src/HOL/Hyperreal/Transcendental.thy
author huffman
Thu Jun 07 02:34:37 2007 +0200 (2007-06-07)
changeset 23286 85e7e043b980
parent 23278 375335bf619f
child 23413 5caa2710dd5b
permissions -rw-r--r--
tuned
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(*  Title       : Transcendental.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998,1999 University of Cambridge
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                  1999,2001 University of Edinburgh
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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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 NthRoot Fact Series EvenOdd Deriv
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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::recpower"
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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_Suc power_commutes)
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qed
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lemma lemma_realpow_diff_sumr:
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  fixes y :: "'a::{recpower,comm_semiring_0}" 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 (auto simp add: setsum_right_distrib lemma_realpow_diff mult_ac
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  simp del: setsum_op_ivl_Suc cong: strong_setsum_cong)
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lemma lemma_realpow_diff_sumr2:
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  fixes y :: "'a::{recpower,comm_ring}" shows
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     "x ^ (Suc n) - y ^ (Suc n) =  
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      (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * y ^ (n - p))"
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apply (induct "n", simp add: power_Suc)
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apply (simp add: power_Suc del: setsum_op_ivl_Suc)
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apply (subst setsum_op_ivl_Suc)
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apply (subst lemma_realpow_diff_sumr)
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apply (simp add: right_distrib del: setsum_op_ivl_Suc)
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apply (subst mult_left_commute [where a="x - y"])
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apply (erule subst)
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apply (simp add: power_Suc ring_eq_simps)
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done
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lemma lemma_realpow_rev_sumr:
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     "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =  
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      (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p))"
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apply (rule setsum_reindex_cong [where f="\<lambda>i. n - i"])
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apply (rule inj_onI, simp)
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apply auto
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apply (rule_tac x="n - x" in image_eqI, simp, simp)
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done
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text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
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x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
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lemma powser_insidea:
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  fixes x z :: "'a::{real_normed_field,banach,recpower}"
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  assumes 1: "summable (\<lambda>n. f n * x ^ n)"
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  assumes 2: "norm z < norm x"
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  shows "summable (\<lambda>n. norm (f n * z ^ n))"
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proof -
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  from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
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  from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
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    by (rule summable_LIMSEQ_zero)
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  hence "convergent (\<lambda>n. f n * x ^ n)"
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    by (rule convergentI)
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  hence "Cauchy (\<lambda>n. f n * x ^ n)"
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    by (simp add: Cauchy_convergent_iff)
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  hence "Bseq (\<lambda>n. f n * x ^ n)"
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    by (rule Cauchy_Bseq)
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  then obtain K where 3: "0 < K" and 4: "\<forall>n. norm (f n * x ^ n) \<le> K"
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    by (simp add: Bseq_def, safe)
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  have "\<exists>N. \<forall>n\<ge>N. norm (norm (f n * z ^ n)) \<le>
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                   K * norm (z ^ n) * inverse (norm (x ^ n))"
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  proof (intro exI allI impI)
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    fix n::nat assume "0 \<le> n"
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    have "norm (norm (f n * z ^ n)) * norm (x ^ n) =
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          norm (f n * x ^ n) * norm (z ^ n)"
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      by (simp add: norm_mult abs_mult)
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    also have "\<dots> \<le> K * norm (z ^ n)"
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      by (simp only: mult_right_mono 4 norm_ge_zero)
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    also have "\<dots> = K * norm (z ^ n) * (inverse (norm (x ^ n)) * norm (x ^ n))"
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      by (simp add: x_neq_0)
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    also have "\<dots> = K * norm (z ^ n) * inverse (norm (x ^ n)) * norm (x ^ n)"
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      by (simp only: mult_assoc)
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    finally show "norm (norm (f n * z ^ n)) \<le>
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                  K * norm (z ^ n) * inverse (norm (x ^ n))"
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      by (simp add: mult_le_cancel_right x_neq_0)
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  qed
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  moreover have "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
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  proof -
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    from 2 have "norm (norm (z * inverse x)) < 1"
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      using x_neq_0
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      by (simp add: nonzero_norm_divide divide_inverse [symmetric])
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    hence "summable (\<lambda>n. norm (z * inverse x) ^ n)"
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      by (rule summable_geometric)
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    hence "summable (\<lambda>n. K * norm (z * inverse x) ^ n)"
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      by (rule summable_mult)
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    thus "summable (\<lambda>n. K * norm (z ^ n) * inverse (norm (x ^ n)))"
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      using x_neq_0
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      by (simp add: norm_mult nonzero_norm_inverse power_mult_distrib
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                    power_inverse norm_power mult_assoc)
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  qed
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  ultimately show "summable (\<lambda>n. norm (f n * z ^ n))"
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    by (rule summable_comparison_test)
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qed
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lemma powser_inside:
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  fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach,recpower}" shows
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     "[| summable (%n. f(n) * (x ^ n)); norm z < norm x |]  
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      ==> summable (%n. f(n) * (z ^ n))"
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by (rule powser_insidea [THEN summable_norm_cancel])
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subsection{*Term-by-Term Differentiability of Power Series*}
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definition
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  diffs :: "(nat => 'a::ring_1) => nat => 'a" where
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  "diffs c = (%n. of_nat (Suc n) * c(Suc n))"
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text{*Lemma about distributing negation over it*}
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lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
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by (simp add: diffs_def)
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text{*Show that we can shift the terms down one*}
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lemma lemma_diffs:
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     "(\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) =  
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      (\<Sum>n=0..<n. of_nat n * c(n) * (x ^ (n - Suc 0))) +  
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      (of_nat n * c(n) * x ^ (n - Suc 0))"
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apply (induct "n")
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apply (auto simp add: mult_assoc add_assoc [symmetric] diffs_def)
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done
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lemma lemma_diffs2:
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     "(\<Sum>n=0..<n. of_nat n * c(n) * (x ^ (n - Suc 0))) =  
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      (\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) -  
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      (of_nat n * c(n) * x ^ (n - Suc 0))"
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by (auto simp add: lemma_diffs)
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lemma diffs_equiv:
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     "summable (%n. (diffs c)(n) * (x ^ n)) ==>  
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      (%n. of_nat n * c(n) * (x ^ (n - Suc 0))) sums  
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         (\<Sum>n. (diffs c)(n) * (x ^ n))"
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apply (subgoal_tac " (%n. of_nat n * c (n) * (x ^ (n - Suc 0))) ----> 0")
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apply (rule_tac [2] LIMSEQ_imp_Suc)
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apply (drule summable_sums) 
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apply (auto simp add: sums_def)
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apply (drule_tac X="(\<lambda>n. \<Sum>n = 0..<n. diffs c n * x ^ n)" in LIMSEQ_diff)
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apply (auto simp add: lemma_diffs2 [symmetric] diffs_def [symmetric])
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apply (simp add: diffs_def summable_LIMSEQ_zero)
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done
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lemma lemma_termdiff1:
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  fixes z :: "'a :: {recpower,comm_ring}" shows
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  "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =  
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   (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p))))"
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by (auto simp add: right_distrib diff_minus power_add [symmetric] mult_ac
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  cong: strong_setsum_cong)
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lemma less_add_one: "m < n ==> (\<exists>d. n = m + d + Suc 0)"
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by (simp add: less_iff_Suc_add)
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lemma sumdiff: "a + b - (c + d) = a - c + b - (d::real)"
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by arith
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lemma sumr_diff_mult_const2:
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  "setsum f {0..<n} - of_nat n * (r::'a::ring_1) = (\<Sum>i = 0..<n. f i - r)"
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by (simp add: setsum_subtractf)
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lemma lemma_termdiff2:
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  fixes h :: "'a :: {recpower,field}"
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  assumes h: "h \<noteq> 0" shows
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  "((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0) =
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   h * (\<Sum>p=0..< n - Suc 0. \<Sum>q=0..< n - Suc 0 - p.
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        (z + h) ^ q * z ^ (n - 2 - q))" (is "?lhs = ?rhs")
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apply (subgoal_tac "h * ?lhs = h * ?rhs", simp add: h)
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apply (simp add: right_diff_distrib diff_divide_distrib h)
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apply (simp only: times_divide_eq_left [symmetric])
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apply (simp add: divide_self [OF h])
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apply (simp add: mult_assoc [symmetric])
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apply (cases "n", simp)
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apply (simp add: lemma_realpow_diff_sumr2 h
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                 right_diff_distrib [symmetric] mult_assoc
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            del: realpow_Suc setsum_op_ivl_Suc of_nat_Suc)
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apply (subst lemma_realpow_rev_sumr)
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apply (subst sumr_diff_mult_const2)
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apply simp
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apply (simp only: lemma_termdiff1 setsum_right_distrib)
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apply (rule setsum_cong [OF refl])
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apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
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apply (clarify)
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apply (simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac
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            del: setsum_op_ivl_Suc realpow_Suc)
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apply (subst mult_assoc [symmetric], subst power_add [symmetric])
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apply (simp add: mult_ac)
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done
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lemma real_setsum_nat_ivl_bounded2:
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  fixes K :: "'a::ordered_semidom"
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  assumes f: "\<And>p::nat. p < n \<Longrightarrow> f p \<le> K"
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  assumes K: "0 \<le> K"
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  shows "setsum f {0..<n-k} \<le> of_nat n * K"
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apply (rule order_trans [OF setsum_mono])
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apply (rule f, simp)
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apply (simp add: mult_right_mono K)
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done
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lemma lemma_termdiff3:
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  fixes h z :: "'a::{real_normed_field,recpower}"
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  assumes 1: "h \<noteq> 0"
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  assumes 2: "norm z \<le> K"
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  assumes 3: "norm (z + h) \<le> K"
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  shows "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0))
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          \<le> of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
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proof -
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  have "norm (((z + h) ^ n - z ^ n) / h - of_nat n * z ^ (n - Suc 0)) =
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        norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
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          (z + h) ^ q * z ^ (n - 2 - q)) * norm h"
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    apply (subst lemma_termdiff2 [OF 1])
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    apply (subst norm_mult)
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    apply (rule mult_commute)
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    done
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  also have "\<dots> \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2)) * norm h"
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  proof (rule mult_right_mono [OF _ norm_ge_zero])
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    from norm_ge_zero 2 have K: "0 \<le> K" by (rule order_trans)
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    have le_Kn: "\<And>i j n. i + j = n \<Longrightarrow> norm ((z + h) ^ i * z ^ j) \<le> K ^ n"
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      apply (erule subst)
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      apply (simp only: norm_mult norm_power power_add)
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      apply (intro mult_mono power_mono 2 3 norm_ge_zero zero_le_power K)
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      done
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    show "norm (\<Sum>p = 0..<n - Suc 0. \<Sum>q = 0..<n - Suc 0 - p.
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              (z + h) ^ q * z ^ (n - 2 - q))
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          \<le> of_nat n * (of_nat (n - Suc 0) * K ^ (n - 2))"
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      apply (intro
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         order_trans [OF norm_setsum]
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         real_setsum_nat_ivl_bounded2
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         mult_nonneg_nonneg
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         zero_le_imp_of_nat
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         zero_le_power K)
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      apply (rule le_Kn, simp)
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      done
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  qed
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  also have "\<dots> = of_nat n * of_nat (n - Suc 0) * K ^ (n - 2) * norm h"
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    by (simp only: mult_assoc)
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  finally show ?thesis .
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qed
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lemma lemma_termdiff4:
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  fixes f :: "'a::{real_normed_field,recpower} \<Rightarrow>
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              'b::real_normed_vector"
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  assumes k: "0 < (k::real)"
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  assumes le: "\<And>h. \<lbrakk>h \<noteq> 0; norm h < k\<rbrakk> \<Longrightarrow> norm (f h) \<le> K * norm h"
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  shows "f -- 0 --> 0"
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proof (simp add: LIM_def, safe)
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  fix r::real assume r: "0 < r"
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  have zero_le_K: "0 \<le> K"
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    apply (cut_tac k)
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    apply (cut_tac h="of_real (k/2)" in le, simp)
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    apply (simp del: of_real_divide)
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    apply (drule order_trans [OF norm_ge_zero])
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    apply (simp add: zero_le_mult_iff)
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    done
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  show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
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  proof (cases)
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    assume "K = 0"
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    with k r le have "0 < k \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < k \<longrightarrow> norm (f x) < r)"
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      by simp
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    thus "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)" ..
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  next
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    assume K_neq_zero: "K \<noteq> 0"
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    with zero_le_K have K: "0 < K" by simp
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    show "\<exists>s. 0 < s \<and> (\<forall>x. x \<noteq> 0 \<and> norm x < s \<longrightarrow> norm (f x) < r)"
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    proof (rule exI, safe)
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      from k r K show "0 < min k (r * inverse K / 2)"
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   278
        by (simp add: mult_pos_pos positive_imp_inverse_positive)
huffman@20860
   279
    next
huffman@23082
   280
      fix x::'a
huffman@23082
   281
      assume x1: "x \<noteq> 0" and x2: "norm x < min k (r * inverse K / 2)"
huffman@23082
   282
      from x2 have x3: "norm x < k" and x4: "norm x < r * inverse K / 2"
huffman@20860
   283
        by simp_all
huffman@23082
   284
      from x1 x3 le have "norm (f x) \<le> K * norm x" by simp
huffman@23082
   285
      also from x4 K have "K * norm x < K * (r * inverse K / 2)"
huffman@20860
   286
        by (rule mult_strict_left_mono)
huffman@20860
   287
      also have "\<dots> = r / 2"
huffman@20860
   288
        using K_neq_zero by simp
huffman@20860
   289
      also have "r / 2 < r"
huffman@20860
   290
        using r by simp
huffman@23082
   291
      finally show "norm (f x) < r" .
huffman@20860
   292
    qed
huffman@20860
   293
  qed
huffman@20860
   294
qed
paulson@15077
   295
paulson@15229
   296
lemma lemma_termdiff5:
huffman@23112
   297
  fixes g :: "'a::{recpower,real_normed_field} \<Rightarrow>
huffman@23082
   298
              nat \<Rightarrow> 'b::banach"
huffman@20860
   299
  assumes k: "0 < (k::real)"
huffman@20860
   300
  assumes f: "summable f"
huffman@23082
   301
  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
   302
  shows "(\<lambda>h. suminf (g h)) -- 0 --> 0"
huffman@20860
   303
proof (rule lemma_termdiff4 [OF k])
huffman@23082
   304
  fix h::'a assume "h \<noteq> 0" and "norm h < k"
huffman@23082
   305
  hence A: "\<forall>n. norm (g h n) \<le> f n * norm h"
huffman@20860
   306
    by (simp add: le)
huffman@23082
   307
  hence "\<exists>N. \<forall>n\<ge>N. norm (norm (g h n)) \<le> f n * norm h"
huffman@20860
   308
    by simp
huffman@23082
   309
  moreover from f have B: "summable (\<lambda>n. f n * norm h)"
huffman@20860
   310
    by (rule summable_mult2)
huffman@23082
   311
  ultimately have C: "summable (\<lambda>n. norm (g h n))"
huffman@20860
   312
    by (rule summable_comparison_test)
huffman@23082
   313
  hence "norm (suminf (g h)) \<le> (\<Sum>n. norm (g h n))"
huffman@23082
   314
    by (rule summable_norm)
huffman@23082
   315
  also from A C B have "(\<Sum>n. norm (g h n)) \<le> (\<Sum>n. f n * norm h)"
huffman@20860
   316
    by (rule summable_le)
huffman@23082
   317
  also from f have "(\<Sum>n. f n * norm h) = suminf f * norm h"
huffman@20860
   318
    by (rule suminf_mult2 [symmetric])
huffman@23082
   319
  finally show "norm (suminf (g h)) \<le> suminf f * norm h" .
huffman@20860
   320
qed
paulson@15077
   321
paulson@15077
   322
paulson@15077
   323
text{* FIXME: Long proofs*}
paulson@15077
   324
paulson@15077
   325
lemma termdiffs_aux:
huffman@23112
   326
  fixes x :: "'a::{recpower,real_normed_field,banach}"
huffman@20849
   327
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
huffman@23082
   328
  assumes 2: "norm x < norm K"
huffman@20860
   329
  shows "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h
huffman@23082
   330
             - of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
huffman@20849
   331
proof -
huffman@20860
   332
  from dense [OF 2]
huffman@23082
   333
  obtain r where r1: "norm x < r" and r2: "r < norm K" by fast
huffman@23082
   334
  from norm_ge_zero r1 have r: "0 < r"
huffman@20860
   335
    by (rule order_le_less_trans)
huffman@20860
   336
  hence r_neq_0: "r \<noteq> 0" by simp
huffman@20860
   337
  show ?thesis
huffman@20849
   338
  proof (rule lemma_termdiff5)
huffman@23082
   339
    show "0 < r - norm x" using r1 by simp
huffman@20849
   340
  next
huffman@23082
   341
    from r r2 have "norm (of_real r::'a) < norm K"
huffman@23082
   342
      by simp
huffman@23082
   343
    with 1 have "summable (\<lambda>n. norm (diffs (diffs c) n * (of_real r ^ n)))"
huffman@20860
   344
      by (rule powser_insidea)
huffman@23082
   345
    hence "summable (\<lambda>n. diffs (diffs (\<lambda>n. norm (c n))) n * r ^ n)"
huffman@23082
   346
      using r
huffman@23082
   347
      by (simp add: diffs_def norm_mult norm_power del: of_nat_Suc)
huffman@23082
   348
    hence "summable (\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))"
huffman@20860
   349
      by (rule diffs_equiv [THEN sums_summable])
huffman@23082
   350
    also have "(\<lambda>n. of_nat n * diffs (\<lambda>n. norm (c n)) n * r ^ (n - Suc 0))
huffman@23082
   351
      = (\<lambda>n. diffs (%m. of_nat (m - Suc 0) * norm (c m) * inverse r) n * (r ^ n))"
huffman@20849
   352
      apply (rule ext)
huffman@20849
   353
      apply (simp add: diffs_def)
huffman@20849
   354
      apply (case_tac n, simp_all add: r_neq_0)
huffman@20849
   355
      done
huffman@20860
   356
    finally have "summable 
huffman@23082
   357
      (\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) * r ^ (n - Suc 0))"
huffman@20860
   358
      by (rule diffs_equiv [THEN sums_summable])
huffman@20860
   359
    also have
huffman@23082
   360
      "(\<lambda>n. of_nat n * (of_nat (n - Suc 0) * norm (c n) * inverse r) *
huffman@20860
   361
           r ^ (n - Suc 0)) =
huffman@23082
   362
       (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))"
huffman@20849
   363
      apply (rule ext)
huffman@20849
   364
      apply (case_tac "n", simp)
huffman@20849
   365
      apply (case_tac "nat", simp)
huffman@20849
   366
      apply (simp add: r_neq_0)
huffman@20849
   367
      done
huffman@20860
   368
    finally show
huffman@23082
   369
      "summable (\<lambda>n. norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2))" .
huffman@20849
   370
  next
huffman@23082
   371
    fix h::'a and n::nat
huffman@20860
   372
    assume h: "h \<noteq> 0"
huffman@23082
   373
    assume "norm h < r - norm x"
huffman@23082
   374
    hence "norm x + norm h < r" by simp
huffman@23082
   375
    with norm_triangle_ineq have xh: "norm (x + h) < r"
huffman@20860
   376
      by (rule order_le_less_trans)
huffman@23082
   377
    show "norm (c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))
huffman@23082
   378
          \<le> norm (c n) * of_nat n * of_nat (n - Suc 0) * r ^ (n - 2) * norm h"
huffman@23082
   379
      apply (simp only: norm_mult mult_assoc)
huffman@23082
   380
      apply (rule mult_left_mono [OF _ norm_ge_zero])
huffman@20860
   381
      apply (simp (no_asm) add: mult_assoc [symmetric])
huffman@20860
   382
      apply (rule lemma_termdiff3)
huffman@20860
   383
      apply (rule h)
huffman@20860
   384
      apply (rule r1 [THEN order_less_imp_le])
huffman@20860
   385
      apply (rule xh [THEN order_less_imp_le])
huffman@20860
   386
      done
huffman@20849
   387
  qed
huffman@20849
   388
qed
webertj@20217
   389
huffman@20860
   390
lemma termdiffs:
huffman@23112
   391
  fixes K x :: "'a::{recpower,real_normed_field,banach}"
huffman@20860
   392
  assumes 1: "summable (\<lambda>n. c n * K ^ n)"
huffman@20860
   393
  assumes 2: "summable (\<lambda>n. (diffs c) n * K ^ n)"
huffman@20860
   394
  assumes 3: "summable (\<lambda>n. (diffs (diffs c)) n * K ^ n)"
huffman@23082
   395
  assumes 4: "norm x < norm K"
huffman@20860
   396
  shows "DERIV (\<lambda>x. \<Sum>n. c n * x ^ n) x :> (\<Sum>n. (diffs c) n * x ^ n)"
huffman@20860
   397
proof (simp add: deriv_def, rule LIM_zero_cancel)
huffman@20860
   398
  show "(\<lambda>h. (suminf (\<lambda>n. c n * (x + h) ^ n) - suminf (\<lambda>n. c n * x ^ n)) / h
huffman@20860
   399
            - suminf (\<lambda>n. diffs c n * x ^ n)) -- 0 --> 0"
huffman@20860
   400
  proof (rule LIM_equal2)
huffman@23082
   401
    show "0 < norm K - norm x" by (simp add: less_diff_eq 4)
huffman@20860
   402
  next
huffman@23082
   403
    fix h :: 'a
huffman@20860
   404
    assume "h \<noteq> 0"
huffman@23082
   405
    assume "norm (h - 0) < norm K - norm x"
huffman@23082
   406
    hence "norm x + norm h < norm K" by simp
huffman@23082
   407
    hence 5: "norm (x + h) < norm K"
huffman@23082
   408
      by (rule norm_triangle_ineq [THEN order_le_less_trans])
huffman@20860
   409
    have A: "summable (\<lambda>n. c n * x ^ n)"
huffman@20860
   410
      by (rule powser_inside [OF 1 4])
huffman@20860
   411
    have B: "summable (\<lambda>n. c n * (x + h) ^ n)"
huffman@20860
   412
      by (rule powser_inside [OF 1 5])
huffman@20860
   413
    have C: "summable (\<lambda>n. diffs c n * x ^ n)"
huffman@20860
   414
      by (rule powser_inside [OF 2 4])
huffman@20860
   415
    show "((\<Sum>n. c n * (x + h) ^ n) - (\<Sum>n. c n * x ^ n)) / h
huffman@20860
   416
             - (\<Sum>n. diffs c n * x ^ n) = 
huffman@23082
   417
          (\<Sum>n. c n * (((x + h) ^ n - x ^ n) / h - of_nat n * x ^ (n - Suc 0)))"
huffman@20860
   418
      apply (subst sums_unique [OF diffs_equiv [OF C]])
huffman@20860
   419
      apply (subst suminf_diff [OF B A])
huffman@20860
   420
      apply (subst suminf_divide [symmetric])
huffman@20860
   421
      apply (rule summable_diff [OF B A])
huffman@20860
   422
      apply (subst suminf_diff)
huffman@20860
   423
      apply (rule summable_divide)
huffman@20860
   424
      apply (rule summable_diff [OF B A])
huffman@20860
   425
      apply (rule sums_summable [OF diffs_equiv [OF C]])
huffman@20860
   426
      apply (rule_tac f="suminf" in arg_cong)
huffman@20860
   427
      apply (rule ext)
huffman@20860
   428
      apply (simp add: ring_eq_simps)
huffman@20860
   429
      done
huffman@20860
   430
  next
huffman@20860
   431
    show "(\<lambda>h. \<Sum>n. c n * (((x + h) ^ n - x ^ n) / h -
huffman@23082
   432
               of_nat n * x ^ (n - Suc 0))) -- 0 --> 0"
huffman@20860
   433
        by (rule termdiffs_aux [OF 3 4])
huffman@20860
   434
  qed
huffman@20860
   435
qed
huffman@20860
   436
paulson@15077
   437
huffman@23043
   438
subsection{*Exponential Function*}
huffman@23043
   439
huffman@23043
   440
definition
huffman@23115
   441
  exp :: "'a \<Rightarrow> 'a::{recpower,real_normed_field,banach}" where
huffman@23115
   442
  "exp x = (\<Sum>n. x ^ n /# real (fact n))"
huffman@23043
   443
huffman@23043
   444
definition
huffman@23043
   445
  sin :: "real => real" where
huffman@23043
   446
  "sin x = (\<Sum>n. (if even(n) then 0 else
huffman@23177
   447
             (-1 ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n)"
huffman@23043
   448
 
huffman@23043
   449
definition
huffman@23043
   450
  cos :: "real => real" where
huffman@23177
   451
  "cos x = (\<Sum>n. (if even(n) then (-1 ^ (n div 2))/(real (fact n)) 
huffman@23043
   452
                            else 0) * x ^ n)"
huffman@23115
   453
huffman@23115
   454
lemma summable_exp_generic:
huffman@23115
   455
  fixes x :: "'a::{real_normed_algebra_1,recpower,banach}"
huffman@23115
   456
  defines S_def: "S \<equiv> \<lambda>n. x ^ n /# real (fact n)"
huffman@23115
   457
  shows "summable S"
huffman@23115
   458
proof -
huffman@23115
   459
  have S_Suc: "\<And>n. S (Suc n) = (x * S n) /# real (Suc n)"
huffman@23115
   460
    unfolding S_def by (simp add: power_Suc del: mult_Suc)
huffman@23115
   461
  obtain r :: real where r0: "0 < r" and r1: "r < 1"
huffman@23115
   462
    using dense [OF zero_less_one] by fast
huffman@23115
   463
  obtain N :: nat where N: "norm x < real N * r"
huffman@23115
   464
    using reals_Archimedean3 [OF r0] by fast
huffman@23115
   465
  from r1 show ?thesis
huffman@23115
   466
  proof (rule ratio_test [rule_format])
huffman@23115
   467
    fix n :: nat
huffman@23115
   468
    assume n: "N \<le> n"
huffman@23115
   469
    have "norm x \<le> real N * r"
huffman@23115
   470
      using N by (rule order_less_imp_le)
huffman@23115
   471
    also have "real N * r \<le> real (Suc n) * r"
huffman@23115
   472
      using r0 n by (simp add: mult_right_mono)
huffman@23115
   473
    finally have "norm x * norm (S n) \<le> real (Suc n) * r * norm (S n)"
huffman@23115
   474
      using norm_ge_zero by (rule mult_right_mono)
huffman@23115
   475
    hence "norm (x * S n) \<le> real (Suc n) * r * norm (S n)"
huffman@23115
   476
      by (rule order_trans [OF norm_mult_ineq])
huffman@23115
   477
    hence "norm (x * S n) / real (Suc n) \<le> r * norm (S n)"
huffman@23115
   478
      by (simp add: pos_divide_le_eq mult_ac)
huffman@23115
   479
    thus "norm (S (Suc n)) \<le> r * norm (S n)"
huffman@23115
   480
      by (simp add: S_Suc norm_scaleR inverse_eq_divide)
huffman@23115
   481
  qed
huffman@23115
   482
qed
huffman@23115
   483
huffman@23115
   484
lemma summable_norm_exp:
huffman@23115
   485
  fixes x :: "'a::{real_normed_algebra_1,recpower,banach}"
huffman@23115
   486
  shows "summable (\<lambda>n. norm (x ^ n /# real (fact n)))"
huffman@23115
   487
proof (rule summable_norm_comparison_test [OF exI, rule_format])
huffman@23115
   488
  show "summable (\<lambda>n. norm x ^ n /# real (fact n))"
huffman@23115
   489
    by (rule summable_exp_generic)
huffman@23115
   490
next
huffman@23115
   491
  fix n show "norm (x ^ n /# real (fact n)) \<le> norm x ^ n /# real (fact n)"
huffman@23115
   492
    by (simp add: norm_scaleR norm_power_ineq)
huffman@23115
   493
qed
huffman@23115
   494
huffman@23043
   495
lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
huffman@23115
   496
by (insert summable_exp_generic [where x=x], simp)
huffman@23043
   497
huffman@23043
   498
lemma summable_sin: 
huffman@23043
   499
     "summable (%n.  
huffman@23043
   500
           (if even n then 0  
huffman@23177
   501
           else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) *  
huffman@23043
   502
                x ^ n)"
huffman@23043
   503
apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
huffman@23043
   504
apply (rule_tac [2] summable_exp)
huffman@23043
   505
apply (rule_tac x = 0 in exI)
huffman@23043
   506
apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
huffman@23043
   507
done
huffman@23043
   508
huffman@23043
   509
lemma summable_cos: 
huffman@23043
   510
      "summable (%n.  
huffman@23043
   511
           (if even n then  
huffman@23177
   512
           -1 ^ (n div 2)/(real (fact n)) else 0) * x ^ n)"
huffman@23043
   513
apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
huffman@23043
   514
apply (rule_tac [2] summable_exp)
huffman@23043
   515
apply (rule_tac x = 0 in exI)
huffman@23043
   516
apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
huffman@23043
   517
done
huffman@23043
   518
huffman@23242
   519
lemma lemma_STAR_sin:
huffman@23043
   520
     "(if even n then 0  
huffman@23177
   521
       else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"
huffman@23043
   522
by (induct "n", auto)
huffman@23043
   523
huffman@23242
   524
lemma lemma_STAR_cos:
huffman@23043
   525
     "0 < n -->  
huffman@23177
   526
      -1 ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
huffman@23043
   527
by (induct "n", auto)
huffman@23043
   528
huffman@23242
   529
lemma lemma_STAR_cos1:
huffman@23043
   530
     "0 < n -->  
huffman@23043
   531
      (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
huffman@23043
   532
by (induct "n", auto)
huffman@23043
   533
huffman@23242
   534
lemma lemma_STAR_cos2:
huffman@23177
   535
  "(\<Sum>n=1..<n. if even n then -1 ^ (n div 2)/(real (fact n)) *  0 ^ n 
huffman@23043
   536
                         else 0) = 0"
huffman@23043
   537
apply (induct "n")
huffman@23043
   538
apply (case_tac [2] "n", auto)
huffman@23043
   539
done
huffman@23043
   540
huffman@23115
   541
lemma exp_converges: "(\<lambda>n. x ^ n /# real (fact n)) sums exp x"
huffman@23115
   542
unfolding exp_def by (rule summable_exp_generic [THEN summable_sums])
huffman@23043
   543
huffman@23043
   544
lemma sin_converges: 
huffman@23043
   545
      "(%n. (if even n then 0  
huffman@23177
   546
            else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) *  
huffman@23043
   547
                 x ^ n) sums sin(x)"
huffman@23112
   548
unfolding sin_def by (rule summable_sin [THEN summable_sums])
huffman@23043
   549
huffman@23043
   550
lemma cos_converges: 
huffman@23043
   551
      "(%n. (if even n then  
huffman@23177
   552
           -1 ^ (n div 2)/(real (fact n))  
huffman@23043
   553
           else 0) * x ^ n) sums cos(x)"
huffman@23112
   554
unfolding cos_def by (rule summable_cos [THEN summable_sums])
huffman@23043
   555
huffman@23043
   556
paulson@15077
   557
subsection{*Formal Derivatives of Exp, Sin, and Cos Series*} 
paulson@15077
   558
paulson@15077
   559
lemma exp_fdiffs: 
paulson@15077
   560
      "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
huffman@23082
   561
by (simp add: diffs_def mult_assoc [symmetric] real_of_nat_def
huffman@23082
   562
         del: mult_Suc of_nat_Suc)
paulson@15077
   563
huffman@23115
   564
lemma diffs_of_real: "diffs (\<lambda>n. of_real (f n)) = (\<lambda>n. of_real (diffs f n))"
huffman@23115
   565
by (simp add: diffs_def)
huffman@23115
   566
paulson@15077
   567
lemma sin_fdiffs: 
paulson@15077
   568
      "diffs(%n. if even n then 0  
huffman@23177
   569
           else -1 ^ ((n - Suc 0) div 2)/(real (fact n)))  
paulson@15077
   570
       = (%n. if even n then  
huffman@23177
   571
                 -1 ^ (n div 2)/(real (fact n))  
paulson@15077
   572
              else 0)"
paulson@15229
   573
by (auto intro!: ext 
huffman@23082
   574
         simp add: diffs_def divide_inverse real_of_nat_def
huffman@23082
   575
         simp del: mult_Suc of_nat_Suc)
paulson@15077
   576
paulson@15077
   577
lemma sin_fdiffs2: 
paulson@15077
   578
       "diffs(%n. if even n then 0  
huffman@23177
   579
           else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) n  
paulson@15077
   580
       = (if even n then  
huffman@23177
   581
                 -1 ^ (n div 2)/(real (fact n))  
paulson@15077
   582
              else 0)"
huffman@23176
   583
by (simp only: sin_fdiffs)
paulson@15077
   584
paulson@15077
   585
lemma cos_fdiffs: 
paulson@15077
   586
      "diffs(%n. if even n then  
huffman@23177
   587
                 -1 ^ (n div 2)/(real (fact n)) else 0)  
paulson@15077
   588
       = (%n. - (if even n then 0  
huffman@23177
   589
           else -1 ^ ((n - Suc 0)div 2)/(real (fact n))))"
paulson@15229
   590
by (auto intro!: ext 
huffman@23082
   591
         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex real_of_nat_def
huffman@23082
   592
         simp del: mult_Suc of_nat_Suc)
paulson@15077
   593
paulson@15077
   594
paulson@15077
   595
lemma cos_fdiffs2: 
paulson@15077
   596
      "diffs(%n. if even n then  
huffman@23177
   597
                 -1 ^ (n div 2)/(real (fact n)) else 0) n 
paulson@15077
   598
       = - (if even n then 0  
huffman@23177
   599
           else -1 ^ ((n - Suc 0)div 2)/(real (fact n)))"
huffman@23176
   600
by (simp only: cos_fdiffs)
paulson@15077
   601
paulson@15077
   602
text{*Now at last we can get the derivatives of exp, sin and cos*}
paulson@15077
   603
paulson@15077
   604
lemma lemma_sin_minus:
nipkow@15546
   605
     "- sin x = (\<Sum>n. - ((if even n then 0 
huffman@23177
   606
                  else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))"
paulson@15077
   607
by (auto intro!: sums_unique sums_minus sin_converges)
paulson@15077
   608
huffman@23115
   609
lemma lemma_exp_ext: "exp = (\<lambda>x. \<Sum>n. x ^ n /# real (fact n))"
paulson@15077
   610
by (auto intro!: ext simp add: exp_def)
paulson@15077
   611
paulson@15077
   612
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
paulson@15229
   613
apply (simp add: exp_def)
paulson@15077
   614
apply (subst lemma_exp_ext)
huffman@23115
   615
apply (subgoal_tac "DERIV (\<lambda>u. \<Sum>n. of_real (inverse (real (fact n))) * u ^ n) x :> (\<Sum>n. diffs (\<lambda>n. of_real (inverse (real (fact n)))) n * x ^ n)")
huffman@23115
   616
apply (rule_tac [2] K = "of_real (1 + norm x)" in termdiffs)
huffman@23115
   617
apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
huffman@23115
   618
apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
huffman@23115
   619
apply (simp del: of_real_add)
paulson@15077
   620
done
paulson@15077
   621
paulson@15077
   622
lemma lemma_sin_ext:
nipkow@15546
   623
     "sin = (%x. \<Sum>n. 
paulson@15077
   624
                   (if even n then 0  
huffman@23177
   625
                       else -1 ^ ((n - Suc 0) div 2)/(real (fact n))) *  
nipkow@15546
   626
                   x ^ n)"
paulson@15077
   627
by (auto intro!: ext simp add: sin_def)
paulson@15077
   628
paulson@15077
   629
lemma lemma_cos_ext:
nipkow@15546
   630
     "cos = (%x. \<Sum>n. 
huffman@23177
   631
                   (if even n then -1 ^ (n div 2)/(real (fact n)) else 0) *
nipkow@15546
   632
                   x ^ n)"
paulson@15077
   633
by (auto intro!: ext simp add: cos_def)
paulson@15077
   634
paulson@15077
   635
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
paulson@15229
   636
apply (simp add: cos_def)
paulson@15077
   637
apply (subst lemma_sin_ext)
paulson@15077
   638
apply (auto simp add: sin_fdiffs2 [symmetric])
paulson@15229
   639
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
webertj@20217
   640
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs)
paulson@15077
   641
done
paulson@15077
   642
paulson@15077
   643
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
paulson@15077
   644
apply (subst lemma_cos_ext)
paulson@15077
   645
apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
paulson@15229
   646
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
webertj@20217
   647
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus)
paulson@15077
   648
done
paulson@15077
   649
huffman@23045
   650
lemma isCont_exp [simp]: "isCont exp x"
huffman@23045
   651
by (rule DERIV_exp [THEN DERIV_isCont])
huffman@23045
   652
huffman@23045
   653
lemma isCont_sin [simp]: "isCont sin x"
huffman@23045
   654
by (rule DERIV_sin [THEN DERIV_isCont])
huffman@23045
   655
huffman@23045
   656
lemma isCont_cos [simp]: "isCont cos x"
huffman@23045
   657
by (rule DERIV_cos [THEN DERIV_isCont])
huffman@23045
   658
paulson@15077
   659
paulson@15077
   660
subsection{*Properties of the Exponential Function*}
paulson@15077
   661
huffman@23278
   662
lemma powser_zero:
huffman@23278
   663
  fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra_1,recpower}"
huffman@23278
   664
  shows "(\<Sum>n. f n * 0 ^ n) = f 0"
paulson@15077
   665
proof -
huffman@23278
   666
  have "(\<Sum>n = 0..<1. f n * 0 ^ n) = (\<Sum>n. f n * 0 ^ n)"
huffman@23115
   667
    by (rule sums_unique [OF series_zero], simp add: power_0_left)
huffman@23278
   668
  thus ?thesis by simp
paulson@15077
   669
qed
paulson@15077
   670
huffman@23278
   671
lemma exp_zero [simp]: "exp 0 = 1"
huffman@23278
   672
unfolding exp_def by (simp add: scaleR_conv_of_real powser_zero)
huffman@23278
   673
huffman@23115
   674
lemma setsum_head2:
huffman@23115
   675
  "m \<le> n \<Longrightarrow> setsum f {m..n} = f m + setsum f {Suc m..n}"
huffman@23115
   676
by (simp add: setsum_head atLeastSucAtMost_greaterThanAtMost)
huffman@23115
   677
huffman@23115
   678
lemma setsum_cl_ivl_Suc2:
huffman@23115
   679
  "(\<Sum>i=m..Suc n. f i) = (if Suc n < m then 0 else f m + (\<Sum>i=m..n. f (Suc i)))"
huffman@23115
   680
by (simp add: setsum_head2 setsum_shift_bounds_cl_Suc_ivl
huffman@23115
   681
         del: setsum_cl_ivl_Suc)
huffman@23115
   682
huffman@23115
   683
lemma exp_series_add:
huffman@23115
   684
  fixes x y :: "'a::{real_field,recpower}"
huffman@23115
   685
  defines S_def: "S \<equiv> \<lambda>x n. x ^ n /# real (fact n)"
huffman@23115
   686
  shows "S (x + y) n = (\<Sum>i=0..n. S x i * S y (n - i))"
huffman@23115
   687
proof (induct n)
huffman@23115
   688
  case 0
huffman@23115
   689
  show ?case
huffman@23115
   690
    unfolding S_def by simp
huffman@23115
   691
next
huffman@23115
   692
  case (Suc n)
huffman@23115
   693
  have S_Suc: "\<And>x n. S x (Suc n) = (x * S x n) /# real (Suc n)"
huffman@23115
   694
    unfolding S_def by (simp add: power_Suc del: mult_Suc)
huffman@23115
   695
  hence times_S: "\<And>x n. x * S x n = real (Suc n) *# S x (Suc n)"
huffman@23115
   696
    by simp
huffman@23115
   697
huffman@23115
   698
  have "real (Suc n) *# S (x + y) (Suc n) = (x + y) * S (x + y) n"
huffman@23115
   699
    by (simp only: times_S)
huffman@23115
   700
  also have "\<dots> = (x + y) * (\<Sum>i=0..n. S x i * S y (n-i))"
huffman@23115
   701
    by (simp only: Suc)
huffman@23115
   702
  also have "\<dots> = x * (\<Sum>i=0..n. S x i * S y (n-i))
huffman@23115
   703
                + y * (\<Sum>i=0..n. S x i * S y (n-i))"
huffman@23115
   704
    by (rule left_distrib)
huffman@23115
   705
  also have "\<dots> = (\<Sum>i=0..n. (x * S x i) * S y (n-i))
huffman@23115
   706
                + (\<Sum>i=0..n. S x i * (y * S y (n-i)))"
huffman@23115
   707
    by (simp only: setsum_right_distrib mult_ac)
huffman@23115
   708
  also have "\<dots> = (\<Sum>i=0..n. real (Suc i) *# (S x (Suc i) * S y (n-i)))
huffman@23115
   709
                + (\<Sum>i=0..n. real (Suc n-i) *# (S x i * S y (Suc n-i)))"
huffman@23115
   710
    by (simp add: times_S Suc_diff_le)
huffman@23115
   711
  also have "(\<Sum>i=0..n. real (Suc i) *# (S x (Suc i) * S y (n-i))) =
huffman@23115
   712
             (\<Sum>i=0..Suc n. real i *# (S x i * S y (Suc n-i)))"
huffman@23115
   713
    by (subst setsum_cl_ivl_Suc2, simp)
huffman@23115
   714
  also have "(\<Sum>i=0..n. real (Suc n-i) *# (S x i * S y (Suc n-i))) =
huffman@23115
   715
             (\<Sum>i=0..Suc n. real (Suc n-i) *# (S x i * S y (Suc n-i)))"
huffman@23115
   716
    by (subst setsum_cl_ivl_Suc, simp)
huffman@23115
   717
  also have "(\<Sum>i=0..Suc n. real i *# (S x i * S y (Suc n-i))) +
huffman@23115
   718
             (\<Sum>i=0..Suc n. real (Suc n-i) *# (S x i * S y (Suc n-i))) =
huffman@23115
   719
             (\<Sum>i=0..Suc n. real (Suc n) *# (S x i * S y (Suc n-i)))"
huffman@23115
   720
    by (simp only: setsum_addf [symmetric] scaleR_left_distrib [symmetric]
huffman@23115
   721
              real_of_nat_add [symmetric], simp)
huffman@23115
   722
  also have "\<dots> = real (Suc n) *# (\<Sum>i=0..Suc n. S x i * S y (Suc n-i))"
huffman@23127
   723
    by (simp only: scaleR_right.setsum)
huffman@23115
   724
  finally show
huffman@23115
   725
    "S (x + y) (Suc n) = (\<Sum>i=0..Suc n. S x i * S y (Suc n - i))"
huffman@23115
   726
    by (simp add: scaleR_cancel_left del: setsum_cl_ivl_Suc)
huffman@23115
   727
qed
huffman@23115
   728
huffman@23115
   729
lemma exp_add: "exp (x + y) = exp x * exp y"
huffman@23115
   730
unfolding exp_def
huffman@23115
   731
by (simp only: Cauchy_product summable_norm_exp exp_series_add)
huffman@23115
   732
huffman@23241
   733
lemma exp_of_real: "exp (of_real x) = of_real (exp x)"
huffman@23241
   734
unfolding exp_def
huffman@23241
   735
apply (subst of_real.suminf)
huffman@23241
   736
apply (rule summable_exp_generic)
huffman@23241
   737
apply (simp add: scaleR_conv_of_real)
huffman@23241
   738
done
huffman@23241
   739
huffman@23115
   740
lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
huffman@22998
   741
apply (drule order_le_imp_less_or_eq, auto)
paulson@15229
   742
apply (simp add: exp_def)
paulson@15077
   743
apply (rule real_le_trans)
paulson@15229
   744
apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
paulson@15077
   745
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_power zero_le_mult_iff)
paulson@15077
   746
done
paulson@15077
   747
huffman@23115
   748
lemma exp_gt_one [simp]: "0 < (x::real) ==> 1 < exp x"
paulson@15077
   749
apply (rule order_less_le_trans)
avigad@17014
   750
apply (rule_tac [2] exp_ge_add_one_self_aux, auto)
paulson@15077
   751
done
paulson@15077
   752
paulson@15077
   753
lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)"
paulson@15077
   754
proof -
paulson@15077
   755
  have "DERIV (exp \<circ> (\<lambda>x. x + y)) x :> exp (x + y) * (1+0)"
huffman@23069
   756
    by (fast intro: DERIV_chain DERIV_add DERIV_exp DERIV_ident DERIV_const) 
paulson@15077
   757
  thus ?thesis by (simp add: o_def)
paulson@15077
   758
qed
paulson@15077
   759
paulson@15077
   760
lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)"
paulson@15077
   761
proof -
paulson@15077
   762
  have "DERIV (exp \<circ> uminus) x :> exp (- x) * - 1"
huffman@23069
   763
    by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_ident)
paulson@15077
   764
  thus ?thesis by (simp add: o_def)
paulson@15077
   765
qed
paulson@15077
   766
paulson@15077
   767
lemma DERIV_exp_exp_zero [simp]: "DERIV (%x. exp (x + y) * exp (- x)) x :> 0"
paulson@15077
   768
proof -
paulson@15077
   769
  have "DERIV (\<lambda>x. exp (x + y) * exp (- x)) x
paulson@15077
   770
       :> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)"
paulson@15077
   771
    by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult) 
huffman@23115
   772
  thus ?thesis by (simp add: mult_commute)
paulson@15077
   773
qed
paulson@15077
   774
huffman@23115
   775
lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y::real)"
paulson@15077
   776
proof -
paulson@15077
   777
  have "\<forall>x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0" by simp
paulson@15077
   778
  hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)" 
paulson@15077
   779
    by (rule DERIV_isconst_all) 
paulson@15077
   780
  thus ?thesis by simp
paulson@15077
   781
qed
paulson@15077
   782
paulson@15077
   783
lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1"
huffman@23115
   784
by (simp add: exp_add [symmetric])
paulson@15077
   785
paulson@15077
   786
lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1"
paulson@15077
   787
by (simp add: mult_commute)
paulson@15077
   788
paulson@15077
   789
paulson@15077
   790
lemma exp_minus: "exp(-x) = inverse(exp(x))"
paulson@15077
   791
by (auto intro: inverse_unique [symmetric])
paulson@15077
   792
paulson@15077
   793
text{*Proof: because every exponential can be seen as a square.*}
huffman@23115
   794
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
paulson@15077
   795
apply (rule_tac t = x in real_sum_of_halves [THEN subst])
paulson@15077
   796
apply (subst exp_add, auto)
paulson@15077
   797
done
paulson@15077
   798
paulson@15077
   799
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
paulson@15077
   800
apply (cut_tac x = x in exp_mult_minus2)
paulson@15077
   801
apply (auto simp del: exp_mult_minus2)
paulson@15077
   802
done
paulson@15077
   803
huffman@23115
   804
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
paulson@15077
   805
by (simp add: order_less_le)
paulson@15077
   806
huffman@23115
   807
lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x::real)"
paulson@15077
   808
by (auto intro: positive_imp_inverse_positive)
paulson@15077
   809
huffman@23115
   810
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
paulson@15229
   811
by auto
paulson@15077
   812
paulson@15077
   813
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
paulson@15251
   814
apply (induct "n")
paulson@15077
   815
apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
paulson@15077
   816
done
paulson@15077
   817
paulson@15077
   818
lemma exp_diff: "exp(x - y) = exp(x)/(exp y)"
paulson@15229
   819
apply (simp add: diff_minus divide_inverse)
paulson@15077
   820
apply (simp (no_asm) add: exp_add exp_minus)
paulson@15077
   821
done
paulson@15077
   822
paulson@15077
   823
paulson@15077
   824
lemma exp_less_mono:
huffman@23115
   825
  fixes x y :: real
paulson@15077
   826
  assumes xy: "x < y" shows "exp x < exp y"
paulson@15077
   827
proof -
paulson@15077
   828
  have "1 < exp (y + - x)"
paulson@15077
   829
    by (rule real_less_sum_gt_zero [THEN exp_gt_one])
paulson@15077
   830
  hence "exp x * inverse (exp x) < exp y * inverse (exp x)"
paulson@15077
   831
    by (auto simp add: exp_add exp_minus)
paulson@15077
   832
  thus ?thesis
nipkow@15539
   833
    by (simp add: divide_inverse [symmetric] pos_less_divide_eq
paulson@15228
   834
             del: divide_self_if)
paulson@15077
   835
qed
paulson@15077
   836
huffman@23115
   837
lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
paulson@15228
   838
apply (simp add: linorder_not_le [symmetric]) 
paulson@15228
   839
apply (auto simp add: order_le_less exp_less_mono) 
paulson@15077
   840
done
paulson@15077
   841
huffman@23115
   842
lemma exp_less_cancel_iff [iff]: "(exp(x::real) < exp(y)) = (x < y)"
paulson@15077
   843
by (auto intro: exp_less_mono exp_less_cancel)
paulson@15077
   844
huffman@23115
   845
lemma exp_le_cancel_iff [iff]: "(exp(x::real) \<le> exp(y)) = (x \<le> y)"
paulson@15077
   846
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
   847
huffman@23115
   848
lemma exp_inj_iff [iff]: "(exp (x::real) = exp y) = (x = y)"
paulson@15077
   849
by (simp add: order_eq_iff)
paulson@15077
   850
huffman@23115
   851
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
paulson@15077
   852
apply (rule IVT)
huffman@23045
   853
apply (auto intro: isCont_exp simp add: le_diff_eq)
paulson@15077
   854
apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)") 
paulson@15077
   855
apply simp 
avigad@17014
   856
apply (rule exp_ge_add_one_self_aux, simp)
paulson@15077
   857
done
paulson@15077
   858
huffman@23115
   859
lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
paulson@15077
   860
apply (rule_tac x = 1 and y = y in linorder_cases)
paulson@15077
   861
apply (drule order_less_imp_le [THEN lemma_exp_total])
paulson@15077
   862
apply (rule_tac [2] x = 0 in exI)
paulson@15077
   863
apply (frule_tac [3] real_inverse_gt_one)
paulson@15077
   864
apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
paulson@15077
   865
apply (rule_tac x = "-x" in exI)
paulson@15077
   866
apply (simp add: exp_minus)
paulson@15077
   867
done
paulson@15077
   868
paulson@15077
   869
paulson@15077
   870
subsection{*Properties of the Logarithmic Function*}
paulson@15077
   871
huffman@23043
   872
definition
huffman@23043
   873
  ln :: "real => real" where
huffman@23043
   874
  "ln x = (THE u. exp u = x)"
huffman@23043
   875
huffman@23043
   876
lemma ln_exp [simp]: "ln (exp x) = x"
paulson@15077
   877
by (simp add: ln_def)
paulson@15077
   878
huffman@22654
   879
lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
huffman@22654
   880
by (auto dest: exp_total)
huffman@22654
   881
huffman@23043
   882
lemma exp_ln_iff [simp]: "(exp (ln x) = x) = (0 < x)"
paulson@15077
   883
apply (auto dest: exp_total)
paulson@15077
   884
apply (erule subst, simp) 
paulson@15077
   885
done
paulson@15077
   886
paulson@15077
   887
lemma ln_mult: "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)"
paulson@15077
   888
apply (rule exp_inj_iff [THEN iffD1])
huffman@22654
   889
apply (simp add: exp_add exp_ln mult_pos_pos)
paulson@15077
   890
done
paulson@15077
   891
paulson@15077
   892
lemma ln_inj_iff[simp]: "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)"
paulson@15077
   893
apply (simp only: exp_ln_iff [symmetric])
paulson@15077
   894
apply (erule subst)+
paulson@15077
   895
apply simp 
paulson@15077
   896
done
paulson@15077
   897
paulson@15077
   898
lemma ln_one[simp]: "ln 1 = 0"
paulson@15077
   899
by (rule exp_inj_iff [THEN iffD1], auto)
paulson@15077
   900
paulson@15077
   901
lemma ln_inverse: "0 < x ==> ln(inverse x) = - ln x"
paulson@15077
   902
apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1])
paulson@15077
   903
apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric])
paulson@15077
   904
done
paulson@15077
   905
paulson@15077
   906
lemma ln_div: 
paulson@15077
   907
    "[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y"
paulson@15229
   908
apply (simp add: divide_inverse)
paulson@15077
   909
apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse)
paulson@15077
   910
done
paulson@15077
   911
paulson@15077
   912
lemma ln_less_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)"
paulson@15077
   913
apply (simp only: exp_ln_iff [symmetric])
paulson@15077
   914
apply (erule subst)+
paulson@15077
   915
apply simp 
paulson@15077
   916
done
paulson@15077
   917
paulson@15077
   918
lemma ln_le_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x \<le> ln y) = (x \<le> y)"
paulson@15077
   919
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
   920
paulson@15077
   921
lemma ln_realpow: "0 < x ==> ln(x ^ n) = real n * ln(x)"
paulson@15077
   922
by (auto dest!: exp_total simp add: exp_real_of_nat_mult [symmetric])
paulson@15077
   923
paulson@15077
   924
lemma ln_add_one_self_le_self [simp]: "0 \<le> x ==> ln(1 + x) \<le> x"
paulson@15077
   925
apply (rule ln_exp [THEN subst])
avigad@17014
   926
apply (rule ln_le_cancel_iff [THEN iffD2]) 
avigad@17014
   927
apply (auto simp add: exp_ge_add_one_self_aux)
paulson@15077
   928
done
paulson@15077
   929
paulson@15077
   930
lemma ln_less_self [simp]: "0 < x ==> ln x < x"
paulson@15077
   931
apply (rule order_less_le_trans)
paulson@15077
   932
apply (rule_tac [2] ln_add_one_self_le_self)
paulson@15077
   933
apply (rule ln_less_cancel_iff [THEN iffD2], auto)
paulson@15077
   934
done
paulson@15077
   935
paulson@15234
   936
lemma ln_ge_zero [simp]:
paulson@15077
   937
  assumes x: "1 \<le> x" shows "0 \<le> ln x"
paulson@15077
   938
proof -
paulson@15077
   939
  have "0 < x" using x by arith
paulson@15077
   940
  hence "exp 0 \<le> exp (ln x)"
huffman@22915
   941
    by (simp add: x)
paulson@15077
   942
  thus ?thesis by (simp only: exp_le_cancel_iff)
paulson@15077
   943
qed
paulson@15077
   944
paulson@15077
   945
lemma ln_ge_zero_imp_ge_one:
paulson@15077
   946
  assumes ln: "0 \<le> ln x" 
paulson@15077
   947
      and x:  "0 < x"
paulson@15077
   948
  shows "1 \<le> x"
paulson@15077
   949
proof -
paulson@15077
   950
  from ln have "ln 1 \<le> ln x" by simp
paulson@15077
   951
  thus ?thesis by (simp add: x del: ln_one) 
paulson@15077
   952
qed
paulson@15077
   953
paulson@15077
   954
lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
paulson@15077
   955
by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
paulson@15077
   956
paulson@15234
   957
lemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)"
paulson@15234
   958
by (insert ln_ge_zero_iff [of x], arith)
paulson@15234
   959
paulson@15077
   960
lemma ln_gt_zero:
paulson@15077
   961
  assumes x: "1 < x" shows "0 < ln x"
paulson@15077
   962
proof -
paulson@15077
   963
  have "0 < x" using x by arith
huffman@22915
   964
  hence "exp 0 < exp (ln x)" by (simp add: x)
paulson@15077
   965
  thus ?thesis  by (simp only: exp_less_cancel_iff)
paulson@15077
   966
qed
paulson@15077
   967
paulson@15077
   968
lemma ln_gt_zero_imp_gt_one:
paulson@15077
   969
  assumes ln: "0 < ln x" 
paulson@15077
   970
      and x:  "0 < x"
paulson@15077
   971
  shows "1 < x"
paulson@15077
   972
proof -
paulson@15077
   973
  from ln have "ln 1 < ln x" by simp
paulson@15077
   974
  thus ?thesis by (simp add: x del: ln_one) 
paulson@15077
   975
qed
paulson@15077
   976
paulson@15077
   977
lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
paulson@15077
   978
by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
paulson@15077
   979
paulson@15234
   980
lemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)"
paulson@15234
   981
by (insert ln_less_zero_iff [of x] ln_gt_zero_iff [of x], arith)
paulson@15077
   982
paulson@15077
   983
lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
paulson@15234
   984
by simp
paulson@15077
   985
paulson@15077
   986
lemma exp_ln_eq: "exp u = x ==> ln x = u"
paulson@15077
   987
by auto
paulson@15077
   988
huffman@23045
   989
lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
huffman@23045
   990
apply (subgoal_tac "isCont ln (exp (ln x))", simp)
huffman@23045
   991
apply (rule isCont_inverse_function [where f=exp], simp_all)
huffman@23045
   992
done
huffman@23045
   993
huffman@23045
   994
lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
huffman@23045
   995
apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
huffman@23045
   996
apply (erule lemma_DERIV_subst [OF DERIV_exp exp_ln])
huffman@23045
   997
apply (simp_all add: abs_if isCont_ln)
huffman@23045
   998
done
huffman@23045
   999
paulson@15077
  1000
paulson@15077
  1001
subsection{*Basic Properties of the Trigonometric Functions*}
paulson@15077
  1002
paulson@15077
  1003
lemma sin_zero [simp]: "sin 0 = 0"
huffman@23278
  1004
unfolding sin_def by (simp add: powser_zero)
paulson@15077
  1005
paulson@15077
  1006
lemma cos_zero [simp]: "cos 0 = 1"
huffman@23278
  1007
unfolding cos_def by (simp add: powser_zero)
paulson@15077
  1008
paulson@15077
  1009
lemma DERIV_sin_sin_mult [simp]:
paulson@15077
  1010
     "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077
  1011
by (rule DERIV_mult, auto)
paulson@15077
  1012
paulson@15077
  1013
lemma DERIV_sin_sin_mult2 [simp]:
paulson@15077
  1014
     "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"
paulson@15077
  1015
apply (cut_tac x = x in DERIV_sin_sin_mult)
paulson@15077
  1016
apply (auto simp add: mult_assoc)
paulson@15077
  1017
done
paulson@15077
  1018
paulson@15077
  1019
lemma DERIV_sin_realpow2 [simp]:
paulson@15077
  1020
     "DERIV (%x. (sin x)\<twosuperior>) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077
  1021
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077
  1022
paulson@15077
  1023
lemma DERIV_sin_realpow2a [simp]:
paulson@15077
  1024
     "DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"
paulson@15077
  1025
by (auto simp add: numeral_2_eq_2)
paulson@15077
  1026
paulson@15077
  1027
lemma DERIV_cos_cos_mult [simp]:
paulson@15077
  1028
     "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077
  1029
by (rule DERIV_mult, auto)
paulson@15077
  1030
paulson@15077
  1031
lemma DERIV_cos_cos_mult2 [simp]:
paulson@15077
  1032
     "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"
paulson@15077
  1033
apply (cut_tac x = x in DERIV_cos_cos_mult)
paulson@15077
  1034
apply (auto simp add: mult_ac)
paulson@15077
  1035
done
paulson@15077
  1036
paulson@15077
  1037
lemma DERIV_cos_realpow2 [simp]:
paulson@15077
  1038
     "DERIV (%x. (cos x)\<twosuperior>) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077
  1039
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077
  1040
paulson@15077
  1041
lemma DERIV_cos_realpow2a [simp]:
paulson@15077
  1042
     "DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"
paulson@15077
  1043
by (auto simp add: numeral_2_eq_2)
paulson@15077
  1044
paulson@15077
  1045
lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
paulson@15077
  1046
by auto
paulson@15077
  1047
paulson@15077
  1048
lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\<twosuperior>) x :> -(2 * cos(x) * sin(x))"
paulson@15077
  1049
apply (rule lemma_DERIV_subst)
paulson@15077
  1050
apply (rule DERIV_cos_realpow2a, auto)
paulson@15077
  1051
done
paulson@15077
  1052
paulson@15077
  1053
(* most useful *)
paulson@15229
  1054
lemma DERIV_cos_cos_mult3 [simp]:
paulson@15229
  1055
     "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
paulson@15077
  1056
apply (rule lemma_DERIV_subst)
paulson@15077
  1057
apply (rule DERIV_cos_cos_mult2, auto)
paulson@15077
  1058
done
paulson@15077
  1059
paulson@15077
  1060
lemma DERIV_sin_circle_all: 
paulson@15077
  1061
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>  
paulson@15077
  1062
             (2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
paulson@15229
  1063
apply (simp only: diff_minus, safe)
paulson@15229
  1064
apply (rule DERIV_add) 
paulson@15077
  1065
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1066
done
paulson@15077
  1067
paulson@15229
  1068
lemma DERIV_sin_circle_all_zero [simp]:
paulson@15229
  1069
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
paulson@15077
  1070
by (cut_tac DERIV_sin_circle_all, auto)
paulson@15077
  1071
paulson@15077
  1072
lemma sin_cos_squared_add [simp]: "((sin x)\<twosuperior>) + ((cos x)\<twosuperior>) = 1"
paulson@15077
  1073
apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all])
paulson@15077
  1074
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1075
done
paulson@15077
  1076
paulson@15077
  1077
lemma sin_cos_squared_add2 [simp]: "((cos x)\<twosuperior>) + ((sin x)\<twosuperior>) = 1"
huffman@23286
  1078
apply (subst add_commute)
paulson@15077
  1079
apply (simp (no_asm) del: realpow_Suc)
paulson@15077
  1080
done
paulson@15077
  1081
paulson@15077
  1082
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
paulson@15077
  1083
apply (cut_tac x = x in sin_cos_squared_add2)
paulson@15077
  1084
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1085
done
paulson@15077
  1086
paulson@15077
  1087
lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
paulson@15229
  1088
apply (rule_tac a1 = "(cos x)\<twosuperior>" in add_right_cancel [THEN iffD1])
paulson@15077
  1089
apply (simp del: realpow_Suc)
paulson@15077
  1090
done
paulson@15077
  1091
paulson@15077
  1092
lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
paulson@15077
  1093
apply (rule_tac a1 = "(sin x)\<twosuperior>" in add_right_cancel [THEN iffD1])
paulson@15077
  1094
apply (simp del: realpow_Suc)
paulson@15077
  1095
done
paulson@15077
  1096
paulson@15077
  1097
lemma real_gt_one_ge_zero_add_less: "[| 1 < x; 0 \<le> y |] ==> 1 < x + (y::real)"
paulson@15077
  1098
by arith
paulson@15077
  1099
paulson@15081
  1100
lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
huffman@23097
  1101
by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
paulson@15077
  1102
paulson@15077
  1103
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
paulson@15077
  1104
apply (insert abs_sin_le_one [of x]) 
huffman@22998
  1105
apply (simp add: abs_le_iff del: abs_sin_le_one) 
paulson@15077
  1106
done
paulson@15077
  1107
paulson@15077
  1108
lemma sin_le_one [simp]: "sin x \<le> 1"
paulson@15077
  1109
apply (insert abs_sin_le_one [of x]) 
huffman@22998
  1110
apply (simp add: abs_le_iff del: abs_sin_le_one) 
paulson@15077
  1111
done
paulson@15077
  1112
paulson@15081
  1113
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
huffman@23097
  1114
by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
paulson@15077
  1115
paulson@15077
  1116
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
paulson@15077
  1117
apply (insert abs_cos_le_one [of x]) 
huffman@22998
  1118
apply (simp add: abs_le_iff del: abs_cos_le_one) 
paulson@15077
  1119
done
paulson@15077
  1120
paulson@15077
  1121
lemma cos_le_one [simp]: "cos x \<le> 1"
paulson@15077
  1122
apply (insert abs_cos_le_one [of x]) 
huffman@22998
  1123
apply (simp add: abs_le_iff del: abs_cos_le_one)
paulson@15077
  1124
done
paulson@15077
  1125
paulson@15077
  1126
lemma DERIV_fun_pow: "DERIV g x :> m ==>  
paulson@15077
  1127
      DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
paulson@15077
  1128
apply (rule lemma_DERIV_subst)
paulson@15229
  1129
apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2)
paulson@15077
  1130
apply (rule DERIV_pow, auto)
paulson@15077
  1131
done
paulson@15077
  1132
paulson@15229
  1133
lemma DERIV_fun_exp:
paulson@15229
  1134
     "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
paulson@15077
  1135
apply (rule lemma_DERIV_subst)
paulson@15077
  1136
apply (rule_tac f = exp in DERIV_chain2)
paulson@15077
  1137
apply (rule DERIV_exp, auto)
paulson@15077
  1138
done
paulson@15077
  1139
paulson@15229
  1140
lemma DERIV_fun_sin:
paulson@15229
  1141
     "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
paulson@15077
  1142
apply (rule lemma_DERIV_subst)
paulson@15077
  1143
apply (rule_tac f = sin in DERIV_chain2)
paulson@15077
  1144
apply (rule DERIV_sin, auto)
paulson@15077
  1145
done
paulson@15077
  1146
paulson@15229
  1147
lemma DERIV_fun_cos:
paulson@15229
  1148
     "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
paulson@15077
  1149
apply (rule lemma_DERIV_subst)
paulson@15077
  1150
apply (rule_tac f = cos in DERIV_chain2)
paulson@15077
  1151
apply (rule DERIV_cos, auto)
paulson@15077
  1152
done
paulson@15077
  1153
huffman@23069
  1154
lemmas DERIV_intros = DERIV_ident DERIV_const DERIV_cos DERIV_cmult 
paulson@15077
  1155
                    DERIV_sin  DERIV_exp  DERIV_inverse DERIV_pow 
paulson@15077
  1156
                    DERIV_add  DERIV_diff  DERIV_mult  DERIV_minus 
paulson@15077
  1157
                    DERIV_inverse_fun DERIV_quotient DERIV_fun_pow 
paulson@15077
  1158
                    DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos 
paulson@15077
  1159
paulson@15077
  1160
(* lemma *)
paulson@15229
  1161
lemma lemma_DERIV_sin_cos_add:
paulson@15229
  1162
     "\<forall>x.  
paulson@15077
  1163
         DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077
  1164
               (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"
paulson@15077
  1165
apply (safe, rule lemma_DERIV_subst)
paulson@15077
  1166
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15077
  1167
  --{*replaces the old @{text DERIV_tac}*}
paulson@15229
  1168
apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
paulson@15077
  1169
done
paulson@15077
  1170
paulson@15077
  1171
lemma sin_cos_add [simp]:
paulson@15077
  1172
     "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077
  1173
      (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
paulson@15077
  1174
apply (cut_tac y = 0 and x = x and y7 = y 
paulson@15077
  1175
       in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])
paulson@15077
  1176
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1177
done
paulson@15077
  1178
paulson@15077
  1179
lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
paulson@15077
  1180
apply (cut_tac x = x and y = y in sin_cos_add)
huffman@22969
  1181
apply (simp del: sin_cos_add)
paulson@15077
  1182
done
paulson@15077
  1183
paulson@15077
  1184
lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
paulson@15077
  1185
apply (cut_tac x = x and y = y in sin_cos_add)
huffman@22969
  1186
apply (simp del: sin_cos_add)
paulson@15077
  1187
done
paulson@15077
  1188
paulson@15085
  1189
lemma lemma_DERIV_sin_cos_minus:
paulson@15085
  1190
    "\<forall>x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"
paulson@15077
  1191
apply (safe, rule lemma_DERIV_subst)
paulson@15077
  1192
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15229
  1193
apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
paulson@15077
  1194
done
paulson@15077
  1195
paulson@15085
  1196
lemma sin_cos_minus [simp]: 
paulson@15085
  1197
    "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"
paulson@15085
  1198
apply (cut_tac y = 0 and x = x 
paulson@15085
  1199
       in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])
huffman@22969
  1200
apply simp
paulson@15077
  1201
done
paulson@15077
  1202
paulson@15077
  1203
lemma sin_minus [simp]: "sin (-x) = -sin(x)"
paulson@15077
  1204
apply (cut_tac x = x in sin_cos_minus)
huffman@22969
  1205
apply (simp del: sin_cos_minus)
paulson@15077
  1206
done
paulson@15077
  1207
paulson@15077
  1208
lemma cos_minus [simp]: "cos (-x) = cos(x)"
paulson@15077
  1209
apply (cut_tac x = x in sin_cos_minus)
huffman@22969
  1210
apply (simp del: sin_cos_minus)
paulson@15077
  1211
done
paulson@15077
  1212
paulson@15077
  1213
lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
huffman@22969
  1214
by (simp add: diff_minus sin_add)
paulson@15077
  1215
paulson@15077
  1216
lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
paulson@15077
  1217
by (simp add: sin_diff mult_commute)
paulson@15077
  1218
paulson@15077
  1219
lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
huffman@22969
  1220
by (simp add: diff_minus cos_add)
paulson@15077
  1221
paulson@15077
  1222
lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
paulson@15077
  1223
by (simp add: cos_diff mult_commute)
paulson@15077
  1224
paulson@15077
  1225
lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
paulson@15077
  1226
by (cut_tac x = x and y = x in sin_add, auto)
paulson@15077
  1227
paulson@15077
  1228
paulson@15077
  1229
lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
paulson@15077
  1230
apply (cut_tac x = x and y = x in cos_add)
huffman@22969
  1231
apply (simp add: power2_eq_square)
paulson@15077
  1232
done
paulson@15077
  1233
paulson@15077
  1234
paulson@15077
  1235
subsection{*The Constant Pi*}
paulson@15077
  1236
huffman@23043
  1237
definition
huffman@23043
  1238
  pi :: "real" where
huffman@23053
  1239
  "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
huffman@23043
  1240
paulson@15077
  1241
text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"}; 
paulson@15077
  1242
   hence define pi.*}
paulson@15077
  1243
paulson@15077
  1244
lemma sin_paired:
huffman@23177
  1245
     "(%n. -1 ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) 
paulson@15077
  1246
      sums  sin x"
paulson@15077
  1247
proof -
paulson@15077
  1248
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1249
            (if even k then 0
huffman@23177
  1250
             else -1 ^ ((k - Suc 0) div 2) / real (fact k)) *
paulson@15077
  1251
            x ^ k) 
huffman@23176
  1252
	sums sin x"
huffman@23176
  1253
    unfolding sin_def
paulson@15077
  1254
    by (rule sin_converges [THEN sums_summable, THEN sums_group], simp) 
huffman@23176
  1255
  thus ?thesis by (simp add: mult_ac)
paulson@15077
  1256
qed
paulson@15077
  1257
paulson@15077
  1258
lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077
  1259
apply (subgoal_tac 
paulson@15077
  1260
       "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
huffman@23177
  1261
              -1 ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1)) 
huffman@23177
  1262
     sums (\<Sum>n. -1 ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
paulson@15077
  1263
 prefer 2
paulson@15077
  1264
 apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1265
apply (rotate_tac 2)
paulson@15077
  1266
apply (drule sin_paired [THEN sums_unique, THEN ssubst])
paulson@15077
  1267
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1268
apply (frule sums_unique)
paulson@15077
  1269
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1270
apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])
paulson@15077
  1271
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1272
apply (erule sums_summable)
paulson@15077
  1273
apply (case_tac "m=0")
paulson@15077
  1274
apply (simp (no_asm_simp))
paulson@15234
  1275
apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x") 
nipkow@15539
  1276
apply (simp only: mult_less_cancel_left, simp)  
nipkow@15539
  1277
apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
paulson@15077
  1278
apply (subgoal_tac "x*x < 2*3", simp) 
paulson@15077
  1279
apply (rule mult_strict_mono)
paulson@15085
  1280
apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc)
paulson@15077
  1281
apply (subst fact_Suc)
paulson@15077
  1282
apply (subst fact_Suc)
paulson@15077
  1283
apply (subst fact_Suc)
paulson@15077
  1284
apply (subst fact_Suc)
paulson@15077
  1285
apply (subst real_of_nat_mult)
paulson@15077
  1286
apply (subst real_of_nat_mult)
paulson@15077
  1287
apply (subst real_of_nat_mult)
paulson@15077
  1288
apply (subst real_of_nat_mult)
nipkow@15539
  1289
apply (simp (no_asm) add: divide_inverse del: fact_Suc)
paulson@15077
  1290
apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
paulson@15077
  1291
apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right) 
paulson@15077
  1292
apply (auto simp add: mult_assoc simp del: fact_Suc)
paulson@15077
  1293
apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right) 
paulson@15077
  1294
apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
paulson@15077
  1295
apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)") 
paulson@15077
  1296
apply (erule ssubst)+
paulson@15077
  1297
apply (auto simp del: fact_Suc)
paulson@15077
  1298
apply (subgoal_tac "0 < x ^ (4 * m) ")
paulson@15077
  1299
 prefer 2 apply (simp only: zero_less_power) 
paulson@15077
  1300
apply (simp (no_asm_simp) add: mult_less_cancel_left)
paulson@15077
  1301
apply (rule mult_strict_mono)
paulson@15077
  1302
apply (simp_all (no_asm_simp))
paulson@15077
  1303
done
paulson@15077
  1304
paulson@15077
  1305
lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077
  1306
by (auto intro: sin_gt_zero)
paulson@15077
  1307
paulson@15077
  1308
lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
paulson@15077
  1309
apply (cut_tac x = x in sin_gt_zero1)
paulson@15077
  1310
apply (auto simp add: cos_squared_eq cos_double)
paulson@15077
  1311
done
paulson@15077
  1312
paulson@15077
  1313
lemma cos_paired:
huffman@23177
  1314
     "(%n. -1 ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
paulson@15077
  1315
proof -
paulson@15077
  1316
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
huffman@23177
  1317
            (if even k then -1 ^ (k div 2) / real (fact k) else 0) *
paulson@15077
  1318
            x ^ k) 
huffman@23176
  1319
        sums cos x"
huffman@23176
  1320
    unfolding cos_def
paulson@15077
  1321
    by (rule cos_converges [THEN sums_summable, THEN sums_group], simp) 
huffman@23176
  1322
  thus ?thesis by (simp add: mult_ac)
paulson@15077
  1323
qed
paulson@15077
  1324
paulson@15077
  1325
declare zero_less_power [simp]
paulson@15077
  1326
paulson@15077
  1327
lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
paulson@15077
  1328
by simp
paulson@15077
  1329
huffman@23053
  1330
lemma cos_two_less_zero [simp]: "cos (2) < 0"
paulson@15077
  1331
apply (cut_tac x = 2 in cos_paired)
paulson@15077
  1332
apply (drule sums_minus)
paulson@15077
  1333
apply (rule neg_less_iff_less [THEN iffD1]) 
nipkow@15539
  1334
apply (frule sums_unique, auto)
nipkow@15539
  1335
apply (rule_tac y =
huffman@23177
  1336
 "\<Sum>n=0..< Suc(Suc(Suc 0)). - (-1 ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
paulson@15481
  1337
       in order_less_trans)
paulson@15077
  1338
apply (simp (no_asm) add: fact_num_eq_if realpow_num_eq_if del: fact_Suc realpow_Suc)
nipkow@15561
  1339
apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
paulson@15077
  1340
apply (rule sumr_pos_lt_pair)
paulson@15077
  1341
apply (erule sums_summable, safe)
paulson@15085
  1342
apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric] 
paulson@15085
  1343
            del: fact_Suc)
paulson@15077
  1344
apply (rule real_mult_inverse_cancel2)
paulson@15077
  1345
apply (rule real_of_nat_fact_gt_zero)+
paulson@15077
  1346
apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
paulson@15077
  1347
apply (subst fact_lemma) 
paulson@15481
  1348
apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
paulson@15481
  1349
apply (simp only: real_of_nat_mult)
huffman@23007
  1350
apply (rule mult_strict_mono, force)
huffman@23007
  1351
  apply (rule_tac [3] real_of_nat_fact_ge_zero)
paulson@15481
  1352
 prefer 2 apply force
paulson@15077
  1353
apply (rule real_of_nat_less_iff [THEN iffD2])
paulson@15077
  1354
apply (rule fact_less_mono, auto)
paulson@15077
  1355
done
huffman@23053
  1356
huffman@23053
  1357
lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
huffman@23053
  1358
lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
paulson@15077
  1359
paulson@15077
  1360
lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
paulson@15077
  1361
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")
paulson@15077
  1362
apply (rule_tac [2] IVT2)
paulson@15077
  1363
apply (auto intro: DERIV_isCont DERIV_cos)
paulson@15077
  1364
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077
  1365
apply (rule ccontr)
paulson@15077
  1366
apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")
paulson@15077
  1367
apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
paulson@15077
  1368
apply (drule_tac f = cos in Rolle)
paulson@15077
  1369
apply (drule_tac [5] f = cos in Rolle)
paulson@15077
  1370
apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
paulson@15077
  1371
apply (drule_tac y1 = xa in order_le_less_trans [THEN sin_gt_zero])
paulson@15077
  1372
apply (assumption, rule_tac y=y in order_less_le_trans, simp_all) 
paulson@15077
  1373
apply (drule_tac y1 = y in order_le_less_trans [THEN sin_gt_zero], assumption, simp_all) 
paulson@15077
  1374
done
paulson@15077
  1375
    
huffman@23053
  1376
lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
paulson@15077
  1377
by (simp add: pi_def)
paulson@15077
  1378
paulson@15077
  1379
lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
huffman@23053
  1380
by (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1381
huffman@23053
  1382
lemma pi_half_gt_zero [simp]: "0 < pi / 2"
huffman@23053
  1383
apply (rule order_le_neq_trans)
huffman@23053
  1384
apply (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1385
apply (rule notI, drule arg_cong [where f=cos], simp)
paulson@15077
  1386
done
paulson@15077
  1387
huffman@23053
  1388
lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
huffman@23053
  1389
lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
paulson@15077
  1390
huffman@23053
  1391
lemma pi_half_less_two [simp]: "pi / 2 < 2"
huffman@23053
  1392
apply (rule order_le_neq_trans)
huffman@23053
  1393
apply (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1394
apply (rule notI, drule arg_cong [where f=cos], simp)
paulson@15077
  1395
done
huffman@23053
  1396
huffman@23053
  1397
lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
huffman@23053
  1398
lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
paulson@15077
  1399
paulson@15077
  1400
lemma pi_gt_zero [simp]: "0 < pi"
huffman@23053
  1401
by (insert pi_half_gt_zero, simp)
huffman@23053
  1402
huffman@23053
  1403
lemma pi_ge_zero [simp]: "0 \<le> pi"
huffman@23053
  1404
by (rule pi_gt_zero [THEN order_less_imp_le])
paulson@15077
  1405
paulson@15077
  1406
lemma pi_neq_zero [simp]: "pi \<noteq> 0"
huffman@22998
  1407
by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym])
paulson@15077
  1408
huffman@23053
  1409
lemma pi_not_less_zero [simp]: "\<not> pi < 0"
huffman@23053
  1410
by (simp add: linorder_not_less)
paulson@15077
  1411
paulson@15077
  1412
lemma minus_pi_half_less_zero [simp]: "-(pi/2) < 0"
paulson@15077
  1413
by auto
paulson@15077
  1414
paulson@15077
  1415
lemma sin_pi_half [simp]: "sin(pi/2) = 1"
paulson@15077
  1416
apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
paulson@15077
  1417
apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
huffman@23053
  1418
apply (simp add: power2_eq_square)
paulson@15077
  1419
done
paulson@15077
  1420
paulson@15077
  1421
lemma cos_pi [simp]: "cos pi = -1"
nipkow@15539
  1422
by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
paulson@15077
  1423
paulson@15077
  1424
lemma sin_pi [simp]: "sin pi = 0"
nipkow@15539
  1425
by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
paulson@15077
  1426
paulson@15077
  1427
lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
paulson@15229
  1428
by (simp add: diff_minus cos_add)
huffman@23053
  1429
declare sin_cos_eq [symmetric, simp]
paulson@15077
  1430
paulson@15077
  1431
lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
paulson@15229
  1432
by (simp add: cos_add)
paulson@15077
  1433
declare minus_sin_cos_eq [symmetric, simp]
paulson@15077
  1434
paulson@15077
  1435
lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
paulson@15229
  1436
by (simp add: diff_minus sin_add)
huffman@23053
  1437
declare cos_sin_eq [symmetric, simp]
paulson@15077
  1438
paulson@15077
  1439
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
paulson@15229
  1440
by (simp add: sin_add)
paulson@15077
  1441
paulson@15077
  1442
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
paulson@15229
  1443
by (simp add: sin_add)
paulson@15077
  1444
paulson@15077
  1445
lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
paulson@15229
  1446
by (simp add: cos_add)
paulson@15077
  1447
paulson@15077
  1448
lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
paulson@15077
  1449
by (simp add: sin_add cos_double)
paulson@15077
  1450
paulson@15077
  1451
lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
paulson@15077
  1452
by (simp add: cos_add cos_double)
paulson@15077
  1453
paulson@15077
  1454
lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
paulson@15251
  1455
apply (induct "n")
paulson@15077
  1456
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1457
done
paulson@15077
  1458
paulson@15383
  1459
lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
paulson@15383
  1460
proof -
paulson@15383
  1461
  have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
paulson@15383
  1462
  also have "... = -1 ^ n" by (rule cos_npi) 
paulson@15383
  1463
  finally show ?thesis .
paulson@15383
  1464
qed
paulson@15383
  1465
paulson@15077
  1466
lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
paulson@15251
  1467
apply (induct "n")
paulson@15077
  1468
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1469
done
paulson@15077
  1470
paulson@15077
  1471
lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
paulson@15383
  1472
by (simp add: mult_commute [of pi]) 
paulson@15077
  1473
paulson@15077
  1474
lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
paulson@15077
  1475
by (simp add: cos_double)
paulson@15077
  1476
paulson@15077
  1477
lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
paulson@15229
  1478
by simp
paulson@15077
  1479
paulson@15077
  1480
lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
paulson@15077
  1481
apply (rule sin_gt_zero, assumption)
paulson@15077
  1482
apply (rule order_less_trans, assumption)
paulson@15077
  1483
apply (rule pi_half_less_two)
paulson@15077
  1484
done
paulson@15077
  1485
paulson@15077
  1486
lemma sin_less_zero: 
paulson@15077
  1487
  assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
paulson@15077
  1488
proof -
paulson@15077
  1489
  have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2) 
paulson@15077
  1490
  thus ?thesis by simp
paulson@15077
  1491
qed
paulson@15077
  1492
paulson@15077
  1493
lemma pi_less_4: "pi < 4"
paulson@15077
  1494
by (cut_tac pi_half_less_two, auto)
paulson@15077
  1495
paulson@15077
  1496
lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1497
apply (cut_tac pi_less_4)
paulson@15077
  1498
apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
paulson@15077
  1499
apply (cut_tac cos_is_zero, safe)
paulson@15077
  1500
apply (rename_tac y z)
paulson@15077
  1501
apply (drule_tac x = y in spec)
paulson@15077
  1502
apply (drule_tac x = "pi/2" in spec, simp) 
paulson@15077
  1503
done
paulson@15077
  1504
paulson@15077
  1505
lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1506
apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15077
  1507
apply (rule cos_minus [THEN subst])
paulson@15077
  1508
apply (rule cos_gt_zero)
paulson@15077
  1509
apply (auto intro: cos_gt_zero)
paulson@15077
  1510
done
paulson@15077
  1511
 
paulson@15077
  1512
lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
paulson@15077
  1513
apply (auto simp add: order_le_less cos_gt_zero_pi)
paulson@15077
  1514
apply (subgoal_tac "x = pi/2", auto) 
paulson@15077
  1515
done
paulson@15077
  1516
paulson@15077
  1517
lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
paulson@15077
  1518
apply (subst sin_cos_eq)
paulson@15077
  1519
apply (rotate_tac 1)
paulson@15077
  1520
apply (drule real_sum_of_halves [THEN ssubst])
paulson@15077
  1521
apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
paulson@15077
  1522
done
paulson@15077
  1523
paulson@15077
  1524
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
paulson@15077
  1525
by (auto simp add: order_le_less sin_gt_zero_pi)
paulson@15077
  1526
paulson@15077
  1527
lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
paulson@15077
  1528
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
paulson@15077
  1529
apply (rule_tac [2] IVT2)
paulson@15077
  1530
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
paulson@15077
  1531
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077
  1532
apply (rule ccontr, auto)
paulson@15077
  1533
apply (drule_tac f = cos in Rolle)
paulson@15077
  1534
apply (drule_tac [5] f = cos in Rolle)
paulson@15077
  1535
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
paulson@15077
  1536
            dest!: DERIV_cos [THEN DERIV_unique] 
paulson@15077
  1537
            simp add: differentiable_def)
paulson@15077
  1538
apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
paulson@15077
  1539
done
paulson@15077
  1540
paulson@15077
  1541
lemma sin_total:
paulson@15077
  1542
     "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
paulson@15077
  1543
apply (rule ccontr)
paulson@15077
  1544
apply (subgoal_tac "\<forall>x. (- (pi/2) \<le> x & x \<le> pi/2 & (sin x = y)) = (0 \<le> (x + pi/2) & (x + pi/2) \<le> pi & (cos (x + pi/2) = -y))")
wenzelm@18585
  1545
apply (erule contrapos_np)
paulson@15077
  1546
apply (simp del: minus_sin_cos_eq [symmetric])
paulson@15077
  1547
apply (cut_tac y="-y" in cos_total, simp) apply simp 
paulson@15077
  1548
apply (erule ex1E)
paulson@15229
  1549
apply (rule_tac a = "x - (pi/2)" in ex1I)
huffman@23286
  1550
apply (simp (no_asm) add: add_assoc)
paulson@15077
  1551
apply (rotate_tac 3)
paulson@15077
  1552
apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all) 
paulson@15077
  1553
done
paulson@15077
  1554
paulson@15077
  1555
lemma reals_Archimedean4:
paulson@15077
  1556
     "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
paulson@15077
  1557
apply (auto dest!: reals_Archimedean3)
paulson@15077
  1558
apply (drule_tac x = x in spec, clarify) 
paulson@15077
  1559
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
paulson@15077
  1560
 prefer 2 apply (erule LeastI) 
paulson@15077
  1561
apply (case_tac "LEAST m::nat. x < real m * y", simp) 
paulson@15077
  1562
apply (subgoal_tac "~ x < real nat * y")
paulson@15077
  1563
 prefer 2 apply (rule not_less_Least, simp, force)  
paulson@15077
  1564
done
paulson@15077
  1565
paulson@15077
  1566
(* Pre Isabelle99-2 proof was simpler- numerals arithmetic 
paulson@15077
  1567
   now causes some unwanted re-arrangements of literals!   *)
paulson@15229
  1568
lemma cos_zero_lemma:
paulson@15229
  1569
     "[| 0 \<le> x; cos x = 0 |] ==>  
paulson@15077
  1570
      \<exists>n::nat. ~even n & x = real n * (pi/2)"
paulson@15077
  1571
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
paulson@15086
  1572
apply (subgoal_tac "0 \<le> x - real n * pi & 
paulson@15086
  1573
                    (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
paulson@15086
  1574
apply (auto simp add: compare_rls) 
paulson@15077
  1575
  prefer 3 apply (simp add: cos_diff) 
paulson@15077
  1576
 prefer 2 apply (simp add: real_of_nat_Suc left_distrib) 
paulson@15077
  1577
apply (simp add: cos_diff)
paulson@15077
  1578
apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
paulson@15077
  1579
apply (rule_tac [2] cos_total, safe)
paulson@15077
  1580
apply (drule_tac x = "x - real n * pi" in spec)
paulson@15077
  1581
apply (drule_tac x = "pi/2" in spec)
paulson@15077
  1582
apply (simp add: cos_diff)
paulson@15229
  1583
apply (rule_tac x = "Suc (2 * n)" in exI)
paulson@15077
  1584
apply (simp add: real_of_nat_Suc left_distrib, auto)
paulson@15077
  1585
done
paulson@15077
  1586
paulson@15229
  1587
lemma sin_zero_lemma:
paulson@15229
  1588
     "[| 0 \<le> x; sin x = 0 |] ==>  
paulson@15077
  1589
      \<exists>n::nat. even n & x = real n * (pi/2)"
paulson@15077
  1590
apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
paulson@15077
  1591
 apply (clarify, rule_tac x = "n - 1" in exI)
paulson@15077
  1592
 apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
paulson@15085
  1593
apply (rule cos_zero_lemma)
paulson@15085
  1594
apply (simp_all add: add_increasing)  
paulson@15077
  1595
done
paulson@15077
  1596
paulson@15077
  1597
paulson@15229
  1598
lemma cos_zero_iff:
paulson@15229
  1599
     "(cos x = 0) =  
paulson@15077
  1600
      ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |    
paulson@15077
  1601
       (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1602
apply (rule iffI)
paulson@15077
  1603
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1604
apply (drule cos_zero_lemma, assumption+)
paulson@15077
  1605
apply (cut_tac x="-x" in cos_zero_lemma, simp, simp) 
paulson@15077
  1606
apply (force simp add: minus_equation_iff [of x]) 
paulson@15077
  1607
apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) 
nipkow@15539
  1608
apply (auto simp add: cos_add)
paulson@15077
  1609
done
paulson@15077
  1610
paulson@15077
  1611
(* ditto: but to a lesser extent *)
paulson@15229
  1612
lemma sin_zero_iff:
paulson@15229
  1613
     "(sin x = 0) =  
paulson@15077
  1614
      ((\<exists>n::nat. even n & (x = real n * (pi/2))) |    
paulson@15077
  1615
       (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1616
apply (rule iffI)
paulson@15077
  1617
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1618
apply (drule sin_zero_lemma, assumption+)
paulson@15077
  1619
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
paulson@15077
  1620
apply (force simp add: minus_equation_iff [of x]) 
nipkow@15539
  1621
apply (auto simp add: even_mult_two_ex)
paulson@15077
  1622
done
paulson@15077
  1623
paulson@15077
  1624
paulson@15077
  1625
subsection{*Tangent*}
paulson@15077
  1626
huffman@23043
  1627
definition
huffman@23043
  1628
  tan :: "real => real" where
huffman@23043
  1629
  "tan x = (sin x)/(cos x)"
huffman@23043
  1630
paulson@15077
  1631
lemma tan_zero [simp]: "tan 0 = 0"
paulson@15077
  1632
by (simp add: tan_def)
paulson@15077
  1633
paulson@15077
  1634
lemma tan_pi [simp]: "tan pi = 0"
paulson@15077
  1635
by (simp add: tan_def)
paulson@15077
  1636
paulson@15077
  1637
lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
paulson@15077
  1638
by (simp add: tan_def)
paulson@15077
  1639
paulson@15077
  1640
lemma tan_minus [simp]: "tan (-x) = - tan x"
paulson@15077
  1641
by (simp add: tan_def minus_mult_left)
paulson@15077
  1642
paulson@15077
  1643
lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
paulson@15077
  1644
by (simp add: tan_def)
paulson@15077
  1645
paulson@15077
  1646
lemma lemma_tan_add1: 
paulson@15077
  1647
      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077
  1648
        ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
paulson@15229
  1649
apply (simp add: tan_def divide_inverse)
paulson@15229
  1650
apply (auto simp del: inverse_mult_distrib 
paulson@15229
  1651
            simp add: inverse_mult_distrib [symmetric] mult_ac)
paulson@15077
  1652
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15229
  1653
apply (auto simp del: inverse_mult_distrib 
paulson@15229
  1654
            simp add: mult_assoc left_diff_distrib cos_add)
paulson@15234
  1655
done  
paulson@15077
  1656
paulson@15077
  1657
lemma add_tan_eq: 
paulson@15077
  1658
      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077
  1659
       ==> tan x + tan y = sin(x + y)/(cos x * cos y)"
paulson@15229
  1660
apply (simp add: tan_def)
paulson@15077
  1661
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15077
  1662
apply (auto simp add: mult_assoc left_distrib)
nipkow@15539
  1663
apply (simp add: sin_add)
paulson@15077
  1664
done
paulson@15077
  1665
paulson@15229
  1666
lemma tan_add:
paulson@15229
  1667
     "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]  
paulson@15077
  1668
      ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
paulson@15077
  1669
apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)
paulson@15077
  1670
apply (simp add: tan_def)
paulson@15077
  1671
done
paulson@15077
  1672
paulson@15229
  1673
lemma tan_double:
paulson@15229
  1674
     "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]  
paulson@15077
  1675
      ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
paulson@15077
  1676
apply (insert tan_add [of x x]) 
paulson@15077
  1677
apply (simp add: mult_2 [symmetric])  
paulson@15077
  1678
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1679
done
paulson@15077
  1680
paulson@15077
  1681
lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
paulson@15229
  1682
by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) 
paulson@15077
  1683
paulson@15077
  1684
lemma tan_less_zero: 
paulson@15077
  1685
  assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
paulson@15077
  1686
proof -
paulson@15077
  1687
  have "0 < tan (- x)" using prems by (simp only: tan_gt_zero) 
paulson@15077
  1688
  thus ?thesis by simp
paulson@15077
  1689
qed
paulson@15077
  1690
paulson@15077
  1691
lemma lemma_DERIV_tan:
paulson@15077
  1692
     "cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"
paulson@15077
  1693
apply (rule lemma_DERIV_subst)
paulson@15077
  1694
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15079
  1695
apply (auto simp add: divide_inverse numeral_2_eq_2)
paulson@15077
  1696
done
paulson@15077
  1697
paulson@15077
  1698
lemma DERIV_tan [simp]: "cos x \<noteq> 0 ==> DERIV tan x :> inverse((cos x)\<twosuperior>)"
paulson@15077
  1699
by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])
paulson@15077
  1700
huffman@23045
  1701
lemma isCont_tan [simp]: "cos x \<noteq> 0 ==> isCont tan x"
huffman@23045
  1702
by (rule DERIV_tan [THEN DERIV_isCont])
huffman@23045
  1703
paulson@15077
  1704
lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
paulson@15077
  1705
apply (subgoal_tac "(\<lambda>x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")
paulson@15229
  1706
apply (simp add: divide_inverse [symmetric])
huffman@22613
  1707
apply (rule LIM_mult)
paulson@15077
  1708
apply (rule_tac [2] inverse_1 [THEN subst])
paulson@15077
  1709
apply (rule_tac [2] LIM_inverse)
paulson@15077
  1710
apply (simp_all add: divide_inverse [symmetric]) 
paulson@15077
  1711
apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric]) 
paulson@15077
  1712
apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
paulson@15077
  1713
done
paulson@15077
  1714
paulson@15077
  1715
lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
paulson@15077
  1716
apply (cut_tac LIM_cos_div_sin)
paulson@15077
  1717
apply (simp only: LIM_def)
paulson@15077
  1718
apply (drule_tac x = "inverse y" in spec, safe, force)
paulson@15077
  1719
apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
paulson@15229
  1720
apply (rule_tac x = "(pi/2) - e" in exI)
paulson@15077
  1721
apply (simp (no_asm_simp))
paulson@15229
  1722
apply (drule_tac x = "(pi/2) - e" in spec)
paulson@15229
  1723
apply (auto simp add: tan_def)
paulson@15077
  1724
apply (rule inverse_less_iff_less [THEN iffD1])
paulson@15079
  1725
apply (auto simp add: divide_inverse)
paulson@15229
  1726
apply (rule real_mult_order) 
paulson@15229
  1727
apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
paulson@15229
  1728
apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute) 
paulson@15077
  1729
done
paulson@15077
  1730
paulson@15077
  1731
lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
huffman@22998
  1732
apply (frule order_le_imp_less_or_eq, safe)
paulson@15077
  1733
 prefer 2 apply force
paulson@15077
  1734
apply (drule lemma_tan_total, safe)
paulson@15077
  1735
apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
paulson@15077
  1736
apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
paulson@15077
  1737
apply (drule_tac y = xa in order_le_imp_less_or_eq)
paulson@15077
  1738
apply (auto dest: cos_gt_zero)
paulson@15077
  1739
done
paulson@15077
  1740
paulson@15077
  1741
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  1742
apply (cut_tac linorder_linear [of 0 y], safe)
paulson@15077
  1743
apply (drule tan_total_pos)
paulson@15077
  1744
apply (cut_tac [2] y="-y" in tan_total_pos, safe)
paulson@15077
  1745
apply (rule_tac [3] x = "-x" in exI)
paulson@15077
  1746
apply (auto intro!: exI)
paulson@15077
  1747
done
paulson@15077
  1748
paulson@15077
  1749
lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  1750
apply (cut_tac y = y in lemma_tan_total1, auto)
paulson@15077
  1751
apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
paulson@15077
  1752
apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
paulson@15077
  1753
apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
paulson@15077
  1754
apply (rule_tac [4] Rolle)
paulson@15077
  1755
apply (rule_tac [2] Rolle)
paulson@15077
  1756
apply (auto intro!: DERIV_tan DERIV_isCont exI 
paulson@15077
  1757
            simp add: differentiable_def)
paulson@15077
  1758
txt{*Now, simulate TRYALL*}
paulson@15077
  1759
apply (rule_tac [!] DERIV_tan asm_rl)
paulson@15077
  1760
apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
huffman@22998
  1761
	    simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym]) 
paulson@15077
  1762
done
paulson@15077
  1763
huffman@23043
  1764
huffman@23043
  1765
subsection {* Inverse Trigonometric Functions *}
huffman@23043
  1766
huffman@23043
  1767
definition
huffman@23043
  1768
  arcsin :: "real => real" where
huffman@23043
  1769
  "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
huffman@23043
  1770
huffman@23043
  1771
definition
huffman@23043
  1772
  arccos :: "real => real" where
huffman@23043
  1773
  "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
huffman@23043
  1774
huffman@23043
  1775
definition     
huffman@23043
  1776
  arctan :: "real => real" where
huffman@23043
  1777
  "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
huffman@23043
  1778
paulson@15229
  1779
lemma arcsin:
paulson@15229
  1780
     "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077
  1781
      ==> -(pi/2) \<le> arcsin y &  
paulson@15077
  1782
           arcsin y \<le> pi/2 & sin(arcsin y) = y"
huffman@23011
  1783
unfolding arcsin_def by (rule theI' [OF sin_total])
huffman@23011
  1784
huffman@23011
  1785
lemma arcsin_pi:
huffman@23011
  1786
     "[| -1 \<le> y; y \<le> 1 |]  
huffman@23011
  1787
      ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
huffman@23011
  1788
apply (drule (1) arcsin)
huffman@23011
  1789
apply (force intro: order_trans)
paulson@15077
  1790
done
paulson@15077
  1791
paulson@15077
  1792
lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
paulson@15077
  1793
by (blast dest: arcsin)
paulson@15077
  1794
      
paulson@15077
  1795
lemma arcsin_bounded:
paulson@15077
  1796
     "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
paulson@15077
  1797
by (blast dest: arcsin)
paulson@15077
  1798
paulson@15077
  1799
lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
paulson@15077
  1800
by (blast dest: arcsin)
paulson@15077
  1801
paulson@15077
  1802
lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
paulson@15077
  1803
by (blast dest: arcsin)
paulson@15077
  1804
paulson@15077
  1805
lemma arcsin_lt_bounded:
paulson@15077
  1806
     "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
paulson@15077
  1807
apply (frule order_less_imp_le)
paulson@15077
  1808
apply (frule_tac y = y in order_less_imp_le)
paulson@15077
  1809
apply (frule arcsin_bounded)
paulson@15077
  1810
apply (safe, simp)
paulson@15077
  1811
apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
paulson@15077
  1812
apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
paulson@15077
  1813
apply (drule_tac [!] f = sin in arg_cong, auto)
paulson@15077
  1814
done
paulson@15077
  1815
paulson@15077
  1816
lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
paulson@15077
  1817
apply (unfold arcsin_def)
huffman@23011
  1818
apply (rule the1_equality)
paulson@15077
  1819
apply (rule sin_total, auto)
paulson@15077
  1820
done
paulson@15077
  1821
huffman@22975
  1822
lemma arccos:
paulson@15229
  1823
     "[| -1 \<le> y; y \<le> 1 |]  
huffman@22975
  1824
      ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
huffman@23011
  1825
unfolding arccos_def by (rule theI' [OF cos_total])
paulson@15077
  1826
huffman@22975
  1827
lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
huffman@22975
  1828
by (blast dest: arccos)
paulson@15077
  1829
      
huffman@22975
  1830
lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
huffman@22975
  1831
by (blast dest: arccos)
paulson@15077
  1832
huffman@22975
  1833
lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
huffman@22975
  1834
by (blast dest: arccos)
paulson@15077
  1835
huffman@22975
  1836
lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
huffman@22975
  1837
by (blast dest: arccos)
paulson@15077
  1838
huffman@22975
  1839
lemma arccos_lt_bounded:
paulson@15229
  1840
     "[| -1 < y; y < 1 |]  
huffman@22975
  1841
      ==> 0 < arccos y & arccos y < pi"
paulson@15077
  1842
apply (frule order_less_imp_le)
paulson@15077
  1843
apply (frule_tac y = y in order_less_imp_le)
huffman@22975
  1844
apply (frule arccos_bounded, auto)
huffman@22975
  1845
apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
paulson@15077
  1846
apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
paulson@15077
  1847
apply (drule_tac [!] f = cos in arg_cong, auto)
paulson@15077
  1848
done
paulson@15077
  1849
huffman@22975
  1850
lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
huffman@22975
  1851
apply (simp add: arccos_def)
huffman@23011
  1852
apply (auto intro!: the1_equality cos_total)
paulson@15077
  1853
done
paulson@15077
  1854
huffman@22975
  1855
lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
huffman@22975
  1856
apply (simp add: arccos_def)
huffman@23011
  1857
apply (auto intro!: the1_equality cos_total)
paulson@15077
  1858
done
paulson@15077
  1859
huffman@23045
  1860
lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<twosuperior>)"
huffman@23045
  1861
apply (subgoal_tac "x\<twosuperior> \<le> 1")
huffman@23052
  1862
apply (rule power2_eq_imp_eq)
huffman@23045
  1863
apply (simp add: cos_squared_eq)
huffman@23045
  1864
apply (rule cos_ge_zero)
huffman@23045
  1865
apply (erule (1) arcsin_lbound)
huffman@23045
  1866
apply (erule (1) arcsin_ubound)
huffman@23045
  1867
apply simp
huffman@23045
  1868
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
huffman@23045
  1869
apply (rule power_mono, simp, simp)
huffman@23045
  1870
done
huffman@23045
  1871
huffman@23045
  1872
lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<twosuperior>)"
huffman@23045
  1873
apply (subgoal_tac "x\<twosuperior> \<le> 1")
huffman@23052
  1874
apply (rule power2_eq_imp_eq)
huffman@23045
  1875
apply (simp add: sin_squared_eq)
huffman@23045
  1876
apply (rule sin_ge_zero)
huffman@23045
  1877
apply (erule (1) arccos_lbound)
huffman@23045
  1878
apply (erule (1) arccos_ubound)
huffman@23045
  1879
apply simp
huffman@23045
  1880
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
huffman@23045
  1881
apply (rule power_mono, simp, simp)
huffman@23045
  1882
done
huffman@23045
  1883
paulson@15077
  1884
lemma arctan [simp]:
paulson@15077
  1885
     "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
huffman@23011
  1886
unfolding arctan_def by (rule theI' [OF tan_total])
paulson@15077
  1887
paulson@15077
  1888
lemma tan_arctan: "tan(arctan y) = y"
paulson@15077
  1889
by auto
paulson@15077
  1890
paulson@15077
  1891
lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
paulson@15077
  1892
by (auto simp only: arctan)
paulson@15077
  1893
paulson@15077
  1894
lemma arctan_lbound: "- (pi/2) < arctan y"
paulson@15077
  1895
by auto
paulson@15077
  1896
paulson@15077
  1897
lemma arctan_ubound: "arctan y < pi/2"
paulson@15077
  1898
by (auto simp only: arctan)
paulson@15077
  1899
paulson@15077
  1900
lemma arctan_tan: 
paulson@15077
  1901
      "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
paulson@15077
  1902
apply (unfold arctan_def)
huffman@23011
  1903
apply (rule the1_equality)
paulson@15077
  1904
apply (rule tan_total, auto)
paulson@15077
  1905
done
paulson@15077
  1906
paulson@15077
  1907
lemma arctan_zero_zero [simp]: "arctan 0 = 0"
paulson@15077
  1908
by (insert arctan_tan [of 0], simp)
paulson@15077
  1909
paulson@15077
  1910
lemma cos_arctan_not_zero [simp]: "cos(arctan x) \<noteq> 0"
paulson@15077
  1911
apply (auto simp add: cos_zero_iff)
paulson@15077
  1912
apply (case_tac "n")
paulson@15077
  1913
apply (case_tac [3] "n")
paulson@15077
  1914
apply (cut_tac [2] y = x in arctan_ubound)
paulson@15077
  1915
apply (cut_tac [4] y = x in arctan_lbound) 
paulson@15077
  1916
apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)
paulson@15077
  1917
done
paulson@15077
  1918
paulson@15077
  1919
lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
paulson@15077
  1920
apply (rule power_inverse [THEN subst])
paulson@15077
  1921
apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
huffman@22960
  1922
apply (auto dest: field_power_not_zero
huffman@20516
  1923
        simp add: power_mult_distrib left_distrib power_divide tan_def 
paulson@15077
  1924
                  mult_assoc power_inverse [symmetric] 
paulson@15077
  1925
        simp del: realpow_Suc)
paulson@15077
  1926
done
paulson@15077
  1927
huffman@23045
  1928
lemma isCont_inverse_function2:
huffman@23045
  1929
  fixes f g :: "real \<Rightarrow> real" shows
huffman@23045
  1930
  "\<lbrakk>a < x; x < b;
huffman@23045
  1931
    \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
huffman@23045
  1932
    \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
huffman@23045
  1933
   \<Longrightarrow> isCont g (f x)"
huffman@23045
  1934
apply (rule isCont_inverse_function
huffman@23045
  1935
       [where f=f and d="min (x - a) (b - x)"])
huffman@23045
  1936
apply (simp_all add: abs_le_iff)
huffman@23045
  1937
done
huffman@23045
  1938
huffman@23045
  1939
lemma isCont_arcsin: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arcsin x"
huffman@23045
  1940
apply (subgoal_tac "isCont arcsin (sin (arcsin x))", simp)
huffman@23045
  1941
apply (rule isCont_inverse_function2 [where f=sin])
huffman@23045
  1942
apply (erule (1) arcsin_lt_bounded [THEN conjunct1])
huffman@23045
  1943
apply (erule (1) arcsin_lt_bounded [THEN conjunct2])
huffman@23045
  1944
apply (fast intro: arcsin_sin, simp)
huffman@23045
  1945
done
huffman@23045
  1946
huffman@23045
  1947
lemma isCont_arccos: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arccos x"
huffman@23045
  1948
apply (subgoal_tac "isCont arccos (cos (arccos x))", simp)
huffman@23045
  1949
apply (rule isCont_inverse_function2 [where f=cos])
huffman@23045
  1950
apply (erule (1) arccos_lt_bounded [THEN conjunct1])
huffman@23045
  1951
apply (erule (1) arccos_lt_bounded [THEN conjunct2])
huffman@23045
  1952
apply (fast intro: arccos_cos, simp)
huffman@23045
  1953
done
huffman@23045
  1954
huffman@23045
  1955
lemma isCont_arctan: "isCont arctan x"
huffman@23045
  1956
apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
huffman@23045
  1957
apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
huffman@23045
  1958
apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
huffman@23045
  1959
apply (erule (1) isCont_inverse_function2 [where f=tan])
huffman@23045
  1960
apply (clarify, rule arctan_tan)
huffman@23045
  1961
apply (erule (1) order_less_le_trans)
huffman@23045
  1962
apply (erule (1) order_le_less_trans)
huffman@23045
  1963
apply (clarify, rule isCont_tan)
huffman@23045
  1964
apply (rule less_imp_neq [symmetric])
huffman@23045
  1965
apply (rule cos_gt_zero_pi)
huffman@23045
  1966
apply (erule (1) order_less_le_trans)
huffman@23045
  1967
apply (erule (1) order_le_less_trans)
huffman@23045
  1968
done
huffman@23045
  1969
huffman@23045
  1970
lemma DERIV_arcsin:
huffman@23045
  1971
  "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<twosuperior>))"
huffman@23045
  1972
apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
huffman@23045
  1973
apply (rule lemma_DERIV_subst [OF DERIV_sin])
huffman@23045
  1974
apply (simp add: cos_arcsin)
huffman@23045
  1975
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
huffman@23045
  1976
apply (rule power_strict_mono, simp, simp, simp)
huffman@23045
  1977
apply assumption
huffman@23045
  1978
apply assumption
huffman@23045
  1979
apply simp
huffman@23045
  1980
apply (erule (1) isCont_arcsin)
huffman@23045
  1981
done
huffman@23045
  1982
huffman@23045
  1983
lemma DERIV_arccos:
huffman@23045
  1984
  "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<twosuperior>))"
huffman@23045
  1985
apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
huffman@23045
  1986
apply (rule lemma_DERIV_subst [OF DERIV_cos])
huffman@23045
  1987
apply (simp add: sin_arccos)
huffman@23045
  1988
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
huffman@23045
  1989
apply (rule power_strict_mono, simp, simp, simp)
huffman@23045
  1990
apply assumption
huffman@23045
  1991
apply assumption
huffman@23045
  1992
apply simp
huffman@23045
  1993
apply (erule (1) isCont_arccos)
huffman@23045
  1994
done
huffman@23045
  1995
huffman@23045
  1996
lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<twosuperior>)"
huffman@23045
  1997
apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
huffman@23045
  1998
apply (rule lemma_DERIV_subst [OF DERIV_tan])
huffman@23045
  1999
apply (rule cos_arctan_not_zero)
huffman@23045
  2000
apply (simp add: power_inverse tan_sec [symmetric])
huffman@23045
  2001
apply (subgoal_tac "0 < 1 + x\<twosuperior>", simp)
huffman@23045
  2002
apply (simp add: add_pos_nonneg)
huffman@23045
  2003
apply (simp, simp, simp, rule isCont_arctan)
huffman@23045
  2004
done
huffman@23045
  2005
huffman@23045
  2006
huffman@23043
  2007
subsection {* More Theorems about Sin and Cos *}
huffman@23043
  2008
huffman@23052
  2009
lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
huffman@23052
  2010
proof -
huffman@23052
  2011
  let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
huffman@23052
  2012
  have nonneg: "0 \<le> ?c"
huffman@23052
  2013
    by (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
huffman@23052
  2014
  have "0 = cos (pi / 4 + pi / 4)"
huffman@23052
  2015
    by simp
huffman@23052
  2016
  also have "cos (pi / 4 + pi / 4) = ?c\<twosuperior> - ?s\<twosuperior>"
huffman@23052
  2017
    by (simp only: cos_add power2_eq_square)
huffman@23052
  2018
  also have "\<dots> = 2 * ?c\<twosuperior> - 1"
huffman@23052
  2019
    by (simp add: sin_squared_eq)
huffman@23052
  2020
  finally have "?c\<twosuperior> = (sqrt 2 / 2)\<twosuperior>"
huffman@23052
  2021
    by (simp add: power_divide)
huffman@23052
  2022
  thus ?thesis
huffman@23052
  2023
    using nonneg by (rule power2_eq_imp_eq) simp
huffman@23052
  2024
qed
huffman@23052
  2025
huffman@23052
  2026
lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"
huffman@23052
  2027
proof -
huffman@23052
  2028
  let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
huffman@23052
  2029
  have pos_c: "0 < ?c"
huffman@23052
  2030
    by (rule cos_gt_zero, simp, simp)
huffman@23052
  2031
  have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
huffman@23066
  2032
    by simp
huffman@23052
  2033
  also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
huffman@23052
  2034
    by (simp only: cos_add sin_add)
huffman@23052
  2035
  also have "\<dots> = ?c * (?c\<twosuperior> - 3 * ?s\<twosuperior>)"
huffman@23052
  2036
    by (simp add: ring_eq_simps power2_eq_square)
huffman@23052
  2037
  finally have "?c\<twosuperior> = (sqrt 3 / 2)\<twosuperior>"
huffman@23052
  2038
    using pos_c by (simp add: sin_squared_eq power_divide)
huffman@23052
  2039
  thus ?thesis
huffman@23052
  2040
    using pos_c [THEN order_less_imp_le]
huffman@23052
  2041
    by (rule power2_eq_imp_eq) simp
huffman@23052
  2042
qed
huffman@23052
  2043
huffman@23052
  2044
lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
huffman@23052
  2045
proof -
huffman@23052
  2046
  have "sin (pi / 4) = cos (pi / 2 - pi / 4)" by (rule sin_cos_eq)
huffman@23052
  2047
  also have "pi / 2 - pi / 4 = pi / 4" by simp
huffman@23052
  2048
  also have "cos (pi / 4) = sqrt 2 / 2" by (rule cos_45)
huffman@23052
  2049
  finally show ?thesis .
huffman@23052
  2050
qed
huffman@23052
  2051
huffman@23052
  2052
lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
huffman@23052
  2053
proof -
huffman@23052
  2054
  have "sin (pi / 3) = cos (pi / 2 - pi / 3)" by (rule sin_cos_eq)
huffman@23052
  2055
  also have "pi / 2 - pi / 3 = pi / 6" by simp
huffman@23052
  2056
  also have "cos (pi / 6) = sqrt 3 / 2" by (rule cos_30)
huffman@23052
  2057
  finally show ?thesis .
huffman@23052
  2058
qed
huffman@23052
  2059
huffman@23052
  2060
lemma cos_60: "cos (pi / 3) = 1 / 2"
huffman@23052
  2061
apply (rule power2_eq_imp_eq)
huffman@23052
  2062
apply (simp add: cos_squared_eq sin_60 power_divide)
huffman@23052
  2063
apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
huffman@23052
  2064
done
huffman@23052
  2065
huffman@23052
  2066
lemma sin_30: "sin (pi / 6) = 1 / 2"
huffman@23052
  2067
proof -
huffman@23052
  2068
  have "sin (pi / 6) = cos (pi / 2 - pi / 6)" by (rule sin_cos_eq)
huffman@23066
  2069
  also have "pi / 2 - pi / 6 = pi / 3" by simp
huffman@23052
  2070
  also have "cos (pi / 3) = 1 / 2" by (rule cos_60)
huffman@23052
  2071
  finally show ?thesis .
huffman@23052
  2072
qed
huffman@23052
  2073
huffman@23052
  2074
lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
huffman@23052
  2075
unfolding tan_def by (simp add: sin_30 cos_30)
huffman@23052
  2076
huffman@23052
  2077
lemma tan_45: "tan (pi / 4) = 1"
huffman@23052
  2078
unfolding tan_def by (simp add: sin_45 cos_45)
huffman@23052
  2079
huffman@23052
  2080
lemma tan_60: "tan (pi / 3) = sqrt 3"
huffman@23052
  2081
unfolding tan_def by (simp add: sin_60 cos_60)
huffman@23052
  2082
paulson@15085
  2083
text{*NEEDED??*}
paulson@15229
  2084
lemma [simp]:
paulson@15229
  2085
     "sin (x + 1 / 2 * real (Suc m) * pi) =  
paulson@15229
  2086
      cos (x + 1 / 2 * real  (m) * pi)"
paulson@15229
  2087
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib, auto)
paulson@15077
  2088
paulson@15085
  2089
text{*NEEDED??*}
paulson@15229
  2090
lemma [simp]:
paulson@15229
  2091
     "sin (x + real (Suc m) * pi / 2) =  
paulson@15229
  2092
      cos (x + real (m) * pi / 2)"
paulson@15229
  2093
by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)
paulson@15077
  2094
paulson@15077
  2095
lemma DERIV_sin_add [simp]: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)"
paulson@15077
  2096
apply (rule lemma_DERIV_subst)
paulson@15077
  2097
apply (rule_tac f = sin and g = "%x. x + k" in DERIV_chain2)
paulson@15077
  2098
apply (best intro!: DERIV_intros intro: DERIV_chain2)+
paulson@15077
  2099
apply (simp (no_asm))
paulson@15077
  2100
done
paulson@15077
  2101
paulson@15383
  2102
lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
paulson@15383
  2103
proof -
paulson@15383
  2104
  have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
paulson@15383
  2105
    by (auto simp add: right_distrib sin_add left_distrib mult_ac)
paulson@15383
  2106
  thus ?thesis
paulson@15383
  2107
    by (simp add: real_of_nat_Suc left_distrib add_divide_distrib 
paulson@15383
  2108
                  mult_commute [of pi])
paulson@15383
  2109
qed
paulson@15077
  2110
paulson@15077
  2111
lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
paulson@15077
  2112
by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
paulson@15077
  2113
paulson@15077
  2114
lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
huffman@23066
  2115
apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
huffman@23066
  2116
apply (subst cos_add, simp)
paulson@15077
  2117
done
paulson@15077
  2118
paulson@15077
  2119
lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
paulson@15077
  2120
by (auto simp add: mult_assoc)
paulson@15077
  2121
paulson@15077
  2122
lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
huffman@23066
  2123
apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
huffman@23066
  2124
apply (subst sin_add, simp)
paulson@15077
  2125
done
paulson@15077
  2126
paulson@15077
  2127
(*NEEDED??*)
paulson@15229
  2128
lemma [simp]:
paulson@15229
  2129
     "cos(x + 1 / 2 * real(Suc m) * pi) = -sin (x + 1 / 2 * real m * pi)"
paulson@15077
  2130
apply (simp only: cos_add sin_add real_of_nat_Suc right_distrib left_distrib minus_mult_right, auto)
paulson@15077
  2131
done
paulson@15077
  2132
paulson@15077
  2133
(*NEEDED??*)
paulson@15077
  2134
lemma [simp]: "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
paulson@15229
  2135
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto)
paulson@15077
  2136
paulson@15077
  2137
lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
paulson@15229
  2138
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto)
paulson@15077
  2139
paulson@15077
  2140
lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
paulson@15077
  2141
apply (rule lemma_DERIV_subst)
paulson@15077
  2142
apply (rule_tac f = cos and g = "%x. x + k" in DERIV_chain2)
paulson@15077
  2143
apply (best intro!: DERIV_intros intro: DERIV_chain2)+
paulson@15077
  2144
apply (simp (no_asm))
paulson@15077
  2145
done
paulson@15077
  2146
paulson@15081
  2147
lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
nipkow@15539
  2148
by (auto simp add: sin_zero_iff even_mult_two_ex)
paulson@15077
  2149
huffman@23115
  2150
lemma exp_eq_one_iff [simp]: "(exp (x::real) = 1) = (x = 0)"
paulson@15077
  2151
apply auto
paulson@15077
  2152
apply (drule_tac f = ln in arg_cong, auto)
paulson@15077
  2153
done
paulson@15077
  2154
paulson@15077
  2155
lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
paulson@15077
  2156
by (cut_tac x = x in sin_cos_squared_add3, auto)
paulson@15077
  2157
paulson@15077
  2158
huffman@22978
  2159
subsection {* Existence of Polar Coordinates *}
paulson@15077
  2160
huffman@22978
  2161
lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<twosuperior> + y\<twosuperior>)\<bar> \<le> 1"
huffman@22978
  2162
apply (rule power2_le_imp_le [OF _ zero_le_one])
huffman@22978
  2163
apply (simp add: abs_divide power_divide divide_le_eq not_sum_power2_lt_zero)
paulson@15077
  2164
done
paulson@15077
  2165
huffman@22978
  2166
lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
huffman@22978
  2167
by (simp add: abs_le_iff)
paulson@15077
  2168
huffman@23045
  2169
lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<twosuperior>)"
huffman@23045
  2170
by (simp add: sin_arccos abs_le_iff)
paulson@15077
  2171
huffman@22978
  2172
lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
paulson@15228
  2173
huffman@23045
  2174
lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
paulson@15077
  2175
paulson@15229
  2176
lemma polar_ex1:
huffman@22978
  2177
     "0 < y ==> \<exists>r a. x = r * cos a & y = r * sin a"
paulson@15229
  2178
apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>)" in exI)
huffman@22978
  2179
apply (rule_tac x = "arccos (x / sqrt (x\<twosuperior> + y\<twosuperior>))" in exI)
huffman@22978
  2180
apply (simp add: cos_arccos_lemma1)
huffman@23045
  2181
apply (simp add: sin_arccos_lemma1)
huffman@23045
  2182
apply (simp add: power_divide)
huffman@23045
  2183
apply (simp add: real_sqrt_mult [symmetric])
huffman@23045
  2184
apply (simp add: right_diff_distrib)
paulson@15077
  2185
done
paulson@15077
  2186
paulson@15229
  2187
lemma polar_ex2:
huffman@22978
  2188
     "y < 0 ==> \<exists>r a. x = r * cos a & y = r * sin a"
huffman@22978
  2189
apply (insert polar_ex1 [where x=x and y="-y"], simp, clarify)
paulson@15077
  2190
apply (rule_tac x = r in exI)
huffman@22978
  2191
apply (rule_tac x = "-a" in exI, simp)
paulson@15077
  2192
done
paulson@15077
  2193
paulson@15077
  2194
lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
huffman@22978
  2195
apply (rule_tac x=0 and y=y in linorder_cases)
huffman@22978
  2196
apply (erule polar_ex1)
huffman@22978
  2197
apply (rule_tac x=x in exI, rule_tac x=0 in exI, simp)
huffman@22978
  2198
apply (erule polar_ex2)
paulson@15077
  2199
done
paulson@15077
  2200
paulson@15077
  2201
huffman@23043
  2202
subsection {* Theorems about Limits *}
huffman@23043
  2203
paulson@15077
  2204
(* need to rename second isCont_inverse *)
paulson@15077
  2205
paulson@15229
  2206
lemma isCont_inv_fun:
huffman@20561
  2207
  fixes f g :: "real \<Rightarrow> real"
huffman@20561
  2208
  shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
paulson@15077
  2209
         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
paulson@15077
  2210
      ==> isCont g (f x)"
huffman@22722
  2211
by (rule isCont_inverse_function)
paulson@15077
  2212
paulson@15077
  2213
lemma isCont_inv_fun_inv:
huffman@20552
  2214
  fixes f g :: "real \<Rightarrow> real"
huffman@20552
  2215
  shows "[| 0 < d;  
paulson@15077
  2216
         \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
paulson@15077
  2217
         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
paulson@15077
  2218
       ==> \<exists>e. 0 < e &  
paulson@15081
  2219
             (\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
paulson@15077
  2220
apply (drule isCont_inj_range)
paulson@15077
  2221
prefer 2 apply (assumption, assumption, auto)
paulson@15077
  2222
apply (rule_tac x = e in exI, auto)
paulson@15077
  2223
apply (rotate_tac 2)
paulson@15077
  2224
apply (drule_tac x = y in spec, auto)
paulson@15077
  2225
done
paulson@15077
  2226
paulson@15077
  2227
paulson@15077
  2228
text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
paulson@15229
  2229
lemma LIM_fun_gt_zero:
huffman@20552
  2230
     "[| f -- c --> (l::real); 0 < l |]  
huffman@20561
  2231
         ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
paulson@15077
  2232
apply (auto simp add: LIM_def)
paulson@15077
  2233
apply (drule_tac x = "l/2" in spec, safe, force)
paulson@15077
  2234
apply (rule_tac x = s in exI)
huffman@22998
  2235
apply (auto simp only: abs_less_iff)
paulson@15077
  2236
done
paulson@15077
  2237
paulson@15229
  2238
lemma LIM_fun_less_zero:
huffman@20552
  2239
     "[| f -- c --> (l::real); l < 0 |]  
huffman@20561
  2240
      ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
paulson@15077
  2241
apply (auto simp add: LIM_def)
paulson@15077
  2242
apply (drule_tac x = "-l/2" in spec, safe, force)
paulson@15077
  2243
apply (rule_tac x = s in exI)
huffman@22998
  2244
apply (auto simp only: abs_less_iff)
paulson@15077
  2245
done
paulson@15077
  2246
paulson@15077
  2247
paulson@15077
  2248
lemma LIM_fun_not_zero:
huffman@20552
  2249
     "[| f -- c --> (l::real); l \<noteq> 0 |] 
huffman@20561
  2250
      ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
paulson@15077
  2251
apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
paulson@15077
  2252
apply (drule LIM_fun_less_zero)
paulson@15241
  2253
apply (drule_tac [3] LIM_fun_gt_zero)
paulson@15241
  2254
apply force+
paulson@15077
  2255
done
webertj@20432
  2256
  
paulson@12196
  2257
end