelzl@29803
  1221
proof -
hoelzl@29803
  1222
  let "?f' x n" = "(-1)^n * (x - 1)^n"
hoelzl@29803
  1223
hoelzl@29803
  1224
  have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
hoelzl@29803
  1225
  proof (rule DERIV_isconst3[where x=x])
hoelzl@29803
  1226
    fix x :: real assume "x \<in> {0 <..< 2}" hence "0 < x" and "x < 2" by auto
hoelzl@29803
  1227
    have "norm (1 - x) < 1" using `0 < x` and `x < 2` by auto
hoelzl@29803
  1228
    have "1 / x = 1 / (1 - (1 - x))" by auto
hoelzl@29803
  1229
    also have "\<dots> = (\<Sum> n. (1 - x)^n)" using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
hoelzl@29803
  1230
    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)
hoelzl@29803
  1231
    finally have "DERIV ln x :> suminf (?f' x)" using DERIV_ln[OF `0 < x`] unfolding real_divide_def by auto
hoelzl@29803
  1232
    moreover
hoelzl@29803
  1233
    have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
hoelzl@29803
  1234
    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
  1235
    proof (rule DERIV_power_series')
hoelzl@29803
  1236
      show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1" using `0 < x` `x < 2` by auto
hoelzl@29803
  1237
      { fix x :: real assume "x \<in> {- 1<..<1}" hence "norm (-x) < 1" by auto
wenzelm@32960
  1238
        show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
huffman@30082
  1239
          unfolding One_nat_def
huffman@35216
  1240
          by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
hoelzl@29803
  1241
      }
hoelzl@29803
  1242
    qed
huffman@30082
  1243
    hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)" unfolding One_nat_def by auto
hoelzl@29803
  1244
    hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)" unfolding DERIV_iff repos .
hoelzl@29803
  1245
    ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
hoelzl@29803
  1246
      by (rule DERIV_diff)
hoelzl@29803
  1247
    thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
hoelzl@29803
  1248
  qed (auto simp add: assms)
hoelzl@29803
  1249
  thus ?thesis by (auto simp add: suminf_zero)
hoelzl@29803
  1250
qed
paulson@15077
  1251
huffman@29164
  1252
subsection {* Sine and Cosine *}
huffman@29164
  1253
huffman@29164
  1254
definition
huffman@31271
  1255
  sin_coeff :: "nat \<Rightarrow> real" where
huffman@31271
  1256
  "sin_coeff = (\<lambda>n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"
huffman@31271
  1257
huffman@31271
  1258
definition
huffman@31271
  1259
  cos_coeff :: "nat \<Rightarrow> real" where
huffman@31271
  1260
  "cos_coeff = (\<lambda>n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"
huffman@31271
  1261
huffman@31271
  1262
definition
huffman@29164
  1263
  sin :: "real => real" where
huffman@31271
  1264
  "sin x = (\<Sum>n. sin_coeff n * x ^ n)"
huffman@31271
  1265
huffman@29164
  1266
definition
huffman@29164
  1267
  cos :: "real => real" where
huffman@31271
  1268
  "cos x = (\<Sum>n. cos_coeff n * x ^ n)"
huffman@31271
  1269
huffman@31271
  1270
lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
huffman@31271
  1271
unfolding sin_coeff_def
huffman@29164
  1272
apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
huffman@29164
  1273
apply (rule_tac [2] summable_exp)
huffman@29164
  1274
apply (rule_tac x = 0 in exI)
huffman@29164
  1275
apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
huffman@29164
  1276
done
huffman@29164
  1277
huffman@31271
  1278
lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"
huffman@31271
  1279
unfolding cos_coeff_def
huffman@29164
  1280
apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
huffman@29164
  1281
apply (rule_tac [2] summable_exp)
huffman@29164
  1282
apply (rule_tac x = 0 in exI)
huffman@29164
  1283
apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
huffman@29164
  1284
done
huffman@29164
  1285
huffman@29164
  1286
lemma lemma_STAR_sin:
huffman@29164
  1287
     "(if even n then 0  
huffman@29164
  1288
       else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"
huffman@29164
  1289
by (induct "n", auto)
huffman@29164
  1290
huffman@29164
  1291
lemma lemma_STAR_cos:
huffman@29164
  1292
     "0 < n -->  
huffman@29164
  1293
      -1 ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
huffman@29164
  1294
by (induct "n", auto)
huffman@29164
  1295
huffman@29164
  1296
lemma lemma_STAR_cos1:
huffman@29164
  1297
     "0 < n -->  
huffman@29164
  1298
      (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
huffman@29164
  1299
by (induct "n", auto)
huffman@29164
  1300
huffman@29164
  1301
lemma lemma_STAR_cos2:
huffman@29164
  1302
  "(\<Sum>n=1..<n. if even n then -1 ^ (n div 2)/(real (fact n)) *  0 ^ n 
huffman@29164
  1303
                         else 0) = 0"
huffman@29164
  1304
apply (induct "n")
huffman@29164
  1305
apply (case_tac [2] "n", auto)
huffman@29164
  1306
done
huffman@29164
  1307
huffman@31271
  1308
lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
huffman@29164
  1309
unfolding sin_def by (rule summable_sin [THEN summable_sums])
huffman@29164
  1310
huffman@31271
  1311
lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
huffman@29164
  1312
unfolding cos_def by (rule summable_cos [THEN summable_sums])
huffman@29164
  1313
huffman@31271
  1314
lemma sin_fdiffs: "diffs sin_coeff = cos_coeff"
huffman@31271
  1315
unfolding sin_coeff_def cos_coeff_def
huffman@29164
  1316
by (auto intro!: ext 
huffman@29164
  1317
         simp add: diffs_def divide_inverse real_of_nat_def of_nat_mult
huffman@29164
  1318
         simp del: mult_Suc of_nat_Suc)
huffman@29164
  1319
huffman@31271
  1320
lemma sin_fdiffs2: "diffs sin_coeff n = cos_coeff n"
huffman@29164
  1321
by (simp only: sin_fdiffs)
huffman@29164
  1322
huffman@31271
  1323
lemma cos_fdiffs: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
huffman@31271
  1324
unfolding sin_coeff_def cos_coeff_def
huffman@29164
  1325
by (auto intro!: ext 
huffman@29164
  1326
         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex real_of_nat_def of_nat_mult
huffman@29164
  1327
         simp del: mult_Suc of_nat_Suc)
huffman@29164
  1328
huffman@31271
  1329
lemma cos_fdiffs2: "diffs cos_coeff n = - sin_coeff n"
huffman@29164
  1330
by (simp only: cos_fdiffs)
huffman@29164
  1331
huffman@29164
  1332
text{*Now at last we can get the derivatives of exp, sin and cos*}
huffman@29164
  1333
huffman@31271
  1334
lemma lemma_sin_minus: "- sin x = (\<Sum>n. - (sin_coeff n * x ^ n))"
huffman@29164
  1335
by (auto intro!: sums_unique sums_minus sin_converges)
huffman@29164
  1336
huffman@31271
  1337
lemma lemma_sin_ext: "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
huffman@29164
  1338
by (auto intro!: ext simp add: sin_def)
huffman@29164
  1339
huffman@31271
  1340
lemma lemma_cos_ext: "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
huffman@29164
  1341
by (auto intro!: ext simp add: cos_def)
huffman@29164
  1342
huffman@29164
  1343
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
huffman@29164
  1344
apply (simp add: cos_def)
huffman@29164
  1345
apply (subst lemma_sin_ext)
huffman@29164
  1346
apply (auto simp add: sin_fdiffs2 [symmetric])
huffman@29164
  1347
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
huffman@29164
  1348
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs)
huffman@29164
  1349
done
huffman@29164
  1350
huffman@29164
  1351
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
huffman@29164
  1352
apply (subst lemma_cos_ext)
huffman@29164
  1353
apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
huffman@29164
  1354
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
huffman@29164
  1355
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus)
huffman@29164
  1356
done
huffman@29164
  1357
huffman@29164
  1358
lemma isCont_sin [simp]: "isCont sin x"
huffman@29164
  1359
by (rule DERIV_sin [THEN DERIV_isCont])
huffman@29164
  1360
huffman@29164
  1361
lemma isCont_cos [simp]: "isCont cos x"
huffman@29164
  1362
by (rule DERIV_cos [THEN DERIV_isCont])
huffman@29164
  1363
huffman@29164
  1364
hoelzl@31880
  1365
declare
hoelzl@31880
  1366
  DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  1367
  DERIV_ln[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  1368
  DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  1369
  DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@31880
  1370
huffman@29164
  1371
subsection {* Properties of Sine and Cosine *}
paulson@15077
  1372
paulson@15077
  1373
lemma sin_zero [simp]: "sin 0 = 0"
huffman@31271
  1374
unfolding sin_def sin_coeff_def by (simp add: powser_zero)
paulson@15077
  1375
paulson@15077
  1376
lemma cos_zero [simp]: "cos 0 = 1"
huffman@31271
  1377
unfolding cos_def cos_coeff_def by (simp add: powser_zero)
paulson@15077
  1378
paulson@15077
  1379
lemma DERIV_sin_sin_mult [simp]:
paulson@15077
  1380
     "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077
  1381
by (rule DERIV_mult, auto)
paulson@15077
  1382
paulson@15077
  1383
lemma DERIV_sin_sin_mult2 [simp]:
paulson@15077
  1384
     "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"
paulson@15077
  1385
apply (cut_tac x = x in DERIV_sin_sin_mult)
paulson@15077
  1386
apply (auto simp add: mult_assoc)
paulson@15077
  1387
done
paulson@15077
  1388
paulson@15077
  1389
lemma DERIV_sin_realpow2 [simp]:
paulson@15077
  1390
     "DERIV (%x. (sin x)\<twosuperior>) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077
  1391
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077
  1392
paulson@15077
  1393
lemma DERIV_sin_realpow2a [simp]:
paulson@15077
  1394
     "DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"
paulson@15077
  1395
by (auto simp add: numeral_2_eq_2)
paulson@15077
  1396
paulson@15077
  1397
lemma DERIV_cos_cos_mult [simp]:
paulson@15077
  1398
     "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077
  1399
by (rule DERIV_mult, auto)
paulson@15077
  1400
paulson@15077
  1401
lemma DERIV_cos_cos_mult2 [simp]:
paulson@15077
  1402
     "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"
paulson@15077
  1403
apply (cut_tac x = x in DERIV_cos_cos_mult)
paulson@15077
  1404
apply (auto simp add: mult_ac)
paulson@15077
  1405
done
paulson@15077
  1406
paulson@15077
  1407
lemma DERIV_cos_realpow2 [simp]:
paulson@15077
  1408
     "DERIV (%x. (cos x)\<twosuperior>) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077
  1409
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077
  1410
paulson@15077
  1411
lemma DERIV_cos_realpow2a [simp]:
paulson@15077
  1412
     "DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"
paulson@15077
  1413
by (auto simp add: numeral_2_eq_2)
paulson@15077
  1414
paulson@15077
  1415
lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
paulson@15077
  1416
by auto
paulson@15077
  1417
paulson@15077
  1418
lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\<twosuperior>) x :> -(2 * cos(x) * sin(x))"
hoelzl@31881
  1419
  by (auto intro!: DERIV_intros)
paulson@15077
  1420
paulson@15077
  1421
(* most useful *)
paulson@15229
  1422
lemma DERIV_cos_cos_mult3 [simp]:
paulson@15229
  1423
     "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
hoelzl@31881
  1424
  by (auto intro!: DERIV_intros)
paulson@15077
  1425
paulson@15077
  1426
lemma DERIV_sin_circle_all: 
paulson@15077
  1427
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>  
paulson@15077
  1428
             (2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
hoelzl@31881
  1429
  by (auto intro!: DERIV_intros)
paulson@15077
  1430
paulson@15229
  1431
lemma DERIV_sin_circle_all_zero [simp]:
paulson@15229
  1432
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
paulson@15077
  1433
by (cut_tac DERIV_sin_circle_all, auto)
paulson@15077
  1434
paulson@15077
  1435
lemma sin_cos_squared_add [simp]: "((sin x)\<twosuperior>) + ((cos x)\<twosuperior>) = 1"
paulson@15077
  1436
apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all])
paulson@15077
  1437
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1438
done
paulson@15077
  1439
paulson@15077
  1440
lemma sin_cos_squared_add2 [simp]: "((cos x)\<twosuperior>) + ((sin x)\<twosuperior>) = 1"
huffman@23286
  1441
apply (subst add_commute)
huffman@30273
  1442
apply (rule sin_cos_squared_add)
paulson@15077
  1443
done
paulson@15077
  1444
paulson@15077
  1445
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
paulson@15077
  1446
apply (cut_tac x = x in sin_cos_squared_add2)
huffman@30273
  1447
apply (simp add: power2_eq_square)
paulson@15077
  1448
done
paulson@15077
  1449
paulson@15077
  1450
lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
paulson@15229
  1451
apply (rule_tac a1 = "(cos x)\<twosuperior>" in add_right_cancel [THEN iffD1])
huffman@30273
  1452
apply simp
paulson@15077
  1453
done
paulson@15077
  1454
paulson@15077
  1455
lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
paulson@15077
  1456
apply (rule_tac a1 = "(sin x)\<twosuperior>" in add_right_cancel [THEN iffD1])
huffman@30273
  1457
apply simp
paulson@15077
  1458
done
paulson@15077
  1459
paulson@15081
  1460
lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
huffman@23097
  1461
by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
paulson@15077
  1462
paulson@15077
  1463
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
paulson@15077
  1464
apply (insert abs_sin_le_one [of x]) 
huffman@22998
  1465
apply (simp add: abs_le_iff del: abs_sin_le_one) 
paulson@15077
  1466
done
paulson@15077
  1467
paulson@15077
  1468
lemma sin_le_one [simp]: "sin x \<le> 1"
paulson@15077
  1469
apply (insert abs_sin_le_one [of x]) 
huffman@22998
  1470
apply (simp add: abs_le_iff del: abs_sin_le_one) 
paulson@15077
  1471
done
paulson@15077
  1472
paulson@15081
  1473
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
huffman@23097
  1474
by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
paulson@15077
  1475
paulson@15077
  1476
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
paulson@15077
  1477
apply (insert abs_cos_le_one [of x]) 
huffman@22998
  1478
apply (simp add: abs_le_iff del: abs_cos_le_one) 
paulson@15077
  1479
done
paulson@15077
  1480
paulson@15077
  1481
lemma cos_le_one [simp]: "cos x \<le> 1"
paulson@15077
  1482
apply (insert abs_cos_le_one [of x]) 
huffman@22998
  1483
apply (simp add: abs_le_iff del: abs_cos_le_one)
paulson@15077
  1484
done
paulson@15077
  1485
paulson@15077
  1486
lemma DERIV_fun_pow: "DERIV g x :> m ==>  
paulson@15077
  1487
      DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
huffman@30082
  1488
unfolding One_nat_def
paulson@15077
  1489
apply (rule lemma_DERIV_subst)
paulson@15229
  1490
apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2)
paulson@15077
  1491
apply (rule DERIV_pow, auto)
paulson@15077
  1492
done
paulson@15077
  1493
paulson@15229
  1494
lemma DERIV_fun_exp:
paulson@15229
  1495
     "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
paulson@15077
  1496
apply (rule lemma_DERIV_subst)
paulson@15077
  1497
apply (rule_tac f = exp in DERIV_chain2)
paulson@15077
  1498
apply (rule DERIV_exp, auto)
paulson@15077
  1499
done
paulson@15077
  1500
paulson@15229
  1501
lemma DERIV_fun_sin:
paulson@15229
  1502
     "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
paulson@15077
  1503
apply (rule lemma_DERIV_subst)
paulson@15077
  1504
apply (rule_tac f = sin in DERIV_chain2)
paulson@15077
  1505
apply (rule DERIV_sin, auto)
paulson@15077
  1506
done
paulson@15077
  1507
paulson@15229
  1508
lemma DERIV_fun_cos:
paulson@15229
  1509
     "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
paulson@15077
  1510
apply (rule lemma_DERIV_subst)
paulson@15077
  1511
apply (rule_tac f = cos in DERIV_chain2)
paulson@15077
  1512
apply (rule DERIV_cos, auto)
paulson@15077
  1513
done
paulson@15077
  1514
paulson@15077
  1515
(* lemma *)
paulson@15229
  1516
lemma lemma_DERIV_sin_cos_add:
paulson@15229
  1517
     "\<forall>x.  
paulson@15077
  1518
         DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077
  1519
               (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"
hoelzl@31881
  1520
  by (auto intro!: DERIV_intros simp add: algebra_simps)
paulson@15077
  1521
paulson@15077
  1522
lemma sin_cos_add [simp]:
paulson@15077
  1523
     "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077
  1524
      (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
paulson@15077
  1525
apply (cut_tac y = 0 and x = x and y7 = y 
paulson@15077
  1526
       in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])
paulson@15077
  1527
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1528
done
paulson@15077
  1529
paulson@15077
  1530
lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
paulson@15077
  1531
apply (cut_tac x = x and y = y in sin_cos_add)
huffman@22969
  1532
apply (simp del: sin_cos_add)
paulson@15077
  1533
done
paulson@15077
  1534
paulson@15077
  1535
lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
paulson@15077
  1536
apply (cut_tac x = x and y = y in sin_cos_add)
huffman@22969
  1537
apply (simp del: sin_cos_add)
paulson@15077
  1538
done
paulson@15077
  1539
paulson@15085
  1540
lemma lemma_DERIV_sin_cos_minus:
paulson@15085
  1541
    "\<forall>x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"
hoelzl@31881
  1542
  by (auto intro!: DERIV_intros simp add: algebra_simps)
hoelzl@31881
  1543
paulson@15077
  1544
huffman@29165
  1545
lemma sin_cos_minus: 
paulson@15085
  1546
    "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"
paulson@15085
  1547
apply (cut_tac y = 0 and x = x 
paulson@15085
  1548
       in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])
huffman@22969
  1549
apply simp
paulson@15077
  1550
done
paulson@15077
  1551
paulson@15077
  1552
lemma sin_minus [simp]: "sin (-x) = -sin(x)"
huffman@29165
  1553
  using sin_cos_minus [where x=x] by simp
paulson@15077
  1554
paulson@15077
  1555
lemma cos_minus [simp]: "cos (-x) = cos(x)"
huffman@29165
  1556
  using sin_cos_minus [where x=x] by simp
paulson@15077
  1557
paulson@15077
  1558
lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
huffman@22969
  1559
by (simp add: diff_minus sin_add)
paulson@15077
  1560
paulson@15077
  1561
lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
paulson@15077
  1562
by (simp add: sin_diff mult_commute)
paulson@15077
  1563
paulson@15077
  1564
lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
huffman@22969
  1565
by (simp add: diff_minus cos_add)
paulson@15077
  1566
paulson@15077
  1567
lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
paulson@15077
  1568
by (simp add: cos_diff mult_commute)
paulson@15077
  1569
paulson@15077
  1570
lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
huffman@29165
  1571
  using sin_add [where x=x and y=x] by simp
paulson@15077
  1572
paulson@15077
  1573
lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
huffman@29165
  1574
  using cos_add [where x=x and y=x]
huffman@29165
  1575
  by (simp add: power2_eq_square)
paulson@15077
  1576
paulson@15077
  1577
huffman@29164
  1578
subsection {* The Constant Pi *}
paulson@15077
  1579
huffman@23043
  1580
definition
huffman@23043
  1581
  pi :: "real" where
huffman@23053
  1582
  "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
huffman@23043
  1583
paulson@15077
  1584
text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"}; 
paulson@15077
  1585
   hence define pi.*}
paulson@15077
  1586
paulson@15077
  1587
lemma sin_paired:
huffman@23177
  1588
     "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) 
paulson@15077
  1589
      sums  sin x"
paulson@15077
  1590
proof -
huffman@31271
  1591
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
huffman@23176
  1592
    unfolding sin_def
paulson@15077
  1593
    by (rule sin_converges [THEN sums_summable, THEN sums_group], simp) 
huffman@31271
  1594
  thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)
paulson@15077
  1595
qed
paulson@15077
  1596
huffman@30273
  1597
text {* FIXME: This is a long, ugly proof! *}
paulson@15077
  1598
lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077
  1599
apply (subgoal_tac 
paulson@15077
  1600
       "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
huffman@23177
  1601
              -1 ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1)) 
huffman@23177
  1602
     sums (\<Sum>n. -1 ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
paulson@15077
  1603
 prefer 2
paulson@15077
  1604
 apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1605
apply (rotate_tac 2)
paulson@15077
  1606
apply (drule sin_paired [THEN sums_unique, THEN ssubst])
huffman@30082
  1607
unfolding One_nat_def
avigad@32047
  1608
apply (auto simp del: fact_Suc)
paulson@15077
  1609
apply (frule sums_unique)
avigad@32047
  1610
apply (auto simp del: fact_Suc)
paulson@15077
  1611
apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])
avigad@32047
  1612
apply (auto simp del: fact_Suc)
paulson@15077
  1613
apply (erule sums_summable)
paulson@15077
  1614
apply (case_tac "m=0")
paulson@15077
  1615
apply (simp (no_asm_simp))
paulson@15234
  1616
apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x") 
nipkow@15539
  1617
apply (simp only: mult_less_cancel_left, simp)  
nipkow@15539
  1618
apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
paulson@15077
  1619
apply (subgoal_tac "x*x < 2*3", simp) 
paulson@15077
  1620
apply (rule mult_strict_mono)
avigad@32047
  1621
apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc)
avigad@32047
  1622
apply (subst fact_Suc)
avigad@32047
  1623
apply (subst fact_Suc)
avigad@32047
  1624
apply (subst fact_Suc)
avigad@32047
  1625
apply (subst fact_Suc)
paulson@15077
  1626
apply (subst real_of_nat_mult)
paulson@15077
  1627
apply (subst real_of_nat_mult)
paulson@15077
  1628
apply (subst real_of_nat_mult)
paulson@15077
  1629
apply (subst real_of_nat_mult)
avigad@32047
  1630
apply (simp (no_asm) add: divide_inverse del: fact_Suc)
avigad@32047
  1631
apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
paulson@15077
  1632
apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right) 
avigad@32047
  1633
apply (auto simp add: mult_assoc simp del: fact_Suc)
paulson@15077
  1634
apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right) 
avigad@32047
  1635
apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
paulson@15077
  1636
apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)") 
paulson@15077
  1637
apply (erule ssubst)+
avigad@32047
  1638
apply (auto simp del: fact_Suc)
paulson@15077
  1639
apply (subgoal_tac "0 < x ^ (4 * m) ")
paulson@15077
  1640
 prefer 2 apply (simp only: zero_less_power) 
paulson@15077
  1641
apply (simp (no_asm_simp) add: mult_less_cancel_left)
paulson@15077
  1642
apply (rule mult_strict_mono)
paulson@15077
  1643
apply (simp_all (no_asm_simp))
paulson@15077
  1644
done
paulson@15077
  1645
paulson@15077
  1646
lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077
  1647
by (auto intro: sin_gt_zero)
paulson@15077
  1648
paulson@15077
  1649
lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
paulson@15077
  1650
apply (cut_tac x = x in sin_gt_zero1)
paulson@15077
  1651
apply (auto simp add: cos_squared_eq cos_double)
paulson@15077
  1652
done
paulson@15077
  1653
paulson@15077
  1654
lemma cos_paired:
huffman@23177
  1655
     "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
paulson@15077
  1656
proof -
huffman@31271
  1657
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
huffman@23176
  1658
    unfolding cos_def
paulson@15077
  1659
    by (rule cos_converges [THEN sums_summable, THEN sums_group], simp) 
huffman@31271
  1660
  thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)
paulson@15077
  1661
qed
paulson@15077
  1662
paulson@15077
  1663
lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
paulson@15077
  1664
by simp
paulson@15077
  1665
huffman@23053
  1666
lemma cos_two_less_zero [simp]: "cos (2) < 0"
paulson@15077
  1667
apply (cut_tac x = 2 in cos_paired)
paulson@15077
  1668
apply (drule sums_minus)
paulson@15077
  1669
apply (rule neg_less_iff_less [THEN iffD1]) 
nipkow@15539
  1670
apply (frule sums_unique, auto)
nipkow@15539
  1671
apply (rule_tac y =
huffman@23177
  1672
 "\<Sum>n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
paulson@15481
  1673
       in order_less_trans)
avigad@32047
  1674
apply (simp (no_asm) add: fact_num_eq_if_nat realpow_num_eq_if del: fact_Suc)
nipkow@15561
  1675
apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
paulson@15077
  1676
apply (rule sumr_pos_lt_pair)
paulson@15077
  1677
apply (erule sums_summable, safe)
huffman@30082
  1678
unfolding One_nat_def
paulson@15085
  1679
apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric] 
avigad@32047
  1680
            del: fact_Suc)
paulson@15077
  1681
apply (rule real_mult_inverse_cancel2)
paulson@15077
  1682
apply (rule real_of_nat_fact_gt_zero)+
avigad@32047
  1683
apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
paulson@15077
  1684
apply (subst fact_lemma) 
avigad@32047
  1685
apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
paulson@15481
  1686
apply (simp only: real_of_nat_mult)
huffman@23007
  1687
apply (rule mult_strict_mono, force)
huffman@27483
  1688
  apply (rule_tac [3] real_of_nat_ge_zero)
paulson@15481
  1689
 prefer 2 apply force
paulson@15077
  1690
apply (rule real_of_nat_less_iff [THEN iffD2])
avigad@32036
  1691
apply (rule fact_less_mono_nat, auto)
paulson@15077
  1692
done
huffman@23053
  1693
huffman@23053
  1694
lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
huffman@23053
  1695
lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
paulson@15077
  1696
paulson@15077
  1697
lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
paulson@15077
  1698
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")
paulson@15077
  1699
apply (rule_tac [2] IVT2)
paulson@15077
  1700
apply (auto intro: DERIV_isCont DERIV_cos)
paulson@15077
  1701
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077
  1702
apply (rule ccontr)
paulson@15077
  1703
apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")
paulson@15077
  1704
apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
paulson@15077
  1705
apply (drule_tac f = cos in Rolle)
paulson@15077
  1706
apply (drule_tac [5] f = cos in Rolle)
paulson@15077
  1707
apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
paulson@33667
  1708
apply (metis order_less_le_trans real_less_def sin_gt_zero)
paulson@33667
  1709
apply (metis order_less_le_trans real_less_def sin_gt_zero)
paulson@15077
  1710
done
hoelzl@31880
  1711
huffman@23053
  1712
lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
paulson@15077
  1713
by (simp add: pi_def)
paulson@15077
  1714
paulson@15077
  1715
lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
huffman@23053
  1716
by (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1717
huffman@23053
  1718
lemma pi_half_gt_zero [simp]: "0 < pi / 2"
huffman@23053
  1719
apply (rule order_le_neq_trans)
huffman@23053
  1720
apply (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1721
apply (rule notI, drule arg_cong [where f=cos], simp)
paulson@15077
  1722
done
paulson@15077
  1723
huffman@23053
  1724
lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
huffman@23053
  1725
lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
paulson@15077
  1726
huffman@23053
  1727
lemma pi_half_less_two [simp]: "pi / 2 < 2"
huffman@23053
  1728
apply (rule order_le_neq_trans)
huffman@23053
  1729
apply (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1730
apply (rule notI, drule arg_cong [where f=cos], simp)
paulson@15077
  1731
done
huffman@23053
  1732
huffman@23053
  1733
lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
huffman@23053
  1734
lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
paulson@15077
  1735
paulson@15077
  1736
lemma pi_gt_zero [simp]: "0 < pi"
huffman@23053
  1737
by (insert pi_half_gt_zero, simp)
huffman@23053
  1738
huffman@23053
  1739
lemma pi_ge_zero [simp]: "0 \<le> pi"
huffman@23053
  1740
by (rule pi_gt_zero [THEN order_less_imp_le])
paulson@15077
  1741
paulson@15077
  1742
lemma pi_neq_zero [simp]: "pi \<noteq> 0"
huffman@22998
  1743
by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym])
paulson@15077
  1744
huffman@23053
  1745
lemma pi_not_less_zero [simp]: "\<not> pi < 0"
huffman@23053
  1746
by (simp add: linorder_not_less)
paulson@15077
  1747
huffman@29165
  1748
lemma minus_pi_half_less_zero: "-(pi/2) < 0"
huffman@29165
  1749
by simp
paulson@15077
  1750
hoelzl@29803
  1751
lemma m2pi_less_pi: "- (2 * pi) < pi"
hoelzl@29803
  1752
proof -
hoelzl@29803
  1753
  have "- (2 * pi) < 0" and "0 < pi" by auto
hoelzl@29803
  1754
  from order_less_trans[OF this] show ?thesis .
hoelzl@29803
  1755
qed
hoelzl@29803
  1756
paulson@15077
  1757
lemma sin_pi_half [simp]: "sin(pi/2) = 1"
paulson@15077
  1758
apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
paulson@15077
  1759
apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
huffman@23053
  1760
apply (simp add: power2_eq_square)
paulson@15077
  1761
done
paulson@15077
  1762
paulson@15077
  1763
lemma cos_pi [simp]: "cos pi = -1"
nipkow@15539
  1764
by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
paulson@15077
  1765
paulson@15077
  1766
lemma sin_pi [simp]: "sin pi = 0"
nipkow@15539
  1767
by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
paulson@15077
  1768
paulson@15077
  1769
lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
paulson@15229
  1770
by (simp add: diff_minus cos_add)
huffman@23053
  1771
declare sin_cos_eq [symmetric, simp]
paulson@15077
  1772
paulson@15077
  1773
lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
paulson@15229
  1774
by (simp add: cos_add)
paulson@15077
  1775
declare minus_sin_cos_eq [symmetric, simp]
paulson@15077
  1776
paulson@15077
  1777
lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
paulson@15229
  1778
by (simp add: diff_minus sin_add)
huffman@23053
  1779
declare cos_sin_eq [symmetric, simp]
paulson@15077
  1780
paulson@15077
  1781
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
paulson@15229
  1782
by (simp add: sin_add)
paulson@15077
  1783
paulson@15077
  1784
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
paulson@15229
  1785
by (simp add: sin_add)
paulson@15077
  1786
paulson@15077
  1787
lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
paulson@15229
  1788
by (simp add: cos_add)
paulson@15077
  1789
paulson@15077
  1790
lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
paulson@15077
  1791
by (simp add: sin_add cos_double)
paulson@15077
  1792
paulson@15077
  1793
lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
paulson@15077
  1794
by (simp add: cos_add cos_double)
paulson@15077
  1795
paulson@15077
  1796
lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
paulson@15251
  1797
apply (induct "n")
paulson@15077
  1798
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1799
done
paulson@15077
  1800
paulson@15383
  1801
lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
paulson@15383
  1802
proof -
paulson@15383
  1803
  have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
paulson@15383
  1804
  also have "... = -1 ^ n" by (rule cos_npi) 
paulson@15383
  1805
  finally show ?thesis .
paulson@15383
  1806
qed
paulson@15383
  1807
paulson@15077
  1808
lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
paulson@15251
  1809
apply (induct "n")
paulson@15077
  1810
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1811
done
paulson@15077
  1812
paulson@15077
  1813
lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
paulson@15383
  1814
by (simp add: mult_commute [of pi]) 
paulson@15077
  1815
paulson@15077
  1816
lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
paulson@15077
  1817
by (simp add: cos_double)
paulson@15077
  1818
paulson@15077
  1819
lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
paulson@15229
  1820
by simp
paulson@15077
  1821
paulson@15077
  1822
lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
paulson@15077
  1823
apply (rule sin_gt_zero, assumption)
paulson@15077
  1824
apply (rule order_less_trans, assumption)
paulson@15077
  1825
apply (rule pi_half_less_two)
paulson@15077
  1826
done
paulson@15077
  1827
paulson@15077
  1828
lemma sin_less_zero: 
paulson@15077
  1829
  assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
paulson@15077
  1830
proof -
paulson@15077
  1831
  have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2) 
paulson@15077
  1832
  thus ?thesis by simp
paulson@15077
  1833
qed
paulson@15077
  1834
paulson@15077
  1835
lemma pi_less_4: "pi < 4"
paulson@15077
  1836
by (cut_tac pi_half_less_two, auto)
paulson@15077
  1837
paulson@15077
  1838
lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1839
apply (cut_tac pi_less_4)
paulson@15077
  1840
apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
paulson@15077
  1841
apply (cut_tac cos_is_zero, safe)
paulson@15077
  1842
apply (rename_tac y z)
paulson@15077
  1843
apply (drule_tac x = y in spec)
paulson@15077
  1844
apply (drule_tac x = "pi/2" in spec, simp) 
paulson@15077
  1845
done
paulson@15077
  1846
paulson@15077
  1847
lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1848
apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15077
  1849
apply (rule cos_minus [THEN subst])
paulson@15077
  1850
apply (rule cos_gt_zero)
paulson@15077
  1851
apply (auto intro: cos_gt_zero)
paulson@15077
  1852
done
paulson@15077
  1853
 
paulson@15077
  1854
lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
paulson@15077
  1855
apply (auto simp add: order_le_less cos_gt_zero_pi)
paulson@15077
  1856
apply (subgoal_tac "x = pi/2", auto) 
paulson@15077
  1857
done
paulson@15077
  1858
paulson@15077
  1859
lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
paulson@15077
  1860
apply (subst sin_cos_eq)
paulson@15077
  1861
apply (rotate_tac 1)
paulson@15077
  1862
apply (drule real_sum_of_halves [THEN ssubst])
paulson@15077
  1863
apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
paulson@15077
  1864
done
paulson@15077
  1865
hoelzl@29803
  1866
hoelzl@29803
  1867
lemma pi_ge_two: "2 \<le> pi"
hoelzl@29803
  1868
proof (rule ccontr)
hoelzl@29803
  1869
  assume "\<not> 2 \<le> pi" hence "pi < 2" by auto
hoelzl@29803
  1870
  have "\<exists>y > pi. y < 2 \<and> y < 2 * pi"
hoelzl@29803
  1871
  proof (cases "2 < 2 * pi")
hoelzl@29803
  1872
    case True with dense[OF `pi < 2`] show ?thesis by auto
hoelzl@29803
  1873
  next
hoelzl@29803
  1874
    case False have "pi < 2 * pi" by auto
hoelzl@29803
  1875
    from dense[OF this] and False show ?thesis by auto
hoelzl@29803
  1876
  qed
hoelzl@29803
  1877
  then obtain y where "pi < y" and "y < 2" and "y < 2 * pi" by blast
hoelzl@29803
  1878
  hence "0 < sin y" using sin_gt_zero by auto
hoelzl@29803
  1879
  moreover 
hoelzl@29803
  1880
  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
  1881
  ultimately show False by auto
hoelzl@29803
  1882
qed
hoelzl@29803
  1883
paulson@15077
  1884
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
paulson@15077
  1885
by (auto simp add: order_le_less sin_gt_zero_pi)
paulson@15077
  1886
paulson@15077
  1887
lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
paulson@15077
  1888
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
paulson@15077
  1889
apply (rule_tac [2] IVT2)
paulson@15077
  1890
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
paulson@15077
  1891
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077
  1892
apply (rule ccontr, auto)
paulson@15077
  1893
apply (drule_tac f = cos in Rolle)
paulson@15077
  1894
apply (drule_tac [5] f = cos in Rolle)
paulson@15077
  1895
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
paulson@15077
  1896
            dest!: DERIV_cos [THEN DERIV_unique] 
paulson@15077
  1897
            simp add: differentiable_def)
paulson@15077
  1898
apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
paulson@15077
  1899
done
paulson@15077
  1900
paulson@15077
  1901
lemma sin_total:
paulson@15077
  1902
     "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
paulson@15077
  1903
apply (rule ccontr)
paulson@15077
  1904
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
  1905
apply (erule contrapos_np)
paulson@15077
  1906
apply (simp del: minus_sin_cos_eq [symmetric])
paulson@15077
  1907
apply (cut_tac y="-y" in cos_total, simp) apply simp 
paulson@15077
  1908
apply (erule ex1E)
paulson@15229
  1909
apply (rule_tac a = "x - (pi/2)" in ex1I)
huffman@23286
  1910
apply (simp (no_asm) add: add_assoc)
paulson@15077
  1911
apply (rotate_tac 3)
paulson@15077
  1912
apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all) 
paulson@15077
  1913
done
paulson@15077
  1914
paulson@15077
  1915
lemma reals_Archimedean4:
paulson@15077
  1916
     "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
paulson@15077
  1917
apply (auto dest!: reals_Archimedean3)
paulson@15077
  1918
apply (drule_tac x = x in spec, clarify) 
paulson@15077
  1919
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
paulson@15077
  1920
 prefer 2 apply (erule LeastI) 
paulson@15077
  1921
apply (case_tac "LEAST m::nat. x < real m * y", simp) 
paulson@15077
  1922
apply (subgoal_tac "~ x < real nat * y")
paulson@15077
  1923
 prefer 2 apply (rule not_less_Least, simp, force)  
paulson@15077
  1924
done
paulson@15077
  1925
paulson@15077
  1926
(* Pre Isabelle99-2 proof was simpler- numerals arithmetic 
paulson@15077
  1927
   now causes some unwanted re-arrangements of literals!   *)
paulson@15229
  1928
lemma cos_zero_lemma:
paulson@15229
  1929
     "[| 0 \<le> x; cos x = 0 |] ==>  
paulson@15077
  1930
      \<exists>n::nat. ~even n & x = real n * (pi/2)"
paulson@15077
  1931
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
paulson@15086
  1932
apply (subgoal_tac "0 \<le> x - real n * pi & 
paulson@15086
  1933
                    (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
nipkow@29667
  1934
apply (auto simp add: algebra_simps real_of_nat_Suc)
nipkow@29667
  1935
 prefer 2 apply (simp add: cos_diff)
paulson@15077
  1936
apply (simp add: cos_diff)
paulson@15077
  1937
apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
paulson@15077
  1938
apply (rule_tac [2] cos_total, safe)
paulson@15077
  1939
apply (drule_tac x = "x - real n * pi" in spec)
paulson@15077
  1940
apply (drule_tac x = "pi/2" in spec)
paulson@15077
  1941
apply (simp add: cos_diff)
paulson@15229
  1942
apply (rule_tac x = "Suc (2 * n)" in exI)
nipkow@29667
  1943
apply (simp add: real_of_nat_Suc algebra_simps, auto)
paulson@15077
  1944
done
paulson@15077
  1945
paulson@15229
  1946
lemma sin_zero_lemma:
paulson@15229
  1947
     "[| 0 \<le> x; sin x = 0 |] ==>  
paulson@15077
  1948
      \<exists>n::nat. even n & x = real n * (pi/2)"
paulson@15077
  1949
apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
paulson@15077
  1950
 apply (clarify, rule_tac x = "n - 1" in exI)
paulson@15077
  1951
 apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
paulson@15085
  1952
apply (rule cos_zero_lemma)
paulson@15085
  1953
apply (simp_all add: add_increasing)  
paulson@15077
  1954
done
paulson@15077
  1955
paulson@15077
  1956
paulson@15229
  1957
lemma cos_zero_iff:
paulson@15229
  1958
     "(cos x = 0) =  
paulson@15077
  1959
      ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |    
paulson@15077
  1960
       (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1961
apply (rule iffI)
paulson@15077
  1962
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1963
apply (drule cos_zero_lemma, assumption+)
paulson@15077
  1964
apply (cut_tac x="-x" in cos_zero_lemma, simp, simp) 
paulson@15077
  1965
apply (force simp add: minus_equation_iff [of x]) 
paulson@15077
  1966
apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) 
nipkow@15539
  1967
apply (auto simp add: cos_add)
paulson@15077
  1968
done
paulson@15077
  1969
paulson@15077
  1970
(* ditto: but to a lesser extent *)
paulson@15229
  1971
lemma sin_zero_iff:
paulson@15229
  1972
     "(sin x = 0) =  
paulson@15077
  1973
      ((\<exists>n::nat. even n & (x = real n * (pi/2))) |    
paulson@15077
  1974
       (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1975
apply (rule iffI)
paulson@15077
  1976
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1977
apply (drule sin_zero_lemma, assumption+)
paulson@15077
  1978
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
paulson@15077
  1979
apply (force simp add: minus_equation_iff [of x]) 
nipkow@15539
  1980
apply (auto simp add: even_mult_two_ex)
paulson@15077
  1981
done
paulson@15077
  1982
hoelzl@29803
  1983
lemma cos_monotone_0_pi: assumes "0 \<le> y" and "y < x" and "x \<le> pi"
hoelzl@29803
  1984
  shows "cos x < cos y"
hoelzl@29803
  1985
proof -
wenzelm@33549
  1986
  have "- (x - y) < 0" using assms by auto
hoelzl@29803
  1987
hoelzl@29803
  1988
  from MVT2[OF `y < x` DERIV_cos[THEN impI, THEN allI]]
hoelzl@29803
  1989
  obtain z where "y < z" and "z < x" and cos_diff: "cos x - cos y = (x - y) * - sin z" by auto
wenzelm@33549
  1990
  hence "0 < z" and "z < pi" using assms by auto
hoelzl@29803
  1991
  hence "0 < sin z" using sin_gt_zero_pi by auto
hoelzl@29803
  1992
  hence "cos x - cos y < 0" unfolding cos_diff minus_mult_commute[symmetric] using `- (x - y) < 0` by (rule mult_pos_neg2)
hoelzl@29803
  1993
  thus ?thesis by auto
hoelzl@29803
  1994
qed
hoelzl@29803
  1995
hoelzl@29803
  1996
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
  1997
proof (cases "y < x")
hoelzl@29803
  1998
  case True show ?thesis using cos_monotone_0_pi[OF `0 \<le> y` True `x \<le> pi`] by auto
hoelzl@29803
  1999
next
hoelzl@29803
  2000
  case False hence "y = x" using `y \<le> x` by auto
hoelzl@29803
  2001
  thus ?thesis by auto
hoelzl@29803
  2002
qed
hoelzl@29803
  2003
hoelzl@29803
  2004
lemma cos_monotone_minus_pi_0: assumes "-pi \<le> y" and "y < x" and "x \<le> 0"
hoelzl@29803
  2005
  shows "cos y < cos x"
hoelzl@29803
  2006
proof -
wenzelm@33549
  2007
  have "0 \<le> -x" and "-x < -y" and "-y \<le> pi" using assms by auto
hoelzl@29803
  2008
  from cos_monotone_0_pi[OF this]
hoelzl@29803
  2009
  show ?thesis unfolding cos_minus .
hoelzl@29803
  2010
qed
hoelzl@29803
  2011
hoelzl@29803
  2012
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
  2013
proof (cases "y < x")
hoelzl@29803
  2014
  case True show ?thesis using cos_monotone_minus_pi_0[OF `-pi \<le> y` True `x \<le> 0`] by auto
hoelzl@29803
  2015
next
hoelzl@29803
  2016
  case False hence "y = x" using `y \<le> x` by auto
hoelzl@29803
  2017
  thus ?thesis by auto
hoelzl@29803
  2018
qed
hoelzl@29803
  2019
hoelzl@29803
  2020
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
  2021
proof -
wenzelm@33549
  2022
  have "0 \<le> y + pi / 2" and "y + pi / 2 \<le> x + pi / 2" and "x + pi /2 \<le> pi"
wenzelm@33549
  2023
    using pi_ge_two and assms by auto
hoelzl@29803
  2024
  from cos_monotone_0_pi'[OF this] show ?thesis unfolding minus_sin_cos_eq[symmetric] by auto
hoelzl@29803
  2025
qed
paulson@15077
  2026
huffman@29164
  2027
subsection {* Tangent *}
paulson@15077
  2028
huffman@23043
  2029
definition
huffman@23043
  2030
  tan :: "real => real" where
huffman@23043
  2031
  "tan x = (sin x)/(cos x)"
huffman@23043
  2032
paulson@15077
  2033
lemma tan_zero [simp]: "tan 0 = 0"
paulson@15077
  2034
by (simp add: tan_def)
paulson@15077
  2035
paulson@15077
  2036
lemma tan_pi [simp]: "tan pi = 0"
paulson@15077
  2037
by (simp add: tan_def)
paulson@15077
  2038
paulson@15077
  2039
lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
paulson@15077
  2040
by (simp add: tan_def)
paulson@15077
  2041
paulson@15077
  2042
lemma tan_minus [simp]: "tan (-x) = - tan x"
paulson@15077
  2043
by (simp add: tan_def minus_mult_left)
paulson@15077
  2044
paulson@15077
  2045
lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
paulson@15077
  2046
by (simp add: tan_def)
paulson@15077
  2047
paulson@15077
  2048
lemma lemma_tan_add1: 
paulson@15077
  2049
      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077
  2050
        ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
paulson@15229
  2051
apply (simp add: tan_def divide_inverse)
paulson@15229
  2052
apply (auto simp del: inverse_mult_distrib 
paulson@15229
  2053
            simp add: inverse_mult_distrib [symmetric] mult_ac)
paulson@15077
  2054
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15229
  2055
apply (auto simp del: inverse_mult_distrib 
paulson@15229
  2056
            simp add: mult_assoc left_diff_distrib cos_add)
nipkow@29667
  2057
done
paulson@15077
  2058
paulson@15077
  2059
lemma add_tan_eq: 
paulson@15077
  2060
      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077
  2061
       ==> tan x + tan y = sin(x + y)/(cos x * cos y)"
paulson@15229
  2062
apply (simp add: tan_def)
paulson@15077
  2063
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15077
  2064
apply (auto simp add: mult_assoc left_distrib)
nipkow@15539
  2065
apply (simp add: sin_add)
paulson@15077
  2066
done
paulson@15077
  2067
paulson@15229
  2068
lemma tan_add:
paulson@15229
  2069
     "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]  
paulson@15077
  2070
      ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
paulson@15077
  2071
apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)
paulson@15077
  2072
apply (simp add: tan_def)
paulson@15077
  2073
done
paulson@15077
  2074
paulson@15229
  2075
lemma tan_double:
paulson@15229
  2076
     "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]  
paulson@15077
  2077
      ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
paulson@15077
  2078
apply (insert tan_add [of x x]) 
paulson@15077
  2079
apply (simp add: mult_2 [symmetric])  
paulson@15077
  2080
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  2081
done
paulson@15077
  2082
paulson@15077
  2083
lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
paulson@15229
  2084
by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) 
paulson@15077
  2085
paulson@15077
  2086
lemma tan_less_zero: 
paulson@15077
  2087
  assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
paulson@15077
  2088
proof -
paulson@15077
  2089
  have "0 < tan (- x)" using prems by (simp only: tan_gt_zero) 
paulson@15077
  2090
  thus ?thesis by simp
paulson@15077
  2091
qed
paulson@15077
  2092
hoelzl@29803
  2093
lemma tan_half: fixes x :: real assumes "- (pi / 2) < x" and "x < pi / 2"
hoelzl@29803
  2094
  shows "tan x = sin (2 * x) / (cos (2 * x) + 1)"
hoelzl@29803
  2095
proof -
hoelzl@29803
  2096
  from cos_gt_zero_pi[OF `- (pi / 2) < x` `x < pi / 2`]
hoelzl@29803
  2097
  have "cos x \<noteq> 0" by auto
hoelzl@29803
  2098
hoelzl@29803
  2099
  have minus_cos_2x: "\<And>X. X - cos (2*x) = X - (cos x) ^ 2 + (sin x) ^ 2" unfolding cos_double by algebra
hoelzl@29803
  2100
hoelzl@29803
  2101
  have "tan x = (tan x + tan x) / 2" by auto
hoelzl@29803
  2102
  also have "\<dots> = sin (x + x) / (cos x * cos x) / 2" unfolding add_tan_eq[OF `cos x \<noteq> 0` `cos x \<noteq> 0`] ..
hoelzl@29803
  2103
  also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + (cos x) ^ 2 + cos (2*x) - cos (2*x))" unfolding divide_divide_eq_left numeral_2_eq_2 by auto
hoelzl@29803
  2104
  also have "\<dots> = sin (2 * x) / ((cos x) ^ 2 + cos (2*x) + (sin x)^2)" unfolding minus_cos_2x by auto
hoelzl@29803
  2105
  also have "\<dots> = sin (2 * x) / (cos (2*x) + 1)" by auto
hoelzl@29803
  2106
  finally show ?thesis .
hoelzl@29803
  2107
qed
hoelzl@29803
  2108
paulson@15077
  2109
lemma lemma_DERIV_tan:
paulson@15077
  2110
     "cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"
hoelzl@31881
  2111
  by (auto intro!: DERIV_intros simp add: field_simps numeral_2_eq_2)
paulson@15077
  2112
paulson@15077
  2113
lemma DERIV_tan [simp]: "cos x \<noteq> 0 ==> DERIV tan x :> inverse((cos x)\<twosuperior>)"
paulson@15077
  2114
by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])
paulson@15077
  2115
huffman@23045
  2116
lemma isCont_tan [simp]: "cos x \<noteq> 0 ==> isCont tan x"
huffman@23045
  2117
by (rule DERIV_tan [THEN DERIV_isCont])
huffman@23045
  2118
paulson@15077
  2119
lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
paulson@15077
  2120
apply (subgoal_tac "(\<lambda>x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")
paulson@15229
  2121
apply (simp add: divide_inverse [symmetric])
huffman@22613
  2122
apply (rule LIM_mult)
paulson@15077
  2123
apply (rule_tac [2] inverse_1 [THEN subst])
paulson@15077
  2124
apply (rule_tac [2] LIM_inverse)
paulson@15077
  2125
apply (simp_all add: divide_inverse [symmetric]) 
paulson@15077
  2126
apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric]) 
paulson@15077
  2127
apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
paulson@15077
  2128
done
paulson@15077
  2129
paulson@15077
  2130
lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
paulson@15077
  2131
apply (cut_tac LIM_cos_div_sin)
huffman@31338
  2132
apply (simp only: LIM_eq)
paulson@15077
  2133
apply (drule_tac x = "inverse y" in spec, safe, force)
paulson@15077
  2134
apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
paulson@15229
  2135
apply (rule_tac x = "(pi/2) - e" in exI)
paulson@15077
  2136
apply (simp (no_asm_simp))
paulson@15229
  2137
apply (drule_tac x = "(pi/2) - e" in spec)
paulson@15229
  2138
apply (auto simp add: tan_def)
paulson@15077
  2139
apply (rule inverse_less_iff_less [THEN iffD1])
paulson@15079
  2140
apply (auto simp add: divide_inverse)
paulson@15229
  2141
apply (rule real_mult_order) 
paulson@15229
  2142
apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
paulson@15229
  2143
apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute) 
paulson@15077
  2144
done
paulson@15077
  2145
paulson@15077
  2146
lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
huffman@22998
  2147
apply (frule order_le_imp_less_or_eq, safe)
paulson@15077
  2148
 prefer 2 apply force
paulson@15077
  2149
apply (drule lemma_tan_total, safe)
paulson@15077
  2150
apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
paulson@15077
  2151
apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
paulson@15077
  2152
apply (drule_tac y = xa in order_le_imp_less_or_eq)
paulson@15077
  2153
apply (auto dest: cos_gt_zero)
paulson@15077
  2154
done
paulson@15077
  2155
paulson@15077
  2156
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  2157
apply (cut_tac linorder_linear [of 0 y], safe)
paulson@15077
  2158
apply (drule tan_total_pos)
paulson@15077
  2159
apply (cut_tac [2] y="-y" in tan_total_pos, safe)
paulson@15077
  2160
apply (rule_tac [3] x = "-x" in exI)
paulson@15077
  2161
apply (auto intro!: exI)
paulson@15077
  2162
done
paulson@15077
  2163
paulson@15077
  2164
lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  2165
apply (cut_tac y = y in lemma_tan_total1, auto)
paulson@15077
  2166
apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
paulson@15077
  2167
apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
paulson@15077
  2168
apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
paulson@15077
  2169
apply (rule_tac [4] Rolle)
paulson@15077
  2170
apply (rule_tac [2] Rolle)
paulson@15077
  2171
apply (auto intro!: DERIV_tan DERIV_isCont exI 
paulson@15077
  2172
            simp add: differentiable_def)
paulson@15077
  2173
txt{*Now, simulate TRYALL*}
paulson@15077
  2174
apply (rule_tac [!] DERIV_tan asm_rl)
paulson@15077
  2175
apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
wenzelm@32960
  2176
            simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym]) 
paulson@15077
  2177
done
paulson@15077
  2178
hoelzl@29803
  2179
lemma tan_monotone: assumes "- (pi / 2) < y" and "y < x" and "x < pi / 2"
hoelzl@29803
  2180
  shows "tan y < tan x"
hoelzl@29803
  2181
proof -
hoelzl@29803
  2182
  have "\<forall> x'. y \<le> x' \<and> x' \<le> x \<longrightarrow> DERIV tan x' :> inverse (cos x'^2)"
hoelzl@29803
  2183
  proof (rule allI, rule impI)
hoelzl@29803
  2184
    fix x' :: real assume "y \<le> x' \<and> x' \<le> x"
wenzelm@33549
  2185
    hence "-(pi/2) < x'" and "x' < pi/2" using assms by auto
hoelzl@29803
  2186
    from cos_gt_zero_pi[OF this]
hoelzl@29803
  2187
    have "cos x' \<noteq> 0" by auto
hoelzl@29803
  2188
    thus "DERIV tan x' :> inverse (cos x'^2)" by (rule DERIV_tan)
hoelzl@29803
  2189
  qed
hoelzl@29803
  2190
  from MVT2[OF `y < x` this] 
hoelzl@29803
  2191
  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
  2192
  hence "- (pi / 2) < z" and "z < pi / 2" using assms by auto
hoelzl@29803
  2193
  hence "0 < cos z" using cos_gt_zero_pi by auto
hoelzl@29803
  2194
  hence inv_pos: "0 < inverse ((cos z)\<twosuperior>)" by auto
hoelzl@29803
  2195
  have "0 < x - y" using `y < x` by auto
hoelzl@29803
  2196
  from real_mult_order[OF this inv_pos]
hoelzl@29803
  2197
  have "0 < tan x - tan y" unfolding tan_diff by auto
hoelzl@29803
  2198
  thus ?thesis by auto
hoelzl@29803
  2199
qed
hoelzl@29803
  2200
hoelzl@29803
  2201
lemma tan_monotone': assumes "- (pi / 2) < y" and "y < pi / 2" and "- (pi / 2) < x" and "x < pi / 2"
hoelzl@29803
  2202
  shows "(y < x) = (tan y < tan x)"
hoelzl@29803
  2203
proof
hoelzl@29803
  2204
  assume "y < x" thus "tan y < tan x" using tan_monotone and `- (pi / 2) < y` and `x < pi / 2` by auto
hoelzl@29803
  2205
next
hoelzl@29803
  2206
  assume "tan y < tan x"
hoelzl@29803
  2207
  show "y < x"
hoelzl@29803
  2208
  proof (rule ccontr)
hoelzl@29803
  2209
    assume "\<not> y < x" hence "x \<le> y" by auto
hoelzl@29803
  2210
    hence "tan x \<le> tan y" 
hoelzl@29803
  2211
    proof (cases "x = y")
hoelzl@29803
  2212
      case True thus ?thesis by auto
hoelzl@29803
  2213
    next
hoelzl@29803
  2214
      case False hence "x < y" using `x \<le> y` by auto
hoelzl@29803
  2215
      from tan_monotone[OF `- (pi/2) < x` this `y < pi / 2`] show ?thesis by auto
hoelzl@29803
  2216
    qed
hoelzl@29803
  2217
    thus False using `tan y < tan x` by auto
hoelzl@29803
  2218
  qed
hoelzl@29803
  2219
qed
hoelzl@29803
  2220
hoelzl@29803
  2221
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
  2222
hoelzl@29803
  2223
lemma tan_periodic_pi[simp]: "tan (x + pi) = tan x" 
hoelzl@29803
  2224
  by (simp add: tan_def)
hoelzl@29803
  2225
hoelzl@29803
  2226
lemma tan_periodic_nat[simp]: fixes n :: nat shows "tan (x + real n * pi) = tan x" 
hoelzl@29803
  2227
proof (induct n arbitrary: x)
hoelzl@29803
  2228
  case (Suc n)
nipkow@31790
  2229
  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 real_add_mult_distrib by auto
hoelzl@29803
  2230
  show ?case unfolding split_pi_off using Suc by auto
hoelzl@29803
  2231
qed auto
hoelzl@29803
  2232
hoelzl@29803
  2233
lemma tan_periodic_int[simp]: fixes i :: int shows "tan (x + real i * pi) = tan x"
hoelzl@29803
  2234
proof (cases "0 \<le> i")
hoelzl@29803
  2235
  case True hence i_nat: "real i = real (nat i)" by auto
hoelzl@29803
  2236
  show ?thesis unfolding i_nat by auto
hoelzl@29803
  2237
next
hoelzl@29803
  2238
  case False hence i_nat: "real i = - real (nat (-i))" by auto
hoelzl@29803
  2239
  have "tan x = tan (x + real i * pi - real i * pi)" by auto
hoelzl@29803
  2240
  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
  2241
  finally show ?thesis by auto
hoelzl@29803
  2242
qed
hoelzl@29803
  2243
hoelzl@29803
  2244
lemma tan_periodic_n[simp]: "tan (x + number_of n * pi) = tan x"
hoelzl@29803
  2245
  using tan_periodic_int[of _ "number_of n" ] unfolding real_number_of .
huffman@23043
  2246
huffman@23043
  2247
subsection {* Inverse Trigonometric Functions *}
huffman@23043
  2248
huffman@23043
  2249
definition
huffman@23043
  2250
  arcsin :: "real => real" where
huffman@23043
  2251
  "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
huffman@23043
  2252
huffman@23043
  2253
definition
huffman@23043
  2254
  arccos :: "real => real" where
huffman@23043
  2255
  "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
huffman@23043
  2256
huffman@23043
  2257
definition     
huffman@23043
  2258
  arctan :: "real => real" where
huffman@23043
  2259
  "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
huffman@23043
  2260
paulson@15229
  2261
lemma arcsin:
paulson@15229
  2262
     "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077
  2263
      ==> -(pi/2) \<le> arcsin y &  
paulson@15077
  2264
           arcsin y \<le> pi/2 & sin(arcsin y) = y"
huffman@23011
  2265
unfolding arcsin_def by (rule theI' [OF sin_total])
huffman@23011
  2266
huffman@23011
  2267
lemma arcsin_pi:
huffman@23011
  2268
     "[| -1 \<le> y; y \<le> 1 |]  
huffman@23011
  2269
      ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
huffman@23011
  2270
apply (drule (1) arcsin)
huffman@23011
  2271
apply (force intro: order_trans)
paulson@15077
  2272
done
paulson@15077
  2273
paulson@15077
  2274
lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
paulson@15077
  2275
by (blast dest: arcsin)
paulson@15077
  2276
      
paulson@15077
  2277
lemma arcsin_bounded:
paulson@15077
  2278
     "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
paulson@15077
  2279
by (blast dest: arcsin)
paulson@15077
  2280
paulson@15077
  2281
lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
paulson@15077
  2282
by (blast dest: arcsin)
paulson@15077
  2283
paulson@15077
  2284
lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
paulson@15077
  2285
by (blast dest: arcsin)
paulson@15077
  2286
paulson@15077
  2287
lemma arcsin_lt_bounded:
paulson@15077
  2288
     "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
paulson@15077
  2289
apply (frule order_less_imp_le)
paulson@15077
  2290
apply (frule_tac y = y in order_less_imp_le)
paulson@15077
  2291
apply (frule arcsin_bounded)
paulson@15077
  2292
apply (safe, simp)
paulson@15077
  2293
apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
paulson@15077
  2294
apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
paulson@15077
  2295
apply (drule_tac [!] f = sin in arg_cong, auto)
paulson@15077
  2296
done
paulson@15077
  2297
paulson@15077
  2298
lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
paulson@15077
  2299
apply (unfold arcsin_def)
huffman@23011
  2300
apply (rule the1_equality)
paulson@15077
  2301
apply (rule sin_total, auto)
paulson@15077
  2302
done
paulson@15077
  2303
huffman@22975
  2304
lemma arccos:
paulson@15229
  2305
     "[| -1 \<le> y; y \<le> 1 |]  
huffman@22975
  2306
      ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
huffman@23011
  2307
unfolding arccos_def by (rule theI' [OF cos_total])
paulson@15077
  2308
huffman@22975
  2309
lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
huffman@22975
  2310
by (blast dest: arccos)
paulson@15077
  2311
      
huffman@22975
  2312
lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
huffman@22975
  2313
by (blast dest: arccos)
paulson@15077
  2314
huffman@22975
  2315
lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
huffman@22975
  2316
by (blast dest: arccos)
paulson@15077
  2317
huffman@22975
  2318
lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
huffman@22975
  2319
by (blast dest: arccos)
paulson@15077
  2320
huffman@22975
  2321
lemma arccos_lt_bounded:
paulson@15229
  2322
     "[| -1 < y; y < 1 |]  
huffman@22975
  2323
      ==> 0 < arccos y & arccos y < pi"
paulson@15077
  2324
apply (frule order_less_imp_le)
paulson@15077
  2325
apply (frule_tac y = y in order_less_imp_le)
huffman@22975
  2326
apply (frule arccos_bounded, auto)
huffman@22975
  2327
apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
paulson@15077
  2328
apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
paulson@15077
  2329
apply (drule_tac [!] f = cos in arg_cong, auto)
paulson@15077
  2330
done
paulson@15077
  2331
huffman@22975
  2332
lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
huffman@22975
  2333
apply (simp add: arccos_def)
huffman@23011
  2334
apply (auto intro!: the1_equality cos_total)
paulson@15077
  2335
done
paulson@15077
  2336
huffman@22975
  2337
lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
huffman@22975
  2338
apply (simp add: arccos_def)
huffman@23011
  2339
apply (auto intro!: the1_equality cos_total)
paulson@15077
  2340
done
paulson@15077
  2341
huffman@23045
  2342
lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<twosuperior>)"
huffman@23045
  2343
apply (subgoal_tac "x\<twosuperior> \<le> 1")
huffman@23052
  2344
apply (rule power2_eq_imp_eq)
huffman@23045
  2345
apply (simp add: cos_squared_eq)
huffman@23045
  2346
apply (rule cos_ge_zero)
huffman@23045
  2347
apply (erule (1) arcsin_lbound)
huffman@23045
  2348
apply (erule (1) arcsin_ubound)
huffman@23045
  2349
apply simp
huffman@23045
  2350
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
huffman@23045
  2351
apply (rule power_mono, simp, simp)
huffman@23045
  2352
done
huffman@23045
  2353
huffman@23045
  2354
lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<twosuperior>)"
huffman@23045
  2355
apply (subgoal_tac "x\<twosuperior> \<le> 1")
huffman@23052
  2356
apply (rule power2_eq_imp_eq)
huffman@23045
  2357
apply (simp add: sin_squared_eq)
huffman@23045
  2358
apply (rule sin_ge_zero)
huffman@23045
  2359
apply (erule (1) arccos_lbound)
huffman@23045
  2360
apply (erule (1) arccos_ubound)
huffman@23045
  2361
apply simp
huffman@23045
  2362
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
huffman@23045
  2363
apply (rule power_mono, simp, simp)
huffman@23045
  2364
done
huffman@23045
  2365
paulson@15077
  2366
lemma arctan [simp]:
paulson@15077
  2367
     "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
huffman@23011
  2368
unfolding arctan_def by (rule theI' [OF tan_total])
paulson@15077
  2369
paulson@15077
  2370
lemma tan_arctan: "tan(arctan y) = y"
paulson@15077
  2371
by auto
paulson@15077
  2372
paulson@15077
  2373
lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
paulson@15077
  2374
by (auto simp only: arctan)
paulson@15077
  2375
paulson@15077
  2376
lemma arctan_lbound: "- (pi/2) < arctan y"
paulson@15077
  2377
by auto
paulson@15077
  2378
paulson@15077
  2379
lemma arctan_ubound: "arctan y < pi/2"
paulson@15077
  2380
by (auto simp only: arctan)
paulson@15077
  2381
paulson@15077
  2382
lemma arctan_tan: 
paulson@15077
  2383
      "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
paulson@15077
  2384
apply (unfold arctan_def)
huffman@23011
  2385
apply (rule the1_equality)
paulson@15077
  2386
apply (rule tan_total, auto)
paulson@15077
  2387
done
paulson@15077
  2388
paulson@15077
  2389
lemma arctan_zero_zero [sim