src/HOL/Hyperreal/Transcendental.thy
author huffman
Wed May 30 02:41:26 2007 +0200 (2007-05-30)
changeset 23127 56ee8105c002
parent 23115 4615b2078592
child 23176 40a760921d94
permissions -rw-r--r--
simplify names of locale interpretations
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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@23043
   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@23043
   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@23043
   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@23043
   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@23043
   519
lemma lemma_STAR_sin [simp]:
huffman@23043
   520
     "(if even n then 0  
huffman@23043
   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@23043
   524
lemma lemma_STAR_cos [simp]:
huffman@23043
   525
     "0 < n -->  
huffman@23043
   526
      (- 1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
huffman@23043
   527
by (induct "n", auto)
huffman@23043
   528
huffman@23043
   529
lemma lemma_STAR_cos1 [simp]:
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@23043
   534
lemma lemma_STAR_cos2 [simp]:
huffman@23043
   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@23043
   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@23043
   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  
paulson@15077
   569
           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n)))  
paulson@15077
   570
       = (%n. if even n then  
paulson@15077
   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  
paulson@15077
   579
           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) n  
paulson@15077
   580
       = (if even n then  
paulson@15077
   581
                 (- 1) ^ (n div 2)/(real (fact n))  
paulson@15077
   582
              else 0)"
paulson@15229
   583
by (auto intro!: ext 
huffman@23082
   584
         simp add: diffs_def divide_inverse real_of_nat_def
huffman@23082
   585
         simp del: mult_Suc of_nat_Suc)
paulson@15077
   586
paulson@15077
   587
lemma cos_fdiffs: 
paulson@15077
   588
      "diffs(%n. if even n then  
paulson@15077
   589
                 (- 1) ^ (n div 2)/(real (fact n)) else 0)  
paulson@15077
   590
       = (%n. - (if even n then 0  
paulson@15077
   591
           else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n))))"
paulson@15229
   592
by (auto intro!: ext 
huffman@23082
   593
         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex real_of_nat_def
huffman@23082
   594
         simp del: mult_Suc of_nat_Suc)
paulson@15077
   595
paulson@15077
   596
paulson@15077
   597
lemma cos_fdiffs2: 
paulson@15077
   598
      "diffs(%n. if even n then  
paulson@15077
   599
                 (- 1) ^ (n div 2)/(real (fact n)) else 0) n 
paulson@15077
   600
       = - (if even n then 0  
paulson@15077
   601
           else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n)))"
paulson@15229
   602
by (auto intro!: ext 
huffman@23082
   603
         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex real_of_nat_def
huffman@23082
   604
         simp del: mult_Suc of_nat_Suc)
paulson@15077
   605
paulson@15077
   606
text{*Now at last we can get the derivatives of exp, sin and cos*}
paulson@15077
   607
paulson@15077
   608
lemma lemma_sin_minus:
nipkow@15546
   609
     "- sin x = (\<Sum>n. - ((if even n then 0 
paulson@15077
   610
                  else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))"
paulson@15077
   611
by (auto intro!: sums_unique sums_minus sin_converges)
paulson@15077
   612
huffman@23115
   613
lemma lemma_exp_ext: "exp = (\<lambda>x. \<Sum>n. x ^ n /# real (fact n))"
paulson@15077
   614
by (auto intro!: ext simp add: exp_def)
paulson@15077
   615
paulson@15077
   616
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
paulson@15229
   617
apply (simp add: exp_def)
paulson@15077
   618
apply (subst lemma_exp_ext)
huffman@23115
   619
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
   620
apply (rule_tac [2] K = "of_real (1 + norm x)" in termdiffs)
huffman@23115
   621
apply (simp_all only: diffs_of_real scaleR_conv_of_real exp_fdiffs)
huffman@23115
   622
apply (rule exp_converges [THEN sums_summable, unfolded scaleR_conv_of_real])+
huffman@23115
   623
apply (simp del: of_real_add)
paulson@15077
   624
done
paulson@15077
   625
paulson@15077
   626
lemma lemma_sin_ext:
nipkow@15546
   627
     "sin = (%x. \<Sum>n. 
paulson@15077
   628
                   (if even n then 0  
paulson@15077
   629
                       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
nipkow@15546
   630
                   x ^ n)"
paulson@15077
   631
by (auto intro!: ext simp add: sin_def)
paulson@15077
   632
paulson@15077
   633
lemma lemma_cos_ext:
nipkow@15546
   634
     "cos = (%x. \<Sum>n. 
paulson@15077
   635
                   (if even n then (- 1) ^ (n div 2)/(real (fact n)) else 0) *
nipkow@15546
   636
                   x ^ n)"
paulson@15077
   637
by (auto intro!: ext simp add: cos_def)
paulson@15077
   638
paulson@15077
   639
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
paulson@15229
   640
apply (simp add: cos_def)
paulson@15077
   641
apply (subst lemma_sin_ext)
paulson@15077
   642
apply (auto simp add: sin_fdiffs2 [symmetric])
paulson@15229
   643
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
webertj@20217
   644
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs)
paulson@15077
   645
done
paulson@15077
   646
paulson@15077
   647
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
paulson@15077
   648
apply (subst lemma_cos_ext)
paulson@15077
   649
apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
paulson@15229
   650
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
webertj@20217
   651
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
   652
done
paulson@15077
   653
huffman@23045
   654
lemma isCont_exp [simp]: "isCont exp x"
huffman@23045
   655
by (rule DERIV_exp [THEN DERIV_isCont])
huffman@23045
   656
huffman@23045
   657
lemma isCont_sin [simp]: "isCont sin x"
huffman@23045
   658
by (rule DERIV_sin [THEN DERIV_isCont])
huffman@23045
   659
huffman@23045
   660
lemma isCont_cos [simp]: "isCont cos x"
huffman@23045
   661
by (rule DERIV_cos [THEN DERIV_isCont])
huffman@23045
   662
paulson@15077
   663
paulson@15077
   664
subsection{*Properties of the Exponential Function*}
paulson@15077
   665
paulson@15077
   666
lemma exp_zero [simp]: "exp 0 = 1"
paulson@15077
   667
proof -
huffman@23115
   668
  have "(\<Sum>n = 0..<1. (0::'a) ^ n /# real (fact n)) =
huffman@23115
   669
        (\<Sum>n. 0 ^ n /# real (fact n))"
huffman@23115
   670
    by (rule sums_unique [OF series_zero], simp add: power_0_left)
huffman@23115
   671
  thus ?thesis by (simp add: exp_def)
paulson@15077
   672
qed
paulson@15077
   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@23115
   733
lemma exp_ge_add_one_self_aux: "0 \<le> (x::real) ==> (1 + x) \<le> exp(x)"
huffman@22998
   734
apply (drule order_le_imp_less_or_eq, auto)
paulson@15229
   735
apply (simp add: exp_def)
paulson@15077
   736
apply (rule real_le_trans)
paulson@15229
   737
apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
paulson@15077
   738
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_power zero_le_mult_iff)
paulson@15077
   739
done
paulson@15077
   740
huffman@23115
   741
lemma exp_gt_one [simp]: "0 < (x::real) ==> 1 < exp x"
paulson@15077
   742
apply (rule order_less_le_trans)
avigad@17014
   743
apply (rule_tac [2] exp_ge_add_one_self_aux, auto)
paulson@15077
   744
done
paulson@15077
   745
paulson@15077
   746
lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)"
paulson@15077
   747
proof -
paulson@15077
   748
  have "DERIV (exp \<circ> (\<lambda>x. x + y)) x :> exp (x + y) * (1+0)"
huffman@23069
   749
    by (fast intro: DERIV_chain DERIV_add DERIV_exp DERIV_ident DERIV_const) 
paulson@15077
   750
  thus ?thesis by (simp add: o_def)
paulson@15077
   751
qed
paulson@15077
   752
paulson@15077
   753
lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)"
paulson@15077
   754
proof -
paulson@15077
   755
  have "DERIV (exp \<circ> uminus) x :> exp (- x) * - 1"
huffman@23069
   756
    by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_ident)
paulson@15077
   757
  thus ?thesis by (simp add: o_def)
paulson@15077
   758
qed
paulson@15077
   759
paulson@15077
   760
lemma DERIV_exp_exp_zero [simp]: "DERIV (%x. exp (x + y) * exp (- x)) x :> 0"
paulson@15077
   761
proof -
paulson@15077
   762
  have "DERIV (\<lambda>x. exp (x + y) * exp (- x)) x
paulson@15077
   763
       :> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)"
paulson@15077
   764
    by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult) 
huffman@23115
   765
  thus ?thesis by (simp add: mult_commute)
paulson@15077
   766
qed
paulson@15077
   767
huffman@23115
   768
lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y::real)"
paulson@15077
   769
proof -
paulson@15077
   770
  have "\<forall>x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0" by simp
paulson@15077
   771
  hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)" 
paulson@15077
   772
    by (rule DERIV_isconst_all) 
paulson@15077
   773
  thus ?thesis by simp
paulson@15077
   774
qed
paulson@15077
   775
paulson@15077
   776
lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1"
huffman@23115
   777
by (simp add: exp_add [symmetric])
paulson@15077
   778
paulson@15077
   779
lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1"
paulson@15077
   780
by (simp add: mult_commute)
paulson@15077
   781
paulson@15077
   782
paulson@15077
   783
lemma exp_minus: "exp(-x) = inverse(exp(x))"
paulson@15077
   784
by (auto intro: inverse_unique [symmetric])
paulson@15077
   785
paulson@15077
   786
text{*Proof: because every exponential can be seen as a square.*}
huffman@23115
   787
lemma exp_ge_zero [simp]: "0 \<le> exp (x::real)"
paulson@15077
   788
apply (rule_tac t = x in real_sum_of_halves [THEN subst])
paulson@15077
   789
apply (subst exp_add, auto)
paulson@15077
   790
done
paulson@15077
   791
paulson@15077
   792
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
paulson@15077
   793
apply (cut_tac x = x in exp_mult_minus2)
paulson@15077
   794
apply (auto simp del: exp_mult_minus2)
paulson@15077
   795
done
paulson@15077
   796
huffman@23115
   797
lemma exp_gt_zero [simp]: "0 < exp (x::real)"
paulson@15077
   798
by (simp add: order_less_le)
paulson@15077
   799
huffman@23115
   800
lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x::real)"
paulson@15077
   801
by (auto intro: positive_imp_inverse_positive)
paulson@15077
   802
huffman@23115
   803
lemma abs_exp_cancel [simp]: "\<bar>exp x::real\<bar> = exp x"
paulson@15229
   804
by auto
paulson@15077
   805
paulson@15077
   806
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
paulson@15251
   807
apply (induct "n")
paulson@15077
   808
apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
paulson@15077
   809
done
paulson@15077
   810
paulson@15077
   811
lemma exp_diff: "exp(x - y) = exp(x)/(exp y)"
paulson@15229
   812
apply (simp add: diff_minus divide_inverse)
paulson@15077
   813
apply (simp (no_asm) add: exp_add exp_minus)
paulson@15077
   814
done
paulson@15077
   815
paulson@15077
   816
paulson@15077
   817
lemma exp_less_mono:
huffman@23115
   818
  fixes x y :: real
paulson@15077
   819
  assumes xy: "x < y" shows "exp x < exp y"
paulson@15077
   820
proof -
paulson@15077
   821
  have "1 < exp (y + - x)"
paulson@15077
   822
    by (rule real_less_sum_gt_zero [THEN exp_gt_one])
paulson@15077
   823
  hence "exp x * inverse (exp x) < exp y * inverse (exp x)"
paulson@15077
   824
    by (auto simp add: exp_add exp_minus)
paulson@15077
   825
  thus ?thesis
nipkow@15539
   826
    by (simp add: divide_inverse [symmetric] pos_less_divide_eq
paulson@15228
   827
             del: divide_self_if)
paulson@15077
   828
qed
paulson@15077
   829
huffman@23115
   830
lemma exp_less_cancel: "exp (x::real) < exp y ==> x < y"
paulson@15228
   831
apply (simp add: linorder_not_le [symmetric]) 
paulson@15228
   832
apply (auto simp add: order_le_less exp_less_mono) 
paulson@15077
   833
done
paulson@15077
   834
huffman@23115
   835
lemma exp_less_cancel_iff [iff]: "(exp(x::real) < exp(y)) = (x < y)"
paulson@15077
   836
by (auto intro: exp_less_mono exp_less_cancel)
paulson@15077
   837
huffman@23115
   838
lemma exp_le_cancel_iff [iff]: "(exp(x::real) \<le> exp(y)) = (x \<le> y)"
paulson@15077
   839
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
   840
huffman@23115
   841
lemma exp_inj_iff [iff]: "(exp (x::real) = exp y) = (x = y)"
paulson@15077
   842
by (simp add: order_eq_iff)
paulson@15077
   843
huffman@23115
   844
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x::real) = y"
paulson@15077
   845
apply (rule IVT)
huffman@23045
   846
apply (auto intro: isCont_exp simp add: le_diff_eq)
paulson@15077
   847
apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)") 
paulson@15077
   848
apply simp 
avigad@17014
   849
apply (rule exp_ge_add_one_self_aux, simp)
paulson@15077
   850
done
paulson@15077
   851
huffman@23115
   852
lemma exp_total: "0 < (y::real) ==> \<exists>x. exp x = y"
paulson@15077
   853
apply (rule_tac x = 1 and y = y in linorder_cases)
paulson@15077
   854
apply (drule order_less_imp_le [THEN lemma_exp_total])
paulson@15077
   855
apply (rule_tac [2] x = 0 in exI)
paulson@15077
   856
apply (frule_tac [3] real_inverse_gt_one)
paulson@15077
   857
apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
paulson@15077
   858
apply (rule_tac x = "-x" in exI)
paulson@15077
   859
apply (simp add: exp_minus)
paulson@15077
   860
done
paulson@15077
   861
paulson@15077
   862
paulson@15077
   863
subsection{*Properties of the Logarithmic Function*}
paulson@15077
   864
huffman@23043
   865
definition
huffman@23043
   866
  ln :: "real => real" where
huffman@23043
   867
  "ln x = (THE u. exp u = x)"
huffman@23043
   868
huffman@23043
   869
lemma ln_exp [simp]: "ln (exp x) = x"
paulson@15077
   870
by (simp add: ln_def)
paulson@15077
   871
huffman@22654
   872
lemma exp_ln [simp]: "0 < x \<Longrightarrow> exp (ln x) = x"
huffman@22654
   873
by (auto dest: exp_total)
huffman@22654
   874
huffman@23043
   875
lemma exp_ln_iff [simp]: "(exp (ln x) = x) = (0 < x)"
paulson@15077
   876
apply (auto dest: exp_total)
paulson@15077
   877
apply (erule subst, simp) 
paulson@15077
   878
done
paulson@15077
   879
paulson@15077
   880
lemma ln_mult: "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)"
paulson@15077
   881
apply (rule exp_inj_iff [THEN iffD1])
huffman@22654
   882
apply (simp add: exp_add exp_ln mult_pos_pos)
paulson@15077
   883
done
paulson@15077
   884
paulson@15077
   885
lemma ln_inj_iff[simp]: "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)"
paulson@15077
   886
apply (simp only: exp_ln_iff [symmetric])
paulson@15077
   887
apply (erule subst)+
paulson@15077
   888
apply simp 
paulson@15077
   889
done
paulson@15077
   890
paulson@15077
   891
lemma ln_one[simp]: "ln 1 = 0"
paulson@15077
   892
by (rule exp_inj_iff [THEN iffD1], auto)
paulson@15077
   893
paulson@15077
   894
lemma ln_inverse: "0 < x ==> ln(inverse x) = - ln x"
paulson@15077
   895
apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1])
paulson@15077
   896
apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric])
paulson@15077
   897
done
paulson@15077
   898
paulson@15077
   899
lemma ln_div: 
paulson@15077
   900
    "[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y"
paulson@15229
   901
apply (simp add: divide_inverse)
paulson@15077
   902
apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse)
paulson@15077
   903
done
paulson@15077
   904
paulson@15077
   905
lemma ln_less_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)"
paulson@15077
   906
apply (simp only: exp_ln_iff [symmetric])
paulson@15077
   907
apply (erule subst)+
paulson@15077
   908
apply simp 
paulson@15077
   909
done
paulson@15077
   910
paulson@15077
   911
lemma ln_le_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x \<le> ln y) = (x \<le> y)"
paulson@15077
   912
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
   913
paulson@15077
   914
lemma ln_realpow: "0 < x ==> ln(x ^ n) = real n * ln(x)"
paulson@15077
   915
by (auto dest!: exp_total simp add: exp_real_of_nat_mult [symmetric])
paulson@15077
   916
paulson@15077
   917
lemma ln_add_one_self_le_self [simp]: "0 \<le> x ==> ln(1 + x) \<le> x"
paulson@15077
   918
apply (rule ln_exp [THEN subst])
avigad@17014
   919
apply (rule ln_le_cancel_iff [THEN iffD2]) 
avigad@17014
   920
apply (auto simp add: exp_ge_add_one_self_aux)
paulson@15077
   921
done
paulson@15077
   922
paulson@15077
   923
lemma ln_less_self [simp]: "0 < x ==> ln x < x"
paulson@15077
   924
apply (rule order_less_le_trans)
paulson@15077
   925
apply (rule_tac [2] ln_add_one_self_le_self)
paulson@15077
   926
apply (rule ln_less_cancel_iff [THEN iffD2], auto)
paulson@15077
   927
done
paulson@15077
   928
paulson@15234
   929
lemma ln_ge_zero [simp]:
paulson@15077
   930
  assumes x: "1 \<le> x" shows "0 \<le> ln x"
paulson@15077
   931
proof -
paulson@15077
   932
  have "0 < x" using x by arith
paulson@15077
   933
  hence "exp 0 \<le> exp (ln x)"
huffman@22915
   934
    by (simp add: x)
paulson@15077
   935
  thus ?thesis by (simp only: exp_le_cancel_iff)
paulson@15077
   936
qed
paulson@15077
   937
paulson@15077
   938
lemma ln_ge_zero_imp_ge_one:
paulson@15077
   939
  assumes ln: "0 \<le> ln x" 
paulson@15077
   940
      and x:  "0 < x"
paulson@15077
   941
  shows "1 \<le> x"
paulson@15077
   942
proof -
paulson@15077
   943
  from ln have "ln 1 \<le> ln x" by simp
paulson@15077
   944
  thus ?thesis by (simp add: x del: ln_one) 
paulson@15077
   945
qed
paulson@15077
   946
paulson@15077
   947
lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
paulson@15077
   948
by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
paulson@15077
   949
paulson@15234
   950
lemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)"
paulson@15234
   951
by (insert ln_ge_zero_iff [of x], arith)
paulson@15234
   952
paulson@15077
   953
lemma ln_gt_zero:
paulson@15077
   954
  assumes x: "1 < x" shows "0 < ln x"
paulson@15077
   955
proof -
paulson@15077
   956
  have "0 < x" using x by arith
huffman@22915
   957
  hence "exp 0 < exp (ln x)" by (simp add: x)
paulson@15077
   958
  thus ?thesis  by (simp only: exp_less_cancel_iff)
paulson@15077
   959
qed
paulson@15077
   960
paulson@15077
   961
lemma ln_gt_zero_imp_gt_one:
paulson@15077
   962
  assumes ln: "0 < ln x" 
paulson@15077
   963
      and x:  "0 < x"
paulson@15077
   964
  shows "1 < x"
paulson@15077
   965
proof -
paulson@15077
   966
  from ln have "ln 1 < ln x" by simp
paulson@15077
   967
  thus ?thesis by (simp add: x del: ln_one) 
paulson@15077
   968
qed
paulson@15077
   969
paulson@15077
   970
lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
paulson@15077
   971
by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
paulson@15077
   972
paulson@15234
   973
lemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)"
paulson@15234
   974
by (insert ln_less_zero_iff [of x] ln_gt_zero_iff [of x], arith)
paulson@15077
   975
paulson@15077
   976
lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
paulson@15234
   977
by simp
paulson@15077
   978
paulson@15077
   979
lemma exp_ln_eq: "exp u = x ==> ln x = u"
paulson@15077
   980
by auto
paulson@15077
   981
huffman@23045
   982
lemma isCont_ln: "0 < x \<Longrightarrow> isCont ln x"
huffman@23045
   983
apply (subgoal_tac "isCont ln (exp (ln x))", simp)
huffman@23045
   984
apply (rule isCont_inverse_function [where f=exp], simp_all)
huffman@23045
   985
done
huffman@23045
   986
huffman@23045
   987
lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
huffman@23045
   988
by simp (* TODO: put in Deriv.thy *)
huffman@23045
   989
huffman@23045
   990
lemma DERIV_ln: "0 < x \<Longrightarrow> DERIV ln x :> inverse x"
huffman@23045
   991
apply (rule DERIV_inverse_function [where f=exp and a=0 and b="x+1"])
huffman@23045
   992
apply (erule lemma_DERIV_subst [OF DERIV_exp exp_ln])
huffman@23045
   993
apply (simp_all add: abs_if isCont_ln)
huffman@23045
   994
done
huffman@23045
   995
paulson@15077
   996
paulson@15077
   997
subsection{*Basic Properties of the Trigonometric Functions*}
paulson@15077
   998
paulson@15077
   999
lemma sin_zero [simp]: "sin 0 = 0"
paulson@15077
  1000
by (auto intro!: sums_unique [symmetric] LIMSEQ_const 
paulson@15077
  1001
         simp add: sin_def sums_def simp del: power_0_left)
paulson@15077
  1002
nipkow@15539
  1003
lemma lemma_series_zero2:
nipkow@15539
  1004
 "(\<forall>m. n \<le> m --> f m = 0) --> f sums setsum f {0..<n}"
paulson@15077
  1005
by (auto intro: series_zero)
paulson@15077
  1006
paulson@15077
  1007
lemma cos_zero [simp]: "cos 0 = 1"
paulson@15229
  1008
apply (simp add: cos_def)
paulson@15077
  1009
apply (rule sums_unique [symmetric])
paulson@15229
  1010
apply (cut_tac n = 1 and f = "(%n. (if even n then (- 1) ^ (n div 2) / (real (fact n)) else 0) * 0 ^ n)" in lemma_series_zero2)
paulson@15077
  1011
apply auto
paulson@15077
  1012
done
paulson@15077
  1013
paulson@15077
  1014
lemma DERIV_sin_sin_mult [simp]:
paulson@15077
  1015
     "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077
  1016
by (rule DERIV_mult, auto)
paulson@15077
  1017
paulson@15077
  1018
lemma DERIV_sin_sin_mult2 [simp]:
paulson@15077
  1019
     "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"
paulson@15077
  1020
apply (cut_tac x = x in DERIV_sin_sin_mult)
paulson@15077
  1021
apply (auto simp add: mult_assoc)
paulson@15077
  1022
done
paulson@15077
  1023
paulson@15077
  1024
lemma DERIV_sin_realpow2 [simp]:
paulson@15077
  1025
     "DERIV (%x. (sin x)\<twosuperior>) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077
  1026
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077
  1027
paulson@15077
  1028
lemma DERIV_sin_realpow2a [simp]:
paulson@15077
  1029
     "DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"
paulson@15077
  1030
by (auto simp add: numeral_2_eq_2)
paulson@15077
  1031
paulson@15077
  1032
lemma DERIV_cos_cos_mult [simp]:
paulson@15077
  1033
     "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077
  1034
by (rule DERIV_mult, auto)
paulson@15077
  1035
paulson@15077
  1036
lemma DERIV_cos_cos_mult2 [simp]:
paulson@15077
  1037
     "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"
paulson@15077
  1038
apply (cut_tac x = x in DERIV_cos_cos_mult)
paulson@15077
  1039
apply (auto simp add: mult_ac)
paulson@15077
  1040
done
paulson@15077
  1041
paulson@15077
  1042
lemma DERIV_cos_realpow2 [simp]:
paulson@15077
  1043
     "DERIV (%x. (cos x)\<twosuperior>) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077
  1044
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077
  1045
paulson@15077
  1046
lemma DERIV_cos_realpow2a [simp]:
paulson@15077
  1047
     "DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"
paulson@15077
  1048
by (auto simp add: numeral_2_eq_2)
paulson@15077
  1049
paulson@15077
  1050
lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
paulson@15077
  1051
by auto
paulson@15077
  1052
paulson@15077
  1053
lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\<twosuperior>) x :> -(2 * cos(x) * sin(x))"
paulson@15077
  1054
apply (rule lemma_DERIV_subst)
paulson@15077
  1055
apply (rule DERIV_cos_realpow2a, auto)
paulson@15077
  1056
done
paulson@15077
  1057
paulson@15077
  1058
(* most useful *)
paulson@15229
  1059
lemma DERIV_cos_cos_mult3 [simp]:
paulson@15229
  1060
     "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
paulson@15077
  1061
apply (rule lemma_DERIV_subst)
paulson@15077
  1062
apply (rule DERIV_cos_cos_mult2, auto)
paulson@15077
  1063
done
paulson@15077
  1064
paulson@15077
  1065
lemma DERIV_sin_circle_all: 
paulson@15077
  1066
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>  
paulson@15077
  1067
             (2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
paulson@15229
  1068
apply (simp only: diff_minus, safe)
paulson@15229
  1069
apply (rule DERIV_add) 
paulson@15077
  1070
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1071
done
paulson@15077
  1072
paulson@15229
  1073
lemma DERIV_sin_circle_all_zero [simp]:
paulson@15229
  1074
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
paulson@15077
  1075
by (cut_tac DERIV_sin_circle_all, auto)
paulson@15077
  1076
paulson@15077
  1077
lemma sin_cos_squared_add [simp]: "((sin x)\<twosuperior>) + ((cos x)\<twosuperior>) = 1"
paulson@15077
  1078
apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all])
paulson@15077
  1079
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1080
done
paulson@15077
  1081
paulson@15077
  1082
lemma sin_cos_squared_add2 [simp]: "((cos x)\<twosuperior>) + ((sin x)\<twosuperior>) = 1"
paulson@15077
  1083
apply (subst real_add_commute)
paulson@15077
  1084
apply (simp (no_asm) del: realpow_Suc)
paulson@15077
  1085
done
paulson@15077
  1086
paulson@15077
  1087
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
paulson@15077
  1088
apply (cut_tac x = x in sin_cos_squared_add2)
paulson@15077
  1089
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1090
done
paulson@15077
  1091
paulson@15077
  1092
lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
paulson@15229
  1093
apply (rule_tac a1 = "(cos 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 cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
paulson@15077
  1098
apply (rule_tac a1 = "(sin x)\<twosuperior>" in add_right_cancel [THEN iffD1])
paulson@15077
  1099
apply (simp del: realpow_Suc)
paulson@15077
  1100
done
paulson@15077
  1101
paulson@15077
  1102
lemma real_gt_one_ge_zero_add_less: "[| 1 < x; 0 \<le> y |] ==> 1 < x + (y::real)"
paulson@15077
  1103
by arith
paulson@15077
  1104
paulson@15081
  1105
lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
huffman@23097
  1106
by (rule power2_le_imp_le, simp_all add: sin_squared_eq)
paulson@15077
  1107
paulson@15077
  1108
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
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@15077
  1113
lemma sin_le_one [simp]: "sin x \<le> 1"
paulson@15077
  1114
apply (insert abs_sin_le_one [of x]) 
huffman@22998
  1115
apply (simp add: abs_le_iff del: abs_sin_le_one) 
paulson@15077
  1116
done
paulson@15077
  1117
paulson@15081
  1118
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
huffman@23097
  1119
by (rule power2_le_imp_le, simp_all add: cos_squared_eq)
paulson@15077
  1120
paulson@15077
  1121
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
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 cos_le_one [simp]: "cos x \<le> 1"
paulson@15077
  1127
apply (insert abs_cos_le_one [of x]) 
huffman@22998
  1128
apply (simp add: abs_le_iff del: abs_cos_le_one)
paulson@15077
  1129
done
paulson@15077
  1130
paulson@15077
  1131
lemma DERIV_fun_pow: "DERIV g x :> m ==>  
paulson@15077
  1132
      DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
paulson@15077
  1133
apply (rule lemma_DERIV_subst)
paulson@15229
  1134
apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2)
paulson@15077
  1135
apply (rule DERIV_pow, auto)
paulson@15077
  1136
done
paulson@15077
  1137
paulson@15229
  1138
lemma DERIV_fun_exp:
paulson@15229
  1139
     "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
paulson@15077
  1140
apply (rule lemma_DERIV_subst)
paulson@15077
  1141
apply (rule_tac f = exp in DERIV_chain2)
paulson@15077
  1142
apply (rule DERIV_exp, auto)
paulson@15077
  1143
done
paulson@15077
  1144
paulson@15229
  1145
lemma DERIV_fun_sin:
paulson@15229
  1146
     "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
paulson@15077
  1147
apply (rule lemma_DERIV_subst)
paulson@15077
  1148
apply (rule_tac f = sin in DERIV_chain2)
paulson@15077
  1149
apply (rule DERIV_sin, auto)
paulson@15077
  1150
done
paulson@15077
  1151
paulson@15229
  1152
lemma DERIV_fun_cos:
paulson@15229
  1153
     "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
paulson@15077
  1154
apply (rule lemma_DERIV_subst)
paulson@15077
  1155
apply (rule_tac f = cos in DERIV_chain2)
paulson@15077
  1156
apply (rule DERIV_cos, auto)
paulson@15077
  1157
done
paulson@15077
  1158
huffman@23069
  1159
lemmas DERIV_intros = DERIV_ident DERIV_const DERIV_cos DERIV_cmult 
paulson@15077
  1160
                    DERIV_sin  DERIV_exp  DERIV_inverse DERIV_pow 
paulson@15077
  1161
                    DERIV_add  DERIV_diff  DERIV_mult  DERIV_minus 
paulson@15077
  1162
                    DERIV_inverse_fun DERIV_quotient DERIV_fun_pow 
paulson@15077
  1163
                    DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos 
paulson@15077
  1164
paulson@15077
  1165
(* lemma *)
paulson@15229
  1166
lemma lemma_DERIV_sin_cos_add:
paulson@15229
  1167
     "\<forall>x.  
paulson@15077
  1168
         DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077
  1169
               (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"
paulson@15077
  1170
apply (safe, rule lemma_DERIV_subst)
paulson@15077
  1171
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15077
  1172
  --{*replaces the old @{text DERIV_tac}*}
paulson@15229
  1173
apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
paulson@15077
  1174
done
paulson@15077
  1175
paulson@15077
  1176
lemma sin_cos_add [simp]:
paulson@15077
  1177
     "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077
  1178
      (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
paulson@15077
  1179
apply (cut_tac y = 0 and x = x and y7 = y 
paulson@15077
  1180
       in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])
paulson@15077
  1181
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1182
done
paulson@15077
  1183
paulson@15077
  1184
lemma sin_add: "sin (x + y) = sin x * cos y + cos 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@15077
  1189
lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
paulson@15077
  1190
apply (cut_tac x = x and y = y in sin_cos_add)
huffman@22969
  1191
apply (simp del: sin_cos_add)
paulson@15077
  1192
done
paulson@15077
  1193
paulson@15085
  1194
lemma lemma_DERIV_sin_cos_minus:
paulson@15085
  1195
    "\<forall>x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"
paulson@15077
  1196
apply (safe, rule lemma_DERIV_subst)
paulson@15077
  1197
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15229
  1198
apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
paulson@15077
  1199
done
paulson@15077
  1200
paulson@15085
  1201
lemma sin_cos_minus [simp]: 
paulson@15085
  1202
    "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"
paulson@15085
  1203
apply (cut_tac y = 0 and x = x 
paulson@15085
  1204
       in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])
huffman@22969
  1205
apply simp
paulson@15077
  1206
done
paulson@15077
  1207
paulson@15077
  1208
lemma sin_minus [simp]: "sin (-x) = -sin(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 cos_minus [simp]: "cos (-x) = cos(x)"
paulson@15077
  1214
apply (cut_tac x = x in sin_cos_minus)
huffman@22969
  1215
apply (simp del: sin_cos_minus)
paulson@15077
  1216
done
paulson@15077
  1217
paulson@15077
  1218
lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
huffman@22969
  1219
by (simp add: diff_minus sin_add)
paulson@15077
  1220
paulson@15077
  1221
lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
paulson@15077
  1222
by (simp add: sin_diff mult_commute)
paulson@15077
  1223
paulson@15077
  1224
lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
huffman@22969
  1225
by (simp add: diff_minus cos_add)
paulson@15077
  1226
paulson@15077
  1227
lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
paulson@15077
  1228
by (simp add: cos_diff mult_commute)
paulson@15077
  1229
paulson@15077
  1230
lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
paulson@15077
  1231
by (cut_tac x = x and y = x in sin_add, auto)
paulson@15077
  1232
paulson@15077
  1233
paulson@15077
  1234
lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
paulson@15077
  1235
apply (cut_tac x = x and y = x in cos_add)
huffman@22969
  1236
apply (simp add: power2_eq_square)
paulson@15077
  1237
done
paulson@15077
  1238
paulson@15077
  1239
paulson@15077
  1240
subsection{*The Constant Pi*}
paulson@15077
  1241
huffman@23043
  1242
definition
huffman@23043
  1243
  pi :: "real" where
huffman@23053
  1244
  "pi = 2 * (THE x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
huffman@23043
  1245
paulson@15077
  1246
text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"}; 
paulson@15077
  1247
   hence define pi.*}
paulson@15077
  1248
paulson@15077
  1249
lemma sin_paired:
paulson@15077
  1250
     "(%n. (- 1) ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) 
paulson@15077
  1251
      sums  sin x"
paulson@15077
  1252
proof -
paulson@15077
  1253
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1254
            (if even k then 0
paulson@15077
  1255
             else (- 1) ^ ((k - Suc 0) div 2) / real (fact k)) *
paulson@15077
  1256
            x ^ k) 
paulson@15077
  1257
	sums
nipkow@15546
  1258
	(\<Sum>n. (if even n then 0
paulson@15077
  1259
		     else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
paulson@15077
  1260
	            x ^ n)" 
paulson@15077
  1261
    by (rule sin_converges [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1262
  thus ?thesis by (simp add: mult_ac sin_def)
paulson@15077
  1263
qed
paulson@15077
  1264
paulson@15077
  1265
lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077
  1266
apply (subgoal_tac 
paulson@15077
  1267
       "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1268
              (- 1) ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1)) 
nipkow@15546
  1269
     sums (\<Sum>n. (- 1) ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
paulson@15077
  1270
 prefer 2
paulson@15077
  1271
 apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1272
apply (rotate_tac 2)
paulson@15077
  1273
apply (drule sin_paired [THEN sums_unique, THEN ssubst])
paulson@15077
  1274
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1275
apply (frule sums_unique)
paulson@15077
  1276
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1277
apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])
paulson@15077
  1278
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1279
apply (erule sums_summable)
paulson@15077
  1280
apply (case_tac "m=0")
paulson@15077
  1281
apply (simp (no_asm_simp))
paulson@15234
  1282
apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x") 
nipkow@15539
  1283
apply (simp only: mult_less_cancel_left, simp)  
nipkow@15539
  1284
apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
paulson@15077
  1285
apply (subgoal_tac "x*x < 2*3", simp) 
paulson@15077
  1286
apply (rule mult_strict_mono)
paulson@15085
  1287
apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc)
paulson@15077
  1288
apply (subst fact_Suc)
paulson@15077
  1289
apply (subst fact_Suc)
paulson@15077
  1290
apply (subst fact_Suc)
paulson@15077
  1291
apply (subst fact_Suc)
paulson@15077
  1292
apply (subst real_of_nat_mult)
paulson@15077
  1293
apply (subst real_of_nat_mult)
paulson@15077
  1294
apply (subst real_of_nat_mult)
paulson@15077
  1295
apply (subst real_of_nat_mult)
nipkow@15539
  1296
apply (simp (no_asm) add: divide_inverse del: fact_Suc)
paulson@15077
  1297
apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
paulson@15077
  1298
apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right) 
paulson@15077
  1299
apply (auto simp add: mult_assoc simp del: fact_Suc)
paulson@15077
  1300
apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right) 
paulson@15077
  1301
apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
paulson@15077
  1302
apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)") 
paulson@15077
  1303
apply (erule ssubst)+
paulson@15077
  1304
apply (auto simp del: fact_Suc)
paulson@15077
  1305
apply (subgoal_tac "0 < x ^ (4 * m) ")
paulson@15077
  1306
 prefer 2 apply (simp only: zero_less_power) 
paulson@15077
  1307
apply (simp (no_asm_simp) add: mult_less_cancel_left)
paulson@15077
  1308
apply (rule mult_strict_mono)
paulson@15077
  1309
apply (simp_all (no_asm_simp))
paulson@15077
  1310
done
paulson@15077
  1311
paulson@15077
  1312
lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077
  1313
by (auto intro: sin_gt_zero)
paulson@15077
  1314
paulson@15077
  1315
lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
paulson@15077
  1316
apply (cut_tac x = x in sin_gt_zero1)
paulson@15077
  1317
apply (auto simp add: cos_squared_eq cos_double)
paulson@15077
  1318
done
paulson@15077
  1319
paulson@15077
  1320
lemma cos_paired:
paulson@15077
  1321
     "(%n. (- 1) ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
paulson@15077
  1322
proof -
paulson@15077
  1323
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1324
            (if even k then (- 1) ^ (k div 2) / real (fact k) else 0) *
paulson@15077
  1325
            x ^ k) 
paulson@15077
  1326
        sums
nipkow@15546
  1327
	(\<Sum>n. (if even n then (- 1) ^ (n div 2) / real (fact n) else 0) *
paulson@15077
  1328
	      x ^ n)" 
paulson@15077
  1329
    by (rule cos_converges [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1330
  thus ?thesis by (simp add: mult_ac cos_def)
paulson@15077
  1331
qed
paulson@15077
  1332
paulson@15077
  1333
declare zero_less_power [simp]
paulson@15077
  1334
paulson@15077
  1335
lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
paulson@15077
  1336
by simp
paulson@15077
  1337
huffman@23053
  1338
lemma cos_two_less_zero [simp]: "cos (2) < 0"
paulson@15077
  1339
apply (cut_tac x = 2 in cos_paired)
paulson@15077
  1340
apply (drule sums_minus)
paulson@15077
  1341
apply (rule neg_less_iff_less [THEN iffD1]) 
nipkow@15539
  1342
apply (frule sums_unique, auto)
nipkow@15539
  1343
apply (rule_tac y =
nipkow@15539
  1344
 "\<Sum>n=0..< Suc(Suc(Suc 0)). - ((- 1) ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
paulson@15481
  1345
       in order_less_trans)
paulson@15077
  1346
apply (simp (no_asm) add: fact_num_eq_if realpow_num_eq_if del: fact_Suc realpow_Suc)
nipkow@15561
  1347
apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
paulson@15077
  1348
apply (rule sumr_pos_lt_pair)
paulson@15077
  1349
apply (erule sums_summable, safe)
paulson@15085
  1350
apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric] 
paulson@15085
  1351
            del: fact_Suc)
paulson@15077
  1352
apply (rule real_mult_inverse_cancel2)
paulson@15077
  1353
apply (rule real_of_nat_fact_gt_zero)+
paulson@15077
  1354
apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
paulson@15077
  1355
apply (subst fact_lemma) 
paulson@15481
  1356
apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
paulson@15481
  1357
apply (simp only: real_of_nat_mult)
huffman@23007
  1358
apply (rule mult_strict_mono, force)
huffman@23007
  1359
  apply (rule_tac [3] real_of_nat_fact_ge_zero)
paulson@15481
  1360
 prefer 2 apply force
paulson@15077
  1361
apply (rule real_of_nat_less_iff [THEN iffD2])
paulson@15077
  1362
apply (rule fact_less_mono, auto)
paulson@15077
  1363
done
huffman@23053
  1364
huffman@23053
  1365
lemmas cos_two_neq_zero [simp] = cos_two_less_zero [THEN less_imp_neq]
huffman@23053
  1366
lemmas cos_two_le_zero [simp] = cos_two_less_zero [THEN order_less_imp_le]
paulson@15077
  1367
paulson@15077
  1368
lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
paulson@15077
  1369
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")
paulson@15077
  1370
apply (rule_tac [2] IVT2)
paulson@15077
  1371
apply (auto intro: DERIV_isCont DERIV_cos)
paulson@15077
  1372
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077
  1373
apply (rule ccontr)
paulson@15077
  1374
apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")
paulson@15077
  1375
apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
paulson@15077
  1376
apply (drule_tac f = cos in Rolle)
paulson@15077
  1377
apply (drule_tac [5] f = cos in Rolle)
paulson@15077
  1378
apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
paulson@15077
  1379
apply (drule_tac y1 = xa in order_le_less_trans [THEN sin_gt_zero])
paulson@15077
  1380
apply (assumption, rule_tac y=y in order_less_le_trans, simp_all) 
paulson@15077
  1381
apply (drule_tac y1 = y in order_le_less_trans [THEN sin_gt_zero], assumption, simp_all) 
paulson@15077
  1382
done
paulson@15077
  1383
    
huffman@23053
  1384
lemma pi_half: "pi/2 = (THE x. 0 \<le> x & x \<le> 2 & cos x = 0)"
paulson@15077
  1385
by (simp add: pi_def)
paulson@15077
  1386
paulson@15077
  1387
lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
huffman@23053
  1388
by (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1389
huffman@23053
  1390
lemma pi_half_gt_zero [simp]: "0 < pi / 2"
huffman@23053
  1391
apply (rule order_le_neq_trans)
huffman@23053
  1392
apply (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1393
apply (rule notI, drule arg_cong [where f=cos], simp)
paulson@15077
  1394
done
paulson@15077
  1395
huffman@23053
  1396
lemmas pi_half_neq_zero [simp] = pi_half_gt_zero [THEN less_imp_neq, symmetric]
huffman@23053
  1397
lemmas pi_half_ge_zero [simp] = pi_half_gt_zero [THEN order_less_imp_le]
paulson@15077
  1398
huffman@23053
  1399
lemma pi_half_less_two [simp]: "pi / 2 < 2"
huffman@23053
  1400
apply (rule order_le_neq_trans)
huffman@23053
  1401
apply (simp add: pi_half cos_is_zero [THEN theI'])
huffman@23053
  1402
apply (rule notI, drule arg_cong [where f=cos], simp)
paulson@15077
  1403
done
huffman@23053
  1404
huffman@23053
  1405
lemmas pi_half_neq_two [simp] = pi_half_less_two [THEN less_imp_neq]
huffman@23053
  1406
lemmas pi_half_le_two [simp] =  pi_half_less_two [THEN order_less_imp_le]
paulson@15077
  1407
paulson@15077
  1408
lemma pi_gt_zero [simp]: "0 < pi"
huffman@23053
  1409
by (insert pi_half_gt_zero, simp)
huffman@23053
  1410
huffman@23053
  1411
lemma pi_ge_zero [simp]: "0 \<le> pi"
huffman@23053
  1412
by (rule pi_gt_zero [THEN order_less_imp_le])
paulson@15077
  1413
paulson@15077
  1414
lemma pi_neq_zero [simp]: "pi \<noteq> 0"
huffman@22998
  1415
by (rule pi_gt_zero [THEN less_imp_neq, THEN not_sym])
paulson@15077
  1416
huffman@23053
  1417
lemma pi_not_less_zero [simp]: "\<not> pi < 0"
huffman@23053
  1418
by (simp add: linorder_not_less)
paulson@15077
  1419
paulson@15077
  1420
lemma minus_pi_half_less_zero [simp]: "-(pi/2) < 0"
paulson@15077
  1421
by auto
paulson@15077
  1422
paulson@15077
  1423
lemma sin_pi_half [simp]: "sin(pi/2) = 1"
paulson@15077
  1424
apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
paulson@15077
  1425
apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
huffman@23053
  1426
apply (simp add: power2_eq_square)
paulson@15077
  1427
done
paulson@15077
  1428
paulson@15077
  1429
lemma cos_pi [simp]: "cos pi = -1"
nipkow@15539
  1430
by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
paulson@15077
  1431
paulson@15077
  1432
lemma sin_pi [simp]: "sin pi = 0"
nipkow@15539
  1433
by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
paulson@15077
  1434
paulson@15077
  1435
lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
paulson@15229
  1436
by (simp add: diff_minus cos_add)
huffman@23053
  1437
declare sin_cos_eq [symmetric, simp]
paulson@15077
  1438
paulson@15077
  1439
lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
paulson@15229
  1440
by (simp add: cos_add)
paulson@15077
  1441
declare minus_sin_cos_eq [symmetric, simp]
paulson@15077
  1442
paulson@15077
  1443
lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
paulson@15229
  1444
by (simp add: diff_minus sin_add)
huffman@23053
  1445
declare cos_sin_eq [symmetric, simp]
paulson@15077
  1446
paulson@15077
  1447
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
paulson@15229
  1448
by (simp add: sin_add)
paulson@15077
  1449
paulson@15077
  1450
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
paulson@15229
  1451
by (simp add: sin_add)
paulson@15077
  1452
paulson@15077
  1453
lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
paulson@15229
  1454
by (simp add: cos_add)
paulson@15077
  1455
paulson@15077
  1456
lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
paulson@15077
  1457
by (simp add: sin_add cos_double)
paulson@15077
  1458
paulson@15077
  1459
lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
paulson@15077
  1460
by (simp add: cos_add cos_double)
paulson@15077
  1461
paulson@15077
  1462
lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
paulson@15251
  1463
apply (induct "n")
paulson@15077
  1464
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1465
done
paulson@15077
  1466
paulson@15383
  1467
lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
paulson@15383
  1468
proof -
paulson@15383
  1469
  have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
paulson@15383
  1470
  also have "... = -1 ^ n" by (rule cos_npi) 
paulson@15383
  1471
  finally show ?thesis .
paulson@15383
  1472
qed
paulson@15383
  1473
paulson@15077
  1474
lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
paulson@15251
  1475
apply (induct "n")
paulson@15077
  1476
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1477
done
paulson@15077
  1478
paulson@15077
  1479
lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
paulson@15383
  1480
by (simp add: mult_commute [of pi]) 
paulson@15077
  1481
paulson@15077
  1482
lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
paulson@15077
  1483
by (simp add: cos_double)
paulson@15077
  1484
paulson@15077
  1485
lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
paulson@15229
  1486
by simp
paulson@15077
  1487
paulson@15077
  1488
lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
paulson@15077
  1489
apply (rule sin_gt_zero, assumption)
paulson@15077
  1490
apply (rule order_less_trans, assumption)
paulson@15077
  1491
apply (rule pi_half_less_two)
paulson@15077
  1492
done
paulson@15077
  1493
paulson@15077
  1494
lemma sin_less_zero: 
paulson@15077
  1495
  assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
paulson@15077
  1496
proof -
paulson@15077
  1497
  have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2) 
paulson@15077
  1498
  thus ?thesis by simp
paulson@15077
  1499
qed
paulson@15077
  1500
paulson@15077
  1501
lemma pi_less_4: "pi < 4"
paulson@15077
  1502
by (cut_tac pi_half_less_two, auto)
paulson@15077
  1503
paulson@15077
  1504
lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1505
apply (cut_tac pi_less_4)
paulson@15077
  1506
apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
paulson@15077
  1507
apply (cut_tac cos_is_zero, safe)
paulson@15077
  1508
apply (rename_tac y z)
paulson@15077
  1509
apply (drule_tac x = y in spec)
paulson@15077
  1510
apply (drule_tac x = "pi/2" in spec, simp) 
paulson@15077
  1511
done
paulson@15077
  1512
paulson@15077
  1513
lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1514
apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15077
  1515
apply (rule cos_minus [THEN subst])
paulson@15077
  1516
apply (rule cos_gt_zero)
paulson@15077
  1517
apply (auto intro: cos_gt_zero)
paulson@15077
  1518
done
paulson@15077
  1519
 
paulson@15077
  1520
lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
paulson@15077
  1521
apply (auto simp add: order_le_less cos_gt_zero_pi)
paulson@15077
  1522
apply (subgoal_tac "x = pi/2", auto) 
paulson@15077
  1523
done
paulson@15077
  1524
paulson@15077
  1525
lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
paulson@15077
  1526
apply (subst sin_cos_eq)
paulson@15077
  1527
apply (rotate_tac 1)
paulson@15077
  1528
apply (drule real_sum_of_halves [THEN ssubst])
paulson@15077
  1529
apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
paulson@15077
  1530
done
paulson@15077
  1531
paulson@15077
  1532
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
paulson@15077
  1533
by (auto simp add: order_le_less sin_gt_zero_pi)
paulson@15077
  1534
paulson@15077
  1535
lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
paulson@15077
  1536
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
paulson@15077
  1537
apply (rule_tac [2] IVT2)
paulson@15077
  1538
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
paulson@15077
  1539
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077
  1540
apply (rule ccontr, auto)
paulson@15077
  1541
apply (drule_tac f = cos in Rolle)
paulson@15077
  1542
apply (drule_tac [5] f = cos in Rolle)
paulson@15077
  1543
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
paulson@15077
  1544
            dest!: DERIV_cos [THEN DERIV_unique] 
paulson@15077
  1545
            simp add: differentiable_def)
paulson@15077
  1546
apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
paulson@15077
  1547
done
paulson@15077
  1548
paulson@15077
  1549
lemma sin_total:
paulson@15077
  1550
     "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
paulson@15077
  1551
apply (rule ccontr)
paulson@15077
  1552
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
  1553
apply (erule contrapos_np)
paulson@15077
  1554
apply (simp del: minus_sin_cos_eq [symmetric])
paulson@15077
  1555
apply (cut_tac y="-y" in cos_total, simp) apply simp 
paulson@15077
  1556
apply (erule ex1E)
paulson@15229
  1557
apply (rule_tac a = "x - (pi/2)" in ex1I)
paulson@15077
  1558
apply (simp (no_asm) add: real_add_assoc)
paulson@15077
  1559
apply (rotate_tac 3)
paulson@15077
  1560
apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all) 
paulson@15077
  1561
done
paulson@15077
  1562
paulson@15077
  1563
lemma reals_Archimedean4:
paulson@15077
  1564
     "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
paulson@15077
  1565
apply (auto dest!: reals_Archimedean3)
paulson@15077
  1566
apply (drule_tac x = x in spec, clarify) 
paulson@15077
  1567
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
paulson@15077
  1568
 prefer 2 apply (erule LeastI) 
paulson@15077
  1569
apply (case_tac "LEAST m::nat. x < real m * y", simp) 
paulson@15077
  1570
apply (subgoal_tac "~ x < real nat * y")
paulson@15077
  1571
 prefer 2 apply (rule not_less_Least, simp, force)  
paulson@15077
  1572
done
paulson@15077
  1573
paulson@15077
  1574
(* Pre Isabelle99-2 proof was simpler- numerals arithmetic 
paulson@15077
  1575
   now causes some unwanted re-arrangements of literals!   *)
paulson@15229
  1576
lemma cos_zero_lemma:
paulson@15229
  1577
     "[| 0 \<le> x; cos x = 0 |] ==>  
paulson@15077
  1578
      \<exists>n::nat. ~even n & x = real n * (pi/2)"
paulson@15077
  1579
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
paulson@15086
  1580
apply (subgoal_tac "0 \<le> x - real n * pi & 
paulson@15086
  1581
                    (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
paulson@15086
  1582
apply (auto simp add: compare_rls) 
paulson@15077
  1583
  prefer 3 apply (simp add: cos_diff) 
paulson@15077
  1584
 prefer 2 apply (simp add: real_of_nat_Suc left_distrib) 
paulson@15077
  1585
apply (simp add: cos_diff)
paulson@15077
  1586
apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
paulson@15077
  1587
apply (rule_tac [2] cos_total, safe)
paulson@15077
  1588
apply (drule_tac x = "x - real n * pi" in spec)
paulson@15077
  1589
apply (drule_tac x = "pi/2" in spec)
paulson@15077
  1590
apply (simp add: cos_diff)
paulson@15229
  1591
apply (rule_tac x = "Suc (2 * n)" in exI)
paulson@15077
  1592
apply (simp add: real_of_nat_Suc left_distrib, auto)
paulson@15077
  1593
done
paulson@15077
  1594
paulson@15229
  1595
lemma sin_zero_lemma:
paulson@15229
  1596
     "[| 0 \<le> x; sin x = 0 |] ==>  
paulson@15077
  1597
      \<exists>n::nat. even n & x = real n * (pi/2)"
paulson@15077
  1598
apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
paulson@15077
  1599
 apply (clarify, rule_tac x = "n - 1" in exI)
paulson@15077
  1600
 apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
paulson@15085
  1601
apply (rule cos_zero_lemma)
paulson@15085
  1602
apply (simp_all add: add_increasing)  
paulson@15077
  1603
done
paulson@15077
  1604
paulson@15077
  1605
paulson@15229
  1606
lemma cos_zero_iff:
paulson@15229
  1607
     "(cos x = 0) =  
paulson@15077
  1608
      ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |    
paulson@15077
  1609
       (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1610
apply (rule iffI)
paulson@15077
  1611
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1612
apply (drule cos_zero_lemma, assumption+)
paulson@15077
  1613
apply (cut_tac x="-x" in cos_zero_lemma, simp, simp) 
paulson@15077
  1614
apply (force simp add: minus_equation_iff [of x]) 
paulson@15077
  1615
apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) 
nipkow@15539
  1616
apply (auto simp add: cos_add)
paulson@15077
  1617
done
paulson@15077
  1618
paulson@15077
  1619
(* ditto: but to a lesser extent *)
paulson@15229
  1620
lemma sin_zero_iff:
paulson@15229
  1621
     "(sin x = 0) =  
paulson@15077
  1622
      ((\<exists>n::nat. even n & (x = real n * (pi/2))) |    
paulson@15077
  1623
       (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1624
apply (rule iffI)
paulson@15077
  1625
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1626
apply (drule sin_zero_lemma, assumption+)
paulson@15077
  1627
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
paulson@15077
  1628
apply (force simp add: minus_equation_iff [of x]) 
nipkow@15539
  1629
apply (auto simp add: even_mult_two_ex)
paulson@15077
  1630
done
paulson@15077
  1631
paulson@15077
  1632
paulson@15077
  1633
subsection{*Tangent*}
paulson@15077
  1634
huffman@23043
  1635
definition
huffman@23043
  1636
  tan :: "real => real" where
huffman@23043
  1637
  "tan x = (sin x)/(cos x)"
huffman@23043
  1638
paulson@15077
  1639
lemma tan_zero [simp]: "tan 0 = 0"
paulson@15077
  1640
by (simp add: tan_def)
paulson@15077
  1641
paulson@15077
  1642
lemma tan_pi [simp]: "tan pi = 0"
paulson@15077
  1643
by (simp add: tan_def)
paulson@15077
  1644
paulson@15077
  1645
lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
paulson@15077
  1646
by (simp add: tan_def)
paulson@15077
  1647
paulson@15077
  1648
lemma tan_minus [simp]: "tan (-x) = - tan x"
paulson@15077
  1649
by (simp add: tan_def minus_mult_left)
paulson@15077
  1650
paulson@15077
  1651
lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
paulson@15077
  1652
by (simp add: tan_def)
paulson@15077
  1653
paulson@15077
  1654
lemma lemma_tan_add1: 
paulson@15077
  1655
      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077
  1656
        ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
paulson@15229
  1657
apply (simp add: tan_def divide_inverse)
paulson@15229
  1658
apply (auto simp del: inverse_mult_distrib 
paulson@15229
  1659
            simp add: inverse_mult_distrib [symmetric] mult_ac)
paulson@15077
  1660
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15229
  1661
apply (auto simp del: inverse_mult_distrib 
paulson@15229
  1662
            simp add: mult_assoc left_diff_distrib cos_add)
paulson@15234
  1663
done  
paulson@15077
  1664
paulson@15077
  1665
lemma add_tan_eq: 
paulson@15077
  1666
      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077
  1667
       ==> tan x + tan y = sin(x + y)/(cos x * cos y)"
paulson@15229
  1668
apply (simp add: tan_def)
paulson@15077
  1669
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15077
  1670
apply (auto simp add: mult_assoc left_distrib)
nipkow@15539
  1671
apply (simp add: sin_add)
paulson@15077
  1672
done
paulson@15077
  1673
paulson@15229
  1674
lemma tan_add:
paulson@15229
  1675
     "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]  
paulson@15077
  1676
      ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
paulson@15077
  1677
apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)
paulson@15077
  1678
apply (simp add: tan_def)
paulson@15077
  1679
done
paulson@15077
  1680
paulson@15229
  1681
lemma tan_double:
paulson@15229
  1682
     "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]  
paulson@15077
  1683
      ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
paulson@15077
  1684
apply (insert tan_add [of x x]) 
paulson@15077
  1685
apply (simp add: mult_2 [symmetric])  
paulson@15077
  1686
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1687
done
paulson@15077
  1688
paulson@15077
  1689
lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
paulson@15229
  1690
by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) 
paulson@15077
  1691
paulson@15077
  1692
lemma tan_less_zero: 
paulson@15077
  1693
  assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
paulson@15077
  1694
proof -
paulson@15077
  1695
  have "0 < tan (- x)" using prems by (simp only: tan_gt_zero) 
paulson@15077
  1696
  thus ?thesis by simp
paulson@15077
  1697
qed
paulson@15077
  1698
paulson@15077
  1699
lemma lemma_DERIV_tan:
paulson@15077
  1700
     "cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"
paulson@15077
  1701
apply (rule lemma_DERIV_subst)
paulson@15077
  1702
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15079
  1703
apply (auto simp add: divide_inverse numeral_2_eq_2)
paulson@15077
  1704
done
paulson@15077
  1705
paulson@15077
  1706
lemma DERIV_tan [simp]: "cos x \<noteq> 0 ==> DERIV tan x :> inverse((cos x)\<twosuperior>)"
paulson@15077
  1707
by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])
paulson@15077
  1708
huffman@23045
  1709
lemma isCont_tan [simp]: "cos x \<noteq> 0 ==> isCont tan x"
huffman@23045
  1710
by (rule DERIV_tan [THEN DERIV_isCont])
huffman@23045
  1711
paulson@15077
  1712
lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
paulson@15077
  1713
apply (subgoal_tac "(\<lambda>x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")
paulson@15229
  1714
apply (simp add: divide_inverse [symmetric])
huffman@22613
  1715
apply (rule LIM_mult)
paulson@15077
  1716
apply (rule_tac [2] inverse_1 [THEN subst])
paulson@15077
  1717
apply (rule_tac [2] LIM_inverse)
paulson@15077
  1718
apply (simp_all add: divide_inverse [symmetric]) 
paulson@15077
  1719
apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric]) 
paulson@15077
  1720
apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
paulson@15077
  1721
done
paulson@15077
  1722
paulson@15077
  1723
lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
paulson@15077
  1724
apply (cut_tac LIM_cos_div_sin)
paulson@15077
  1725
apply (simp only: LIM_def)
paulson@15077
  1726
apply (drule_tac x = "inverse y" in spec, safe, force)
paulson@15077
  1727
apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
paulson@15229
  1728
apply (rule_tac x = "(pi/2) - e" in exI)
paulson@15077
  1729
apply (simp (no_asm_simp))
paulson@15229
  1730
apply (drule_tac x = "(pi/2) - e" in spec)
paulson@15229
  1731
apply (auto simp add: tan_def)
paulson@15077
  1732
apply (rule inverse_less_iff_less [THEN iffD1])
paulson@15079
  1733
apply (auto simp add: divide_inverse)
paulson@15229
  1734
apply (rule real_mult_order) 
paulson@15229
  1735
apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
paulson@15229
  1736
apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute) 
paulson@15077
  1737
done
paulson@15077
  1738
paulson@15077
  1739
lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
huffman@22998
  1740
apply (frule order_le_imp_less_or_eq, safe)
paulson@15077
  1741
 prefer 2 apply force
paulson@15077
  1742
apply (drule lemma_tan_total, safe)
paulson@15077
  1743
apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
paulson@15077
  1744
apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
paulson@15077
  1745
apply (drule_tac y = xa in order_le_imp_less_or_eq)
paulson@15077
  1746
apply (auto dest: cos_gt_zero)
paulson@15077
  1747
done
paulson@15077
  1748
paulson@15077
  1749
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  1750
apply (cut_tac linorder_linear [of 0 y], safe)
paulson@15077
  1751
apply (drule tan_total_pos)
paulson@15077
  1752
apply (cut_tac [2] y="-y" in tan_total_pos, safe)
paulson@15077
  1753
apply (rule_tac [3] x = "-x" in exI)
paulson@15077
  1754
apply (auto intro!: exI)
paulson@15077
  1755
done
paulson@15077
  1756
paulson@15077
  1757
lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  1758
apply (cut_tac y = y in lemma_tan_total1, auto)
paulson@15077
  1759
apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
paulson@15077
  1760
apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
paulson@15077
  1761
apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
paulson@15077
  1762
apply (rule_tac [4] Rolle)
paulson@15077
  1763
apply (rule_tac [2] Rolle)
paulson@15077
  1764
apply (auto intro!: DERIV_tan DERIV_isCont exI 
paulson@15077
  1765
            simp add: differentiable_def)
paulson@15077
  1766
txt{*Now, simulate TRYALL*}
paulson@15077
  1767
apply (rule_tac [!] DERIV_tan asm_rl)
paulson@15077
  1768
apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
huffman@22998
  1769
	    simp add: cos_gt_zero_pi [THEN less_imp_neq, THEN not_sym]) 
paulson@15077
  1770
done
paulson@15077
  1771
huffman@23043
  1772
huffman@23043
  1773
subsection {* Inverse Trigonometric Functions *}
huffman@23043
  1774
huffman@23043
  1775
definition
huffman@23043
  1776
  arcsin :: "real => real" where
huffman@23043
  1777
  "arcsin y = (THE x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
huffman@23043
  1778
huffman@23043
  1779
definition
huffman@23043
  1780
  arccos :: "real => real" where
huffman@23043
  1781
  "arccos y = (THE x. 0 \<le> x & x \<le> pi & cos x = y)"
huffman@23043
  1782
huffman@23043
  1783
definition     
huffman@23043
  1784
  arctan :: "real => real" where
huffman@23043
  1785
  "arctan y = (THE x. -(pi/2) < x & x < pi/2 & tan x = y)"
huffman@23043
  1786
paulson@15229
  1787
lemma arcsin:
paulson@15229
  1788
     "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077
  1789
      ==> -(pi/2) \<le> arcsin y &  
paulson@15077
  1790
           arcsin y \<le> pi/2 & sin(arcsin y) = y"
huffman@23011
  1791
unfolding arcsin_def by (rule theI' [OF sin_total])
huffman@23011
  1792
huffman@23011
  1793
lemma arcsin_pi:
huffman@23011
  1794
     "[| -1 \<le> y; y \<le> 1 |]  
huffman@23011
  1795
      ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
huffman@23011
  1796
apply (drule (1) arcsin)
huffman@23011
  1797
apply (force intro: order_trans)
paulson@15077
  1798
done
paulson@15077
  1799
paulson@15077
  1800
lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
paulson@15077
  1801
by (blast dest: arcsin)
paulson@15077
  1802
      
paulson@15077
  1803
lemma arcsin_bounded:
paulson@15077
  1804
     "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
paulson@15077
  1805
by (blast dest: arcsin)
paulson@15077
  1806
paulson@15077
  1807
lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
paulson@15077
  1808
by (blast dest: arcsin)
paulson@15077
  1809
paulson@15077
  1810
lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
paulson@15077
  1811
by (blast dest: arcsin)
paulson@15077
  1812
paulson@15077
  1813
lemma arcsin_lt_bounded:
paulson@15077
  1814
     "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
paulson@15077
  1815
apply (frule order_less_imp_le)
paulson@15077
  1816
apply (frule_tac y = y in order_less_imp_le)
paulson@15077
  1817
apply (frule arcsin_bounded)
paulson@15077
  1818
apply (safe, simp)
paulson@15077
  1819
apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
paulson@15077
  1820
apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
paulson@15077
  1821
apply (drule_tac [!] f = sin in arg_cong, auto)
paulson@15077
  1822
done
paulson@15077
  1823
paulson@15077
  1824
lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
paulson@15077
  1825
apply (unfold arcsin_def)
huffman@23011
  1826
apply (rule the1_equality)
paulson@15077
  1827
apply (rule sin_total, auto)
paulson@15077
  1828
done
paulson@15077
  1829
huffman@22975
  1830
lemma arccos:
paulson@15229
  1831
     "[| -1 \<le> y; y \<le> 1 |]  
huffman@22975
  1832
      ==> 0 \<le> arccos y & arccos y \<le> pi & cos(arccos y) = y"
huffman@23011
  1833
unfolding arccos_def by (rule theI' [OF cos_total])
paulson@15077
  1834
huffman@22975
  1835
lemma cos_arccos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arccos y) = y"
huffman@22975
  1836
by (blast dest: arccos)
paulson@15077
  1837
      
huffman@22975
  1838
lemma arccos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y & arccos y \<le> pi"
huffman@22975
  1839
by (blast dest: arccos)
paulson@15077
  1840
huffman@22975
  1841
lemma arccos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arccos y"
huffman@22975
  1842
by (blast dest: arccos)
paulson@15077
  1843
huffman@22975
  1844
lemma arccos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arccos y \<le> pi"
huffman@22975
  1845
by (blast dest: arccos)
paulson@15077
  1846
huffman@22975
  1847
lemma arccos_lt_bounded:
paulson@15229
  1848
     "[| -1 < y; y < 1 |]  
huffman@22975
  1849
      ==> 0 < arccos y & arccos y < pi"
paulson@15077
  1850
apply (frule order_less_imp_le)
paulson@15077
  1851
apply (frule_tac y = y in order_less_imp_le)
huffman@22975
  1852
apply (frule arccos_bounded, auto)
huffman@22975
  1853
apply (drule_tac y = "arccos y" in order_le_imp_less_or_eq)
paulson@15077
  1854
apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
paulson@15077
  1855
apply (drule_tac [!] f = cos in arg_cong, auto)
paulson@15077
  1856
done
paulson@15077
  1857
huffman@22975
  1858
lemma arccos_cos: "[|0 \<le> x; x \<le> pi |] ==> arccos(cos x) = x"
huffman@22975
  1859
apply (simp add: arccos_def)
huffman@23011
  1860
apply (auto intro!: the1_equality cos_total)
paulson@15077
  1861
done
paulson@15077
  1862
huffman@22975
  1863
lemma arccos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arccos(cos x) = -x"
huffman@22975
  1864
apply (simp add: arccos_def)
huffman@23011
  1865
apply (auto intro!: the1_equality cos_total)
paulson@15077
  1866
done
paulson@15077
  1867
huffman@23045
  1868
lemma cos_arcsin: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> cos (arcsin x) = sqrt (1 - x\<twosuperior>)"
huffman@23045
  1869
apply (subgoal_tac "x\<twosuperior> \<le> 1")
huffman@23052
  1870
apply (rule power2_eq_imp_eq)
huffman@23045
  1871
apply (simp add: cos_squared_eq)
huffman@23045
  1872
apply (rule cos_ge_zero)
huffman@23045
  1873
apply (erule (1) arcsin_lbound)
huffman@23045
  1874
apply (erule (1) arcsin_ubound)
huffman@23045
  1875
apply simp
huffman@23045
  1876
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
huffman@23045
  1877
apply (rule power_mono, simp, simp)
huffman@23045
  1878
done
huffman@23045
  1879
huffman@23045
  1880
lemma sin_arccos: "\<lbrakk>-1 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> sin (arccos x) = sqrt (1 - x\<twosuperior>)"
huffman@23045
  1881
apply (subgoal_tac "x\<twosuperior> \<le> 1")
huffman@23052
  1882
apply (rule power2_eq_imp_eq)
huffman@23045
  1883
apply (simp add: sin_squared_eq)
huffman@23045
  1884
apply (rule sin_ge_zero)
huffman@23045
  1885
apply (erule (1) arccos_lbound)
huffman@23045
  1886
apply (erule (1) arccos_ubound)
huffman@23045
  1887
apply simp
huffman@23045
  1888
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> \<le> 1\<twosuperior>", simp)
huffman@23045
  1889
apply (rule power_mono, simp, simp)
huffman@23045
  1890
done
huffman@23045
  1891
paulson@15077
  1892
lemma arctan [simp]:
paulson@15077
  1893
     "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
huffman@23011
  1894
unfolding arctan_def by (rule theI' [OF tan_total])
paulson@15077
  1895
paulson@15077
  1896
lemma tan_arctan: "tan(arctan y) = y"
paulson@15077
  1897
by auto
paulson@15077
  1898
paulson@15077
  1899
lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
paulson@15077
  1900
by (auto simp only: arctan)
paulson@15077
  1901
paulson@15077
  1902
lemma arctan_lbound: "- (pi/2) < arctan y"
paulson@15077
  1903
by auto
paulson@15077
  1904
paulson@15077
  1905
lemma arctan_ubound: "arctan y < pi/2"
paulson@15077
  1906
by (auto simp only: arctan)
paulson@15077
  1907
paulson@15077
  1908
lemma arctan_tan: 
paulson@15077
  1909
      "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
paulson@15077
  1910
apply (unfold arctan_def)
huffman@23011
  1911
apply (rule the1_equality)
paulson@15077
  1912
apply (rule tan_total, auto)
paulson@15077
  1913
done
paulson@15077
  1914
paulson@15077
  1915
lemma arctan_zero_zero [simp]: "arctan 0 = 0"
paulson@15077
  1916
by (insert arctan_tan [of 0], simp)
paulson@15077
  1917
paulson@15077
  1918
lemma cos_arctan_not_zero [simp]: "cos(arctan x) \<noteq> 0"
paulson@15077
  1919
apply (auto simp add: cos_zero_iff)
paulson@15077
  1920
apply (case_tac "n")
paulson@15077
  1921
apply (case_tac [3] "n")
paulson@15077
  1922
apply (cut_tac [2] y = x in arctan_ubound)
paulson@15077
  1923
apply (cut_tac [4] y = x in arctan_lbound) 
paulson@15077
  1924
apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)
paulson@15077
  1925
done
paulson@15077
  1926
paulson@15077
  1927
lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
paulson@15077
  1928
apply (rule power_inverse [THEN subst])
paulson@15077
  1929
apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
huffman@22960
  1930
apply (auto dest: field_power_not_zero
huffman@20516
  1931
        simp add: power_mult_distrib left_distrib power_divide tan_def 
paulson@15077
  1932
                  mult_assoc power_inverse [symmetric] 
paulson@15077
  1933
        simp del: realpow_Suc)
paulson@15077
  1934
done
paulson@15077
  1935
huffman@23045
  1936
lemma isCont_inverse_function2:
huffman@23045
  1937
  fixes f g :: "real \<Rightarrow> real" shows
huffman@23045
  1938
  "\<lbrakk>a < x; x < b;
huffman@23045
  1939
    \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
huffman@23045
  1940
    \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
huffman@23045
  1941
   \<Longrightarrow> isCont g (f x)"
huffman@23045
  1942
apply (rule isCont_inverse_function
huffman@23045
  1943
       [where f=f and d="min (x - a) (b - x)"])
huffman@23045
  1944
apply (simp_all add: abs_le_iff)
huffman@23045
  1945
done
huffman@23045
  1946
huffman@23045
  1947
lemma isCont_arcsin: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arcsin x"
huffman@23045
  1948
apply (subgoal_tac "isCont arcsin (sin (arcsin x))", simp)
huffman@23045
  1949
apply (rule isCont_inverse_function2 [where f=sin])
huffman@23045
  1950
apply (erule (1) arcsin_lt_bounded [THEN conjunct1])
huffman@23045
  1951
apply (erule (1) arcsin_lt_bounded [THEN conjunct2])
huffman@23045
  1952
apply (fast intro: arcsin_sin, simp)
huffman@23045
  1953
done
huffman@23045
  1954
huffman@23045
  1955
lemma isCont_arccos: "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> isCont arccos x"
huffman@23045
  1956
apply (subgoal_tac "isCont arccos (cos (arccos x))", simp)
huffman@23045
  1957
apply (rule isCont_inverse_function2 [where f=cos])
huffman@23045
  1958
apply (erule (1) arccos_lt_bounded [THEN conjunct1])
huffman@23045
  1959
apply (erule (1) arccos_lt_bounded [THEN conjunct2])
huffman@23045
  1960
apply (fast intro: arccos_cos, simp)
huffman@23045
  1961
done
huffman@23045
  1962
huffman@23045
  1963
lemma isCont_arctan: "isCont arctan x"
huffman@23045
  1964
apply (rule arctan_lbound [of x, THEN dense, THEN exE], clarify)
huffman@23045
  1965
apply (rule arctan_ubound [of x, THEN dense, THEN exE], clarify)
huffman@23045
  1966
apply (subgoal_tac "isCont arctan (tan (arctan x))", simp)
huffman@23045
  1967
apply (erule (1) isCont_inverse_function2 [where f=tan])
huffman@23045
  1968
apply (clarify, rule arctan_tan)
huffman@23045
  1969
apply (erule (1) order_less_le_trans)
huffman@23045
  1970
apply (erule (1) order_le_less_trans)
huffman@23045
  1971
apply (clarify, rule isCont_tan)
huffman@23045
  1972
apply (rule less_imp_neq [symmetric])
huffman@23045
  1973
apply (rule cos_gt_zero_pi)
huffman@23045
  1974
apply (erule (1) order_less_le_trans)
huffman@23045
  1975
apply (erule (1) order_le_less_trans)
huffman@23045
  1976
done
huffman@23045
  1977
huffman@23045
  1978
lemma DERIV_arcsin:
huffman@23045
  1979
  "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arcsin x :> inverse (sqrt (1 - x\<twosuperior>))"
huffman@23045
  1980
apply (rule DERIV_inverse_function [where f=sin and a="-1" and b="1"])
huffman@23045
  1981
apply (rule lemma_DERIV_subst [OF DERIV_sin])
huffman@23045
  1982
apply (simp add: cos_arcsin)
huffman@23045
  1983
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
huffman@23045
  1984
apply (rule power_strict_mono, simp, simp, simp)
huffman@23045
  1985
apply assumption
huffman@23045
  1986
apply assumption
huffman@23045
  1987
apply simp
huffman@23045
  1988
apply (erule (1) isCont_arcsin)
huffman@23045
  1989
done
huffman@23045
  1990
huffman@23045
  1991
lemma DERIV_arccos:
huffman@23045
  1992
  "\<lbrakk>-1 < x; x < 1\<rbrakk> \<Longrightarrow> DERIV arccos x :> inverse (- sqrt (1 - x\<twosuperior>))"
huffman@23045
  1993
apply (rule DERIV_inverse_function [where f=cos and a="-1" and b="1"])
huffman@23045
  1994
apply (rule lemma_DERIV_subst [OF DERIV_cos])
huffman@23045
  1995
apply (simp add: sin_arccos)
huffman@23045
  1996
apply (subgoal_tac "\<bar>x\<bar>\<twosuperior> < 1\<twosuperior>", simp)
huffman@23045
  1997
apply (rule power_strict_mono, simp, simp, simp)
huffman@23045
  1998
apply assumption
huffman@23045
  1999
apply assumption
huffman@23045
  2000
apply simp
huffman@23045
  2001
apply (erule (1) isCont_arccos)
huffman@23045
  2002
done
huffman@23045
  2003
huffman@23045
  2004
lemma DERIV_arctan: "DERIV arctan x :> inverse (1 + x\<twosuperior>)"
huffman@23045
  2005
apply (rule DERIV_inverse_function [where f=tan and a="x - 1" and b="x + 1"])
huffman@23045
  2006
apply (rule lemma_DERIV_subst [OF DERIV_tan])
huffman@23045
  2007
apply (rule cos_arctan_not_zero)
huffman@23045
  2008
apply (simp add: power_inverse tan_sec [symmetric])
huffman@23045
  2009
apply (subgoal_tac "0 < 1 + x\<twosuperior>", simp)
huffman@23045
  2010
apply (simp add: add_pos_nonneg)
huffman@23045
  2011
apply (simp, simp, simp, rule isCont_arctan)
huffman@23045
  2012
done
huffman@23045
  2013
huffman@23045
  2014
huffman@23043
  2015
subsection {* More Theorems about Sin and Cos *}
huffman@23043
  2016
huffman@23052
  2017
lemma cos_45: "cos (pi / 4) = sqrt 2 / 2"
huffman@23052
  2018
proof -
huffman@23052
  2019
  let ?c = "cos (pi / 4)" and ?s = "sin (pi / 4)"
huffman@23052
  2020
  have nonneg: "0 \<le> ?c"
huffman@23052
  2021
    by (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
huffman@23052
  2022
  have "0 = cos (pi / 4 + pi / 4)"
huffman@23052
  2023
    by simp
huffman@23052
  2024
  also have "cos (pi / 4 + pi / 4) = ?c\<twosuperior> - ?s\<twosuperior>"
huffman@23052
  2025
    by (simp only: cos_add power2_eq_square)
huffman@23052
  2026
  also have "\<dots> = 2 * ?c\<twosuperior> - 1"
huffman@23052
  2027
    by (simp add: sin_squared_eq)
huffman@23052
  2028
  finally have "?c\<twosuperior> = (sqrt 2 / 2)\<twosuperior>"
huffman@23052
  2029
    by (simp add: power_divide)
huffman@23052
  2030
  thus ?thesis
huffman@23052
  2031
    using nonneg by (rule power2_eq_imp_eq) simp
huffman@23052
  2032
qed
huffman@23052
  2033
huffman@23052
  2034
lemma cos_30: "cos (pi / 6) = sqrt 3 / 2"
huffman@23052
  2035
proof -
huffman@23052
  2036
  let ?c = "cos (pi / 6)" and ?s = "sin (pi / 6)"
huffman@23052
  2037
  have pos_c: "0 < ?c"
huffman@23052
  2038
    by (rule cos_gt_zero, simp, simp)
huffman@23052
  2039
  have "0 = cos (pi / 6 + pi / 6 + pi / 6)"
huffman@23066
  2040
    by simp
huffman@23052
  2041
  also have "\<dots> = (?c * ?c - ?s * ?s) * ?c - (?s * ?c + ?c * ?s) * ?s"
huffman@23052
  2042
    by (simp only: cos_add sin_add)
huffman@23052
  2043
  also have "\<dots> = ?c * (?c\<twosuperior> - 3 * ?s\<twosuperior>)"
huffman@23052
  2044
    by (simp add: ring_eq_simps power2_eq_square)
huffman@23052
  2045
  finally have "?c\<twosuperior> = (sqrt 3 / 2)\<twosuperior>"
huffman@23052
  2046
    using pos_c by (simp add: sin_squared_eq power_divide)
huffman@23052
  2047
  thus ?thesis
huffman@23052
  2048
    using pos_c [THEN order_less_imp_le]
huffman@23052
  2049
    by (rule power2_eq_imp_eq) simp
huffman@23052
  2050
qed
huffman@23052
  2051
huffman@23052
  2052
lemma sin_45: "sin (pi / 4) = sqrt 2 / 2"
huffman@23052
  2053
proof -
huffman@23052
  2054
  have "sin (pi / 4) = cos (pi / 2 - pi / 4)" by (rule sin_cos_eq)
huffman@23052
  2055
  also have "pi / 2 - pi / 4 = pi / 4" by simp
huffman@23052
  2056
  also have "cos (pi / 4) = sqrt 2 / 2" by (rule cos_45)
huffman@23052
  2057
  finally show ?thesis .
huffman@23052
  2058
qed
huffman@23052
  2059
huffman@23052
  2060
lemma sin_60: "sin (pi / 3) = sqrt 3 / 2"
huffman@23052
  2061
proof -
huffman@23052
  2062
  have "sin (pi / 3) = cos (pi / 2 - pi / 3)" by (rule sin_cos_eq)
huffman@23052
  2063
  also have "pi / 2 - pi / 3 = pi / 6" by simp
huffman@23052
  2064
  also have "cos (pi / 6) = sqrt 3 / 2" by (rule cos_30)
huffman@23052
  2065
  finally show ?thesis .
huffman@23052
  2066
qed
huffman@23052
  2067
huffman@23052
  2068
lemma cos_60: "cos (pi / 3) = 1 / 2"
huffman@23052
  2069
apply (rule power2_eq_imp_eq)
huffman@23052
  2070
apply (simp add: cos_squared_eq sin_60 power_divide)
huffman@23052
  2071
apply (rule cos_ge_zero, rule order_trans [where y=0], simp_all)
huffman@23052
  2072
done
huffman@23052
  2073
huffman@23052
  2074
lemma sin_30: "sin (pi / 6) = 1 / 2"
huffman@23052
  2075
proof -
huffman@23052
  2076
  have "sin (pi / 6) = cos (pi / 2 - pi / 6)" by (rule sin_cos_eq)
huffman@23066
  2077
  also have "pi / 2 - pi / 6 = pi / 3" by simp
huffman@23052
  2078
  also have "cos (pi / 3) = 1 / 2" by (rule cos_60)
huffman@23052
  2079
  finally show ?thesis .
huffman@23052
  2080
qed
huffman@23052
  2081
huffman@23052
  2082
lemma tan_30: "tan (pi / 6) = 1 / sqrt 3"
huffman@23052
  2083
unfolding tan_def by (simp add: sin_30 cos_30)
huffman@23052
  2084
huffman@23052
  2085
lemma tan_45: "tan (pi / 4) = 1"
huffman@23052
  2086
unfolding tan_def by (simp add: sin_45 cos_45)
huffman@23052
  2087
huffman@23052
  2088
lemma tan_60: "tan (pi / 3) = sqrt 3"
huffman@23052
  2089
unfolding tan_def by (simp add: sin_60 cos_60)
huffman@23052
  2090
paulson@15085
  2091
text{*NEEDED??*}
paulson@15229
  2092
lemma [simp]:
paulson@15229
  2093
     "sin (x + 1 / 2 * real (Suc m) * pi) =  
paulson@15229
  2094
      cos (x + 1 / 2 * real  (m) * pi)"
paulson@15229
  2095
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib, auto)
paulson@15077
  2096
paulson@15085
  2097
text{*NEEDED??*}
paulson@15229
  2098
lemma [simp]:
paulson@15229
  2099
     "sin (x + real (Suc m) * pi / 2) =  
paulson@15229
  2100
      cos (x + real (m) * pi / 2)"
paulson@15229
  2101
by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)
paulson@15077
  2102
paulson@15077
  2103
lemma DERIV_sin_add [simp]: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)"
paulson@15077
  2104
apply (rule lemma_DERIV_subst)
paulson@15077
  2105
apply (rule_tac f = sin and g = "%x. x + k" in DERIV_chain2)
paulson@15077
  2106
apply (best intro!: DERIV_intros intro: DERIV_chain2)+
paulson@15077
  2107
apply (simp (no_asm))
paulson@15077
  2108
done
paulson@15077
  2109
paulson@15383
  2110
lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
paulson@15383
  2111
proof -
paulson@15383
  2112
  have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
paulson@15383
  2113
    by (auto simp add: right_distrib sin_add left_distrib mult_ac)
paulson@15383
  2114
  thus ?thesis
paulson@15383
  2115
    by (simp add: real_of_nat_Suc left_distrib add_divide_distrib 
paulson@15383
  2116
                  mult_commute [of pi])
paulson@15383
  2117
qed
paulson@15077
  2118
paulson@15077
  2119
lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
paulson@15077
  2120
by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
paulson@15077
  2121
paulson@15077
  2122
lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
huffman@23066
  2123
apply (subgoal_tac "cos (pi + pi/2) = 0", simp)
huffman@23066
  2124
apply (subst cos_add, simp)
paulson@15077
  2125
done
paulson@15077
  2126
paulson@15077
  2127
lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
paulson@15077
  2128
by (auto simp add: mult_assoc)
paulson@15077
  2129
paulson@15077
  2130
lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
huffman@23066
  2131
apply (subgoal_tac "sin (pi + pi/2) = - 1", simp)
huffman@23066
  2132
apply (subst sin_add, simp)
paulson@15077
  2133
done
paulson@15077
  2134
paulson@15077
  2135
(*NEEDED??*)
paulson@15229
  2136
lemma [simp]:
paulson@15229
  2137
     "cos(x + 1 / 2 * real(Suc m) * pi) = -sin (x + 1 / 2 * real m * pi)"
paulson@15077
  2138
apply (simp only: cos_add sin_add real_of_nat_Suc right_distrib left_distrib minus_mult_right, auto)
paulson@15077
  2139
done
paulson@15077
  2140
paulson@15077
  2141
(*NEEDED??*)
paulson@15077
  2142
lemma [simp]: "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
paulson@15229
  2143
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto)
paulson@15077
  2144
paulson@15077
  2145
lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
paulson@15229
  2146
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto)
paulson@15077
  2147
paulson@15077
  2148
lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
paulson@15077
  2149
apply (rule lemma_DERIV_subst)
paulson@15077
  2150
apply (rule_tac f = cos and g = "%x. x + k" in DERIV_chain2)
paulson@15077
  2151
apply (best intro!: DERIV_intros intro: DERIV_chain2)+
paulson@15077
  2152
apply (simp (no_asm))
paulson@15077
  2153
done
paulson@15077
  2154
paulson@15081
  2155
lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
nipkow@15539
  2156
by (auto simp add: sin_zero_iff even_mult_two_ex)
paulson@15077
  2157
huffman@23115
  2158
lemma exp_eq_one_iff [simp]: "(exp (x::real) = 1) = (x = 0)"
paulson@15077
  2159
apply auto
paulson@15077
  2160
apply (drule_tac f = ln in arg_cong, auto)
paulson@15077
  2161
done
paulson@15077
  2162
paulson@15077
  2163
lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
paulson@15077
  2164
by (cut_tac x = x in sin_cos_squared_add3, auto)
paulson@15077
  2165
paulson@15077
  2166
huffman@22978
  2167
subsection {* Existence of Polar Coordinates *}
paulson@15077
  2168
huffman@22978
  2169
lemma cos_x_y_le_one: "\<bar>x / sqrt (x\<twosuperior> + y\<twosuperior>)\<bar> \<le> 1"
huffman@22978
  2170
apply (rule power2_le_imp_le [OF _ zero_le_one])
huffman@22978
  2171
apply (simp add: abs_divide power_divide divide_le_eq not_sum_power2_lt_zero)
paulson@15077
  2172
done
paulson@15077
  2173
huffman@22978
  2174
lemma cos_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> cos (arccos y) = y"
huffman@22978
  2175
by (simp add: abs_le_iff)
paulson@15077
  2176
huffman@23045
  2177
lemma sin_arccos_abs: "\<bar>y\<bar> \<le> 1 \<Longrightarrow> sin (arccos y) = sqrt (1 - y\<twosuperior>)"
huffman@23045
  2178
by (simp add: sin_arccos abs_le_iff)
paulson@15077
  2179
huffman@22978
  2180
lemmas cos_arccos_lemma1 = cos_arccos_abs [OF cos_x_y_le_one]
paulson@15228
  2181
huffman@23045
  2182
lemmas sin_arccos_lemma1 = sin_arccos_abs [OF cos_x_y_le_one]
paulson@15077
  2183
paulson@15229
  2184
lemma polar_ex1:
huffman@22978
  2185
     "0 < y ==> \<exists>r a. x = r * cos a & y = r * sin a"
paulson@15229
  2186
apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>)" in exI)
huffman@22978
  2187
apply (rule_tac x = "arccos (x / sqrt (x\<twosuperior> + y\<twosuperior>))" in exI)
huffman@22978
  2188
apply (simp add: cos_arccos_lemma1)
huffman@23045
  2189
apply (simp add: sin_arccos_lemma1)
huffman@23045
  2190
apply (simp add: power_divide)
huffman@23045
  2191
apply (simp add: real_sqrt_mult [symmetric])
huffman@23045
  2192
apply (simp add: right_diff_distrib)
paulson@15077
  2193
done
paulson@15077
  2194
paulson@15229
  2195
lemma polar_ex2:
huffman@22978
  2196
     "y < 0 ==> \<exists>r a. x = r * cos a & y = r * sin a"
huffman@22978
  2197
apply (insert polar_ex1 [where x=x and y="-y"], simp, clarify)
paulson@15077
  2198
apply (rule_tac x = r in exI)
huffman@22978
  2199
apply (rule_tac x = "-a" in exI, simp)
paulson@15077
  2200
done
paulson@15077
  2201
paulson@15077
  2202
lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
huffman@22978
  2203
apply (rule_tac x=0 and y=y in linorder_cases)
huffman@22978
  2204
apply (erule polar_ex1)
huffman@22978
  2205
apply (rule_tac x=x in exI, rule_tac x=0 in exI, simp)
huffman@22978
  2206
apply (erule polar_ex2)
paulson@15077
  2207
done
paulson@15077
  2208
paulson@15077
  2209
huffman@23043
  2210
subsection {* Theorems about Limits *}
huffman@23043
  2211
paulson@15077
  2212
(* need to rename second isCont_inverse *)
paulson@15077
  2213
paulson@15229
  2214
lemma isCont_inv_fun:
huffman@20561
  2215
  fixes f g :: "real \<Rightarrow> real"
huffman@20561
  2216
  shows "[| 0 < d; \<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
      ==> isCont g (f x)"
huffman@22722
  2219
by (rule isCont_inverse_function)
paulson@15077
  2220
paulson@15077
  2221
lemma isCont_inv_fun_inv:
huffman@20552
  2222
  fixes f g :: "real \<Rightarrow> real"
huffman@20552
  2223
  shows "[| 0 < d;  
paulson@15077
  2224
         \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
paulson@15077
  2225
         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
paulson@15077
  2226
       ==> \<exists>e. 0 < e &  
paulson@15081
  2227
             (\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
paulson@15077
  2228
apply (drule isCont_inj_range)
paulson@15077
  2229
prefer 2 apply (assumption, assumption, auto)
paulson@15077
  2230
apply (rule_tac x = e in exI, auto)
paulson@15077
  2231
apply (rotate_tac 2)
paulson@15077
  2232
apply (drule_tac x = y in spec, auto)
paulson@15077
  2233
done
paulson@15077
  2234
paulson@15077
  2235
paulson@15077
  2236
text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
paulson@15229
  2237
lemma LIM_fun_gt_zero:
huffman@20552
  2238
     "[| f -- c --> (l::real); 0 < l |]  
huffman@20561
  2239
         ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
paulson@15077
  2240
apply (auto simp add: LIM_def)
paulson@15077
  2241
apply (drule_tac x = "l/2" in spec, safe, force)
paulson@15077
  2242
apply (rule_tac x = s in exI)
huffman@22998
  2243
apply (auto simp only: abs_less_iff)
paulson@15077
  2244
done
paulson@15077
  2245
paulson@15229
  2246
lemma LIM_fun_less_zero:
huffman@20552
  2247
     "[| f -- c --> (l::real); l < 0 |]  
huffman@20561
  2248
      ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
paulson@15077
  2249
apply (auto simp add: LIM_def)
paulson@15077
  2250
apply (drule_tac x = "-l/2" in spec, safe, force)
paulson@15077
  2251
apply (rule_tac x = s in exI)
huffman@22998
  2252
apply (auto simp only: abs_less_iff)
paulson@15077
  2253
done
paulson@15077
  2254
paulson@15077
  2255
paulson@15077
  2256
lemma LIM_fun_not_zero:
huffman@20552
  2257
     "[| f -- c --> (l::real); l \<noteq> 0 |] 
huffman@20561
  2258
      ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
paulson@15077
  2259
apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
paulson@15077
  2260
apply (drule LIM_fun_less_zero)
paulson@15241
  2261
apply (drule_tac [3] LIM_fun_gt_zero)
paulson@15241
  2262
apply force+
paulson@15077
  2263
done
webertj@20432
  2264
  
paulson@12196
  2265
end