1.1 --- a/src/HOL/Decision_Procs/Approximation.thy Fri Aug 19 07:45:22 2011 -0700
1.2 +++ b/src/HOL/Decision_Procs/Approximation.thy Fri Aug 19 08:39:43 2011 -0700
1.3 @@ -839,7 +839,8 @@
1.4 cos_eq: "cos x = (\<Sum> i = 0 ..< 2 * n. (if even(i) then (-1 ^ (i div 2))/(real (fact i)) else 0) * (real x) ^ i)
1.5 + (cos (t + 1/2 * (2 * n) * pi) / real (fact (2*n))) * (real x)^(2*n)"
1.6 (is "_ = ?SUM + ?rest / ?fact * ?pow")
1.7 - using Maclaurin_cos_expansion2[OF `0 < real x` `0 < 2 * n`] by auto
1.8 + using Maclaurin_cos_expansion2[OF `0 < real x` `0 < 2 * n`]
1.9 + unfolding cos_coeff_def by auto
1.10
1.11 have "cos t * -1^n = cos t * cos (n * pi) + sin t * sin (n * pi)" by auto
1.12 also have "\<dots> = cos (t + n * pi)" using cos_add by auto
1.13 @@ -951,7 +952,8 @@
1.14 sin_eq: "sin x = (\<Sum> i = 0 ..< 2 * n + 1. (if even(i) then 0 else (-1 ^ ((i - Suc 0) div 2))/(real (fact i))) * (real x) ^ i)
1.15 + (sin (t + 1/2 * (2 * n + 1) * pi) / real (fact (2*n + 1))) * (real x)^(2*n + 1)"
1.16 (is "_ = ?SUM + ?rest / ?fact * ?pow")
1.17 - using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < real x`] by auto
1.18 + using Maclaurin_sin_expansion3[OF `0 < 2 * n + 1` `0 < real x`]
1.19 + unfolding sin_coeff_def by auto
1.20
1.21 have "?rest = cos t * -1^n" unfolding sin_add cos_add real_of_nat_add left_distrib right_distrib by auto
1.22 moreover
2.1 --- a/src/HOL/MacLaurin.thy Fri Aug 19 07:45:22 2011 -0700
2.2 +++ b/src/HOL/MacLaurin.thy Fri Aug 19 08:39:43 2011 -0700
2.3 @@ -417,32 +417,32 @@
2.4 cos (x + real (m) * pi / 2)"
2.5 by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)
2.6
2.7 +lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
2.8 + unfolding sin_coeff_def by simp (* TODO: move *)
2.9 +
2.10 lemma Maclaurin_sin_expansion2:
2.11 "\<exists>t. abs t \<le> abs x &
2.12 sin x =
2.13 - (\<Sum>m=0..<n. (if even m then 0
2.14 - else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
2.15 - x ^ m)
2.16 + (\<Sum>m=0..<n. sin_coeff m * x ^ m)
2.17 + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2.18 apply (cut_tac f = sin and n = n and x = x
2.19 and diff = "%n x. sin (x + 1/2*real n * pi)" in Maclaurin_all_lt_objl)
2.20 apply safe
2.21 apply (simp (no_asm))
2.22 apply (simp (no_asm) add: sin_expansion_lemma)
2.23 -apply (case_tac "n", clarify, simp, simp add: lemma_STAR_sin)
2.24 +apply (subst (asm) setsum_0', clarify, case_tac "a", simp, simp)
2.25 +apply (cases n, simp, simp)
2.26 apply (rule ccontr, simp)
2.27 apply (drule_tac x = x in spec, simp)
2.28 apply (erule ssubst)
2.29 apply (rule_tac x = t in exI, simp)
2.30 apply (rule setsum_cong[OF refl])
2.31 -apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
2.32 +apply (auto simp add: sin_coeff_def sin_zero_iff odd_Suc_mult_two_ex)
2.33 done
2.34
2.35 lemma Maclaurin_sin_expansion:
2.36 "\<exists>t. sin x =
2.37 - (\<Sum>m=0..<n. (if even m then 0
2.38 - else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
2.39 - x ^ m)
2.40 + (\<Sum>m=0..<n. sin_coeff m * x ^ m)
2.41 + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2.42 apply (insert Maclaurin_sin_expansion2 [of x n])
2.43 apply (blast intro: elim:)
2.44 @@ -452,9 +452,7 @@
2.45 "[| n > 0; 0 < x |] ==>
2.46 \<exists>t. 0 < t & t < x &
2.47 sin x =
2.48 - (\<Sum>m=0..<n. (if even m then 0
2.49 - else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
2.50 - x ^ m)
2.51 + (\<Sum>m=0..<n. sin_coeff m * x ^ m)
2.52 + ((sin(t + 1/2 * real(n) *pi) / real (fact n)) * x ^ n)"
2.53 apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin_objl)
2.54 apply safe
2.55 @@ -463,16 +461,14 @@
2.56 apply (erule ssubst)
2.57 apply (rule_tac x = t in exI, simp)
2.58 apply (rule setsum_cong[OF refl])
2.59 -apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
2.60 +apply (auto simp add: sin_coeff_def sin_zero_iff odd_Suc_mult_two_ex)
2.61 done
2.62
2.63 lemma Maclaurin_sin_expansion4:
2.64 "0 < x ==>
2.65 \<exists>t. 0 < t & t \<le> x &
2.66 sin x =
2.67 - (\<Sum>m=0..<n. (if even m then 0
2.68 - else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
2.69 - x ^ m)
2.70 + (\<Sum>m=0..<n. sin_coeff m * x ^ m)
2.71 + ((sin(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2.72 apply (cut_tac f = sin and n = n and h = x and diff = "%n x. sin (x + 1/2*real (n) *pi)" in Maclaurin2_objl)
2.73 apply safe
2.74 @@ -481,15 +477,17 @@
2.75 apply (erule ssubst)
2.76 apply (rule_tac x = t in exI, simp)
2.77 apply (rule setsum_cong[OF refl])
2.78 -apply (auto simp add: sin_zero_iff odd_Suc_mult_two_ex)
2.79 +apply (auto simp add: sin_coeff_def sin_zero_iff odd_Suc_mult_two_ex)
2.80 done
2.81
2.82
2.83 subsection{*Maclaurin Expansion for Cosine Function*}
2.84
2.85 +lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
2.86 + unfolding cos_coeff_def by simp (* TODO: move *)
2.87 +
2.88 lemma sumr_cos_zero_one [simp]:
2.89 - "(\<Sum>m=0..<(Suc n).
2.90 - (if even m then -1 ^ (m div 2)/(real (fact m)) else 0) * 0 ^ m) = 1"
2.91 + "(\<Sum>m=0..<(Suc n). cos_coeff m * 0 ^ m) = 1"
2.92 by (induct "n", auto)
2.93
2.94 lemma cos_expansion_lemma:
2.95 @@ -499,10 +497,7 @@
2.96 lemma Maclaurin_cos_expansion:
2.97 "\<exists>t. abs t \<le> abs x &
2.98 cos x =
2.99 - (\<Sum>m=0..<n. (if even m
2.100 - then -1 ^ (m div 2)/(real (fact m))
2.101 - else 0) *
2.102 - x ^ m)
2.103 + (\<Sum>m=0..<n. cos_coeff m * x ^ m)
2.104 + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2.105 apply (cut_tac f = cos and n = n and x = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_all_lt_objl)
2.106 apply safe
2.107 @@ -515,17 +510,14 @@
2.108 apply (erule ssubst)
2.109 apply (rule_tac x = t in exI, simp)
2.110 apply (rule setsum_cong[OF refl])
2.111 -apply (auto simp add: cos_zero_iff even_mult_two_ex)
2.112 +apply (auto simp add: cos_coeff_def cos_zero_iff even_mult_two_ex)
2.113 done
2.114
2.115 lemma Maclaurin_cos_expansion2:
2.116 "[| 0 < x; n > 0 |] ==>
2.117 \<exists>t. 0 < t & t < x &
2.118 cos x =
2.119 - (\<Sum>m=0..<n. (if even m
2.120 - then -1 ^ (m div 2)/(real (fact m))
2.121 - else 0) *
2.122 - x ^ m)
2.123 + (\<Sum>m=0..<n. cos_coeff m * x ^ m)
2.124 + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2.125 apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_objl)
2.126 apply safe
2.127 @@ -534,17 +526,14 @@
2.128 apply (erule ssubst)
2.129 apply (rule_tac x = t in exI, simp)
2.130 apply (rule setsum_cong[OF refl])
2.131 -apply (auto simp add: cos_zero_iff even_mult_two_ex)
2.132 +apply (auto simp add: cos_coeff_def cos_zero_iff even_mult_two_ex)
2.133 done
2.134
2.135 lemma Maclaurin_minus_cos_expansion:
2.136 "[| x < 0; n > 0 |] ==>
2.137 \<exists>t. x < t & t < 0 &
2.138 cos x =
2.139 - (\<Sum>m=0..<n. (if even m
2.140 - then -1 ^ (m div 2)/(real (fact m))
2.141 - else 0) *
2.142 - x ^ m)
2.143 + (\<Sum>m=0..<n. cos_coeff m * x ^ m)
2.144 + ((cos(t + 1/2 * real (n) *pi) / real (fact n)) * x ^ n)"
2.145 apply (cut_tac f = cos and n = n and h = x and diff = "%n x. cos (x + 1/2*real (n) *pi)" in Maclaurin_minus_objl)
2.146 apply safe
2.147 @@ -553,7 +542,7 @@
2.148 apply (erule ssubst)
2.149 apply (rule_tac x = t in exI, simp)
2.150 apply (rule setsum_cong[OF refl])
2.151 -apply (auto simp add: cos_zero_iff even_mult_two_ex)
2.152 +apply (auto simp add: cos_coeff_def cos_zero_iff even_mult_two_ex)
2.153 done
2.154
2.155 (* ------------------------------------------------------------------------- *)
2.156 @@ -565,8 +554,8 @@
2.157 by auto
2.158
2.159 lemma Maclaurin_sin_bound:
2.160 - "abs(sin x - (\<Sum>m=0..<n. (if even m then 0 else (-1 ^ ((m - Suc 0) div 2)) / real (fact m)) *
2.161 - x ^ m)) \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
2.162 + "abs(sin x - (\<Sum>m=0..<n. sin_coeff m * x ^ m))
2.163 + \<le> inverse(real (fact n)) * \<bar>x\<bar> ^ n"
2.164 proof -
2.165 have "!! x (y::real). x \<le> 1 \<Longrightarrow> 0 \<le> y \<Longrightarrow> x * y \<le> 1 * y"
2.166 by (rule_tac mult_right_mono,simp_all)
2.167 @@ -592,6 +581,7 @@
2.168 apply (safe dest!: mod_eqD, simp_all)
2.169 done
2.170 show ?thesis
2.171 + unfolding sin_coeff_def
2.172 apply (subst t2)
2.173 apply (rule sin_bound_lemma)
2.174 apply (rule setsum_cong[OF refl])