src/HOL/Transcendental.thy
author blanchet
Wed Feb 12 08:37:06 2014 +0100 (2014-02-12)
changeset 55417 01fbfb60c33e
parent 54576 e877eec2b698
child 55719 cdddd073bff8
permissions -rw-r--r--
adapted to 'xxx_{case,rec}' renaming, to new theorem names, and to new variable names in theorems
* * *
more transition of 'xxx_rec' to 'rec_xxx' and same for case
* * *
compile
* * *
'rename_tac's to avoid referring to generated names
* * *
more robust scripts with 'rename_tac'
* * *
'where' -> 'of'
* * *
'where' -> 'of'
* * *
renamed 'xxx_rec' to 'rec_xxx'
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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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    Author:     Jeremy Avigad
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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}"
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  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 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}"
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  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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proof (induct n)
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  case 0 show ?case
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    by simp
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next
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  case (Suc n)
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  have "x ^ Suc (Suc n) - y ^ Suc (Suc n) = x * (x * x ^ n) - y * (y * y ^ n)"
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    by simp
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  also have "... = y * (x ^ (Suc n) - y ^ (Suc n)) + (x - y) * (x * x ^ n)"
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    by (simp add: algebra_simps)
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  also have "... = y * ((x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
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    by (simp only: Suc)
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  also have "... = (x - y) * (y * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))) + (x - y) * (x * x ^ n)"
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    by (simp only: mult_left_commute)
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  also have "... = (x - y) * (\<Sum>p = 0..<Suc (Suc n). x ^ p * y ^ (Suc n - p))"
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    by (simp add: setsum_op_ivl_Suc [where n = "Suc n"] distrib_left lemma_realpow_diff_sumr
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             del: setsum_op_ivl_Suc)
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  finally show ?case .
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qed
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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, auto)
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  apply (metis atLeastLessThan_iff diff_diff_cancel diff_less_Suc imageI le0 less_Suc_eq_le)
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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_div_algebra"
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  assumes 1: "summable (\<lambda>n. f n * x ^ n)"
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    and 2: "norm z < norm x"
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  shows "summable (\<lambda>n. norm (f n * z ^ n))"
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proof -
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  from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
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  from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
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    by (rule summable_LIMSEQ_zero)
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  hence "convergent (\<lambda>n. f n * x ^ n)"
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    by (rule convergentI)
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  hence "Cauchy (\<lambda>n. f n * x ^ n)"
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    by (rule convergent_Cauchy)
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  hence "Bseq (\<lambda>n. f n * x ^ n)"
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    by (rule Cauchy_Bseq)
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  then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
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    by (simp add: Bseq_def, safe)
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  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
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                   K * norm (z ^ n) * inverse (norm (x ^ n))"
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  proof (intro exI allI impI)
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    fix n::nat
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    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: norm_mult nonzero_norm_inverse divide_inverse [where 'a=real, 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_div_algebra,banach}"
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  shows
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    "summable (\<lambda>n. f n * (x ^ n)) \<Longrightarrow> norm z < norm x \<Longrightarrow>
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      summable (\<lambda>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:
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  fixes f :: "nat \<Rightarrow> real"
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  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 0
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  then show ?case by simp
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next
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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))"
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    by auto
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  finally show ?case .
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qed
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lemma sums_if':
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  fixes g :: "nat \<Rightarrow> real"
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  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
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  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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  {
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    fix m
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    assume "m \<ge> 2 * no"
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    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)"
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      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
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      show ?thesis
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        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
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      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]]
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        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:
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  fixes g :: "nat \<Rightarrow> real"
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  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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  {
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    fix B T E
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    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
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  } 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"
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    using sums_if'[OF `g sums x`] .
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  {
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    have "?s 0 = 0" by auto
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    have Suc_m1: "\<And> n. Suc n - 1 = n" by auto
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    have if_eq: "\<And>B T E. (if \<not> B then T else E) = (if B then E else T)" by auto
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    have "?s sums y" using sums_if'[OF `f sums y`] .
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    from this[unfolded sums_def, THEN LIMSEQ_Suc]
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    have "(\<lambda> n. if even n then f (n div 2) else 0) sums y"
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      unfolding sums_def setsum_shift_lb_Suc0_0_upt[where f="?s", OF `?s 0 = 0`, symmetric]
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                image_Suc_atLeastLessThan[symmetric] setsum_reindex[OF inj_Suc, unfolded comp_def]
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                even_Suc Suc_m1 if_eq .
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  }
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  from sums_add[OF g_sums this] 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 (rule nested_sequence_unique)
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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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  show "\<forall>n. ?f n \<le> ?f (Suc n)"
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  proof
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    fix n
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    show "?f n \<le> ?f (Suc n)" using mono[of "2*n"] by auto
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  qed
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  show "\<forall>n. ?g (Suc n) \<le> ?g n"
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  proof
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    fix n
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    show "?g (Suc n) \<le> ?g n" using mono[of "Suc (2*n)"]
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      unfolding One_nat_def by auto
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  qed
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  show "\<forall>n. ?f n \<le> ?g n"
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  proof
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    fix n
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    show "?f n \<le> ?g n" using fg_diff a_pos
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      unfolding One_nat_def by auto
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  qed
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  show "(\<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
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    assume "0 < r"
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    with `a ----> 0`[THEN LIMSEQ_D] obtain N where "\<And> n. n \<ge> N \<Longrightarrow> norm (a n - 0) < r"
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      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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qed
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lemma summable_Leibniz':
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  fixes a :: "nat \<Rightarrow> real"
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  assumes a_zero: "a ----> 0"
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    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 = "\<lambda>n. (-1)^n * a n"
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  let ?P = "\<lambda>n. \<Sum>i=0..<n. ?S i"
wenzelm@53079
   266
  let ?f = "\<lambda>n. ?P (2 * n)"
wenzelm@53079
   267
  let ?g = "\<lambda>n. ?P (2 * n + 1)"
wenzelm@53079
   268
  obtain l :: real
wenzelm@53079
   269
    where below_l: "\<forall> n. ?f n \<le> l"
wenzelm@53079
   270
      and "?f ----> l"
wenzelm@53079
   271
      and above_l: "\<forall> n. l \<le> ?g n"
wenzelm@53079
   272
      and "?g ----> l"
hoelzl@29803
   273
    using sums_alternating_upper_lower[OF a_monotone a_pos a_zero] by blast
hoelzl@41970
   274
wenzelm@53079
   275
  let ?Sa = "\<lambda>m. \<Sum> n = 0..<m. ?S n"
hoelzl@29803
   276
  have "?Sa ----> l"
hoelzl@29803
   277
  proof (rule LIMSEQ_I)
wenzelm@53079
   278
    fix r :: real
wenzelm@53079
   279
    assume "0 < r"
hoelzl@41970
   280
    with `?f ----> l`[THEN LIMSEQ_D]
hoelzl@29803
   281
    obtain f_no where f: "\<And> n. n \<ge> f_no \<Longrightarrow> norm (?f n - l) < r" by auto
hoelzl@29803
   282
hoelzl@41970
   283
    from `0 < r` `?g ----> l`[THEN LIMSEQ_D]
hoelzl@29803
   284
    obtain g_no where g: "\<And> n. n \<ge> g_no \<Longrightarrow> norm (?g n - l) < r" by auto
hoelzl@29803
   285
wenzelm@53079
   286
    {
wenzelm@53079
   287
      fix n :: nat
wenzelm@53079
   288
      assume "n \<ge> (max (2 * f_no) (2 * g_no))"
wenzelm@53079
   289
      hence "n \<ge> 2 * f_no" and "n \<ge> 2 * g_no" by auto
hoelzl@29803
   290
      have "norm (?Sa n - l) < r"
hoelzl@29803
   291
      proof (cases "even n")
wenzelm@53079
   292
        case True
wenzelm@53079
   293
        from even_nat_div_two_times_two[OF this]
wenzelm@53079
   294
        have n_eq: "2 * (n div 2) = n"
wenzelm@53079
   295
          unfolding numeral_2_eq_2[symmetric] by auto
wenzelm@53079
   296
        with `n \<ge> 2 * f_no` have "n div 2 \<ge> f_no"
wenzelm@53079
   297
          by auto
wenzelm@53079
   298
        from f[OF this] show ?thesis
wenzelm@53079
   299
          unfolding n_eq atLeastLessThanSuc_atLeastAtMost .
hoelzl@29803
   300
      next
wenzelm@53079
   301
        case False
wenzelm@53079
   302
        hence "even (n - 1)" by simp
wenzelm@32960
   303
        from even_nat_div_two_times_two[OF this]
wenzelm@53079
   304
        have n_eq: "2 * ((n - 1) div 2) = n - 1"
wenzelm@53079
   305
          unfolding numeral_2_eq_2[symmetric] by auto
wenzelm@53079
   306
        hence range_eq: "n - 1 + 1 = n"
wenzelm@53079
   307
          using odd_pos[OF False] by auto
wenzelm@53079
   308
wenzelm@53079
   309
        from n_eq `n \<ge> 2 * g_no` have "(n - 1) div 2 \<ge> g_no"
wenzelm@53079
   310
          by auto
wenzelm@53079
   311
        from g[OF this] show ?thesis
wenzelm@53079
   312
          unfolding n_eq atLeastLessThanSuc_atLeastAtMost range_eq .
hoelzl@29803
   313
      qed
hoelzl@29803
   314
    }
wenzelm@53079
   315
    thus "\<exists>no. \<forall>n \<ge> no. norm (?Sa n - l) < r" by blast
hoelzl@29803
   316
  qed
wenzelm@53079
   317
  hence sums_l: "(\<lambda>i. (-1)^i * a i) sums l"
wenzelm@53079
   318
    unfolding sums_def atLeastLessThanSuc_atLeastAtMost[symmetric] .
hoelzl@29803
   319
  thus "summable ?S" using summable_def by auto
hoelzl@29803
   320
hoelzl@29803
   321
  have "l = suminf ?S" using sums_unique[OF sums_l] .
hoelzl@29803
   322
wenzelm@53079
   323
  fix n
wenzelm@53079
   324
  show "suminf ?S \<le> ?g n"
wenzelm@53079
   325
    unfolding sums_unique[OF sums_l, symmetric] using above_l by auto
wenzelm@53079
   326
  show "?f n \<le> suminf ?S"
wenzelm@53079
   327
    unfolding sums_unique[OF sums_l, symmetric] using below_l by auto
wenzelm@53079
   328
  show "?g ----> suminf ?S"
wenzelm@53079
   329
    using `?g ----> l` `l = suminf ?S` by auto
wenzelm@53079
   330
  show "?f ----> suminf ?S"
wenzelm@53079
   331
    using `?f ----> l` `l = suminf ?S` by auto
hoelzl@29803
   332
qed
hoelzl@29803
   333
wenzelm@53079
   334
theorem summable_Leibniz:
wenzelm@53079
   335
  fixes a :: "nat \<Rightarrow> real"
hoelzl@29803
   336
  assumes a_zero: "a ----> 0" and "monoseq a"
hoelzl@29803
   337
  shows "summable (\<lambda> n. (-1)^n * a n)" (is "?summable")
wenzelm@53079
   338
    and "0 < a 0 \<longrightarrow>
wenzelm@53079
   339
      (\<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")
wenzelm@53079
   340
    and "a 0 < 0 \<longrightarrow>
wenzelm@53079
   341
      (\<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")
wenzelm@53079
   342
    and "(\<lambda>n. \<Sum>i=0..<2*n. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?f")
wenzelm@53079
   343
    and "(\<lambda>n. \<Sum>i=0..<2*n+1. -1^i*a i) ----> (\<Sum>i. -1^i*a i)" (is "?g")
hoelzl@29803
   344
proof -
hoelzl@29803
   345
  have "?summable \<and> ?pos \<and> ?neg \<and> ?f \<and> ?g"
hoelzl@29803
   346
  proof (cases "(\<forall> n. 0 \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m)")
hoelzl@29803
   347
    case True
wenzelm@53079
   348
    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> a n \<le> a m" and ge0: "\<And> n. 0 \<le> a n"
wenzelm@53079
   349
      by auto
wenzelm@53079
   350
    {
wenzelm@53079
   351
      fix n
wenzelm@53079
   352
      have "a (Suc n) \<le> a n"
wenzelm@53079
   353
        using ord[where n="Suc n" and m=n] by auto
wenzelm@53079
   354
    } note mono = this
wenzelm@53079
   355
    note leibniz = summable_Leibniz'[OF `a ----> 0` ge0]
hoelzl@29803
   356
    from leibniz[OF mono]
hoelzl@29803
   357
    show ?thesis using `0 \<le> a 0` by auto
hoelzl@29803
   358
  next
hoelzl@29803
   359
    let ?a = "\<lambda> n. - a n"
hoelzl@29803
   360
    case False
hoelzl@29803
   361
    with monoseq_le[OF `monoseq a` `a ----> 0`]
hoelzl@29803
   362
    have "(\<forall> n. a n \<le> 0) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)" by auto
wenzelm@53079
   363
    hence ord: "\<And>n m. m \<le> n \<Longrightarrow> ?a n \<le> ?a m" and ge0: "\<And> n. 0 \<le> ?a n"
wenzelm@53079
   364
      by auto
wenzelm@53079
   365
    {
wenzelm@53079
   366
      fix n
wenzelm@53079
   367
      have "?a (Suc n) \<le> ?a n" using ord[where n="Suc n" and m=n]
wenzelm@53079
   368
        by auto
wenzelm@53079
   369
    } note monotone = this
wenzelm@53079
   370
    note leibniz =
wenzelm@53079
   371
      summable_Leibniz'[OF _ ge0, of "\<lambda>x. x",
wenzelm@53079
   372
        OF tendsto_minus[OF `a ----> 0`, unfolded minus_zero] monotone]
wenzelm@53079
   373
    have "summable (\<lambda> n. (-1)^n * ?a n)"
wenzelm@53079
   374
      using leibniz(1) by auto
wenzelm@53079
   375
    then obtain l where "(\<lambda> n. (-1)^n * ?a n) sums l"
wenzelm@53079
   376
      unfolding summable_def by auto
wenzelm@53079
   377
    from this[THEN sums_minus] have "(\<lambda> n. (-1)^n * a n) sums -l"
wenzelm@53079
   378
      by auto
hoelzl@29803
   379
    hence ?summable unfolding summable_def by auto
hoelzl@29803
   380
    moreover
wenzelm@53079
   381
    have "\<And>a b :: real. \<bar>- a - - b\<bar> = \<bar>a - b\<bar>"
wenzelm@53079
   382
      unfolding minus_diff_minus by auto
hoelzl@41970
   383
hoelzl@29803
   384
    from suminf_minus[OF leibniz(1), unfolded mult_minus_right minus_minus]
wenzelm@53079
   385
    have move_minus: "(\<Sum>n. - (-1 ^ n * a n)) = - (\<Sum>n. -1 ^ n * a n)"
wenzelm@53079
   386
      by auto
hoelzl@29803
   387
hoelzl@29803
   388
    have ?pos using `0 \<le> ?a 0` by auto
wenzelm@53079
   389
    moreover have ?neg
wenzelm@53079
   390
      using leibniz(2,4)
wenzelm@53079
   391
      unfolding mult_minus_right setsum_negf move_minus neg_le_iff_le
wenzelm@53079
   392
      by auto
wenzelm@53079
   393
    moreover have ?f and ?g
wenzelm@53079
   394
      using leibniz(3,5)[unfolded mult_minus_right setsum_negf move_minus, THEN tendsto_minus_cancel]
wenzelm@53079
   395
      by auto
hoelzl@29803
   396
    ultimately show ?thesis by auto
hoelzl@29803
   397
  qed
paulson@54576
   398
  then show ?summable and ?pos and ?neg and ?f and ?g 
paulson@54573
   399
    by safe
hoelzl@29803
   400
qed
paulson@15077
   401
huffman@29164
   402
subsection {* Term-by-Term Differentiability of Power Series *}
huffman@23043
   403
wenzelm@53079
   404
definition diffs :: "(nat => 'a::ring_1) => nat => 'a"
wenzelm@53079
   405
  where "diffs c = (\<lambda>n. of_nat (Suc n) * c(Suc n))"
paulson@15077
   406
paulson@15077
   407
text{*Lemma about distributing negation over it*}
wenzelm@53079
   408
lemma diffs_minus: "diffs (\<lambda>n. - c n) = (\<lambda>n. - diffs c n)"
wenzelm@53079
   409
  by (simp add: diffs_def)
paulson@15077
   410
huffman@29163
   411
lemma sums_Suc_imp:
huffman@29163
   412
  assumes f: "f 0 = 0"
huffman@29163
   413
  shows "(\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
wenzelm@53079
   414
  unfolding sums_def
wenzelm@53079
   415
  apply (rule LIMSEQ_imp_Suc)
wenzelm@53079
   416
  apply (subst setsum_shift_lb_Suc0_0_upt [where f=f, OF f, symmetric])
wenzelm@53079
   417
  apply (simp only: setsum_shift_bounds_Suc_ivl)
wenzelm@53079
   418
  done
paulson@15077
   419
paulson@15229
   420
lemma diffs_equiv:
hoelzl@41970
   421
  fixes x :: "'a::{real_normed_vector, ring_1}"
wenzelm@53079
   422
  shows "summable (\<lambda>n. (diffs c)(n) * (x ^ n)) \<Longrightarrow>
wenzelm@53079
   423
      (\<lambda>n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums
nipkow@15546
   424
         (\<Sum>n. (diffs c)(n) * (x ^ n))"
wenzelm@53079
   425
  unfolding diffs_def
paulson@54573
   426
  by (simp add: summable_sums sums_Suc_imp)
paulson@15077
   427
paulson@15077
   428
lemma lemma_termdiff1:
haftmann@31017
   429
  fixes z :: "'a :: {monoid_mult,comm_ring}" shows
hoelzl@41970
   430
  "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =
huffman@23082
   431
   (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
wenzelm@53079
   432
  by (auto simp add: algebra_simps power_add [symmetric])
paulson@15077
   433
huffman@23082
   434
lemma sumr_diff_mult_const2:
huffman@23082
   435
  "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
wenzelm@53079
   436
  by (simp add: setsum_subtractf)
huffman@23082
   437
paulson@15229
   438
lemma lemma_termdiff2:
haftmann@31017
   439
  fixes h :: "'a :: {field}"
wenzelm@53079
   440
  assumes h: "h \<noteq> 0"
wenzelm@53079
   441
  shows
wenzelm@53079
   442
    "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
wenzelm@53079
   443
     h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
wenzelm@53079
   444
          (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
wenzelm@53079
   445
  apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
wenzelm@53079
   446
  apply (simp add: right_diff_distrib diff_divide_distrib h)
wenzelm@53079
   447
  apply (simp add: mult_assoc [symmetric])
wenzelm@53079
   448
  apply (cases "n", simp)
wenzelm@53079
   449
  apply (simp add: lemma_realpow_diff_sumr2 h
wenzelm@53079
   450
                   right_diff_distrib [symmetric] mult_assoc
wenzelm@53079
   451
              del: power_Suc setsum_op_ivl_Suc of_nat_Suc)
wenzelm@53079
   452
  apply (subst lemma_realpow_rev_sumr)
wenzelm@53079
   453
  apply (subst sumr_diff_mult_const2)
wenzelm@53079
   454
  apply simp
wenzelm@53079
   455
  apply (simp only: lemma_termdiff1 setsum_right_distrib)
wenzelm@53079
   456
  apply (rule setsum_cong [OF refl])
haftmann@54230
   457
  apply (simp add: less_iff_Suc_add)
wenzelm@53079
   458
  apply (clarify)
wenzelm@53079
   459
  apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
wenzelm@53079
   460
              del: setsum_op_ivl_Suc power_Suc)
wenzelm@53079
   461
  apply (subst mult_assoc [symmetric], subst power_add [symmetric])
wenzelm@53079
   462
  apply (simp add: mult_ac)
wenzelm@53079
   463
  done
huffman@20860
   464
huffman@20860
   465
lemma real_setsum_nat_ivl_bounded2:
haftmann@35028
   466
  fixes K :: "'a::linordered_semidom"
huffman@23082
   467
  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
wenzelm@53079
   468
    and K: "0 \<le> K"
huffman@23082
   469
  shows "setsum f {0..<n-k} \<le> of_nat n * K"
wenzelm@53079
   470
  apply (rule order_trans [OF setsum_mono])
wenzelm@53079
   471
  apply (rule f, simp)
wenzelm@53079
   472
  apply (simp add: mult_right_mono K)
wenzelm@53079
   473
  done
paulson@15077
   474
paulson@15229
   475
lemma lemma_termdiff3:
haftmann@31017
   476
  fixes h z :: "'a::{real_normed_field}"
huffman@20860
   477
  assumes 1: "h \<noteq> 0"
wenzelm@53079
   478
    and 2: "norm z \<le> K"
wenzelm@53079
   479
    and 3: "norm (z + h) \<le> K"
huffman@23082
   480
  shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
huffman@23082
   481
          \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
huffman@20860
   482
proof -
huffman@23082
   483
  have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
huffman@23082
   484
        norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
huffman@23082
   485
          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
paulson@54573
   486
    by (metis (lifting, no_types) lemma_termdiff2 [OF 1] mult_commute norm_mult)
huffman@23082
   487
  also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
huffman@23082
   488
  proof (rule mult_right_mono [OF _ norm_ge_zero])
wenzelm@53079
   489
    from norm_ge_zero 2 have K: "0 \<le> K"
wenzelm@53079
   490
      by (rule order_trans)
huffman@23082
   491
    have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
huffman@20860
   492
      apply (erule subst)
huffman@23082
   493
      apply (simp only: norm_mult norm_power power_add)
huffman@23082
   494
      apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
huffman@20860
   495
      done
wenzelm@53079
   496
    show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p. (z + h) ^ q * z ^ (n - 2 - q))
huffman@23082
   497
          \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
huffman@20860
   498
      apply (intro
huffman@23082
   499
         order_trans [OF norm_setsum]
huffman@20860
   500
         real_setsum_nat_ivl_bounded2
huffman@20860
   501
         mult_nonneg_nonneg
huffman@47489
   502
         of_nat_0_le_iff
huffman@20860
   503
         zero_le_power K)
huffman@20860
   504
      apply (rule le_Kn, simp)
huffman@20860
   505
      done
huffman@20860
   506
  qed
huffman@23082
   507
  also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
huffman@20860
   508
    by (simp only: mult_assoc)
huffman@20860
   509
  finally show ?thesis .
huffman@20860
   510
qed
paulson@15077
   511
huffman@20860
   512
lemma lemma_termdiff4:
haftmann@31017
   513
  fixes f :: "'a::{real_normed_field} \<Rightarrow>
huffman@23082
   514
              'b::real_normed_vector"
huffman@20860
   515
  assumes k: "0 < (k::real)"
wenzelm@53079
   516
    and le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
huffman@20860
   517
  shows "f -- 0 --> 0"
wenzelm@53079
   518
  unfolding LIM_eq diff_0_right
wenzelm@53079
   519
proof safe
huffman@29163
   520
  let ?h = "of_real (k / 2)::'a"
huffman@29163
   521
  have "?h \<noteq> 0" and "norm ?h < k" using k by simp_all
huffman@29163
   522
  hence "norm (f ?h) \<le> K * norm ?h" by (rule le)
huffman@29163
   523
  hence "0 \<le> K * norm ?h" by (rule order_trans [OF norm_ge_zero])
huffman@29163
   524
  hence zero_le_K: "0 \<le> K" using k by (simp add: zero_le_mult_iff)
huffman@29163
   525
wenzelm@53079
   526
  fix r::real
wenzelm@53079
   527
  assume r: "0 < r"
huffman@23082
   528
  show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
wenzelm@53079
   529
  proof cases
huffman@20860
   530
    assume "K = 0"
huffman@23082
   531
    with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
huffman@20860
   532
      by simp
huffman@23082
   533
    thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
huffman@20860
   534
  next
huffman@20860
   535
    assume K_neq_zero: "K \<noteq> 0"
huffman@20860
   536
    with zero_le_K have K: "0 < K" by simp
huffman@23082
   537
    show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
huffman@20860
   538
    proof (rule exI, safe)
wenzelm@53079
   539
      from k r K
wenzelm@53079
   540
      show "0 < min k (r * inverse K / 2)"
huffman@20860
   541
        by (simp add: mult_pos_pos positive_imp_inverse_positive)
huffman@20860
   542
    next
huffman@23082
   543
      fix x::'a
huffman@23082
   544
      assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
huffman@23082
   545
      from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
huffman@20860
   546
        by simp_all
huffman@23082
   547
      from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
huffman@23082
   548
      also from x4 K have "K * norm x < K * (r * inverse K / 2)"
huffman@20860
   549
        by (rule mult_strict_left_mono)
huffman@20860
   550
      also have "\<dots> = r / 2"
huffman@20860
   551
        using K_neq_zero by simp
huffman@20860
   552
      also have "r / 2 < r"
huffman@20860
   553
        using r by simp
huffman@23082
   554
      finally show "norm (f x) < r" .
huffman@20860
   555
    qed
huffman@20860
   556
  qed
huffman@20860
   557
qed
paulson@15077
   558
paulson@15229
   559
lemma lemma_termdiff5:
wenzelm@53079
   560
  fixes g :: "'a::real_normed_field \<Rightarrow> nat \<Rightarrow> 'b::banach"
huffman@20860
   561
  assumes k: "0 < (k::real)"
huffman@20860
   562
  assumes f: "summable f"
huffman@23082
   563
  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
   564
  shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
huffman@20860
   565
proof (rule lemma_termdiff4 [OF k])
wenzelm@53079
   566
  fix h::'a
wenzelm@53079
   567
  assume "h \<noteq> 0" and "norm h < k"
huffman@23082
   568
  hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
huffman@20860
   569
    by (simp add: le)
huffman@23082
   570
  hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
huffman@20860
   571
    by simp
huffman@23082
   572
  moreover from f have B: "summable (\<lambda>n. f n * norm h)"
huffman@20860
   573
    by (rule summable_mult2)
huffman@23082
   574
  ultimately have C: "summable (\<lambda>n. norm (g h n))"
huffman@20860
   575
    by (rule summable_comparison_test)
huffman@23082
   576
  hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
huffman@23082
   577
    by (rule summable_norm)
huffman@23082
   578
  also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
huffman@20860
   579
    by (rule summable_le)
huffman@23082
   580
  also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
huffman@20860
   581
    by (rule suminf_mult2 [symmetric])
huffman@23082
   582
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
huffman@20860
   583
qed
paulson@15077
   584
paulson@15077
   585
paulson@15077
   586
text{* FIXME: Long proofs*}
paulson@15077
   587
paulson@15077
   588
lemma termdiffs_aux:
haftmann@31017
   589
  fixes x :: "'a::{real_normed_field,banach}"
huffman@20849
   590
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
wenzelm@53079
   591
    and 2: "norm x < norm K"
huffman@20860
   592
  shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
huffman@23082
   593
             - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
huffman@20849
   594
proof -
huffman@20860
   595
  from dense [OF 2]
huffman@23082
   596
  obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
huffman@23082
   597
  from norm_ge_zero r1 have r: "0 < r"
huffman@20860
   598
    by (rule order_le_less_trans)
huffman@20860
   599
  hence r_neq_0: "r \<noteq> 0" by simp
huffman@20860
   600
  show ?thesis
huffman@20849
   601
  proof (rule lemma_termdiff5)
huffman@23082
   602
    show "0 < r - norm x" using r1 by simp
huffman@23082
   603
    from r r2 have "norm (of_real r::'a) < norm K"
huffman@23082
   604
      by simp
huffman@23082
   605
    with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
huffman@20860
   606
      by (rule powser_insidea)
huffman@23082
   607
    hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
huffman@23082
   608
      using r
huffman@23082
   609
      by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
huffman@23082
   610
    hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
huffman@20860
   611
      by (rule diffs_equiv [THEN sums_summable])
wenzelm@53079
   612
    also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0)) =
wenzelm@53079
   613
      (\<lambda>n. diffs (\<lambda>m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
huffman@20849
   614
      apply (rule ext)
huffman@20849
   615
      apply (simp add: diffs_def)
huffman@20849
   616
      apply (case_tac n, simp_all add: r_neq_0)
huffman@20849
   617
      done
hoelzl@41970
   618
    finally have "summable
huffman@23082
   619
      (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
huffman@20860
   620
      by (rule diffs_equiv [THEN sums_summable])
huffman@20860
   621
    also have
huffman@23082
   622
      "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
huffman@20860
   623
           r ^ (n - Suc 0)) =
huffman@23082
   624
       (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
huffman@20849
   625
      apply (rule ext)
huffman@20849
   626
      apply (case_tac "n", simp)
blanchet@55417
   627
      apply (rename_tac nat)
huffman@20849
   628
      apply (case_tac "nat", simp)
huffman@20849
   629
      apply (simp add: r_neq_0)
huffman@20849
   630
      done
wenzelm@53079
   631
    finally
wenzelm@53079
   632
    show "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
huffman@20849
   633
  next
huffman@23082
   634
    fix h::'a and n::nat
huffman@20860
   635
    assume h: "h \<noteq> 0"
huffman@23082
   636
    assume "norm h < r - norm x"
huffman@23082
   637
    hence "norm x + norm h < r" by simp
huffman@23082
   638
    with norm_triangle_ineq have xh: "norm (x + h) < r"
huffman@20860
   639
      by (rule order_le_less_trans)
huffman@23082
   640
    show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
huffman@23082
   641
          \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
huffman@23082
   642
      apply (simp only: norm_mult mult_assoc)
huffman@23082
   643
      apply (rule mult_left_mono [OF _ norm_ge_zero])
paulson@54575
   644
      apply (simp add: mult_assoc [symmetric])
paulson@54575
   645
      apply (metis h lemma_termdiff3 less_eq_real_def r1 xh)
huffman@20860
   646
      done
huffman@20849
   647
  qed
huffman@20849
   648
qed
webertj@20217
   649
huffman@20860
   650
lemma termdiffs:
haftmann@31017
   651
  fixes K x :: "'a::{real_normed_field,banach}"
huffman@20860
   652
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
paulson@54575
   653
      and 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
paulson@54575
   654
      and 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
paulson@54575
   655
      and 4: "norm x < norm K"
huffman@20860
   656
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
wenzelm@53079
   657
  unfolding deriv_def
huffman@29163
   658
proof (rule LIM_zero_cancel)
huffman@20860
   659
  show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
huffman@20860
   660
            - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
huffman@20860
   661
  proof (rule LIM_equal2)
huffman@29163
   662
    show "0 < norm K - norm x" using 4 by (simp add: less_diff_eq)
huffman@20860
   663
  next
huffman@23082
   664
    fix h :: 'a
huffman@20860
   665
    assume "h \<noteq> 0"
huffman@23082
   666
    assume "norm (h - 0) < norm K - norm x"
huffman@23082
   667
    hence "norm x + norm h < norm K" by simp
huffman@23082
   668
    hence 5: "norm (x + h) < norm K"
huffman@23082
   669
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
huffman@20860
   670
    have A: "summable (\<lambda>n. c n * x ^ n)"
huffman@20860
   671
      by (rule powser_inside [OF 1 4])
huffman@20860
   672
    have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
huffman@20860
   673
      by (rule powser_inside [OF 1 5])
huffman@20860
   674
    have C: "summable (\<lambda>n. diffs c n * x ^ n)"
huffman@20860
   675
      by (rule powser_inside [OF 2 4])
paulson@54575
   676
    let ?dp = "(\<Sum>n. of_nat n * c n * x ^ (n - Suc 0))"
paulson@54575
   677
    have "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h - (\<Sum>n. diffs c n * x ^ n) =
paulson@54575
   678
          ((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h - ?dp"  
paulson@54575
   679
      by (metis sums_unique [OF diffs_equiv [OF C]])
paulson@54575
   680
    also have "... = (\<Sum>n. c n * (x + h) ^ n - c n * x ^ n) / h - ?dp"  
paulson@54575
   681
      by (metis suminf_diff [OF B A])
paulson@54575
   682
    also have "... = (\<Sum>n. (c n * (x + h) ^ n - c n * x ^ n) / h)  - ?dp"
paulson@54575
   683
      by (metis suminf_divide [OF summable_diff [OF B A]] )  
paulson@54575
   684
    also have "... = (\<Sum>n. (c n * (x + h) ^ n - c n * x ^ n) / h - of_nat n * c n * x ^ (n - Suc 0))"
huffman@20860
   685
      apply (subst suminf_diff)
paulson@54575
   686
      apply (auto intro: summable_divide summable_diff [OF B A] sums_summable [OF diffs_equiv [OF C]])
huffman@20860
   687
      done
paulson@54575
   688
    also have "... = (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
paulson@54575
   689
      by (simp add: algebra_simps)
paulson@54575
   690
    finally show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
paulson@54575
   691
                   - (\<Sum>n. diffs c n * x ^ n) =
paulson@54575
   692
                  (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))" . 
huffman@20860
   693
  next
wenzelm@53079
   694
    show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
wenzelm@53079
   695
      by (rule termdiffs_aux [OF 3 4])
huffman@20860
   696
  qed
huffman@20860
   697
qed
huffman@20860
   698
paulson@15077
   699
hoelzl@29803
   700
subsection {* Derivability of power series *}
hoelzl@29803
   701
wenzelm@53079
   702
lemma DERIV_series':
wenzelm@53079
   703
  fixes f :: "real \<Rightarrow> nat \<Rightarrow> real"
hoelzl@29803
   704
  assumes DERIV_f: "\<And> n. DERIV (\<lambda> x. f x n) x0 :> (f' x0 n)"
wenzelm@53079
   705
    and allf_summable: "\<And> x. x \<in> {a <..< b} \<Longrightarrow> summable (f x)" and x0_in_I: "x0 \<in> {a <..< b}"
wenzelm@53079
   706
    and "summable (f' x0)"
wenzelm@53079
   707
    and "summable L"
wenzelm@53079
   708
    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
   709
  shows "DERIV (\<lambda> x. suminf (f x)) x0 :> (suminf (f' x0))"
hoelzl@29803
   710
  unfolding deriv_def
hoelzl@29803
   711
proof (rule LIM_I)
wenzelm@53079
   712
  fix r :: real
wenzelm@53079
   713
  assume "0 < r" hence "0 < r/3" by auto
hoelzl@29803
   714
hoelzl@41970
   715
  obtain N_L where N_L: "\<And> n. N_L \<le> n \<Longrightarrow> \<bar> \<Sum> i. L (i + n) \<bar> < r/3"
hoelzl@29803
   716
    using suminf_exist_split[OF `0 < r/3` `summable L`] by auto
hoelzl@29803
   717
hoelzl@41970
   718
  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
   719
    using suminf_exist_split[OF `0 < r/3` `summable (f' x0)`] by auto
hoelzl@29803
   720
hoelzl@29803
   721
  let ?N = "Suc (max N_L N_f')"
hoelzl@29803
   722
  have "\<bar> \<Sum> i. f' x0 (i + ?N) \<bar> < r/3" (is "?f'_part < r/3") and
hoelzl@29803
   723
    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
   724
wenzelm@53079
   725
  let ?diff = "\<lambda>i x. (f (x0 + x) i - f x0 i) / x"
hoelzl@29803
   726
hoelzl@29803
   727
  let ?r = "r / (3 * real ?N)"
hoelzl@29803
   728
  have "0 < 3 * real ?N" by auto
hoelzl@29803
   729
  from divide_pos_pos[OF `0 < r` this]
hoelzl@29803
   730
  have "0 < ?r" .
hoelzl@29803
   731
hoelzl@29803
   732
  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
   733
  def S' \<equiv> "Min (?s ` { 0 ..< ?N })"
hoelzl@29803
   734
hoelzl@29803
   735
  have "0 < S'" unfolding S'_def
hoelzl@29803
   736
  proof (rule iffD2[OF Min_gr_iff])
wenzelm@53079
   737
    show "\<forall>x \<in> (?s ` { 0 ..< ?N }). 0 < x"
wenzelm@53079
   738
    proof
wenzelm@53079
   739
      fix x
wenzelm@53079
   740
      assume "x \<in> ?s ` {0..<?N}"
wenzelm@53079
   741
      then obtain n where "x = ?s n" and "n \<in> {0..<?N}"
wenzelm@53079
   742
        using image_iff[THEN iffD1] by blast
hoelzl@41970
   743
      from DERIV_D[OF DERIV_f[where n=n], THEN LIM_D, OF `0 < ?r`, unfolded real_norm_def]
wenzelm@53079
   744
      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)"
wenzelm@53079
   745
        by auto
wenzelm@53079
   746
      have "0 < ?s n" by (rule someI2[where a=s]) (auto simp add: s_bound)
hoelzl@29803
   747
      thus "0 < x" unfolding `x = ?s n` .
hoelzl@29803
   748
    qed
hoelzl@29803
   749
  qed auto
hoelzl@29803
   750
hoelzl@29803
   751
  def S \<equiv> "min (min (x0 - a) (b - x0)) S'"
wenzelm@53079
   752
  hence "0 < S" and S_a: "S \<le> x0 - a" and S_b: "S \<le> b - x0"
wenzelm@53079
   753
    and "S \<le> S'" using x0_in_I and `0 < S'`
hoelzl@29803
   754
    by auto
hoelzl@29803
   755
wenzelm@53079
   756
  {
wenzelm@53079
   757
    fix x
wenzelm@53079
   758
    assume "x \<noteq> 0" and "\<bar> x \<bar> < S"
wenzelm@53079
   759
    hence x_in_I: "x0 + x \<in> { a <..< b }"
wenzelm@53079
   760
      using S_a S_b by auto
hoelzl@41970
   761
hoelzl@29803
   762
    note diff_smbl = summable_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
hoelzl@29803
   763
    note div_smbl = summable_divide[OF diff_smbl]
hoelzl@29803
   764
    note all_smbl = summable_diff[OF div_smbl `summable (f' x0)`]
hoelzl@29803
   765
    note ign = summable_ignore_initial_segment[where k="?N"]
hoelzl@29803
   766
    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
   767
    note div_shft_smbl = summable_divide[OF diff_shft_smbl]
hoelzl@29803
   768
    note all_shft_smbl = summable_diff[OF div_smbl ign[OF `summable (f' x0)`]]
hoelzl@29803
   769
wenzelm@53079
   770
    {
wenzelm@53079
   771
      fix n
hoelzl@41970
   772
      have "\<bar> ?diff (n + ?N) x \<bar> \<le> L (n + ?N) * \<bar> (x0 + x) - x0 \<bar> / \<bar> x \<bar>"
wenzelm@53079
   773
        using divide_right_mono[OF L_def[OF x_in_I x0_in_I] abs_ge_zero]
wenzelm@53079
   774
        unfolding abs_divide .
wenzelm@53079
   775
      hence "\<bar> (\<bar>?diff (n + ?N) x \<bar>) \<bar> \<le> L (n + ?N)"
wenzelm@53079
   776
        using `x \<noteq> 0` by auto
hoelzl@29803
   777
    } note L_ge = summable_le2[OF allI[OF this] ign[OF `summable L`]]
hoelzl@29803
   778
    from order_trans[OF summable_rabs[OF conjunct1[OF L_ge]] L_ge[THEN conjunct2]]
hoelzl@29803
   779
    have "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> (\<Sum> i. L (i + ?N))" .
wenzelm@53079
   780
    hence "\<bar> \<Sum> i. ?diff (i + ?N) x \<bar> \<le> r / 3" (is "?L_part \<le> r/3")
wenzelm@53079
   781
      using L_estimate by auto
wenzelm@53079
   782
wenzelm@53079
   783
    have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> \<le>
wenzelm@53079
   784
      (\<Sum>n \<in> { 0 ..< ?N}. \<bar>?diff n x - f' x0 n \<bar>)" ..
hoelzl@29803
   785
    also have "\<dots> < (\<Sum>n \<in> { 0 ..< ?N}. ?r)"
hoelzl@29803
   786
    proof (rule setsum_strict_mono)
wenzelm@53079
   787
      fix n
wenzelm@53079
   788
      assume "n \<in> { 0 ..< ?N}"
wenzelm@53079
   789
      have "\<bar>x\<bar> < S" using `\<bar>x\<bar> < S` .
hoelzl@29803
   790
      also have "S \<le> S'" using `S \<le> S'` .
hoelzl@41970
   791
      also have "S' \<le> ?s n" unfolding S'_def
hoelzl@29803
   792
      proof (rule Min_le_iff[THEN iffD2])
wenzelm@53079
   793
        have "?s n \<in> (?s ` {0..<?N}) \<and> ?s n \<le> ?s n"
wenzelm@53079
   794
          using `n \<in> { 0 ..< ?N}` by auto
wenzelm@32960
   795
        thus "\<exists> a \<in> (?s ` {0..<?N}). a \<le> ?s n" by blast
hoelzl@29803
   796
      qed auto
wenzelm@53079
   797
      finally have "\<bar>x\<bar> < ?s n" .
hoelzl@29803
   798
hoelzl@29803
   799
      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
   800
      have "\<forall>x. x \<noteq> 0 \<and> \<bar>x\<bar> < ?s n \<longrightarrow> \<bar>?diff n x - f' x0 n\<bar> < ?r" .
wenzelm@53079
   801
      with `x \<noteq> 0` and `\<bar>x\<bar> < ?s n` show "\<bar>?diff n x - f' x0 n\<bar> < ?r"
wenzelm@53079
   802
        by blast
hoelzl@29803
   803
    qed auto
wenzelm@53079
   804
    also have "\<dots> = of_nat (card {0 ..< ?N}) * ?r"
wenzelm@53079
   805
      by (rule setsum_constant)
wenzelm@53079
   806
    also have "\<dots> = real ?N * ?r"
wenzelm@53079
   807
      unfolding real_eq_of_nat by auto
hoelzl@29803
   808
    also have "\<dots> = r/3" by auto
hoelzl@29803
   809
    finally have "\<bar>\<Sum>n \<in> { 0 ..< ?N}. ?diff n x - f' x0 n \<bar> < r / 3" (is "?diff_part < r / 3") .
hoelzl@29803
   810
hoelzl@29803
   811
    from suminf_diff[OF allf_summable[OF x_in_I] allf_summable[OF x0_in_I]]
wenzelm@53079
   812
    have "\<bar>(suminf (f (x0 + x)) - (suminf (f x0))) / x - suminf (f' x0)\<bar> =
wenzelm@53079
   813
        \<bar>\<Sum>n. ?diff n x - f' x0 n\<bar>"
wenzelm@53079
   814
      unfolding suminf_diff[OF div_smbl `summable (f' x0)`, symmetric]
wenzelm@53079
   815
      using suminf_divide[OF diff_smbl, symmetric] by auto
wenzelm@53079
   816
    also have "\<dots> \<le> ?diff_part + \<bar> (\<Sum>n. ?diff (n + ?N) x) - (\<Sum> n. f' x0 (n + ?N)) \<bar>"
wenzelm@53079
   817
      unfolding suminf_split_initial_segment[OF all_smbl, where k="?N"]
wenzelm@53079
   818
      unfolding suminf_diff[OF div_shft_smbl ign[OF `summable (f' x0)`]]
wenzelm@53079
   819
      by (rule abs_triangle_ineq)
wenzelm@53079
   820
    also have "\<dots> \<le> ?diff_part + ?L_part + ?f'_part"
wenzelm@53079
   821
      using abs_triangle_ineq4 by auto
hoelzl@41970
   822
    also have "\<dots> < r /3 + r/3 + r/3"
huffman@36842
   823
      using `?diff_part < r/3` `?L_part \<le> r/3` and `?f'_part < r/3`
huffman@36842
   824
      by (rule add_strict_mono [OF add_less_le_mono])
wenzelm@53079
   825
    finally have "\<bar>(suminf (f (x0 + x)) - suminf (f x0)) / x - suminf (f' x0)\<bar> < r"
hoelzl@29803
   826
      by auto
wenzelm@53079
   827
  }
wenzelm@53079
   828
  thus "\<exists> s > 0. \<forall> x. x \<noteq> 0 \<and> norm (x - 0) < s \<longrightarrow>
wenzelm@53079
   829
      norm (((\<Sum>n. f (x0 + x) n) - (\<Sum>n. f x0 n)) / x - (\<Sum>n. f' x0 n)) < r"
wenzelm@53079
   830
    using `0 < S` unfolding real_norm_def diff_0_right by blast
hoelzl@29803
   831
qed
hoelzl@29803
   832
wenzelm@53079
   833
lemma DERIV_power_series':
wenzelm@53079
   834
  fixes f :: "nat \<Rightarrow> real"
hoelzl@29803
   835
  assumes converges: "\<And> x. x \<in> {-R <..< R} \<Longrightarrow> summable (\<lambda> n. f n * real (Suc n) * x^n)"
wenzelm@53079
   836
    and x0_in_I: "x0 \<in> {-R <..< R}" and "0 < R"
hoelzl@29803
   837
  shows "DERIV (\<lambda> x. (\<Sum> n. f n * x^(Suc n))) x0 :> (\<Sum> n. f n * real (Suc n) * x0^n)"
hoelzl@29803
   838
  (is "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))")
hoelzl@29803
   839
proof -
wenzelm@53079
   840
  {
wenzelm@53079
   841
    fix R'
wenzelm@53079
   842
    assume "0 < R'" and "R' < R" and "-R' < x0" and "x0 < R'"
wenzelm@53079
   843
    hence "x0 \<in> {-R' <..< R'}" and "R' \<in> {-R <..< R}" and "x0 \<in> {-R <..< R}"
wenzelm@53079
   844
      by auto
hoelzl@29803
   845
    have "DERIV (\<lambda> x. (suminf (?f x))) x0 :> (suminf (?f' x0))"
hoelzl@29803
   846
    proof (rule DERIV_series')
hoelzl@29803
   847
      show "summable (\<lambda> n. \<bar>f n * real (Suc n) * R'^n\<bar>)"
hoelzl@29803
   848
      proof -
wenzelm@53079
   849
        have "(R' + R) / 2 < R" and "0 < (R' + R) / 2"
wenzelm@53079
   850
          using `0 < R'` `0 < R` `R' < R` by auto
wenzelm@53079
   851
        hence in_Rball: "(R' + R) / 2 \<in> {-R <..< R}"
wenzelm@53079
   852
          using `R' < R` by auto
wenzelm@53079
   853
        have "norm R' < norm ((R' + R) / 2)"
wenzelm@53079
   854
          using `0 < R'` `0 < R` `R' < R` by auto
wenzelm@53079
   855
        from powser_insidea[OF converges[OF in_Rball] this] show ?thesis
wenzelm@53079
   856
          by auto
hoelzl@29803
   857
      qed
wenzelm@53079
   858
      {
wenzelm@53079
   859
        fix n x y
wenzelm@53079
   860
        assume "x \<in> {-R' <..< R'}" and "y \<in> {-R' <..< R'}"
wenzelm@32960
   861
        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
   862
        proof -
wenzelm@53079
   863
          have "\<bar>f n * x ^ (Suc n) - f n * y ^ (Suc n)\<bar> =
wenzelm@53079
   864
            (\<bar>f n\<bar> * \<bar>x-y\<bar>) * \<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar>"
wenzelm@53079
   865
            unfolding right_diff_distrib[symmetric] lemma_realpow_diff_sumr2 abs_mult
wenzelm@53079
   866
            by auto
hoelzl@41970
   867
          also have "\<dots> \<le> (\<bar>f n\<bar> * \<bar>x-y\<bar>) * (\<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>)"
wenzelm@32960
   868
          proof (rule mult_left_mono)
wenzelm@53079
   869
            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>)"
wenzelm@53079
   870
              by (rule setsum_abs)
wenzelm@32960
   871
            also have "\<dots> \<le> (\<Sum>p = 0..<Suc n. R' ^ n)"
wenzelm@32960
   872
            proof (rule setsum_mono)
wenzelm@53079
   873
              fix p
wenzelm@53079
   874
              assume "p \<in> {0..<Suc n}"
wenzelm@53079
   875
              hence "p \<le> n" by auto
wenzelm@53079
   876
              {
wenzelm@53079
   877
                fix n
wenzelm@53079
   878
                fix x :: real
wenzelm@53079
   879
                assume "x \<in> {-R'<..<R'}"
wenzelm@32960
   880
                hence "\<bar>x\<bar> \<le> R'"  by auto
wenzelm@53079
   881
                hence "\<bar>x^n\<bar> \<le> R'^n"
wenzelm@53079
   882
                  unfolding power_abs by (rule power_mono, auto)
wenzelm@53079
   883
              }
wenzelm@53079
   884
              from mult_mono[OF this[OF `x \<in> {-R'<..<R'}`, of p] this[OF `y \<in> {-R'<..<R'}`, of "n-p"]] `0 < R'`
wenzelm@53079
   885
              have "\<bar>x^p * y^(n-p)\<bar> \<le> R'^p * R'^(n-p)"
wenzelm@53079
   886
                unfolding abs_mult by auto
wenzelm@53079
   887
              thus "\<bar>x^p * y^(n-p)\<bar> \<le> R'^n"
wenzelm@53079
   888
                unfolding power_add[symmetric] using `p \<le> n` by auto
wenzelm@32960
   889
            qed
wenzelm@53079
   890
            also have "\<dots> = real (Suc n) * R' ^ n"
wenzelm@53079
   891
              unfolding setsum_constant card_atLeastLessThan real_of_nat_def by auto
wenzelm@53079
   892
            finally show "\<bar>\<Sum>p = 0..<Suc n. x ^ p * y ^ (n - p)\<bar> \<le> \<bar>real (Suc n)\<bar> * \<bar>R' ^ n\<bar>"
wenzelm@53079
   893
              unfolding abs_real_of_nat_cancel abs_of_nonneg[OF zero_le_power[OF less_imp_le[OF `0 < R'`]]] .
wenzelm@53079
   894
            show "0 \<le> \<bar>f n\<bar> * \<bar>x - y\<bar>"
wenzelm@53079
   895
              unfolding abs_mult[symmetric] by auto
wenzelm@32960
   896
          qed
wenzelm@53079
   897
          also have "\<dots> = \<bar>f n * real (Suc n) * R' ^ n\<bar> * \<bar>x - y\<bar>"
wenzelm@53079
   898
            unfolding abs_mult mult_assoc[symmetric] by algebra
wenzelm@32960
   899
          finally show ?thesis .
wenzelm@53079
   900
        qed
wenzelm@53079
   901
      }
wenzelm@53079
   902
      {
wenzelm@53079
   903
        fix n
wenzelm@53079
   904
        show "DERIV (\<lambda> x. ?f x n) x0 :> (?f' x0 n)"
wenzelm@53079
   905
          by (auto intro!: DERIV_intros simp del: power_Suc)
wenzelm@53079
   906
      }
wenzelm@53079
   907
      {
wenzelm@53079
   908
        fix x
wenzelm@53079
   909
        assume "x \<in> {-R' <..< R'}"
wenzelm@53079
   910
        hence "R' \<in> {-R <..< R}" and "norm x < norm R'"
wenzelm@53079
   911
          using assms `R' < R` by auto
wenzelm@32960
   912
        have "summable (\<lambda> n. f n * x^n)"
wenzelm@32960
   913
        proof (rule summable_le2[THEN conjunct1, OF _ powser_insidea[OF converges[OF `R' \<in> {-R <..< R}`] `norm x < norm R'`]], rule allI)
wenzelm@32960
   914
          fix n
wenzelm@53079
   915
          have le: "\<bar>f n\<bar> * 1 \<le> \<bar>f n\<bar> * real (Suc n)"
wenzelm@53079
   916
            by (rule mult_left_mono) auto
wenzelm@53079
   917
          show "\<bar>f n * x ^ n\<bar> \<le> norm (f n * real (Suc n) * x ^ n)"
wenzelm@53079
   918
            unfolding real_norm_def abs_mult
wenzelm@53079
   919
            by (rule mult_right_mono) (auto simp add: le[unfolded mult_1_right])
wenzelm@32960
   920
        qed
huffman@36777
   921
        from this[THEN summable_mult2[where c=x], unfolded mult_assoc, unfolded mult_commute]
wenzelm@53079
   922
        show "summable (?f x)" by auto
wenzelm@53079
   923
      }
wenzelm@53079
   924
      show "summable (?f' x0)"
wenzelm@53079
   925
        using converges[OF `x0 \<in> {-R <..< R}`] .
wenzelm@53079
   926
      show "x0 \<in> {-R' <..< R'}"
wenzelm@53079
   927
        using `x0 \<in> {-R' <..< R'}` .
hoelzl@29803
   928
    qed
hoelzl@29803
   929
  } note for_subinterval = this
hoelzl@29803
   930
  let ?R = "(R + \<bar>x0\<bar>) / 2"
hoelzl@29803
   931
  have "\<bar>x0\<bar> < ?R" using assms by auto
hoelzl@29803
   932
  hence "- ?R < x0"
hoelzl@29803
   933
  proof (cases "x0 < 0")
hoelzl@29803
   934
    case True
hoelzl@29803
   935
    hence "- x0 < ?R" using `\<bar>x0\<bar> < ?R` by auto
hoelzl@29803
   936
    thus ?thesis unfolding neg_less_iff_less[symmetric, of "- x0"] by auto
hoelzl@29803
   937
  next
hoelzl@29803
   938
    case False
hoelzl@29803
   939
    have "- ?R < 0" using assms by auto
hoelzl@41970
   940
    also have "\<dots> \<le> x0" using False by auto
hoelzl@29803
   941
    finally show ?thesis .
hoelzl@29803
   942
  qed
wenzelm@53079
   943
  hence "0 < ?R" "?R < R" "- ?R < x0" and "x0 < ?R"
wenzelm@53079
   944
    using assms by auto
hoelzl@29803
   945
  from for_subinterval[OF this]
hoelzl@29803
   946
  show ?thesis .
hoelzl@29803
   947
qed
chaieb@29695
   948
wenzelm@53079
   949
huffman@29164
   950
subsection {* Exponential Function *}
huffman@23043
   951
wenzelm@53079
   952
definition exp :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
wenzelm@53079
   953
  where "exp = (\<lambda>x. \<Sum>n. x ^ n /\<^sub>R real (fact n))"
huffman@23043
   954
huffman@23115
   955
lemma summable_exp_generic:
haftmann@31017
   956
  fixes x :: "'a::{real_normed_algebra_1,banach}"
haftmann@25062
   957
  defines S_def: "S \<equiv> \<lambda>n. x ^ n /\<^sub>R real (fact n)"
huffman@23115
   958
  shows "summable S"
huffman@23115
   959
proof -
haftmann@25062
   960
  have S_Suc: "\<And>n. S (Suc n) = (x * S n) /\<^sub>R real (Suc n)"
huffman@30273
   961
    unfolding S_def by (simp del: mult_Suc)
huffman@23115
   962
  obtain r :: real where r0: "0 < r" and r1: "r < 1"
huffman@23115
   963
    using dense [OF zero_less_one] by fast
huffman@23115
   964
  obtain N :: nat where N: "norm x < real N * r"
huffman@23115
   965
    using reals_Archimedean3 [OF r0] by fast
huffman@23115
   966
  from r1 show ?thesis
huffman@23115
   967
  proof (rule ratio_test [rule_format])
huffman@23115
   968
    fix n :: nat
huffman@23115
   969
    assume n: "N \<le> n"
huffman@23115
   970
    have "norm x \<le> real N * r"
huffman@23115
   971
      using N by (rule order_less_imp_le)
huffman@23115
   972
    also have "real N * r \<le> real (Suc n) * r"
huffman@23115
   973
      using r0 n by (simp add: mult_right_mono)
huffman@23115
   974
    finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
huffman@23115
   975
      using norm_ge_zero by (rule mult_right_mono)
huffman@23115
   976
    hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
huffman@23115
   977
      by (rule order_trans [OF norm_mult_ineq])
huffman@23115
   978
    hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
huffman@23115
   979
      by (simp add: pos_divide_le_eq mult_ac)
huffman@23115
   980
    thus "norm (S (Suc n)) \<le> r * norm (S n)"
huffman@35216
   981
      by (simp add: S_Suc inverse_eq_divide)
huffman@23115
   982
  qed
huffman@23115
   983
qed
huffman@23115
   984
huffman@23115
   985
lemma summable_norm_exp:
haftmann@31017
   986
  fixes x :: "'a::{real_normed_algebra_1,banach}"
haftmann@25062
   987
  shows "summable (\<lambda>n. norm (x ^ n /\<^sub>R real (fact n)))"
huffman@23115
   988
proof (rule summable_norm_comparison_test [OF exI, rule_format])
haftmann@25062
   989
  show "summable (\<lambda>n. norm x ^ n /\<^sub>R real (fact n))"
huffman@23115
   990
    by (rule summable_exp_generic)
wenzelm@53079
   991
  fix n
wenzelm@53079
   992
  show "norm (x ^ n /\<^sub>R real (fact n)) \<le> norm x ^ n /\<^sub>R real (fact n)"
huffman@35216
   993
    by (simp add: norm_power_ineq)
huffman@23115
   994
qed
huffman@23115
   995
wenzelm@53079
   996
lemma summable_exp: "summable (\<lambda>n. inverse (real (fact n)) * x ^ n)"
wenzelm@53079
   997
  using summable_exp_generic [where x=x] by simp
huffman@23043
   998
haftmann@25062
   999
lemma exp_converges: "(\<lambda>n. x ^ n /\<^sub>R real (fact n)) sums exp x"
wenzelm@53079
  1000
  unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
huffman@23043
  1001
huffman@23043
  1002
hoelzl@41970
  1003
lemma exp_fdiffs:
wenzelm@53079
  1004
      "diffs (\<lambda>n. inverse(real (fact n))) = (\<lambda>n. inverse(real (fact n)))"
wenzelm@53079
  1005
  by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def of_nat_mult
wenzelm@53079
  1006
        del: mult_Suc of_nat_Suc)
paulson@15077
  1007
huffman@23115
  1008
lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
wenzelm@53079
  1009
  by (simp add: diffs_def)
huffman@23115
  1010
paulson@15077
  1011
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
wenzelm@53079
  1012
  unfolding exp_def scaleR_conv_of_real
wenzelm@53079
  1013
  apply (rule DERIV_cong)
wenzelm@53079
  1014
  apply (rule termdiffs [where K="of_real (1 + norm x)"])
wenzelm@53079
  1015
  apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
wenzelm@53079
  1016
  apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
wenzelm@53079
  1017
  apply (simp del: of_real_add)
wenzelm@53079
  1018
  done
paulson@15077
  1019
hoelzl@51527
  1020
declare DERIV_exp[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@51527
  1021
huffman@44311
  1022
lemma isCont_exp: "isCont exp x"
huffman@44311
  1023
  by (rule DERIV_exp [THEN DERIV_isCont])
huffman@44311
  1024
huffman@44311
  1025
lemma isCont_exp' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. exp (f x)) a"
huffman@44311
  1026
  by (rule isCont_o2 [OF _ isCont_exp])
huffman@44311
  1027
huffman@44311
  1028
lemma tendsto_exp [tendsto_intros]:
huffman@44311
  1029
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. exp (f x)) ---> exp a) F"
huffman@44311
  1030
  by (rule isCont_tendsto_compose [OF isCont_exp])
huffman@23045
  1031
wenzelm@53079
  1032
lemma continuous_exp [continuous_intros]:
wenzelm@53079
  1033
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. exp (f x))"
hoelzl@51478
  1034
  unfolding continuous_def by (rule tendsto_exp)
hoelzl@51478
  1035
wenzelm@53079
  1036
lemma continuous_on_exp [continuous_on_intros]:
wenzelm@53079
  1037
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. exp (f x))"
hoelzl@51478
  1038
  unfolding continuous_on_def by (auto intro: tendsto_exp)
hoelzl@51478
  1039
wenzelm@53079
  1040
huffman@29167
  1041
subsubsection {* Properties of the Exponential Function *}
paulson@15077
  1042
huffman@23278
  1043
lemma powser_zero:
haftmann@31017
  1044
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1}"
huffman@23278
  1045
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
paulson@15077
  1046
proof -
huffman@23278
  1047
  have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
huffman@23115
  1048
    by (rule sums_unique [OF series_zero], simp add: power_0_left)
huffman@30082
  1049
  thus ?thesis unfolding One_nat_def by simp
paulson@15077
  1050
qed
paulson@15077
  1051
huffman@23278
  1052
lemma exp_zero [simp]: "exp 0 = 1"
wenzelm@53079
  1053
  unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
huffman@23278
  1054
huffman@23115
  1055
lemma setsum_cl_ivl_Suc2:
huffman@23115
  1056
  "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
wenzelm@53079
  1057
  by (simp add: setsum_head_Suc setsum_shift_bounds_cl_Suc_ivl
wenzelm@53079
  1058
           del: setsum_cl_ivl_Suc)
huffman@23115
  1059
huffman@23115
  1060
lemma exp_series_add:
haftmann@31017
  1061
  fixes x y :: "'a::{real_field}"
haftmann@25062
  1062
  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /\<^sub>R real (fact n)"
huffman@23115
  1063
  shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
huffman@23115
  1064
proof (induct n)
huffman@23115
  1065
  case 0
huffman@23115
  1066
  show ?case
huffman@23115
  1067
    unfolding S_def by simp
huffman@23115
  1068
next
huffman@23115
  1069
  case (Suc n)
haftmann@25062
  1070
  have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /\<^sub>R real (Suc n)"
huffman@30273
  1071
    unfolding S_def by (simp del: mult_Suc)
haftmann@25062
  1072
  hence times_S: "\<And>x n. x * S x n = real (Suc n) *\<^sub>R S x (Suc n)"
huffman@23115
  1073
    by simp
huffman@23115
  1074
haftmann@25062
  1075
  have "real (Suc n) *\<^sub>R S (x + y) (Suc n) = (x + y) * S (x + y) n"
huffman@23115
  1076
    by (simp only: times_S)
huffman@23115
  1077
  also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
huffman@23115
  1078
    by (simp only: Suc)
huffman@23115
  1079
  also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
huffman@23115
  1080
                + y * (\<Sum>i=0..n. S x i * S y (n-i))"
webertj@49962
  1081
    by (rule distrib_right)
huffman@23115
  1082
  also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
huffman@23115
  1083
                + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
huffman@23115
  1084
    by (simp only: setsum_right_distrib mult_ac)
haftmann@25062
  1085
  also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i)))
haftmann@25062
  1086
                + (\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
  1087
    by (simp add: times_S Suc_diff_le)
haftmann@25062
  1088
  also have "(\<Sum>i=0..n. real (Suc i) *\<^sub>R (S x (Suc i) * S y (n-i))) =
haftmann@25062
  1089
             (\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
  1090
    by (subst setsum_cl_ivl_Suc2, simp)
haftmann@25062
  1091
  also have "(\<Sum>i=0..n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
haftmann@25062
  1092
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
  1093
    by (subst setsum_cl_ivl_Suc, simp)
haftmann@25062
  1094
  also have "(\<Sum>i=0..Suc n. real i *\<^sub>R (S x i * S y (Suc n-i))) +
haftmann@25062
  1095
             (\<Sum>i=0..Suc n. real (Suc n-i) *\<^sub>R (S x i * S y (Suc n-i))) =
haftmann@25062
  1096
             (\<Sum>i=0..Suc n. real (Suc n) *\<^sub>R (S x i * S y (Suc n-i)))"
huffman@23115
  1097
    by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
huffman@23115
  1098
              real_of_nat_add [symmetric], simp)
haftmann@25062
  1099
  also have "\<dots> = real (Suc n) *\<^sub>R (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
huffman@23127
  1100
    by (simp only: scaleR_right.setsum)
huffman@23115
  1101
  finally show
huffman@23115
  1102
    "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
huffman@35216
  1103
    by (simp del: setsum_cl_ivl_Suc)
huffman@23115
  1104
qed
huffman@23115
  1105
huffman@23115
  1106
lemma exp_add: "exp (x + y) = exp x * exp y"
wenzelm@53079
  1107
  unfolding exp_def
wenzelm@53079
  1108
  by (simp only: Cauchy_product summable_norm_exp exp_series_add)
huffman@23115
  1109
huffman@29170
  1110
lemma mult_exp_exp: "exp x * exp y = exp (x + y)"
wenzelm@53079
  1111
  by (rule exp_add [symmetric])
huffman@29170
  1112
huffman@23241
  1113
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
wenzelm@53079
  1114
  unfolding exp_def
wenzelm@53079
  1115
  apply (subst suminf_of_real)
wenzelm@53079
  1116
  apply (rule summable_exp_generic)
wenzelm@53079
  1117
  apply (simp add: scaleR_conv_of_real)
wenzelm@53079
  1118
  done
huffman@23241
  1119
huffman@29170
  1120
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
huffman@29170
  1121
proof
huffman@29170
  1122
  have "exp x * exp (- x) = 1" by (simp add: mult_exp_exp)
huffman@29170
  1123
  also assume "exp x = 0"
huffman@29170
  1124
  finally show "False" by simp
paulson@15077
  1125
qed
paulson@15077
  1126
huffman@29170
  1127
lemma exp_minus: "exp (- x) = inverse (exp x)"
wenzelm@53079
  1128
  by (rule inverse_unique [symmetric], simp add: mult_exp_exp)
paulson@15077
  1129
huffman@29170
  1130
lemma exp_diff: "exp (x - y) = exp x / exp y"
haftmann@54230
  1131
  using exp_add [of x "- y"] by (simp add: exp_minus divide_inverse)
paulson@15077
  1132
huffman@29167
  1133
huffman@29167
  1134
subsubsection {* Properties of the Exponential Function on Reals *}
huffman@29167
  1135
huffman@29170
  1136
text {* Comparisons of @{term "exp x"} with zero. *}
huffman@29167
  1137
huffman@29167
  1138
text{*Proof: because every exponential can be seen as a square.*}
huffman@29167
  1139
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
huffman@29167
  1140
proof -
huffman@29167
  1141
  have "0 \<le> exp (x/2) * exp (x/2)" by simp
huffman@29167
  1142
  thus ?thesis by (simp add: exp_add [symmetric])
huffman@29167
  1143
qed
huffman@29167
  1144
huffman@23115
  1145
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
wenzelm@53079
  1146
  by (simp add: order_less_le)
paulson@15077
  1147
huffman@29170
  1148
lemma not_exp_less_zero [simp]: "\<not> exp (x::real) < 0"
wenzelm@53079
  1149
  by (simp add: not_less)
huffman@29170
  1150
huffman@29170
  1151
lemma not_exp_le_zero [simp]: "\<not> exp (x::real) \<le> 0"
wenzelm@53079
  1152
  by (simp add: not_le)
paulson@15077
  1153
huffman@23115
  1154
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
wenzelm@53079
  1155
  by simp
paulson@15077
  1156
paulson@15077
  1157
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
wenzelm@53079
  1158
  by (induct n) (auto simp add: real_of_nat_Suc distrib_left exp_add mult_commute)
paulson@15077
  1159
huffman@29170
  1160
text {* Strict monotonicity of exponential. *}
huffman@29170
  1161
paulson@54575
  1162
lemma exp_ge_add_one_self_aux: 
paulson@54575
  1163
  assumes "0 \<le> (x::real)" shows "1+x \<le> exp(x)"
paulson@54575
  1164
using order_le_imp_less_or_eq [OF assms]
paulson@54575
  1165
proof 
paulson@54575
  1166
  assume "0 < x"
paulson@54575
  1167
  have "1+x \<le> (\<Sum>n = 0..<2. inverse (real (fact n)) * x ^ n)"
paulson@54575
  1168
    by (auto simp add: numeral_2_eq_2)
paulson@54575
  1169
  also have "... \<le> (\<Sum>n. inverse (real (fact n)) * x ^ n)"
paulson@54575
  1170
    apply (rule series_pos_le [OF summable_exp])
paulson@54575
  1171
    using `0 < x`
paulson@54575
  1172
    apply (auto  simp add:  zero_le_mult_iff)
paulson@54575
  1173
    done
paulson@54575
  1174
  finally show "1+x \<le> exp x" 
paulson@54575
  1175
    by (simp add: exp_def)
paulson@54575
  1176
next
paulson@54575
  1177
  assume "0 = x"
paulson@54575
  1178
  then show "1 + x \<le> exp x"
paulson@54575
  1179
    by auto
paulson@54575
  1180
qed
huffman@29170
  1181
huffman@29170
  1182
lemma exp_gt_one: "0 < (x::real) \<Longrightarrow> 1 < exp x"
huffman@29170
  1183
proof -
huffman@29170
  1184
  assume x: "0 < x"
huffman@29170
  1185
  hence "1 < 1 + x" by simp
huffman@29170
  1186
  also from x have "1 + x \<le> exp x"
huffman@29170
  1187
    by (simp add: exp_ge_add_one_self_aux)
huffman@29170
  1188
  finally show ?thesis .
huffman@29170
  1189
qed
huffman@29170
  1190
paulson@15077
  1191
lemma exp_less_mono:
huffman@23115
  1192
  fixes x y :: real
wenzelm@53079
  1193
  assumes "x < y"
wenzelm@53079
  1194
  shows "exp x < exp y"
paulson@15077
  1195
proof -
huffman@29165
  1196
  from `x < y` have "0 < y - x" by simp
huffman@29165
  1197
  hence "1 < exp (y - x)" by (rule exp_gt_one)
huffman@29165
  1198
  hence "1 < exp y / exp x" by (simp only: exp_diff)
huffman@29165
  1199
  thus "exp x < exp y" by simp
paulson@15077
  1200
qed
paulson@15077
  1201
wenzelm@53079
  1202
lemma exp_less_cancel: "exp (x::real) < exp y \<Longrightarrow> x < y"
paulson@54575
  1203
  unfolding linorder_not_le [symmetric]
paulson@54575
  1204
  by (auto simp add: order_le_less exp_less_mono)
paulson@15077
  1205
huffman@29170
  1206
lemma exp_less_cancel_iff [iff]: "exp (x::real) < exp y \<longleftrightarrow> x < y"
wenzelm@53079
  1207
  by (auto intro: exp_less_mono exp_less_cancel)
paulson@15077
  1208
huffman@29170
  1209
lemma exp_le_cancel_iff [iff]: "exp (x::real) \<le> exp y \<longleftrightarrow> x \<le> y"
wenzelm@53079
  1210
  by (auto simp add: linorder_not_less [symmetric])
paulson@15077
  1211
huffman@29170
  1212
lemma exp_inj_iff [iff]: "exp (x::real) = exp y \<longleftrightarrow> x = y"
wenzelm@53079
  1213
  by (simp add: order_eq_iff)
paulson@15077
  1214
huffman@29170
  1215
text {* Comparisons of @{term "exp x"} with one. *}
huffman@29170
  1216
huffman@29170
  1217
lemma one_less_exp_iff [simp]: "1 < exp (x::real) \<longleftrightarrow> 0 < x"
huffman@29170
  1218
  using exp_less_cancel_iff [where x=0 and y=x] by simp
huffman@29170
  1219
huffman@29170
  1220
lemma exp_less_one_iff [simp]: "exp (x::real) < 1 \<longleftrightarrow> x < 0"
huffman@29170
  1221
  using exp_less_cancel_iff [where x=x and y=0] by simp
huffman@29170
  1222
huffman@29170
  1223
lemma one_le_exp_iff [simp]: "1 \<le> exp (x::real) \<longleftrightarrow> 0 \<le> x"
huffman@29170
  1224
  using exp_le_cancel_iff [where x=0 and y=x] by simp
huffman@29170
  1225
huffman@29170
  1226
lemma exp_le_one_iff [simp]: "exp (x::real) \<le> 1 \<longleftrightarrow> x \<le> 0"
huffman@29170
  1227
  using exp_le_cancel_iff [where x=x and y=0] by simp
huffman@29170
  1228
huffman@29170
  1229
lemma exp_eq_one_iff [simp]: "exp (x::real) = 1 \<longleftrightarrow> x = 0"
huffman@29170
  1230
  using exp_inj_iff [where x=x and y=0] by simp
huffman@29170
  1231
wenzelm@53079
  1232
lemma lemma_exp_total: "1 \<le> y \<Longrightarrow> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
huffman@44755
  1233
proof (rule IVT)
huffman@44755
  1234
  assume "1 \<le> y"
huffman@44755
  1235
  hence "0 \<le> y - 1" by simp
huffman@44755
  1236
  hence "1 + (y - 1) \<le> exp (y - 1)" by (rule exp_ge_add_one_self_aux)
huffman@44755
  1237
  thus "y \<le> exp (y - 1)" by simp
huffman@44755
  1238
qed (simp_all add: le_diff_eq)
paulson@15077
  1239
wenzelm@53079
  1240
lemma exp_total: "0 < (y::real) \<Longrightarrow> \<exists>x. exp x = y"
huffman@44755
  1241
proof (rule linorder_le_cases [of 1 y])
wenzelm@53079
  1242
  assume "1 \<le> y"
wenzelm@53079
  1243
  thus "\<exists>x. exp x = y" by (fast dest: lemma_exp_total)
huffman@44755
  1244
next
huffman@44755
  1245
  assume "0 < y" and "y \<le> 1"
huffman@44755
  1246
  hence "1 \<le> inverse y" by (simp add: one_le_inverse_iff)
huffman@44755
  1247
  then obtain x where "exp x = inverse y" by (fast dest: lemma_exp_total)
huffman@44755
  1248
  hence "exp (- x) = y" by (simp add: exp_minus)
huffman@44755
  1249
  thus "\<exists>x. exp x = y" ..
huffman@44755
  1250
qed
paulson@15077
  1251
paulson@15077
  1252
huffman@29164
  1253
subsection {* Natural Logarithm *}
paulson@15077
  1254
wenzelm@53079
  1255
definition ln :: "real \<Rightarrow> real"
wenzelm@53079
  1256
  where "ln x = (THE u. exp u = x)"
huffman@23043
  1257
huffman@23043
  1258
lemma ln_exp [simp]: "ln (exp x) = x"
huffman@44308
  1259
  by (simp add: ln_def)
paulson@15077
  1260
huffman@22654
  1261
lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
huffman@44308
  1262
  by (auto dest: exp_total)
huffman@22654
  1263
huffman@29171
  1264
lemma exp_ln_iff [simp]: "exp (ln x) = x \<longleftrightarrow> 0 < x"
huffman@44308
  1265
  by (metis exp_gt_zero exp_ln)
paulson@15077
  1266
huffman@29171
  1267
lemma ln_unique: "exp y = x \<Longrightarrow> ln x = y"
huffman@44308
  1268
  by (erule subst, rule ln_exp)
huffman@29171
  1269
huffman@29171
  1270
lemma ln_one [simp]: "ln 1 = 0"
wenzelm@53079
  1271
  by (rule ln_unique) simp
wenzelm@53079
  1272
wenzelm@53079
  1273
lemma ln_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x * y) = ln x + ln y"
wenzelm@53079
  1274
  by (rule ln_unique) (simp add: exp_add)
huffman@29171
  1275
huffman@29171
  1276
lemma ln_inverse: "0 < x \<Longrightarrow> ln (inverse x) = - ln x"
wenzelm@53079
  1277
  by (rule ln_unique) (simp add: exp_minus)
wenzelm@53079
  1278
wenzelm@53079
  1279
lemma ln_div: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln (x / y) = ln x - ln y"
wenzelm@53079
  1280
  by (rule ln_unique) (simp add: exp_diff)
paulson@15077
  1281
huffman@29171
  1282
lemma ln_realpow: "0 < x \<Longrightarrow> ln (x ^ n) = real n * ln x"
wenzelm@53079
  1283
  by (rule ln_unique) (simp add: exp_real_of_nat_mult)
wenzelm@53079
  1284
wenzelm@53079
  1285
lemma ln_less_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x < ln y \<longleftrightarrow> x < y"
wenzelm@53079
  1286
  by (subst exp_less_cancel_iff [symmetric]) simp
wenzelm@53079
  1287
wenzelm@53079
  1288
lemma ln_le_cancel_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x \<le> ln y \<longleftrightarrow> x \<le> y"
huffman@44308
  1289
  by (simp add: linorder_not_less [symmetric])
huffman@29171
  1290
wenzelm@53079
  1291
lemma ln_inj_iff [simp]: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> ln x = ln y \<longleftrightarrow> x = y"
huffman@44308
  1292
  by (simp add: order_eq_iff)
huffman@29171
  1293
huffman@29171
  1294
lemma ln_add_one_self_le_self [simp]: "0 \<le> x \<Longrightarrow> ln (1 + x) \<le> x"
huffman@44308
  1295
  apply (rule exp_le_cancel_iff [THEN iffD1])
huffman@44308
  1296
  apply (simp add: exp_ge_add_one_self_aux)
huffman@44308
  1297
  done
paulson@15077
  1298
huffman@29171
  1299
lemma ln_less_self [simp]: "0 < x \<Longrightarrow> ln x < x"
huffman@44308
  1300
  by (rule order_less_le_trans [where y="ln (1 + x)"]) simp_all
huffman@44308
  1301
huffman@44308
  1302
lemma ln_ge_zero [simp]: "1 \<le> x \<Longrightarrow> 0 \<le> ln x"
huffman@44308
  1303
  using ln_le_cancel_iff [of 1 x] by simp
huffman@44308
  1304
wenzelm@53079
  1305
lemma ln_ge_zero_imp_ge_one: "0 \<le> ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> x"
huffman@44308
  1306
  using ln_le_cancel_iff [of 1 x] by simp
huffman@44308
  1307
wenzelm@53079
  1308
lemma ln_ge_zero_iff [simp]: "0 < x \<Longrightarrow> 0 \<le> ln x \<longleftrightarrow> 1 \<le> x"
huffman@44308
  1309
  using ln_le_cancel_iff [of 1 x] by simp
huffman@44308
  1310
wenzelm@53079
  1311
lemma ln_less_zero_iff [simp]: "0 < x \<Longrightarrow> ln x < 0 \<longleftrightarrow> x < 1"
huffman@44308
  1312
  using ln_less_cancel_iff [of x 1] by simp
huffman@44308
  1313
huffman@44308
  1314
lemma ln_gt_zero: "1 < x \<Longrightarrow> 0 < ln x"
huffman@44308
  1315
  using ln_less_cancel_iff [of 1 x] by simp
huffman@44308
  1316
wenzelm@53079
  1317
lemma ln_gt_zero_imp_gt_one: "0 < ln x \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
huffman@44308
  1318
  using ln_less_cancel_iff [of 1 x] by simp
huffman@44308
  1319
wenzelm@53079
  1320
lemma ln_gt_zero_iff [simp]: "0 < x \<Longrightarrow> 0 < ln x \<longleftrightarrow> 1 < x"
huffman@44308
  1321
  using ln_less_cancel_iff [of 1 x] by simp
huffman@44308
  1322
wenzelm@53079
  1323
lemma ln_eq_zero_iff [simp]: "0 < x \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1"
huffman@44308
  1324
  using ln_inj_iff [of x 1] by simp
huffman@44308
  1325
wenzelm@53079
  1326
lemma ln_less_zero: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> ln x < 0"
huffman@44308
  1327
  by simp
paulson@15077
  1328
huffman@23045
  1329
lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
huffman@44308
  1330
  apply (subgoal_tac "isCont ln (exp (ln x))", simp)
huffman@44308
  1331
  apply (rule isCont_inverse_function [where f=exp], simp_all)
huffman@44308
  1332
  done
huffman@23045
  1333
huffman@45915
  1334
lemma tendsto_ln [tendsto_intros]:
wenzelm@53079
  1335
  "(f ---> a) F \<Longrightarrow> 0 < a \<Longrightarrow> ((\<lambda>x. ln (f x)) ---> ln a) F"
huffman@45915
  1336
  by (rule isCont_tendsto_compose [OF isCont_ln])
huffman@45915
  1337
hoelzl@51478
  1338
lemma continuous_ln:
hoelzl@51478
  1339
  "continuous F f \<Longrightarrow> 0 < f (Lim F (\<lambda>x. x)) \<Longrightarrow> continuous F (\<lambda>x. ln (f x))"
hoelzl@51478
  1340
  unfolding continuous_def by (rule tendsto_ln)
hoelzl@51478
  1341
hoelzl@51478
  1342
lemma isCont_ln' [continuous_intros]:
hoelzl@51478
  1343
  "continuous (at x) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x) (\<lambda>x. ln (f x))"
hoelzl@51478
  1344
  unfolding continuous_at by (rule tendsto_ln)
hoelzl@51478
  1345
hoelzl@51478
  1346
lemma continuous_within_ln [continuous_intros]:
hoelzl@51478
  1347
  "continuous (at x within s) f \<Longrightarrow> 0 < f x \<Longrightarrow> continuous (at x within s) (\<lambda>x. ln (f x))"
hoelzl@51478
  1348
  unfolding continuous_within by (rule tendsto_ln)
hoelzl@51478
  1349
hoelzl@51478
  1350
lemma continuous_on_ln [continuous_on_intros]:
hoelzl@51478
  1351
  "continuous_on s f \<Longrightarrow> (\<forall>x\<in>s. 0 < f x) \<Longrightarrow> continuous_on s (\<lambda>x. ln (f x))"
hoelzl@51478
  1352
  unfolding continuous_on_def by (auto intro: tendsto_ln)
hoelzl@51478
  1353
huffman@23045
  1354
lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
huffman@44308
  1355
  apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
paulson@54576
  1356
  apply (auto intro: DERIV_cong [OF DERIV_exp exp_ln] isCont_ln)
huffman@44308
  1357
  done
huffman@23045
  1358
wenzelm@53079
  1359
lemma DERIV_ln_divide: "0 < x \<Longrightarrow> DERIV ln x :> 1 / x"
paulson@33667
  1360
  by (rule DERIV_ln[THEN DERIV_cong], simp, simp add: divide_inverse)
paulson@33667
  1361
hoelzl@51527
  1362
declare DERIV_ln_divide[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@51527
  1363
wenzelm@53079
  1364
lemma ln_series:
wenzelm@53079
  1365
  assumes "0 < x" and "x < 2"
wenzelm@53079
  1366
  shows "ln x = (\<Sum> n. (-1)^n * (1 / real (n + 1)) * (x - 1)^(Suc n))"
wenzelm@53079
  1367
  (is "ln x = suminf (?f (x - 1))")
hoelzl@29803
  1368
proof -
wenzelm@53079
  1369
  let ?f' = "\<lambda>x n. (-1)^n * (x - 1)^n"
hoelzl@29803
  1370
hoelzl@29803
  1371
  have "ln x - suminf (?f (x - 1)) = ln 1 - suminf (?f (1 - 1))"
hoelzl@29803
  1372
  proof (rule DERIV_isconst3[where x=x])
wenzelm@53079
  1373
    fix x :: real
wenzelm@53079
  1374
    assume "x \<in> {0 <..< 2}"
wenzelm@53079
  1375
    hence "0 < x" and "x < 2" by auto
wenzelm@53079
  1376
    have "norm (1 - x) < 1"
wenzelm@53079
  1377
      using `0 < x` and `x < 2` by auto
hoelzl@29803
  1378
    have "1 / x = 1 / (1 - (1 - x))" by auto
wenzelm@53079
  1379
    also have "\<dots> = (\<Sum> n. (1 - x)^n)"
wenzelm@53079
  1380
      using geometric_sums[OF `norm (1 - x) < 1`] by (rule sums_unique)
wenzelm@53079
  1381
    also have "\<dots> = suminf (?f' x)"
wenzelm@53079
  1382
      unfolding power_mult_distrib[symmetric]
wenzelm@53079
  1383
      by (rule arg_cong[where f=suminf], rule arg_cong[where f="op ^"], auto)
wenzelm@53079
  1384
    finally have "DERIV ln x :> suminf (?f' x)"
wenzelm@53079
  1385
      using DERIV_ln[OF `0 < x`] unfolding divide_inverse by auto
hoelzl@29803
  1386
    moreover
hoelzl@29803
  1387
    have repos: "\<And> h x :: real. h - 1 + x = h + x - 1" by auto
wenzelm@53079
  1388
    have "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :>
wenzelm@53079
  1389
      (\<Sum>n. (-1)^n * (1 / real (n + 1)) * real (Suc n) * (x - 1) ^ n)"
hoelzl@29803
  1390
    proof (rule DERIV_power_series')
wenzelm@53079
  1391
      show "x - 1 \<in> {- 1<..<1}" and "(0 :: real) < 1"
wenzelm@53079
  1392
        using `0 < x` `x < 2` by auto
wenzelm@53079
  1393
      fix x :: real
wenzelm@53079
  1394
      assume "x \<in> {- 1<..<1}"
wenzelm@53079
  1395
      hence "norm (-x) < 1" by auto
wenzelm@53079
  1396
      show "summable (\<lambda>n. -1 ^ n * (1 / real (n + 1)) * real (Suc n) * x ^ n)"
wenzelm@53079
  1397
        unfolding One_nat_def
wenzelm@53079
  1398
        by (auto simp add: power_mult_distrib[symmetric] summable_geometric[OF `norm (-x) < 1`])
hoelzl@29803
  1399
    qed
wenzelm@53079
  1400
    hence "DERIV (\<lambda>x. suminf (?f x)) (x - 1) :> suminf (?f' x)"
wenzelm@53079
  1401
      unfolding One_nat_def by auto
wenzelm@53079
  1402
    hence "DERIV (\<lambda>x. suminf (?f (x - 1))) x :> suminf (?f' x)"
wenzelm@53079
  1403
      unfolding DERIV_iff repos .
hoelzl@29803
  1404
    ultimately have "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> (suminf (?f' x) - suminf (?f' x))"
hoelzl@29803
  1405
      by (rule DERIV_diff)
hoelzl@29803
  1406
    thus "DERIV (\<lambda>x. ln x - suminf (?f (x - 1))) x :> 0" by auto
hoelzl@29803
  1407
  qed (auto simp add: assms)
huffman@44289
  1408
  thus ?thesis by auto
hoelzl@29803
  1409
qed
paulson@15077
  1410
hoelzl@50326
  1411
lemma exp_first_two_terms: "exp x = 1 + x + (\<Sum> n. inverse(fact (n+2)) * (x ^ (n+2)))"
hoelzl@50326
  1412
proof -
wenzelm@53079
  1413
  have "exp x = suminf (\<lambda>n. inverse(fact n) * (x ^ n))"
hoelzl@50326
  1414
    by (simp add: exp_def)
hoelzl@50326
  1415
  also from summable_exp have "... = (\<Sum> n::nat = 0 ..< 2. inverse(fact n) * (x ^ n)) +
hoelzl@50326
  1416
      (\<Sum> n. inverse(fact(n+2)) * (x ^ (n+2)))" (is "_ = ?a + _")
hoelzl@50326
  1417
    by (rule suminf_split_initial_segment)
hoelzl@50326
  1418
  also have "?a = 1 + x"
hoelzl@50326
  1419
    by (simp add: numeral_2_eq_2)
hoelzl@50326
  1420
  finally show ?thesis .
hoelzl@50326
  1421
qed
hoelzl@50326
  1422
wenzelm@53079
  1423
lemma exp_bound: "0 <= (x::real) \<Longrightarrow> x <= 1 \<Longrightarrow> exp x <= 1 + x + x\<^sup>2"
hoelzl@50326
  1424
proof -
hoelzl@50326
  1425
  assume a: "0 <= x"
hoelzl@50326
  1426
  assume b: "x <= 1"
wenzelm@53079
  1427
  {
wenzelm@53079
  1428
    fix n :: nat
hoelzl@50326
  1429
    have "2 * 2 ^ n \<le> fact (n + 2)"
wenzelm@53079
  1430
      by (induct n) simp_all
hoelzl@50326
  1431
    hence "real ((2::nat) * 2 ^ n) \<le> real (fact (n + 2))"
hoelzl@50326
  1432
      by (simp only: real_of_nat_le_iff)
hoelzl@50326
  1433
    hence "2 * 2 ^ n \<le> real (fact (n + 2))"
hoelzl@50326
  1434
      by simp
hoelzl@50326
  1435
    hence "inverse (fact (n + 2)) \<le> inverse (2 * 2 ^ n)"
hoelzl@50326
  1436
      by (rule le_imp_inverse_le) simp
hoelzl@50326
  1437
    hence "inverse (fact (n + 2)) \<le> 1/2 * (1/2)^n"
wenzelm@53079
  1438
      by (simp add: power_inverse)
wenzelm@53015
  1439
    hence "inverse (fact (n + 2)) * (x^n * x\<^sup>2) \<le> 1/2 * (1/2)^n * (1 * x\<^sup>2)"
hoelzl@50326
  1440
      by (rule mult_mono)
hoelzl@50326
  1441
        (rule mult_mono, simp_all add: power_le_one a b mult_nonneg_nonneg)
wenzelm@53015
  1442
    hence "inverse (fact (n + 2)) * x ^ (n + 2) \<le> (x\<^sup>2/2) * ((1/2)^n)"
hoelzl@50326
  1443
      unfolding power_add by (simp add: mult_ac del: fact_Suc) }
hoelzl@50326
  1444
  note aux1 = this
wenzelm@53015
  1445
  have "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums (x\<^sup>2 / 2 * (1 / (1 - 1 / 2)))"
hoelzl@50326
  1446
    by (intro sums_mult geometric_sums, simp)
wenzelm@53076
  1447
  hence aux2: "(\<lambda>n. x\<^sup>2 / 2 * (1 / 2) ^ n) sums x\<^sup>2"
hoelzl@50326
  1448
    by simp
wenzelm@53079
  1449
  have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <= x\<^sup>2"
hoelzl@50326
  1450
  proof -
wenzelm@53079
  1451
    have "suminf (\<lambda>n. inverse(fact (n+2)) * (x ^ (n+2))) <=
wenzelm@53079
  1452
        suminf (\<lambda>n. (x\<^sup>2/2) * ((1/2)^n))"
hoelzl@50326
  1453
      apply (rule summable_le)
hoelzl@50326
  1454
      apply (rule allI, rule aux1)
hoelzl@50326
  1455
      apply (rule summable_exp [THEN summable_ignore_initial_segment])
hoelzl@50326
  1456
      by (rule sums_summable, rule aux2)
wenzelm@53076
  1457
    also have "... = x\<^sup>2"
hoelzl@50326
  1458
      by (rule sums_unique [THEN sym], rule aux2)
hoelzl@50326
  1459
    finally show ?thesis .
hoelzl@50326
  1460
  qed
hoelzl@50326
  1461
  thus ?thesis unfolding exp_first_two_terms by auto
hoelzl@50326
  1462
qed
hoelzl@50326
  1463
wenzelm@53079
  1464
lemma ln_one_minus_pos_upper_bound: "0 <= x \<Longrightarrow> x < 1 \<Longrightarrow> ln (1 - x) <= - x"
hoelzl@50326
  1465
proof -
hoelzl@50326
  1466
  assume a: "0 <= (x::real)" and b: "x < 1"
wenzelm@53076
  1467
  have "(1 - x) * (1 + x + x\<^sup>2) = (1 - x^3)"
hoelzl@50326
  1468
    by (simp add: algebra_simps power2_eq_square power3_eq_cube)
hoelzl@50326
  1469
  also have "... <= 1"
hoelzl@50326
  1470
    by (auto simp add: a)
wenzelm@53076
  1471
  finally have "(1 - x) * (1 + x + x\<^sup>2) <= 1" .
wenzelm@53015
  1472
  moreover have c: "0 < 1 + x + x\<^sup>2"
hoelzl@50326
  1473
    by (simp add: add_pos_nonneg a)
wenzelm@53076
  1474
  ultimately have "1 - x <= 1 / (1 + x + x\<^sup>2)"
hoelzl@50326
  1475
    by (elim mult_imp_le_div_pos)
hoelzl@50326
  1476
  also have "... <= 1 / exp x"
paulson@54576
  1477
    by (metis a abs_one b exp_bound exp_gt_zero frac_le less_eq_real_def real_sqrt_abs 
paulson@54576
  1478
              real_sqrt_pow2_iff real_sqrt_power)
hoelzl@50326
  1479
  also have "... = exp (-x)"
hoelzl@50326
  1480
    by (auto simp add: exp_minus divide_inverse)
hoelzl@50326
  1481
  finally have "1 - x <= exp (- x)" .
hoelzl@50326
  1482
  also have "1 - x = exp (ln (1 - x))"
paulson@54576
  1483
    by (metis b diff_0 exp_ln_iff less_iff_diff_less_0 minus_diff_eq)
hoelzl@50326
  1484
  finally have "exp (ln (1 - x)) <= exp (- x)" .
hoelzl@50326
  1485
  thus ?thesis by (auto simp only: exp_le_cancel_iff)
hoelzl@50326
  1486
qed
hoelzl@50326
  1487
hoelzl@50326
  1488
lemma exp_ge_add_one_self [simp]: "1 + (x::real) <= exp x"
hoelzl@50326
  1489
  apply (case_tac "0 <= x")
hoelzl@50326
  1490
  apply (erule exp_ge_add_one_self_aux)
hoelzl@50326
  1491
  apply (case_tac "x <= -1")
hoelzl@50326
  1492
  apply (subgoal_tac "1 + x <= 0")
hoelzl@50326
  1493
  apply (erule order_trans)
hoelzl@50326
  1494
  apply simp
hoelzl@50326
  1495
  apply simp
hoelzl@50326
  1496
  apply (subgoal_tac "1 + x = exp(ln (1 + x))")
hoelzl@50326
  1497
  apply (erule ssubst)
hoelzl@50326
  1498
  apply (subst exp_le_cancel_iff)
hoelzl@50326
  1499
  apply (subgoal_tac "ln (1 - (- x)) <= - (- x)")
hoelzl@50326
  1500
  apply simp
hoelzl@50326
  1501
  apply (rule ln_one_minus_pos_upper_bound)
hoelzl@50326
  1502
  apply auto
hoelzl@50326
  1503
done
hoelzl@50326
  1504
wenzelm@53079
  1505
lemma ln_one_plus_pos_lower_bound: "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> x - x\<^sup>2 <= ln (1 + x)"
hoelzl@51527
  1506
proof -
hoelzl@51527
  1507
  assume a: "0 <= x" and b: "x <= 1"
wenzelm@53076
  1508
  have "exp (x - x\<^sup>2) = exp x / exp (x\<^sup>2)"
hoelzl@51527
  1509
    by (rule exp_diff)
wenzelm@53076
  1510
  also have "... <= (1 + x + x\<^sup>2) / exp (x \<^sup>2)"
paulson@54576
  1511
    by (metis a b divide_right_mono exp_bound exp_ge_zero)
wenzelm@53076
  1512
  also have "... <= (1 + x + x\<^sup>2) / (1 + x\<^sup>2)"
paulson@54576
  1513
    by (simp add: a divide_left_mono mult_pos_pos add_pos_nonneg)
hoelzl@51527
  1514
  also from a have "... <= 1 + x"
hoelzl@51527
  1515
    by (simp add: field_simps add_strict_increasing zero_le_mult_iff)
wenzelm@53076
  1516
  finally have "exp (x - x\<^sup>2) <= 1 + x" .
hoelzl@51527
  1517
  also have "... = exp (ln (1 + x))"
hoelzl@51527
  1518
  proof -
hoelzl@51527
  1519
    from a have "0 < 1 + x" by auto
hoelzl@51527
  1520
    thus ?thesis
hoelzl@51527
  1521
      by (auto simp only: exp_ln_iff [THEN sym])
hoelzl@51527
  1522
  qed
wenzelm@53076
  1523
  finally have "exp (x - x\<^sup>2) <= exp (ln (1 + x))" .
paulson@54576
  1524
  thus ?thesis
paulson@54576
  1525
    by (metis exp_le_cancel_iff) 
hoelzl@51527
  1526
qed
hoelzl@51527
  1527
wenzelm@53079
  1528
lemma ln_one_minus_pos_lower_bound:
wenzelm@53079
  1529
  "0 <= x \<Longrightarrow> x <= (1 / 2) \<Longrightarrow> - x - 2 * x\<^sup>2 <= ln (1 - x)"
hoelzl@51527
  1530
proof -
hoelzl@51527
  1531
  assume a: "0 <= x" and b: "x <= (1 / 2)"
wenzelm@53079
  1532
  from b have c: "x < 1" by auto
hoelzl@51527
  1533
  then have "ln (1 - x) = - ln (1 + x / (1 - x))"
paulson@54576
  1534
    apply (subst ln_inverse [symmetric])
paulson@54576
  1535
    apply (simp add: field_simps)
paulson@54576
  1536
    apply (rule arg_cong [where f=ln])
paulson@54576
  1537
    apply (simp add: field_simps)
paulson@54576
  1538
    done
hoelzl@51527
  1539
  also have "- (x / (1 - x)) <= ..."
wenzelm@53079
  1540
  proof -
hoelzl@51527
  1541
    have "ln (1 + x / (1 - x)) <= x / (1 - x)"
hoelzl@51527
  1542
      apply (rule ln_add_one_self_le_self)
hoelzl@51527
  1543
      apply (rule divide_nonneg_pos)
wenzelm@53079
  1544
      using a c apply auto
wenzelm@53079
  1545
      done
hoelzl@51527
  1546
    thus ?thesis
hoelzl@51527
  1547
      by auto
hoelzl@51527
  1548
  qed
hoelzl@51527
  1549
  also have "- (x / (1 - x)) = -x / (1 - x)"
hoelzl@51527
  1550
    by auto
hoelzl@51527
  1551
  finally have d: "- x / (1 - x) <= ln (1 - x)" .
hoelzl@51527
  1552
  have "0 < 1 - x" using a b by simp
wenzelm@53076
  1553
  hence e: "-x - 2 * x\<^sup>2 <= - x / (1 - x)"
hoelzl@51527
  1554
    using mult_right_le_one_le[of "x*x" "2*x"] a b
wenzelm@53079
  1555
    by (simp add: field_simps power2_eq_square)
wenzelm@53076
  1556
  from e d show "- x - 2 * x\<^sup>2 <= ln (1 - x)"
hoelzl@51527
  1557
    by (rule order_trans)
hoelzl@51527
  1558
qed
hoelzl@51527
  1559
wenzelm@53079
  1560
lemma ln_add_one_self_le_self2: "-1 < x \<Longrightarrow> ln(1 + x) <= x"
hoelzl@51527
  1561
  apply (subgoal_tac "ln (1 + x) \<le> ln (exp x)", simp)
hoelzl@51527
  1562
  apply (subst ln_le_cancel_iff)
hoelzl@51527
  1563
  apply auto
wenzelm@53079
  1564
  done
hoelzl@51527
  1565
hoelzl@51527
  1566
lemma abs_ln_one_plus_x_minus_x_bound_nonneg:
wenzelm@53079
  1567
  "0 <= x \<Longrightarrow> x <= 1 \<Longrightarrow> abs(ln (1 + x) - x) <= x\<^sup>2"
hoelzl@51527
  1568
proof -
hoelzl@51527
  1569
  assume x: "0 <= x"
hoelzl@51527
  1570
  assume x1: "x <= 1"
hoelzl@51527
  1571
  from x have "ln (1 + x) <= x"
hoelzl@51527
  1572
    by (rule ln_add_one_self_le_self)
wenzelm@53079
  1573
  then have "ln (1 + x) - x <= 0"
hoelzl@51527
  1574
    by simp
hoelzl@51527
  1575
  then have "abs(ln(1 + x) - x) = - (ln(1 + x) - x)"
hoelzl@51527
  1576
    by (rule abs_of_nonpos)
wenzelm@53079
  1577
  also have "... = x - ln (1 + x)"
hoelzl@51527
  1578
    by simp
wenzelm@53076
  1579
  also have "... <= x\<^sup>2"
hoelzl@51527
  1580
  proof -
wenzelm@53076
  1581
    from x x1 have "x - x\<^sup>2 <= ln (1 + x)"
hoelzl@51527
  1582
      by (intro ln_one_plus_pos_lower_bound)
hoelzl@51527
  1583
    thus ?thesis
hoelzl@51527
  1584
      by simp
hoelzl@51527
  1585
  qed
hoelzl@51527
  1586
  finally show ?thesis .
hoelzl@51527
  1587
qed
hoelzl@51527
  1588
hoelzl@51527
  1589
lemma abs_ln_one_plus_x_minus_x_bound_nonpos:
wenzelm@53079
  1590
  "-(1 / 2) <= x \<Longrightarrow> x <= 0 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
hoelzl@51527
  1591
proof -
hoelzl@51527
  1592
  assume a: "-(1 / 2) <= x"
hoelzl@51527
  1593
  assume b: "x <= 0"
wenzelm@53079
  1594
  have "abs(ln (1 + x) - x) = x - ln(1 - (-x))"
hoelzl@51527
  1595
    apply (subst abs_of_nonpos)
hoelzl@51527
  1596
    apply simp
hoelzl@51527
  1597
    apply (rule ln_add_one_self_le_self2)
hoelzl@51527
  1598
    using a apply auto
hoelzl@51527
  1599
    done
wenzelm@53076
  1600
  also have "... <= 2 * x\<^sup>2"
wenzelm@53076
  1601
    apply (subgoal_tac "- (-x) - 2 * (-x)\<^sup>2 <= ln (1 - (-x))")
hoelzl@51527
  1602
    apply (simp add: algebra_simps)
hoelzl@51527
  1603
    apply (rule ln_one_minus_pos_lower_bound)
hoelzl@51527
  1604
    using a b apply auto
hoelzl@51527
  1605
    done
hoelzl@51527
  1606
  finally show ?thesis .
hoelzl@51527
  1607
qed
hoelzl@51527
  1608
hoelzl@51527
  1609
lemma abs_ln_one_plus_x_minus_x_bound:
wenzelm@53079
  1610
    "abs x <= 1 / 2 \<Longrightarrow> abs(ln (1 + x) - x) <= 2 * x\<^sup>2"
hoelzl@51527
  1611
  apply (case_tac "0 <= x")
hoelzl@51527
  1612
  apply (rule order_trans)
hoelzl@51527
  1613
  apply (rule abs_ln_one_plus_x_minus_x_bound_nonneg)
hoelzl@51527
  1614
  apply auto
hoelzl@51527
  1615
  apply (rule abs_ln_one_plus_x_minus_x_bound_nonpos)
hoelzl@51527
  1616
  apply auto
wenzelm@53079
  1617
  done
wenzelm@53079
  1618
wenzelm@53079
  1619
lemma ln_x_over_x_mono: "exp 1 <= x \<Longrightarrow> x <= y \<Longrightarrow> (ln y / y) <= (ln x / x)"
hoelzl@51527
  1620
proof -
hoelzl@51527
  1621
  assume x: "exp 1 <= x" "x <= y"
hoelzl@51527
  1622
  moreover have "0 < exp (1::real)" by simp
hoelzl@51527
  1623
  ultimately have a: "0 < x" and b: "0 < y"
hoelzl@51527
  1624
    by (fast intro: less_le_trans order_trans)+
hoelzl@51527
  1625
  have "x * ln y - x * ln x = x * (ln y - ln x)"
hoelzl@51527
  1626
    by (simp add: algebra_simps)
hoelzl@51527
  1627
  also have "... = x * ln(y / x)"
hoelzl@51527
  1628
    by (simp only: ln_div a b)
hoelzl@51527
  1629
  also have "y / x = (x + (y - x)) / x"
hoelzl@51527
  1630
    by simp
hoelzl@51527
  1631
  also have "... = 1 + (y - x) / x"
hoelzl@51527
  1632
    using x a by (simp add: field_simps)
hoelzl@51527
  1633
  also have "x * ln(1 + (y - x) / x) <= x * ((y - x) / x)"
hoelzl@51527
  1634
    apply (rule mult_left_mono)
hoelzl@51527
  1635
    apply (rule ln_add_one_self_le_self)
hoelzl@51527
  1636
    apply (rule divide_nonneg_pos)
hoelzl@51527
  1637
    using x a apply simp_all
hoelzl@51527
  1638
    done
hoelzl@51527
  1639
  also have "... = y - x" using a by simp
hoelzl@51527
  1640
  also have "... = (y - x) * ln (exp 1)" by simp
hoelzl@51527
  1641
  also have "... <= (y - x) * ln x"
hoelzl@51527
  1642
    apply (rule mult_left_mono)
hoelzl@51527
  1643
    apply (subst ln_le_cancel_iff)
hoelzl@51527
  1644
    apply fact
hoelzl@51527
  1645
    apply (rule a)
hoelzl@51527
  1646
    apply (rule x)
hoelzl@51527
  1647
    using x apply simp
hoelzl@51527
  1648
    done
hoelzl@51527
  1649
  also have "... = y * ln x - x * ln x"
hoelzl@51527
  1650
    by (rule left_diff_distrib)
hoelzl@51527
  1651
  finally have "x * ln y <= y * ln x"
hoelzl@51527
  1652
    by arith
hoelzl@51527
  1653
  then have "ln y <= (y * ln x) / x" using a by (simp add: field_simps)
hoelzl@51527
  1654
  also have "... = y * (ln x / x)" by simp
hoelzl@51527
  1655
  finally show ?thesis using b by (simp add: field_simps)
hoelzl@51527
  1656
qed
hoelzl@51527
  1657
wenzelm@53079
  1658
lemma ln_le_minus_one: "0 < x \<Longrightarrow> ln x \<le> x - 1"
hoelzl@51527
  1659
  using exp_ge_add_one_self[of "ln x"] by simp
hoelzl@51527
  1660
hoelzl@51527
  1661
lemma ln_eq_minus_one:
wenzelm@53079
  1662
  assumes "0 < x" "ln x = x - 1"
wenzelm@53079
  1663
  shows "x = 1"
hoelzl@51527
  1664
proof -
wenzelm@53079
  1665
  let ?l = "\<lambda>y. ln y - y + 1"
hoelzl@51527
  1666
  have D: "\<And>x. 0 < x \<Longrightarrow> DERIV ?l x :> (1 / x - 1)"
hoelzl@51527
  1667
    by (auto intro!: DERIV_intros)
hoelzl@51527
  1668
hoelzl@51527
  1669
  show ?thesis
hoelzl@51527
  1670
  proof (cases rule: linorder_cases)
hoelzl@51527
  1671
    assume "x < 1"
hoelzl@51527
  1672
    from dense[OF `x < 1`] obtain a where "x < a" "a < 1" by blast
hoelzl@51527
  1673
    from `x < a` have "?l x < ?l a"
hoelzl@51527
  1674
    proof (rule DERIV_pos_imp_increasing, safe)
wenzelm@53079
  1675
      fix y
wenzelm@53079
  1676
      assume "x \<le> y" "y \<le> a"
hoelzl@51527
  1677
      with `0 < x` `a < 1` have "0 < 1 / y - 1" "0 < y"
hoelzl@51527
  1678
        by (auto simp: field_simps)
hoelzl@51527
  1679
      with D show "\<exists>z. DERIV ?l y :> z \<and> 0 < z"
hoelzl@51527
  1680
        by auto
hoelzl@51527
  1681
    qed
hoelzl@51527
  1682
    also have "\<dots> \<le> 0"
hoelzl@51527
  1683
      using ln_le_minus_one `0 < x` `x < a` by (auto simp: field_simps)
hoelzl@51527
  1684
    finally show "x = 1" using assms by auto
hoelzl@51527
  1685
  next
hoelzl@51527
  1686
    assume "1 < x"
wenzelm@53079
  1687
    from dense[OF this] obtain a where "1 < a" "a < x" by blast
hoelzl@51527
  1688
    from `a < x` have "?l x < ?l a"
hoelzl@51527
  1689
    proof (rule DERIV_neg_imp_decreasing, safe)
wenzelm@53079
  1690
      fix y
wenzelm@53079
  1691
      assume "a \<le> y" "y \<le> x"
hoelzl@51527
  1692
      with `1 < a` have "1 / y - 1 < 0" "0 < y"
hoelzl@51527
  1693
        by (auto simp: field_simps)
hoelzl@51527
  1694
      with D show "\<exists>z. DERIV ?l y :> z \<and> z < 0"
hoelzl@51527
  1695
        by blast
hoelzl@51527
  1696
    qed
hoelzl@51527
  1697
    also have "\<dots> \<le> 0"
hoelzl@51527
  1698
      using ln_le_minus_one `1 < a` by (auto simp: field_simps)
hoelzl@51527
  1699
    finally show "x = 1" using assms by auto
wenzelm@53079
  1700
  next
wenzelm@53079
  1701
    assume "x = 1"
wenzelm@53079
  1702
    then show ?thesis by simp
wenzelm@53079
  1703
  qed
hoelzl@51527
  1704
qed
hoelzl@51527
  1705
hoelzl@50326
  1706
lemma exp_at_bot: "(exp ---> (0::real)) at_bot"
hoelzl@50326
  1707
  unfolding tendsto_Zfun_iff
hoelzl@50326
  1708
proof (rule ZfunI, simp add: eventually_at_bot_dense)
hoelzl@50326
  1709
  fix r :: real assume "0 < r"
wenzelm@53079
  1710
  {
wenzelm@53079
  1711
    fix x
wenzelm@53079
  1712
    assume "x < ln r"
hoelzl@50326
  1713
    then have "exp x < exp (ln r)"
hoelzl@50326
  1714
      by simp
hoelzl@50326
  1715
    with `0 < r` have "exp x < r"
wenzelm@53079
  1716
      by simp
wenzelm@53079
  1717
  }
hoelzl@50326
  1718
  then show "\<exists>k. \<forall>n<k. exp n < r" by auto
hoelzl@50326
  1719
qed
hoelzl@50326
  1720
hoelzl@50326
  1721
lemma exp_at_top: "LIM x at_top. exp x :: real :> at_top"
hoelzl@50346
  1722
  by (rule filterlim_at_top_at_top[where Q="\<lambda>x. True" and P="\<lambda>x. 0 < x" and g="ln"])
hoelzl@50346
  1723
     (auto intro: eventually_gt_at_top)
hoelzl@50326
  1724
hoelzl@50326
  1725
lemma ln_at_0: "LIM x at_right 0. ln x :> at_bot"
hoelzl@50346
  1726
  by (rule filterlim_at_bot_at_right[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
hoelzl@51641
  1727
     (auto simp: eventually_at_filter)
hoelzl@50326
  1728
hoelzl@50326
  1729
lemma ln_at_top: "LIM x at_top. ln x :> at_top"
hoelzl@50346
  1730
  by (rule filterlim_at_top_at_top[where Q="\<lambda>x. 0 < x" and P="\<lambda>x. True" and g="exp"])
hoelzl@50346
  1731
     (auto intro: eventually_gt_at_top)
hoelzl@50326
  1732
hoelzl@50347
  1733
lemma tendsto_power_div_exp_0: "((\<lambda>x. x ^ k / exp x) ---> (0::real)) at_top"
hoelzl@50347
  1734
proof (induct k)
wenzelm@53079
  1735
  case 0
hoelzl@50347
  1736
  show "((\<lambda>x. x ^ 0 / exp x) ---> (0::real)) at_top"
hoelzl@50347
  1737
    by (simp add: inverse_eq_divide[symmetric])
hoelzl@50347
  1738
       (metis filterlim_compose[OF tendsto_inverse_0] exp_at_top filterlim_mono
hoelzl@50347
  1739
              at_top_le_at_infinity order_refl)
hoelzl@50347
  1740
next
hoelzl@50347
  1741
  case (Suc k)
hoelzl@50347
  1742
  show ?case
hoelzl@50347
  1743
  proof (rule lhospital_at_top_at_top)
hoelzl@50347
  1744
    show "eventually (\<lambda>x. DERIV (\<lambda>x. x ^ Suc k) x :> (real (Suc k) * x^k)) at_top"
hoelzl@50347
  1745
      by eventually_elim (intro DERIV_intros, simp, simp)
hoelzl@50347
  1746
    show "eventually (\<lambda>x. DERIV exp x :> exp x) at_top"
hoelzl@50347
  1747
      by eventually_elim (auto intro!: DERIV_intros)
hoelzl@50347
  1748
    show "eventually (\<lambda>x. exp x \<noteq> 0) at_top"
hoelzl@50347
  1749
      by auto
hoelzl@50347
  1750
    from tendsto_mult[OF tendsto_const Suc, of "real (Suc k)"]
hoelzl@50347
  1751
    show "((\<lambda>x. real (Suc k) * x ^ k / exp x) ---> 0) at_top"
hoelzl@50347
  1752
      by simp
hoelzl@50347
  1753
  qed (rule exp_at_top)
hoelzl@50347
  1754
qed
hoelzl@50347
  1755
hoelzl@51527
  1756
wenzelm@53079
  1757
definition powr :: "[real,real] => real"  (infixr "powr" 80)
wenzelm@53079
  1758
  -- {*exponentation with real exponent*}
wenzelm@53079
  1759
  where "x powr a = exp(a * ln x)"
wenzelm@53079
  1760
wenzelm@53079
  1761
definition log :: "[real,real] => real"
wenzelm@53079
  1762
  -- {*logarithm of @{term x} to base @{term a}*}
wenzelm@53079
  1763
  where "log a x = ln x / ln a"
hoelzl@51527
  1764
hoelzl@51527
  1765
hoelzl@51527
  1766
lemma tendsto_log [tendsto_intros]:
hoelzl@51527
  1767
  "\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a; a \<noteq> 1; 0 < b\<rbrakk> \<Longrightarrow> ((\<lambda>x. log (f x) (g x)) ---> log a b) F"
hoelzl@51527
  1768
  unfolding log_def by (intro tendsto_intros) auto
hoelzl@51527
  1769
hoelzl@51527
  1770
lemma continuous_log:
wenzelm@53079
  1771
  assumes "continuous F f"
wenzelm@53079
  1772
    and "continuous F g"
wenzelm@53079
  1773
    and "0 < f (Lim F (\<lambda>x. x))"
wenzelm@53079
  1774
    and "f (Lim F (\<lambda>x. x)) \<noteq> 1"
wenzelm@53079
  1775
    and "0 < g (Lim F (\<lambda>x. x))"
hoelzl@51527
  1776
  shows "continuous F (\<lambda>x. log (f x) (g x))"
hoelzl@51527
  1777
  using assms unfolding continuous_def by (rule tendsto_log)
hoelzl@51527
  1778
hoelzl@51527
  1779
lemma continuous_at_within_log[continuous_intros]:
wenzelm@53079
  1780
  assumes "continuous (at a within s) f"
wenzelm@53079
  1781
    and "continuous (at a within s) g"
wenzelm@53079
  1782
    and "0 < f a"
wenzelm@53079
  1783
    and "f a \<noteq> 1"
wenzelm@53079
  1784
    and "0 < g a"
hoelzl@51527
  1785
  shows "continuous (at a within s) (\<lambda>x. log (f x) (g x))"
hoelzl@51527
  1786
  using assms unfolding continuous_within by (rule tendsto_log)
hoelzl@51527
  1787
hoelzl@51527
  1788
lemma isCont_log[continuous_intros, simp]:
hoelzl@51527
  1789
  assumes "isCont f a" "isCont g a" "0 < f a" "f a \<noteq> 1" "0 < g a"
hoelzl@51527
  1790
  shows "isCont (\<lambda>x. log (f x) (g x)) a"
hoelzl@51527
  1791
  using assms unfolding continuous_at by (rule tendsto_log)
hoelzl@51527
  1792
hoelzl@51527
  1793
lemma continuous_on_log[continuous_on_intros]:
wenzelm@53079
  1794
  assumes "continuous_on s f" "continuous_on s g"
wenzelm@53079
  1795
    and "\<forall>x\<in>s. 0 < f x" "\<forall>x\<in>s. f x \<noteq> 1" "\<forall>x\<in>s. 0 < g x"
hoelzl@51527
  1796
  shows "continuous_on s (\<lambda>x. log (f x) (g x))"
hoelzl@51527
  1797
  using assms unfolding continuous_on_def by (fast intro: tendsto_log)
hoelzl@51527
  1798
hoelzl@51527
  1799
lemma powr_one_eq_one [simp]: "1 powr a = 1"
wenzelm@53079
  1800
  by (simp add: powr_def)
hoelzl@51527
  1801
hoelzl@51527
  1802
lemma powr_zero_eq_one [simp]: "x powr 0 = 1"
wenzelm@53079
  1803
  by (simp add: powr_def)
hoelzl@51527
  1804
hoelzl@51527
  1805
lemma powr_one_gt_zero_iff [simp]: "(x powr 1 = x) = (0 < x)"
wenzelm@53079
  1806
  by (simp add: powr_def)
hoelzl@51527
  1807
declare powr_one_gt_zero_iff [THEN iffD2, simp]
hoelzl@51527
  1808
wenzelm@53079
  1809
lemma powr_mult: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x * y) powr a = (x powr a) * (y powr a)"
wenzelm@53079
  1810
  by (simp add: powr_def exp_add [symmetric] ln_mult distrib_left)
hoelzl@51527
  1811
hoelzl@51527
  1812
lemma powr_gt_zero [simp]: "0 < x powr a"
wenzelm@53079
  1813
  by (simp add: powr_def)
hoelzl@51527
  1814
hoelzl@51527
  1815
lemma powr_ge_pzero [simp]: "0 <= x powr y"
wenzelm@53079
  1816
  by (rule order_less_imp_le, rule powr_gt_zero)
hoelzl@51527
  1817
hoelzl@51527
  1818
lemma powr_not_zero [simp]: "x powr a \<noteq> 0"
wenzelm@53079
  1819
  by (simp add: powr_def)
wenzelm@53079
  1820
wenzelm@53079
  1821
lemma powr_divide: "0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (x / y) powr a = (x powr a) / (y powr a)"
wenzelm@53079
  1822
  apply (simp add: divide_inverse positive_imp_inverse_positive powr_mult)
wenzelm@53079
  1823
  apply (simp add: powr_def exp_minus [symmetric] exp_add [symmetric] ln_inverse)
wenzelm@53079
  1824
  done
hoelzl@51527
  1825
hoelzl@51527
  1826
lemma powr_divide2: "x powr a / x powr b = x powr (a - b)"
hoelzl@51527
  1827
  apply (simp add: powr_def)
hoelzl@51527
  1828
  apply (subst exp_diff [THEN sym])
hoelzl@51527
  1829
  apply (simp add: left_diff_distrib)
wenzelm@53079
  1830
  done
hoelzl@51527
  1831
hoelzl@51527
  1832
lemma powr_add: "x powr (a + b) = (x powr a) * (x powr b)"
wenzelm@53079
  1833
  by (simp add: powr_def exp_add [symmetric] distrib_right)
wenzelm@53079
  1834
wenzelm@53079
  1835
lemma powr_mult_base: "0 < x \<Longrightarrow>x * x powr y = x powr (1 + y)"
wenzelm@53079
  1836
  using assms by (auto simp: powr_add)
hoelzl@51527
  1837
hoelzl@51527
  1838
lemma powr_powr: "(x powr a) powr b = x powr (a * b)"
wenzelm@53079
  1839
  by (simp add: powr_def)
hoelzl@51527
  1840
hoelzl@51527
  1841
lemma powr_powr_swap: "(x powr a) powr b = (x powr b) powr a"
wenzelm@53079
  1842
  by (simp add: powr_powr mult_commute)
hoelzl@51527
  1843
hoelzl@51527
  1844
lemma powr_minus: "x powr (-a) = inverse (x powr a)"
wenzelm@53079
  1845
  by (simp add: powr_def exp_minus [symmetric])
hoelzl@51527
  1846
hoelzl@51527
  1847
lemma powr_minus_divide: "x powr (-a) = 1/(x powr a)"
wenzelm@53079
  1848
  by (simp add: divide_inverse powr_minus)
wenzelm@53079
  1849
wenzelm@53079
  1850
lemma powr_less_mono: "a < b \<Longrightarrow> 1 < x \<Longrightarrow> x powr a < x powr b"
wenzelm@53079
  1851
  by (simp add: powr_def)
wenzelm@53079
  1852
wenzelm@53079
  1853
lemma powr_less_cancel: "x powr a < x powr b \<Longrightarrow> 1 < x \<Longrightarrow> a < b"
wenzelm@53079
  1854
  by (simp add: powr_def)
wenzelm@53079
  1855
wenzelm@53079
  1856
lemma powr_less_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a < x powr b) = (a < b)"
wenzelm@53079
  1857
  by (blast intro: powr_less_cancel powr_less_mono)
wenzelm@53079
  1858
wenzelm@53079
  1859
lemma powr_le_cancel_iff [simp]: "1 < x \<Longrightarrow> (x powr a \<le> x powr b) = (a \<le> b)"
wenzelm@53079
  1860
  by (simp add: linorder_not_less [symmetric])
hoelzl@51527
  1861
hoelzl@51527
  1862
lemma log_ln: "ln x = log (exp(1)) x"
wenzelm@53079
  1863
  by (simp add: log_def)
wenzelm@53079
  1864
wenzelm@53079
  1865
lemma DERIV_log:
wenzelm@53079
  1866
  assumes "x > 0"
wenzelm@53079
  1867
  shows "DERIV (\<lambda>y. log b y) x :> 1 / (ln b * x)"
hoelzl@51527
  1868
proof -
hoelzl@51527
  1869
  def lb \<equiv> "1 / ln b"
hoelzl@51527
  1870
  moreover have "DERIV (\<lambda>y. lb * ln y) x :> lb / x"
hoelzl@51527
  1871
    using `x > 0` by (auto intro!: DERIV_intros)
hoelzl@51527
  1872
  ultimately show ?thesis
hoelzl@51527
  1873
    by (simp add: log_def)
hoelzl@51527
  1874
qed
hoelzl@51527
  1875
hoelzl@51527
  1876
lemmas DERIV_log[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@51527
  1877
wenzelm@53079
  1878
lemma powr_log_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> a powr (log a x) = x"
wenzelm@53079
  1879
  by (simp add: powr_def log_def)
wenzelm@53079
  1880
wenzelm@53079
  1881
lemma log_powr_cancel [simp]: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> log a (a powr y) = y"
wenzelm@53079
  1882
  by (simp add: log_def powr_def)
wenzelm@53079
  1883
wenzelm@53079
  1884
lemma log_mult:
wenzelm@53079
  1885
  "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow>
wenzelm@53079
  1886
    log a (x * y) = log a x + log a y"
wenzelm@53079
  1887
  by (simp add: log_def ln_mult divide_inverse distrib_right)
wenzelm@53079
  1888
wenzelm@53079
  1889
lemma log_eq_div_ln_mult_log:
wenzelm@53079
  1890
  "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < b \<Longrightarrow> b \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow>
wenzelm@53079
  1891
    log a x = (ln b/ln a) * log b x"
wenzelm@53079
  1892
  by (simp add: log_def divide_inverse)
hoelzl@51527
  1893
hoelzl@51527
  1894
text{*Base 10 logarithms*}
wenzelm@53079
  1895
lemma log_base_10_eq1: "0 < x \<Longrightarrow> log 10 x = (ln (exp 1) / ln 10) * ln x"
wenzelm@53079
  1896
  by (simp add: log_def)
wenzelm@53079
  1897
wenzelm@53079
  1898
lemma log_base_10_eq2: "0 < x \<Longrightarrow> log 10 x = (log 10 (exp 1)) * ln x"
wenzelm@53079
  1899
  by (simp add: log_def)
hoelzl@51527
  1900
hoelzl@51527
  1901
lemma log_one [simp]: "log a 1 = 0"
wenzelm@53079
  1902
  by (simp add: log_def)
hoelzl@51527
  1903
hoelzl@51527
  1904
lemma log_eq_one [simp]: "[| 0 < a; a \<noteq> 1 |] ==> log a a = 1"
wenzelm@53079
  1905
  by (simp add: log_def)
wenzelm@53079
  1906
wenzelm@53079
  1907
lemma log_inverse: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> log a (inverse x) = - log a x"
wenzelm@53079
  1908
  apply (rule_tac a1 = "log a x" in add_left_cancel [THEN iffD1])
wenzelm@53079
  1909
  apply (simp add: log_mult [symmetric])
wenzelm@53079
  1910
  done
wenzelm@53079
  1911
wenzelm@53079
  1912
lemma log_divide: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a (x/y) = log a x - log a y"
wenzelm@53079
  1913
  by (simp add: log_mult divide_inverse log_inverse)
hoelzl@51527
  1914
hoelzl@51527
  1915
lemma log_less_cancel_iff [simp]:
wenzelm@53079
  1916
  "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> log a x < log a y \<longleftrightarrow> x < y"
wenzelm@53079
  1917
  apply safe
wenzelm@53079
  1918
  apply (rule_tac [2] powr_less_cancel)
wenzelm@53079
  1919
  apply (drule_tac a = "log a x" in powr_less_mono, auto)
wenzelm@53079
  1920
  done
wenzelm@53079
  1921
wenzelm@53079
  1922
lemma log_inj:
wenzelm@53079
  1923
  assumes "1 < b"
wenzelm@53079
  1924
  shows "inj_on (log b) {0 <..}"
hoelzl@51527
  1925
proof (rule inj_onI, simp)
wenzelm@53079
  1926
  fix x y
wenzelm@53079
  1927
  assume pos: "0 < x" "0 < y" and *: "log b x = log b y"
hoelzl@51527
  1928
  show "x = y"
hoelzl@51527
  1929
  proof (cases rule: linorder_cases)
wenzelm@53079
  1930
    assume "x = y"
wenzelm@53079
  1931
    then show ?thesis by simp
wenzelm@53079
  1932
  next
hoelzl@51527
  1933
    assume "x < y" hence "log b x < log b y"
hoelzl@51527
  1934
      using log_less_cancel_iff[OF `1 < b`] pos by simp
wenzelm@53079
  1935
    then show ?thesis using * by simp
hoelzl@51527
  1936
  next
hoelzl@51527
  1937
    assume "y < x" hence "log b y < log b x"
hoelzl@51527
  1938
      using log_less_cancel_iff[OF `1 < b`] pos by simp
wenzelm@53079
  1939
    then show ?thesis using * by simp
wenzelm@53079
  1940
  qed
hoelzl@51527
  1941
qed
hoelzl@51527
  1942
hoelzl@51527
  1943
lemma log_le_cancel_iff [simp]:
wenzelm@53079
  1944
  "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < y \<Longrightarrow> (log a x \<le> log a y) = (x \<le> y)"
wenzelm@53079
  1945
  by (simp add: linorder_not_less [symmetric])
hoelzl@51527
  1946
hoelzl@51527
  1947
lemma zero_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 < log a x \<longleftrightarrow> 1 < x"
hoelzl@51527
  1948
  using log_less_cancel_iff[of a 1 x] by simp
hoelzl@51527
  1949
hoelzl@51527
  1950
lemma zero_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 0 \<le> log a x \<longleftrightarrow> 1 \<le> x"
hoelzl@51527
  1951
  using log_le_cancel_iff[of a 1 x] by simp
hoelzl@51527
  1952
hoelzl@51527
  1953
lemma log_less_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 0 \<longleftrightarrow> x < 1"
hoelzl@51527
  1954
  using log_less_cancel_iff[of a x 1] by simp
hoelzl@51527
  1955
hoelzl@51527
  1956
lemma log_le_zero_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 0 \<longleftrightarrow> x \<le> 1"
hoelzl@51527
  1957
  using log_le_cancel_iff[of a x 1] by simp
hoelzl@51527
  1958
hoelzl@51527
  1959
lemma one_less_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 < log a x \<longleftrightarrow> a < x"
hoelzl@51527
  1960
  using log_less_cancel_iff[of a a x] by simp
hoelzl@51527
  1961
hoelzl@51527
  1962
lemma one_le_log_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> 1 \<le> log a x \<longleftrightarrow> a \<le> x"
hoelzl@51527
  1963
  using log_le_cancel_iff[of a a x] by simp
hoelzl@51527
  1964
hoelzl@51527
  1965
lemma log_less_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x < 1 \<longleftrightarrow> x < a"
hoelzl@51527
  1966
  using log_less_cancel_iff[of a x a] by simp
hoelzl@51527
  1967
hoelzl@51527
  1968
lemma log_le_one_cancel_iff[simp]: "1 < a \<Longrightarrow> 0 < x \<Longrightarrow> log a x \<le> 1 \<longleftrightarrow> x \<le> a"
hoelzl@51527
  1969
  using log_le_cancel_iff[of a x a] by simp
hoelzl@51527
  1970
hoelzl@51527
  1971
lemma powr_realpow: "0 < x ==> x powr (real n) = x^n"
wenzelm@53079
  1972
  apply (induct n)
wenzelm@53079
  1973
  apply simp
hoelzl@51527
  1974
  apply (subgoal_tac "real(Suc n) = real n + 1")
hoelzl@51527
  1975
  apply (erule ssubst)
hoelzl@51527
  1976
  apply (subst powr_add, simp, simp)
wenzelm@53079
  1977
  done
hoelzl@51527
  1978
haftmann@54489
  1979
lemma powr_realpow_numeral: "0 < x \<Longrightarrow> x powr (numeral n :: real) = x ^ (numeral n)"
haftmann@54489
  1980
  unfolding real_of_nat_numeral [symmetric] by (rule powr_realpow)
noschinl@52139
  1981
hoelzl@51527
  1982
lemma powr_realpow2: "0 <= x ==> 0 < n ==> x^n = (if (x = 0) then 0 else x powr (real n))"
hoelzl@51527
  1983
  apply (case_tac "x = 0", simp, simp)
hoelzl@51527
  1984
  apply (rule powr_realpow [THEN sym], simp)
wenzelm@53079
  1985
  done
hoelzl@51527
  1986
hoelzl@51527
  1987
lemma powr_int:
hoelzl@51527
  1988
  assumes "x > 0"
hoelzl@51527
  1989
  shows "x powr i = (if i \<ge> 0 then x ^ nat i else 1 / x ^ nat (-i))"
wenzelm@53079
  1990
proof (cases "i < 0")
wenzelm@53079
  1991
  case True
hoelzl@51527
  1992
  have r: "x powr i = 1 / x powr (-i)" by (simp add: powr_minus field_simps)
hoelzl@51527
  1993
  show ?thesis using `i < 0` `x > 0` by (simp add: r field_simps powr_realpow[symmetric])
wenzelm@53079
  1994
next
wenzelm@53079
  1995
  case False
wenzelm@53079
  1996
  then show ?thesis by (simp add: assms powr_realpow[symmetric])
wenzelm@53079
  1997
qed
hoelzl@51527
  1998
haftmann@54489
  1999
lemma powr_one: "0 < x \<Longrightarrow> x powr 1 = x"
haftmann@54489
  2000
  using powr_realpow [of x 1] by simp
haftmann@54489
  2001
haftmann@54489
  2002
lemma powr_numeral: "0 < x \<Longrightarrow> x powr numeral n = x ^ numeral n"
haftmann@54489
  2003
  by (fact powr_realpow_numeral)
haftmann@54489
  2004
haftmann@54489
  2005
lemma powr_neg_one: "0 < x \<Longrightarrow> x powr - 1 = 1 / x"
haftmann@54489
  2006
  using powr_int [of x "- 1"] by simp
haftmann@54489
  2007
haftmann@54489
  2008
lemma powr_neg_numeral: "0 < x \<Longrightarrow> x powr - numeral n = 1 / x ^ numeral n"
haftmann@54489
  2009
  using powr_int [of x "- numeral n"] by simp
hoelzl@51527
  2010
wenzelm@53079
  2011
lemma root_powr_inverse: "0 < n \<Longrightarrow> 0 < x \<Longrightarrow> root n x = x powr (1/n)"
hoelzl@51527
  2012
  by (rule real_root_pos_unique) (auto simp: powr_realpow[symmetric] powr_powr)
hoelzl@51527
  2013
hoelzl@51527
  2014
lemma ln_powr: "0 < x ==> 0 < y ==> ln(x powr y) = y * ln x"
wenzelm@53079
  2015
  unfolding powr_def by simp
hoelzl@51527
  2016
hoelzl@51527
  2017
lemma log_powr: "0 < x ==> 0 \<le> y ==> log b (x powr y) = y * log b x"
wenzelm@53079
  2018
  apply (cases "y = 0")
hoelzl@51527
  2019
  apply force
hoelzl@51527
  2020
  apply (auto simp add: log_def ln_powr field_simps)
wenzelm@53079
  2021
  done
hoelzl@51527
  2022
hoelzl@51527
  2023
lemma log_nat_power: "0 < x ==> log b (x^n) = real n * log b x"
hoelzl@51527
  2024
  apply (subst powr_realpow [symmetric])
hoelzl@51527
  2025
  apply (auto simp add: log_powr)
wenzelm@53079
  2026
  done
hoelzl@51527
  2027
hoelzl@51527
  2028
lemma ln_bound: "1 <= x ==> ln x <= x"
hoelzl@51527
  2029
  apply (subgoal_tac "ln(1 + (x - 1)) <= x - 1")
hoelzl@51527
  2030
  apply simp
hoelzl@51527
  2031
  apply (rule ln_add_one_self_le_self, simp)
wenzelm@53079
  2032
  done
hoelzl@51527
  2033
hoelzl@51527
  2034
lemma powr_mono: "a <= b ==> 1 <= x ==> x powr a <= x powr b"
wenzelm@53079
  2035
  apply (cases "x = 1", simp)
wenzelm@53079
  2036
  apply (cases "a = b", simp)
hoelzl@51527
  2037
  apply (rule order_less_imp_le)
hoelzl@51527
  2038
  apply (rule powr_less_mono, auto)
wenzelm@53079
  2039
  done
hoelzl@51527
  2040
hoelzl@51527
  2041
lemma ge_one_powr_ge_zero: "1 <= x ==> 0 <= a ==> 1 <= x powr a"
hoelzl@51527
  2042
  apply (subst powr_zero_eq_one [THEN sym])
hoelzl@51527
  2043
  apply (rule powr_mono, assumption+)
wenzelm@53079
  2044
  done
wenzelm@53079
  2045
wenzelm@53079
  2046
lemma powr_less_mono2: "0 < a ==> 0 < x ==> x < y ==> x powr a < y powr a"
hoelzl@51527
  2047
  apply (unfold powr_def)
hoelzl@51527
  2048
  apply (rule exp_less_mono)
hoelzl@51527
  2049
  apply (rule mult_strict_left_mono)
hoelzl@51527
  2050
  apply (subst ln_less_cancel_iff, assumption)
hoelzl@51527
  2051
  apply (rule order_less_trans)
hoelzl@51527
  2052
  prefer 2
hoelzl@51527
  2053
  apply assumption+
wenzelm@53079
  2054
  done
wenzelm@53079
  2055
wenzelm@53079
  2056
lemma powr_less_mono2_neg: "a < 0 ==> 0 < x ==> x < y ==> y powr a < x powr a"
hoelzl@51527
  2057
  apply (unfold powr_def)
hoelzl@51527
  2058
  apply (rule exp_less_mono)
hoelzl@51527
  2059
  apply (rule mult_strict_left_mono_neg)
hoelzl@51527
  2060
  apply (subst ln_less_cancel_iff)
hoelzl@51527
  2061
  apply assumption
hoelzl@51527
  2062
  apply (rule order_less_trans)
hoelzl@51527
  2063
  prefer 2
hoelzl@51527
  2064
  apply assumption+
wenzelm@53079
  2065
  done
hoelzl@51527
  2066
hoelzl@51527
  2067
lemma powr_mono2: "0 <= a ==> 0 < x ==> x <= y ==> x powr a <= y powr a"
hoelzl@51527
  2068
  apply (case_tac "a = 0", simp)
hoelzl@51527
  2069
  apply (case_tac "x = y", simp)
paulson@54575
  2070
  apply (metis less_eq_real_def powr_less_mono2)
wenzelm@53079
  2071
  done
wenzelm@53079
  2072
wenzelm@53079
  2073
lemma powr_inj: "0 < a \<Longrightarrow> a \<noteq> 1 \<Longrightarrow> a powr x = a powr y \<longleftrightarrow> x = y"
hoelzl@51527
  2074
  unfolding powr_def exp_inj_iff by simp
hoelzl@51527
  2075
hoelzl@51527
  2076
lemma ln_powr_bound: "1 <= x ==> 0 < a ==> ln x <= (x powr a) / a"
paulson@54575
  2077
  by (metis less_eq_real_def ln_less_self mult_imp_le_div_pos ln_powr mult_commute 
paulson@54575
  2078
            order.strict_trans2 powr_gt_zero zero_less_one)
hoelzl@51527
  2079
hoelzl@51527
  2080
lemma ln_powr_bound2:
hoelzl@51527
  2081
  assumes "1 < x" and "0 < a"
hoelzl@51527
  2082
  shows "(ln x) powr a <= (a powr a) * x"
hoelzl@51527
  2083
proof -
hoelzl@51527
  2084
  from assms have "ln x <= (x powr (1 / a)) / (1 / a)"
paulson@54575
  2085
    by (metis less_eq_real_def ln_powr_bound zero_less_divide_1_iff)
hoelzl@51527
  2086
  also have "... = a * (x powr (1 / a))"
hoelzl@51527
  2087
    by simp
hoelzl@51527
  2088
  finally have "(ln x) powr a <= (a * (x powr (1 / a))) powr a"
paulson@54575
  2089
    by (metis assms less_imp_le ln_gt_zero powr_mono2)
hoelzl@51527
  2090
  also have "... = (a powr a) * ((x powr (1 / a)) powr a)"
paulson@54575
  2091
    by (metis assms(2) powr_mult powr_gt_zero)
hoelzl@51527
  2092
  also have "(x powr (1 / a)) powr a = x powr ((1 / a) * a)"
hoelzl@51527
  2093
    by (rule powr_powr)
paulson@54575
  2094
  also have "... = x" using assms
paulson@54575
  2095
    by auto
hoelzl@51527
  2096
  finally show ?thesis .
hoelzl@51527
  2097
qed
hoelzl@51527
  2098
hoelzl@51527
  2099
lemma tendsto_powr [tendsto_intros]:
hoelzl@51527
  2100
  "\<lbrakk>(f ---> a) F; (g ---> b) F; 0 < a\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x powr g x) ---> a powr b) F"
hoelzl@51527
  2101
  unfolding powr_def by (intro tendsto_intros)
hoelzl@51527
  2102
hoelzl@51527
  2103
lemma continuous_powr:
wenzelm@53079
  2104
  assumes "continuous F f"
wenzelm@53079
  2105
    and "continuous F g"
wenzelm@53079
  2106
    and "0 < f (Lim F (\<lambda>x. x))"
hoelzl@51527
  2107
  shows "continuous F (\<lambda>x. (f x) powr (g x))"
hoelzl@51527
  2108
  using assms unfolding continuous_def by (rule tendsto_powr)
hoelzl@51527
  2109
hoelzl@51527
  2110
lemma continuous_at_within_powr[continuous_intros]:
wenzelm@53079
  2111
  assumes "continuous (at a within s) f"
wenzelm@53079
  2112
    and "continuous (at a within s) g"
wenzelm@53079
  2113
    and "0 < f a"
hoelzl@51527
  2114
  shows "continuous (at a within s) (\<lambda>x. (f x) powr (g x))"
hoelzl@51527
  2115
  using assms unfolding continuous_within by (rule tendsto_powr)
hoelzl@51527
  2116
hoelzl@51527
  2117
lemma isCont_powr[continuous_intros, simp]:
hoelzl@51527
  2118
  assumes "isCont f a" "isCont g a" "0 < f a"
hoelzl@51527
  2119
  shows "isCont (\<lambda>x. (f x) powr g x) a"
hoelzl@51527
  2120
  using assms unfolding continuous_at by (rule tendsto_powr)
hoelzl@51527
  2121
hoelzl@51527
  2122
lemma continuous_on_powr[continuous_on_intros]:
hoelzl@51527
  2123
  assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. 0 < f x"
hoelzl@51527
  2124
  shows "continuous_on s (\<lambda>x. (f x) powr (g x))"
hoelzl@51527
  2125
  using assms unfolding continuous_on_def by (fast intro: tendsto_powr)
hoelzl@51527
  2126
hoelzl@51527
  2127
(* FIXME: generalize by replacing d by with g x and g ---> d? *)
hoelzl@51527
  2128
lemma tendsto_zero_powrI:
hoelzl@51527
  2129
  assumes "eventually (\<lambda>x. 0 < f x ) F" and "(f ---> 0) F"
wenzelm@53079
  2130
    and "0 < d"
hoelzl@51527
  2131
  shows "((\<lambda>x. f x powr d) ---> 0) F"
hoelzl@51527
  2132
proof (rule tendstoI)
hoelzl@51527
  2133
  fix e :: real assume "0 < e"
hoelzl@51527
  2134
  def Z \<equiv> "e powr (1 / d)"
hoelzl@51527
  2135
  with `0 < e` have "0 < Z" by simp
hoelzl@51527
  2136
  with assms have "eventually (\<lambda>x. 0 < f x \<and> dist (f x) 0 < Z) F"
hoelzl@51527
  2137
    by (intro eventually_conj tendstoD)
hoelzl@51527
  2138
  moreover
hoelzl@51527
  2139
  from assms have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> x powr d < Z powr d"
hoelzl@51527
  2140
    by (intro powr_less_mono2) (auto simp: dist_real_def)
hoelzl@51527
  2141
  with assms `0 < e` have "\<And>x. 0 < x \<and> dist x 0 < Z \<Longrightarrow> dist (x powr d) 0 < e"
hoelzl@51527
  2142
    unfolding dist_real_def Z_def by (auto simp: powr_powr)
hoelzl@51527
  2143
  ultimately
hoelzl@51527
  2144
  show "eventually (\<lambda>x. dist (f x powr d) 0 < e) F" by (rule eventually_elim1)
hoelzl@51527
  2145
qed
hoelzl@51527
  2146
hoelzl@51527
  2147
lemma tendsto_neg_powr:
wenzelm@53079
  2148
  assumes "s < 0"
wenzelm@53079
  2149
    and "LIM x F. f x :> at_top"
hoelzl@51527
  2150
  shows "((\<lambda>x. f x powr s) ---> 0) F"
hoelzl@51527
  2151
proof (rule tendstoI)
hoelzl@51527
  2152
  fix e :: real assume "0 < e"
hoelzl@51527
  2153
  def Z \<equiv> "e powr (1 / s)"
hoelzl@51527
  2154
  from assms have "eventually (\<lambda>x. Z < f x) F"
hoelzl@51527
  2155
    by (simp add: filterlim_at_top_dense)
hoelzl@51527
  2156
  moreover
hoelzl@51527
  2157
  from assms have "\<And>x. Z < x \<Longrightarrow> x powr s < Z powr s"
hoelzl@51527
  2158
    by (auto simp: Z_def intro!: powr_less_mono2_neg)
hoelzl@51527
  2159
  with assms `0 < e` have "\<And>x. Z < x \<Longrightarrow> dist (x powr s) 0 < e"
hoelzl@51527
  2160
    by (simp add: powr_powr Z_def dist_real_def)
hoelzl@51527
  2161
  ultimately
hoelzl@51527
  2162
  show "eventually (\<lambda>x. dist (f x powr s) 0 < e) F" by (rule eventually_elim1)
hoelzl@51527
  2163
qed
hoelzl@51527
  2164
huffman@29164
  2165
subsection {* Sine and Cosine *}
huffman@29164
  2166
huffman@44308
  2167
definition sin_coeff :: "nat \<Rightarrow> real" where
huffman@31271
  2168
  "sin_coeff = (\<lambda>n. if even n then 0 else -1 ^ ((n - Suc 0) div 2) / real (fact n))"
huffman@31271
  2169
huffman@44308
  2170
definition cos_coeff :: "nat \<Rightarrow> real" where
huffman@31271
  2171
  "cos_coeff = (\<lambda>n. if even n then (-1 ^ (n div 2)) / real (fact n) else 0)"
huffman@31271
  2172
wenzelm@53079
  2173
definition sin :: "real \<Rightarrow> real"
wenzelm@53079
  2174
  where "sin = (\<lambda>x. \<Sum>n. sin_coeff n * x ^ n)"
wenzelm@53079
  2175
wenzelm@53079
  2176
definition cos :: "real \<Rightarrow> real"
wenzelm@53079
  2177
  where "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
huffman@31271
  2178
huffman@44319
  2179
lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
huffman@44319
  2180
  unfolding sin_coeff_def by simp
huffman@44319
  2181
huffman@44319
  2182
lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
huffman@44319
  2183
  unfolding cos_coeff_def by simp
huffman@44319
  2184
huffman@44319
  2185
lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
huffman@44319
  2186
  unfolding cos_coeff_def sin_coeff_def
huffman@44319
  2187
  by (simp del: mult_Suc)
huffman@44319
  2188
huffman@44319
  2189
lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
huffman@44319
  2190
  unfolding cos_coeff_def sin_coeff_def
huffman@44319
  2191
  by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)
huffman@44319
  2192
huffman@31271
  2193
lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
wenzelm@53079
  2194
  unfolding sin_coeff_def
wenzelm@53079
  2195
  apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
wenzelm@53079
  2196
  apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
wenzelm@53079
  2197
  done
huffman@29164
  2198
huffman@31271
  2199
lemma summable_cos: "summable (\<lambda>n. cos_coeff n * x ^ n)"
wenzelm@53079
  2200
  unfolding cos_coeff_def
wenzelm@53079
  2201
  apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
wenzelm@53079
  2202
  apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
wenzelm@53079
  2203
  done
huffman@29164
  2204
huffman@31271
  2205
lemma sin_converges: "(\<lambda>n. sin_coeff n * x ^ n) sums sin(x)"
wenzelm@53079
  2206
  unfolding sin_def by (rule summable_sin [THEN summable_sums])
huffman@29164
  2207
huffman@31271
  2208
lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
wenzelm@53079
  2209
  unfolding cos_def by (rule summable_cos [THEN summable_sums])
huffman@29164
  2210
huffman@44319
  2211
lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
huffman@44319
  2212
  by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
huffman@44319
  2213
huffman@44319
  2214
lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
huffman@44319
  2215
  by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
huffman@29164
  2216
huffman@29164
  2217
text{*Now at last we can get the derivatives of exp, sin and cos*}
huffman@29164
  2218
huffman@29164
  2219
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
huffman@44319
  2220
  unfolding sin_def cos_def
huffman@44319
  2221
  apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
huffman@44319
  2222
  apply (simp_all add: diffs_sin_coeff diffs_cos_coeff
huffman@44319
  2223
    summable_minus summable_sin summable_cos)
huffman@44319
  2224
  done
huffman@29164
  2225
hoelzl@51527
  2226
declare DERIV_sin[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@51527
  2227
huffman@29164
  2228
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
huffman@44319
  2229
  unfolding cos_def sin_def
huffman@44319
  2230
  apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
huffman@44319
  2231
  apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus
huffman@44319
  2232
    summable_minus summable_sin summable_cos suminf_minus)
huffman@44319
  2233
  done
huffman@29164
  2234
hoelzl@51527
  2235
declare DERIV_cos[THEN DERIV_chain2, THEN DERIV_cong, DERIV_intros]
hoelzl@51527
  2236
huffman@44311
  2237
lemma isCont_sin: "isCont sin x"
huffman@44311
  2238
  by (rule DERIV_sin [THEN DERIV_isCont])
huffman@44311
  2239
huffman@44311
  2240
lemma isCont_cos: "isCont cos x"
huffman@44311
  2241
  by (rule DERIV_cos [THEN DERIV_isCont])
huffman@44311
  2242
huffman@44311
  2243
lemma isCont_sin' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. sin (f x)) a"
huffman@44311
  2244
  by (rule isCont_o2 [OF _ isCont_sin])
huffman@44311
  2245
huffman@44311
  2246
lemma isCont_cos' [simp]: "isCont f a \<Longrightarrow> isCont (\<lambda>x. cos (f x)) a"
huffman@44311
  2247
  by (rule isCont_o2 [OF _ isCont_cos])
huffman@44311
  2248
huffman@44311
  2249
lemma tendsto_sin [tendsto_intros]:
huffman@44311
  2250
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. sin (f x)) ---> sin a) F"
huffman@44311
  2251
  by (rule isCont_tendsto_compose [OF isCont_sin])
huffman@44311
  2252
huffman@44311
  2253
lemma tendsto_cos [tendsto_intros]:
huffman@44311
  2254
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. cos (f x)) ---> cos a) F"
huffman@44311
  2255
  by (rule isCont_tendsto_compose [OF isCont_cos])
huffman@29164
  2256
hoelzl@51478
  2257
lemma continuous_sin [continuous_intros]:
hoelzl@51478
  2258
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. sin (f x))"
hoelzl@51478
  2259
  unfolding continuous_def by (rule tendsto_sin)
hoelzl@51478
  2260
hoelzl@51478
  2261
lemma continuous_on_sin [continuous_on_intros]:
hoelzl@51478
  2262
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. sin (f x))"
hoelzl@51478
  2263
  unfolding continuous_on_def by (auto intro: tendsto_sin)
hoelzl@51478
  2264
hoelzl@51478
  2265
lemma continuous_cos [continuous_intros]:
hoelzl@51478
  2266
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. cos (f x))"
hoelzl@51478
  2267
  unfolding continuous_def by (rule tendsto_cos)
hoelzl@51478
  2268
hoelzl@51478
  2269
lemma continuous_on_cos [continuous_on_intros]:
hoelzl@51478
  2270
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. cos (f x))"
hoelzl@51478
  2271
  unfolding continuous_on_def by (auto intro: tendsto_cos)
hoelzl@51478
  2272
huffman@29164
  2273
subsection {* Properties of Sine and Cosine *}
paulson@15077
  2274
paulson@15077
  2275
lemma sin_zero [simp]: "sin 0 = 0"
huffman@44311
  2276
  unfolding sin_def sin_coeff_def by (simp add: powser_zero)
paulson@15077
  2277
paulson@15077
  2278
lemma cos_zero [simp]: "cos 0 = 1"
huffman@44311
  2279
  unfolding cos_def cos_coeff_def by (simp add: powser_zero)
paulson@15077
  2280
wenzelm@53015
  2281
lemma sin_cos_squared_add [simp]: "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1"
huffman@44308
  2282
proof -
wenzelm@53015
  2283
  have "\<forall>x. DERIV (\<lambda>x. (sin x)\<^sup>2 + (cos x)\<^sup>2) x :> 0"
huffman@44308
  2284
    by (auto intro!: DERIV_intros)
wenzelm@53015
  2285
  hence "(sin x)\<^sup>2 + (cos x)\<^sup>2 = (sin 0)\<^sup>2 + (cos 0)\<^sup>2"
huffman@44308
  2286
    by (rule DERIV_isconst_all)
wenzelm@53015
  2287
  thus "(sin x)\<^sup>2 + (cos x)\<^sup>2 = 1" by simp
huffman@44308
  2288
qed
huffman@44308
  2289
wenzelm@53015
  2290
lemma sin_cos_squared_add2 [simp]: "(cos x)\<^sup>2 + (sin x)\<^sup>2 = 1"
huffman@44308
  2291
  by (subst add_commute, rule sin_cos_squared_add)
paulson@15077
  2292
paulson@15077
  2293
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
huffman@44308
  2294
  using sin_cos_squared_add2 [unfolded power2_eq_square] .
paulson@15077
  2295
wenzelm@53015
  2296
lemma sin_squared_eq: "(sin x)\<^sup>2 = 1 - (cos x)\<^sup>2"
huffman@44308
  2297
  unfolding eq_diff_eq by (rule sin_cos_squared_add)
paulson@15077
  2298
wenzelm@53015
  2299
lemma cos_squared_eq: "(cos x)\<^sup>2 = 1 - (sin x)\<^sup>2"
huffman@44308
  2300
  unfolding eq_diff_eq by (rule sin_cos_squared_add2)
paulson@15077
  2301
paulson@15081
  2302
lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
huffman@44308
  2303
  by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
paulson@15077
  2304
paulson@15077
  2305
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
huffman@44308
  2306
  using abs_sin_le_one [of x] unfolding abs_le_iff by simp
paulson@15077
  2307
paulson@15077
  2308
lemma sin_le_one [simp]: "sin x \<le> 1"
huffman@44308
  2309
  using abs_sin_le_one [of x] unfolding abs_le_iff by simp
paulson@15077
  2310
paulson@15081
  2311
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
huffman@44308
  2312
  by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
paulson@15077
  2313
paulson@15077
  2314
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
huffman@44308
  2315
  using abs_cos_le_one [of x] unfolding abs_le_iff by simp
paulson@15077
  2316
paulson@15077
  2317
lemma cos_le_one [simp]: "cos x \<le> 1"
huffman@44308
  2318
  using abs_cos_le_one [of x] unfolding abs_le_iff by simp
paulson@15077
  2319
hoelzl@41970
  2320
lemma DERIV_fun_pow: "DERIV g x :> m ==>
wenzelm@53079
  2321
      DERIV (\<lambda>x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
huffman@44311
  2322
  by (auto intro!: DERIV_intros)
paulson@15077
  2323
paulson@15229
  2324
lemma DERIV_fun_exp:
wenzelm@53079
  2325
     "DERIV g x :> m ==> DERIV (\<lambda>x. exp(g x)) x :> exp(g x) * m"
huffman@44311
  2326
  by (auto intro!: DERIV_intros)
paulson@15077
  2327
paulson@15229
  2328
lemma DERIV_fun_sin:
wenzelm@53079
  2329
     "DERIV g x :> m ==> DERIV (\<lambda>x. sin(g x)) x :> cos(g x) * m"
huffman@44311
  2330
  by (auto intro!: DERIV_intros)
paulson@15077
  2331
paulson@15229
  2332
lemma DERIV_fun_cos:
wenzelm@53079
  2333
     "DERIV g x :> m ==> DERIV (\<lambda>x. cos(g x)) x :> -sin(g x) * m"
huffman@44311
  2334
  by (auto intro!: DERIV_intros)
paulson@15077
  2335
huffman@44308
  2336
lemma sin_cos_add_lemma:
wenzelm@53079
  2337
  "(sin (x + y) - (sin x * cos y + cos x * sin y))\<^sup>2 +
wenzelm@53079
  2338
    (cos (x + y) - (cos x * cos y - sin x * sin y))\<^sup>2 = 0"
huffman@44308
  2339
  (is "?f x = 0")
huffman@44308
  2340
proof -
huffman@44308
  2341
  have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
huffman@44308
  2342
    by (auto intro!: DERIV_intros simp add: algebra_simps)
huffman@44308
  2343
  hence "?f x = ?f 0"
huffman@44308
  2344
    by (rule DERIV_isconst_all)
huffman@44308
  2345
  thus ?thesis by simp
huffman@44308
  2346
qed
paulson@15077
  2347
paulson@15077
  2348
lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
huffman@44308
  2349
  using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
paulson@15077
  2350
paulson@15077
  2351
lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
huffman@44308
  2352
  using sin_cos_add_lemma unfolding realpow_two_sum_zero_iff by simp
huffman@44308
  2353
huffman@44308
  2354
lemma sin_cos_minus_lemma:
wenzelm@53015
  2355
  "(sin(-x) + sin(x))\<^sup>2 + (cos(-x) - cos(x))\<^sup>2 = 0" (is "?f x = 0")
huffman@44308
  2356
proof -
huffman@44308
  2357
  have "\<forall>x. DERIV (\<lambda>x. ?f x) x :> 0"
huffman@44308
  2358
    by (auto intro!: DERIV_intros simp add: algebra_simps)
huffman@44308
  2359
  hence "?f x = ?f 0"
huffman@44308
  2360
    by (rule DERIV_isconst_all)
huffman@44308
  2361
  thus ?thesis by simp
huffman@44308
  2362
qed
paulson@15077
  2363
paulson@15077
  2364
lemma sin_minus [simp]: "sin (-x) = -sin(x)"
huffman@44308
  2365
  using sin_cos_minus_lemma [where x=x] by simp
paulson@15077
  2366
paulson@15077
  2367
lemma cos_minus [simp]: "cos (-x) = cos(x)"
huffman@44308
  2368
  using sin_cos_minus_lemma [where x=x] by simp
paulson@15077
  2369
paulson@15077
  2370
lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
haftmann@54230
  2371
  using sin_add [of x "- y"] by simp
paulson@15077
  2372
paulson@15077
  2373
lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
huffman@44308
  2374
  by (simp add: sin_diff mult_commute)
paulson@15077
  2375
paulson@15077
  2376
lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
haftmann@54230
  2377
  using cos_add [of x "- y"] by simp
paulson@15077
  2378
paulson@15077
  2379
lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
huffman@44308
  2380
  by (simp add: cos_diff mult_commute)
paulson@15077
  2381
paulson@15077
  2382
lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
huffman@29165
  2383
  using sin_add [where x=x and y=x] by simp
paulson@15077
  2384
wenzelm@53015
  2385
lemma cos_double: "cos(2* x) = ((cos x)\<^sup>2) - ((sin x)\<^sup>2)"
huffman@29165
  2386
  using cos_add [where x=x and y=x]
huffman@29165
  2387
  by (simp add: power2_eq_square)
paulson@15077
  2388
paulson@15077
  2389
huffman@29164
  2390
subsection {* The Constant Pi *}
paulson@15077
  2391
wenzelm@53079
  2392
definition pi :: real
wenzelm@53079
  2393
  where "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
huffman@23043
  2394
hoelzl@41970
  2395
text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
paulson@15077
  2396
   hence define pi.*}
paulson@15077
  2397
paulson@15077
  2398
lemma sin_paired:
wenzelm@53079
  2399
  "(\<lambda>n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) sums  sin x"
paulson@15077
  2400
proof -
huffman@31271
  2401
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. sin_coeff k * x ^ k) sums sin x"
huffman@44727
  2402
    by (rule sin_converges [THEN sums_group], simp)
huffman@31271
  2403
  thus ?thesis unfolding One_nat_def sin_coeff_def by (simp add: mult_ac)
paulson@15077
  2404
qed
paulson@15077
  2405
huffman@44728
  2406
lemma sin_gt_zero:
wenzelm@53079
  2407
  assumes "0 < x" and "x < 2"
wenzelm@53079
  2408
  shows "0 < sin x"
huffman@44728
  2409
proof -
huffman@44728
  2410
  let ?f = "\<lambda>n. \<Sum>k = n*2..<n*2+2. -1 ^ k / real (fact (2*k+1)) * x^(2*k+1)"
huffman@44728
  2411
  have pos: "\<forall>n. 0 < ?f n"
huffman@44728
  2412
  proof
huffman@44728
  2413
    fix n :: nat
huffman@44728
  2414
    let ?k2 = "real (Suc (Suc (4 * n)))"
huffman@44728
  2415
    let ?k3 = "real (Suc (Suc (Suc (4 * n))))"
huffman@44728
  2416
    have "x * x < ?k2 * ?k3"
huffman@44728
  2417
      using assms by (intro mult_strict_mono', simp_all)
huffman@44728
  2418
    hence "x * x * x * x ^ (n * 4) < ?k2 * ?k3 * x * x ^ (n * 4)"
huffman@44728
  2419
      by (intro mult_strict_right_mono zero_less_power `0 < x`)
huffman@44728
  2420
    thus "0 < ?f n"
huffman@44728
  2421
      by (simp del: mult_Suc,
huffman@44728
  2422
        simp add: less_divide_eq mult_pos_pos field_simps del: mult_Suc)
huffman@44728
  2423
  qed
huffman@44728
  2424
  have sums: "?f sums sin x"
huffman@44728
  2425
    by (rule sin_paired [THEN sums_group], simp)
huffman@44728
  2426
  show "0 < sin x"
huffman@44728
  2427
    unfolding sums_unique [OF sums]
huffman@44728
  2428
    using sums_summable [OF sums] pos
huffman@44728
  2429
    by (rule suminf_gt_zero)
huffman@44728
  2430
qed
paulson@15077
  2431
wenzelm@53079
  2432
lemma cos_double_less_one: "0 < x \<Longrightarrow> x < 2 \<Longrightarrow> cos (2 * x) < 1"
wenzelm@53079
  2433
  using sin_gt_zero [where x = x] by (auto simp add: cos_squared_eq cos_double)
wenzelm@53079
  2434
wenzelm@53079
  2435
lemma cos_paired: "(\<lambda>n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
paulson@15077
  2436
proof -
huffman@31271
  2437
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2. cos_coeff k * x ^ k) sums cos x"
huffman@44727
  2438
    by (rule cos_converges [THEN sums_group], simp)
huffman@31271
  2439
  thus ?thesis unfolding cos_coeff_def by (simp add: mult_ac)
paulson@15077
  2440
qed
paulson@15077
  2441
huffman@36824
  2442
lemma real_mult_inverse_cancel:
hoelzl@41970
  2443
     "[|(0::real) < x; 0 < x1; x1 * y < x * u |]
huffman@36824
  2444
      ==> inverse x * y < inverse x1 * u"
paulson@54575
  2445
  by (metis field_divide_inverse mult_commute mult_assoc pos_divide_less_eq pos_less_divide_eq)
huffman@36824
  2446
huffman@36824
  2447
lemma real_mult_inverse_cancel2:
huffman@36824
  2448
     "[|(0::real) < x;0 < x1; x1 * y < x * u |] ==> y * inverse x < u * inverse x1"
wenzelm@53079
  2449
  by (auto dest: real_mult_inverse_cancel simp add: mult_ac)
huffman@36824
  2450
haftmann@53602
  2451
lemmas realpow_num_eq_if = power_eq_if
haftmann@53602
  2452
haftmann@53602
  2453
lemma cos_two_less_zero [simp]:
haftmann@53602
  2454
  "cos 2 < 0"
haftmann@53602
  2455
proof -
haftmann@53602
  2456
  note fact_Suc [simp del]
haftmann@53602
  2457
  from cos_paired
haftmann@53602
  2458
  have "(\<lambda>n. - (-1 ^ n / real (fact (2 * n)) * 2 ^ (2 * n))) sums - cos 2"
haftmann@53602
  2459
    by (rule sums_minus)
haftmann@53602
  2460
  then have *: "(\<lambda>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n)))) sums - cos 2"
haftmann@53602
  2461
    by simp
haftmann@53602
  2462
  then have **: "summable (\<lambda>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
haftmann@53602
  2463
    by (rule sums_summable)
haftmann@53602
  2464
  have "0 < (\<Sum>n = 0..<Suc (Suc (Suc 0)). - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
haftmann@53602
  2465
    by (simp add: fact_num_eq_if_nat realpow_num_eq_if)
haftmann@53602
  2466
  moreover have "(\<Sum>n = 0..<Suc (Suc (Suc 0)). - (-1 ^ n  * 2 ^ (2 * n) / real (fact (2 * n))))
haftmann@53602
  2467
    < (\<Sum>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
haftmann@53602
  2468
  proof -
haftmann@53602
  2469
    { fix d
haftmann@53602
  2470
      have "4 * real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))
haftmann@53602
  2471
       < real (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))) *
haftmann@53602
  2472
           fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))"
haftmann@53602
  2473
        by (simp only: real_of_nat_mult) (auto intro!: mult_strict_mono fact_less_mono_nat)
haftmann@53602
  2474
      then have "4 * real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))
haftmann@53602
  2475
        < real (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d))))))))))"
haftmann@53602
  2476
        by (simp only: fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
haftmann@53602
  2477
      then have "4 * inverse (real (fact (Suc (Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))))
haftmann@53602
  2478
        < inverse (real (fact (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))))"
haftmann@53602
  2479
        by (simp add: inverse_eq_divide less_divide_eq)
haftmann@53602
  2480
    }
haftmann@53602
  2481
    note *** = this
haftmann@54230
  2482
    have [simp]: "\<And>x y::real. 0 < x - y \<longleftrightarrow> y < x" by arith
haftmann@53602
  2483
    from ** show ?thesis by (rule sumr_pos_lt_pair)
haftmann@54230
  2484
      (simp add: divide_inverse mult_assoc [symmetric] ***)
haftmann@53602
  2485
  qed
haftmann@53602
  2486
  ultimately have "0 < (\<Sum>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
haftmann@53602
  2487
    by (rule order_less_trans)
haftmann@53602
  2488
  moreover from * have "- cos 2 = (\<Sum>n. - (-1 ^ n * 2 ^ (2 * n) / real (fact (2 * n))))"
haftmann@53602
  2489
    by (rule sums_unique)
haftmann@53602
  2490
  ultimately have "0 < - cos 2" by simp
haftmann@53602
  2491
  then show ?thesis by simp
haftmann@53602
  2492
qed
huffman@23053
  2493
huffman@23053
  2494
lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
huffman@23053
  2495
lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
paulson@15077
  2496
wenzelm@53079
  2497
lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 \<and> cos x = 0"
huffman@44730
  2498
proof (rule ex_ex1I)
huffman@44730