1.1 --- a/src/HOL/Complex.thy	Fri Aug 19 18:08:05 2011 -0700
     1.2 +++ b/src/HOL/Complex.thy	Fri Aug 19 18:42:41 2011 -0700
     1.3 @@ -606,14 +606,6 @@
     1.4  abbreviation expi :: "complex \<Rightarrow> complex"
     1.5    where "expi \<equiv> exp"
     1.6  
     1.7 -lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
     1.8 -  unfolding cos_coeff_def sin_coeff_def
     1.9 -  by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)
    1.10 -
    1.11 -lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
    1.12 -  unfolding cos_coeff_def sin_coeff_def
    1.13 -  by (simp del: mult_Suc)
    1.14 -
    1.15  lemma expi_imaginary: "expi (Complex 0 b) = cis b"
    1.16  proof (rule complex_eqI)
    1.17    { fix n have "Complex 0 b ^ n =

     2.1 --- a/src/HOL/MacLaurin.thy	Fri Aug 19 18:08:05 2011 -0700
     2.2 +++ b/src/HOL/MacLaurin.thy	Fri Aug 19 18:42:41 2011 -0700
     2.3 @@ -417,9 +417,6 @@
     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 @@ -486,9 +483,6 @@
    2.14  
    2.15  subsection{*Maclaurin Expansion for Cosine Function*}
    2.16  
    2.17 -lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
    2.18 -  unfolding cos_coeff_def by simp (* TODO: move *)
    2.19 -
    2.20  lemma sumr_cos_zero_one [simp]:
    2.21    "(\<Sum>m=0..<(Suc n). cos_coeff m * 0 ^ m) = 1"
    2.22  by (induct "n", auto)

     3.1 --- a/src/HOL/Transcendental.thy	Fri Aug 19 18:08:05 2011 -0700
     3.2 +++ b/src/HOL/Transcendental.thy	Fri Aug 19 18:42:41 2011 -0700
     3.3 @@ -1220,6 +1220,20 @@
     3.4  definition cos :: "real \<Rightarrow> real" where
     3.5    "cos = (\<lambda>x. \<Sum>n. cos_coeff n * x ^ n)"
     3.6  
     3.7 +lemma sin_coeff_0 [simp]: "sin_coeff 0 = 0"
     3.8 +  unfolding sin_coeff_def by simp
     3.9 +
    3.10 +lemma cos_coeff_0 [simp]: "cos_coeff 0 = 1"
    3.11 +  unfolding cos_coeff_def by simp
    3.12 +
    3.13 +lemma sin_coeff_Suc: "sin_coeff (Suc n) = cos_coeff n / real (Suc n)"
    3.14 +  unfolding cos_coeff_def sin_coeff_def
    3.15 +  by (simp del: mult_Suc)
    3.16 +
    3.17 +lemma cos_coeff_Suc: "cos_coeff (Suc n) = - sin_coeff n / real (Suc n)"
    3.18 +  unfolding cos_coeff_def sin_coeff_def
    3.19 +  by (simp del: mult_Suc, auto simp add: odd_Suc_mult_two_ex)
    3.20 +
    3.21  lemma summable_sin: "summable (\<lambda>n. sin_coeff n * x ^ n)"
    3.22  unfolding sin_coeff_def
    3.23  apply (rule summable_comparison_test [OF _ summable_exp [where x="\<bar>x\<bar>"]])
    3.24 @@ -1238,42 +1252,27 @@
    3.25  lemma cos_converges: "(\<lambda>n. cos_coeff n * x ^ n) sums cos(x)"
    3.26  unfolding cos_def by (rule summable_cos [THEN summable_sums])
    3.27  
    3.28 -lemma sin_fdiffs: "diffs sin_coeff = cos_coeff"
    3.29 -unfolding sin_coeff_def cos_coeff_def
    3.30 -by (auto intro!: ext
    3.31 -         simp add: diffs_def divide_inverse real_of_nat_def of_nat_mult
    3.32 -         simp del: mult_Suc of_nat_Suc)
    3.33 -
    3.34 -lemma sin_fdiffs2: "diffs sin_coeff n = cos_coeff n"
    3.35 -by (simp only: sin_fdiffs)
    3.36 -
    3.37 -lemma cos_fdiffs: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
    3.38 -unfolding sin_coeff_def cos_coeff_def
    3.39 -by (auto intro!: ext
    3.40 -         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex real_of_nat_def of_nat_mult
    3.41 -         simp del: mult_Suc of_nat_Suc)
    3.42 -
    3.43 -lemma cos_fdiffs2: "diffs cos_coeff n = - sin_coeff n"
    3.44 -by (simp only: cos_fdiffs)
    3.45 +lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
    3.46 +  by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)
    3.47 +
    3.48 +lemma diffs_cos_coeff: "diffs cos_coeff = (\<lambda>n. - sin_coeff n)"
    3.49 +  by (simp add: diffs_def cos_coeff_Suc real_of_nat_def del: of_nat_Suc)
    3.50  
    3.51  text{*Now at last we can get the derivatives of exp, sin and cos*}
    3.52  
    3.53 -lemma lemma_sin_minus: "- sin x = (\<Sum>n. - (sin_coeff n * x ^ n))"
    3.54 -by (auto intro!: sums_unique sums_minus sin_converges)
    3.55 -
    3.56  lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
    3.57 -unfolding sin_def cos_def
    3.58 -apply (auto simp add: sin_fdiffs2 [symmetric])
    3.59 -apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
    3.60 -apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs)
    3.61 -done
    3.62 +  unfolding sin_def cos_def
    3.63 +  apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
    3.64 +  apply (simp_all add: diffs_sin_coeff diffs_cos_coeff
    3.65 +    summable_minus summable_sin summable_cos)
    3.66 +  done
    3.67  
    3.68  lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
    3.69 -unfolding cos_def
    3.70 -apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
    3.71 -apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
    3.72 -apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus)
    3.73 -done
    3.74 +  unfolding cos_def sin_def
    3.75 +  apply (rule DERIV_cong, rule termdiffs [where K="1 + \<bar>x\<bar>"])
    3.76 +  apply (simp_all add: diffs_sin_coeff diffs_cos_coeff diffs_minus
    3.77 +    summable_minus summable_sin summable_cos suminf_minus)
    3.78 +  done
    3.79  
    3.80  lemma isCont_sin: "isCont sin x"
    3.81    by (rule DERIV_sin [THEN DERIV_isCont])