author paulson Tue, 31 Mar 2015 15:00:03 +0100 changeset 59862 44b3f4fa33ca parent 59860 a979fc5db526 child 59863 30519ff3dffb
New material and binomial fix
 src/HOL/Binomial.thy file | annotate | diff | comparison | revisions src/HOL/Complex.thy file | annotate | diff | comparison | revisions src/HOL/Deriv.thy file | annotate | diff | comparison | revisions src/HOL/Library/Convex.thy file | annotate | diff | comparison | revisions src/HOL/Library/Formal_Power_Series.thy file | annotate | diff | comparison | revisions src/HOL/Multivariate_Analysis/Complex_Transcendental.thy file | annotate | diff | comparison | revisions src/HOL/Transcendental.thy file | annotate | diff | comparison | revisions
--- a/src/HOL/Binomial.thy	Tue Mar 31 00:21:07 2015 +0200
+++ b/src/HOL/Binomial.thy	Tue Mar 31 15:00:03 2015 +0100
@@ -514,8 +514,7 @@
unfolding pochhammer_eq_0_iff by auto

lemma pochhammer_minus:
-  assumes kn: "k \<le> n"
-  shows "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k"
+    "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k"
proof (cases k)
case 0
then show ?thesis by simp
@@ -531,16 +530,15 @@
qed

lemma pochhammer_minus':
-  assumes kn: "k \<le> n"
-  shows "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
-  unfolding pochhammer_minus[OF kn, where b=b]
+    "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
+  unfolding pochhammer_minus[where b=b]
unfolding mult.assoc[symmetric]
by simp

lemma pochhammer_same: "pochhammer (- of_nat n) n =
((- 1) ^ n :: 'a::{semiring_char_0,comm_ring_1,semiring_no_zero_divisors}) * (fact n)"
-  unfolding pochhammer_minus[OF le_refl[of n]]
+  unfolding pochhammer_minus

--- a/src/HOL/Complex.thy	Tue Mar 31 00:21:07 2015 +0200
+++ b/src/HOL/Complex.thy	Tue Mar 31 15:00:03 2015 +0100
@@ -93,10 +93,10 @@
lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x"

-lemma Re_power_real: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n "
+lemma Re_power_real [simp]: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n "
by (induct n) simp_all

-lemma Im_power_real: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0"
+lemma Im_power_real [simp]: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0"
by (induct n) simp_all

subsection {* Scalar Multiplication *}
@@ -823,6 +823,9 @@
lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>"

+lemma of_real_sqrt: "x \<ge> 0 \<Longrightarrow> of_real (sqrt x) = csqrt (of_real x)"
+  by (simp add: complex_eq_iff norm_complex_def)
+
lemma csqrt_0 [simp]: "csqrt 0 = 0"
by simp

--- a/src/HOL/Deriv.thy	Tue Mar 31 00:21:07 2015 +0200
+++ b/src/HOL/Deriv.thy	Tue Mar 31 15:00:03 2015 +0100
@@ -561,6 +561,10 @@
\<Longrightarrow> (f has_field_derivative f') (at x within t)"

+lemma has_field_derivative_at_within:
+    "(f has_field_derivative f') (at x) \<Longrightarrow> (f has_field_derivative f') (at x within s)"
+  using DERIV_subset by blast
+
abbreviation (input)
DERIV :: "('a::real_normed_field \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"
("(DERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
--- a/src/HOL/Library/Convex.thy	Tue Mar 31 00:21:07 2015 +0200
+++ b/src/HOL/Library/Convex.thy	Tue Mar 31 15:00:03 2015 +0100
@@ -121,6 +121,9 @@
then show "convex {a<..<b}" by (simp only: convex_Int 3 4)
qed

+lemma convex_Reals: "convex Reals"
+  by (simp add: convex_def scaleR_conv_of_real)
+
subsection {* Explicit expressions for convexity in terms of arbitrary sums. *}

lemma convex_setsum:
--- a/src/HOL/Library/Formal_Power_Series.thy	Tue Mar 31 00:21:07 2015 +0200
+++ b/src/HOL/Library/Formal_Power_Series.thy	Tue Mar 31 15:00:03 2015 +0100
@@ -2555,9 +2555,7 @@
let ?KM = "{(k,m). k + m \<le> n}"
let ?f = "\<lambda>s. UNION {(0::nat)..s} (\<lambda>i. {(i,s - i)})"
have th0: "?KM = UNION {0..n} ?f"
-    apply presburger (* FIXME: slow! *)
-    done
+    by (auto simp add: set_eq_iff Bex_def)
show "?l = ?r "
unfolding th0
apply (subst setsum.UNION_disjoint)
@@ -3230,11 +3228,11 @@
then obtain m where m: "n = Suc m" by (cases n) auto
from k0 obtain h where h: "k = Suc h" by (cases k) auto
{
-        assume kn: "k = n"
+        assume "k = n"
then have "b gchoose (n - k) =
(?m1 n * ?p b n * ?m1 k * ?p (of_nat n) k) / (?f n * pochhammer (b - of_nat n + 1) k)"
-          using kn pochhammer_minus'[where k=k and n=n and b=b]
+          using pochhammer_minus'[where k=k and b=b]
using bn0
done
@@ -3357,10 +3355,10 @@
apply (auto simp add: of_nat_diff algebra_simps)
done
have th0: "pochhammer (- (?a + ?b)) n = (- 1)^n * pochhammer (c - a) n"
-    unfolding pochhammer_minus[OF le_refl]
+    unfolding pochhammer_minus
have th1: "pochhammer (- ?b) n = (- 1)^n * pochhammer c n"
-    unfolding pochhammer_minus[OF le_refl]
+    unfolding pochhammer_minus
by simp
have nz: "pochhammer c n \<noteq> 0" using c
@@ -3794,8 +3792,9 @@
with LIMSEQ_inverse_realpow_zero[of 2, simplified, simplified filterlim_iff,
THEN spec, of "\<lambda>x. x < e"]
have "eventually (\<lambda>i. inverse (2 ^ i) < e) sequentially"
+        unfolding eventually_nhds
apply safe
-        apply (auto simp: eventually_nhds) --{*slow*}
+        apply auto --{*slow*}
done
then obtain i where "inverse (2 ^ i) < e" by (auto simp: eventually_sequentially)
have "eventually (\<lambda>x. M i \<le> x) sequentially" by (auto simp: eventually_sequentially)
--- a/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Tue Mar 31 00:21:07 2015 +0200
+++ b/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Tue Mar 31 15:00:03 2015 +0100
@@ -42,7 +42,7 @@

theorem exp_Euler: "exp(ii * z) = cos(z) + ii * sin(z)"
proof -
-  have "(\<lambda>n. (cos_coeff n + ii * sin_coeff n) * z^n)
+  have "(\<lambda>n. (cos_coeff n + ii * sin_coeff n) * z^n)
= (\<lambda>n. (ii * z) ^ n /\<^sub>R (fact n))"
proof
fix n
@@ -74,13 +74,11 @@

subsection{*Relationships between real and complex trig functions*}

-declare sin_of_real [simp] cos_of_real [simp]
-
lemma real_sin_eq [simp]:
fixes x::real
shows "Re(sin(of_real x)) = sin x"
-
+
lemma real_cos_eq [simp]:
fixes x::real
shows "Re(cos(of_real x)) = cos x"
@@ -100,10 +98,10 @@
by (rule exp_converges)
finally have "(\<lambda>n. cnj (z ^ n /\<^sub>R (fact n))) sums (exp (cnj z))" .
moreover have "(\<lambda>n. cnj (z ^ n /\<^sub>R (fact n))) sums (cnj (exp z))"
-    by (metis exp_converges sums_cnj)
+    by (metis exp_converges sums_cnj)
ultimately show ?thesis
using sums_unique2
-    by blast
+    by blast
qed

lemma cnj_sin: "cnj(sin z) = sin(cnj z)"
@@ -112,15 +110,6 @@
lemma cnj_cos: "cnj(cos z) = cos(cnj z)"
by (simp add: cos_exp_eq exp_cnj field_simps)

-lemma exp_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> exp z \<in> \<real>"
-  by (metis Reals_cases Reals_of_real exp_of_real)
-
-lemma sin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> sin z \<in> \<real>"
-  by (metis Reals_cases Reals_of_real sin_of_real)
-
-lemma cos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cos z \<in> \<real>"
-  by (metis Reals_cases Reals_of_real cos_of_real)
-
lemma complex_differentiable_at_sin: "sin complex_differentiable at z"
using DERIV_sin complex_differentiable_def by blast

@@ -141,7 +130,7 @@

subsection{* Get a nice real/imaginary separation in Euler's formula.*}

-lemma Euler: "exp(z) = of_real(exp(Re z)) *
+lemma Euler: "exp(z) = of_real(exp(Re z)) *
(of_real(cos(Im z)) + ii * of_real(sin(Im z)))"

@@ -156,11 +145,20 @@

lemma Im_cos: "Im(cos z) = sin(Re z) * (exp(-(Im z)) - exp(Im z)) / 2"
by (simp add: cos_exp_eq field_simps Im_divide Im_exp)
-
+
+lemma Re_sin_pos: "0 < Re z \<Longrightarrow> Re z < pi \<Longrightarrow> Re (sin z) > 0"
+  by (auto simp: Re_sin Im_sin add_pos_pos sin_gt_zero)
+
+lemma Im_sin_nonneg: "Re z = 0 \<Longrightarrow> 0 \<le> Im z \<Longrightarrow> 0 \<le> Im (sin z)"
+  by (simp add: Re_sin Im_sin algebra_simps)
+
+lemma Im_sin_nonneg2: "Re z = pi \<Longrightarrow> Im z \<le> 0 \<Longrightarrow> 0 \<le> Im (sin z)"
+  by (simp add: Re_sin Im_sin algebra_simps)
+
subsection{*More on the Polar Representation of Complex Numbers*}

lemma exp_Complex: "exp(Complex r t) = of_real(exp r) * Complex (cos t) (sin t)"

lemma exp_eq_1: "exp z = 1 \<longleftrightarrow> Re(z) = 0 \<and> (\<exists>n::int. Im(z) = of_int (2 * n) * pi)"
apply auto
@@ -183,7 +181,7 @@
lemma exp_complex_eqI: "abs(Im w - Im z) < 2*pi \<Longrightarrow> exp w = exp z \<Longrightarrow> w = z"
by (auto simp: exp_eq abs_mult)

-lemma exp_integer_2pi:
+lemma exp_integer_2pi:
assumes "n \<in> Ints"
shows "exp((2 * n * pi) * ii) = 1"
proof -
@@ -208,10 +206,10 @@
done
qed

-lemma exp_i_ne_1:
+lemma exp_i_ne_1:
assumes "0 < x" "x < 2*pi"
shows "exp(\<i> * of_real x) \<noteq> 1"
-proof
+proof
assume "exp (\<i> * of_real x) = 1"
then have "exp (\<i> * of_real x) = exp 0"
by simp
@@ -225,18 +223,18 @@
by (cases n) (auto simp: zero_less_mult_iff mult_less_0_iff)
qed

-lemma sin_eq_0:
+lemma sin_eq_0:
fixes z::complex
shows "sin z = 0 \<longleftrightarrow> (\<exists>n::int. z = of_real(n * pi))"
by (simp add: sin_exp_eq exp_eq of_real_numeral)

-lemma cos_eq_0:
+lemma cos_eq_0:
fixes z::complex
shows "cos z = 0 \<longleftrightarrow> (\<exists>n::int. z = of_real(n * pi) + of_real pi/2)"
using sin_eq_0 [of "z - of_real pi/2"]

-lemma cos_eq_1:
+lemma cos_eq_1:
fixes z::complex
shows "cos z = 1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi))"
proof -
@@ -248,7 +246,7 @@
by simp
show ?thesis
by (auto simp: sin_eq_0 of_real_numeral)
-qed
+qed

lemma csin_eq_1:
fixes z::complex
@@ -275,15 +273,15 @@
done
finally show ?thesis .
-qed
+qed

-lemma ccos_eq_minus1:
+lemma ccos_eq_minus1:
fixes z::complex
shows "cos z = -1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi) + pi)"
using csin_eq_1 [of "z - of_real pi/2"]
apply (simp add: algebra_simps of_real_numeral equation_minus_iff)
-  done
+  done

lemma sin_eq_1: "sin x = 1 \<longleftrightarrow> (\<exists>n::int. x = (2 * n + 1 / 2) * pi)"
(is "_ = ?rhs")
@@ -301,7 +299,7 @@
also have "... = ?rhs"
by (auto simp: algebra_simps)
finally show ?thesis .
-qed
+qed

lemma sin_eq_minus1: "sin x = -1 \<longleftrightarrow> (\<exists>n::int. x = (2*n + 3/2) * pi)"  (is "_ = ?rhs")
proof -
@@ -317,7 +315,7 @@
also have "... = ?rhs"
by (auto simp: algebra_simps)
finally show ?thesis .
-qed
+qed

lemma cos_eq_minus1: "cos x = -1 \<longleftrightarrow> (\<exists>n::int. x = (2*n + 1) * pi)"
(is "_ = ?rhs")
@@ -334,10 +332,10 @@
also have "... = ?rhs"
by (auto simp: algebra_simps)
finally show ?thesis .
-qed
+qed

lemma dist_exp_ii_1: "norm(exp(ii * of_real t) - 1) = 2 * abs(sin(t / 2))"
-  apply (simp add: exp_Euler cmod_def power2_diff algebra_simps)
+  apply (simp add: exp_Euler cmod_def power2_diff sin_of_real cos_of_real algebra_simps)
using cos_double_sin [of "t/2"]
done
@@ -369,7 +367,7 @@

lemmas cos_ii_times = cosh_complex [symmetric]

-lemma norm_cos_squared:
+lemma norm_cos_squared:
"norm(cos z) ^ 2 = cos(Re z) ^ 2 + (exp(Im z) - inverse(exp(Im z))) ^ 2 / 4"
apply (cases z)
@@ -387,12 +385,12 @@
apply (simp only: left_diff_distrib [symmetric] power_mult_distrib)
apply (simp add: power2_eq_square algebra_simps divide_simps)
-  done
+  done

lemma exp_uminus_Im: "exp (- Im z) \<le> exp (cmod z)"
using abs_Im_le_cmod linear order_trans by fastforce

-lemma norm_cos_le:
+lemma norm_cos_le:
fixes z::complex
shows "norm(cos z) \<le> exp(norm z)"
proof -
@@ -405,7 +403,7 @@
done
qed

-lemma norm_cos_plus1_le:
+lemma norm_cos_plus1_le:
fixes z::complex
shows "norm(1 + cos z) \<le> 2 * exp(norm z)"
proof -
@@ -431,12 +429,12 @@

subsection{* Taylor series for complex exponential, sine and cosine.*}

-context
+context
begin

declare power_Suc [simp del]

-lemma Taylor_exp:
+lemma Taylor_exp:
"norm(exp z - (\<Sum>k\<le>n. z ^ k / (fact k))) \<le> exp\<bar>Re z\<bar> * (norm z) ^ (Suc n) / (fact n)"
proof (rule complex_taylor [of _ n "\<lambda>k. exp" "exp\<bar>Re z\<bar>" 0 z, simplified])
show "convex (closed_segment 0 z)"
@@ -459,11 +457,11 @@
show "z \<in> closed_segment 0 z"
using of_real_1 zero_le_one by blast
-qed
+qed

-lemma
+lemma
assumes "0 \<le> u" "u \<le> 1"
-  shows cmod_sin_le_exp: "cmod (sin (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>"
+  shows cmod_sin_le_exp: "cmod (sin (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>"
and cmod_cos_le_exp: "cmod (cos (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>"
proof -
have mono: "\<And>u w z::real. w \<le> u \<Longrightarrow> z \<le> u \<Longrightarrow> w + z \<le> u*2"
@@ -485,16 +483,16 @@
apply (meson less_eq_real_def mult_nonneg_nonpos neg_0_le_iff_le order_trans)
done
qed
-
-lemma Taylor_sin:
-  "norm(sin z - (\<Sum>k\<le>n. complex_of_real (sin_coeff k) * z ^ k))
+
+lemma Taylor_sin:
+  "norm(sin z - (\<Sum>k\<le>n. complex_of_real (sin_coeff k) * z ^ k))
\<le> exp\<bar>Im z\<bar> * (norm z) ^ (Suc n) / (fact n)"
proof -
have mono: "\<And>u w z::real. w \<le> u \<Longrightarrow> z \<le> u \<Longrightarrow> w + z \<le> u*2"
by arith
have *: "cmod (sin z -
(\<Sum>i\<le>n. (-1) ^ (i div 2) * (if even i then sin 0 else cos 0) * z ^ i / (fact i)))
-           \<le> exp \<bar>Im z\<bar> * cmod z ^ Suc n / (fact n)"
+           \<le> exp \<bar>Im z\<bar> * cmod z ^ Suc n / (fact n)"
proof (rule complex_taylor [of "closed_segment 0 z" n "\<lambda>k x. (-1)^(k div 2) * (if even k then sin x else cos x)" "exp\<bar>Im z\<bar>" 0 z,
simplified])
show "convex (closed_segment 0 z)"
@@ -519,7 +517,7 @@
show "z \<in> closed_segment 0 z"
using of_real_1 zero_le_one by blast
-  qed
+  qed
have **: "\<And>k. complex_of_real (sin_coeff k) * z ^ k
= (-1)^(k div 2) * (if even k then sin 0 else cos 0) * z^k / of_nat (fact k)"
by (auto simp: sin_coeff_def elim!: oddE)
@@ -529,15 +527,15 @@
done
qed

-lemma Taylor_cos:
-  "norm(cos z - (\<Sum>k\<le>n. complex_of_real (cos_coeff k) * z ^ k))
+lemma Taylor_cos:
+  "norm(cos z - (\<Sum>k\<le>n. complex_of_real (cos_coeff k) * z ^ k))
\<le> exp\<bar>Im z\<bar> * (norm z) ^ Suc n / (fact n)"
proof -
have mono: "\<And>u w z::real. w \<le> u \<Longrightarrow> z \<le> u \<Longrightarrow> w + z \<le> u*2"
by arith
have *: "cmod (cos z -
(\<Sum>i\<le>n. (-1) ^ (Suc i div 2) * (if even i then cos 0 else sin 0) * z ^ i / (fact i)))
-           \<le> exp \<bar>Im z\<bar> * cmod z ^ Suc n / (fact n)"
+           \<le> exp \<bar>Im z\<bar> * cmod z ^ Suc n / (fact n)"
proof (rule complex_taylor [of "closed_segment 0 z" n "\<lambda>k x. (-1)^(Suc k div 2) * (if even k then cos x else sin x)" "exp\<bar>Im z\<bar>" 0 z,
simplified])
show "convex (closed_segment 0 z)"
@@ -563,7 +561,7 @@
show "z \<in> closed_segment 0 z"
using of_real_1 zero_le_one by blast
-  qed
+  qed
have **: "\<And>k. complex_of_real (cos_coeff k) * z ^ k
= (-1)^(Suc k div 2) * (if even k then cos 0 else sin 0) * z^k / of_nat (fact k)"
by (auto simp: cos_coeff_def elim!: evenE)
@@ -910,6 +908,21 @@
apply (auto simp: exp_of_nat_mult [symmetric])
done

+subsection{*The Unwinding Number and the Ln-product Formula*}
+
+text{*Note that in this special case the unwinding number is -1, 0 or 1.*}
+
+definition unwinding :: "complex \<Rightarrow> complex" where
+   "unwinding(z) = (z - Ln(exp z)) / (of_real(2*pi) * ii)"
+
+lemma unwinding_2pi: "(2*pi) * ii * unwinding(z) = z - Ln(exp z)"
+
+lemma Ln_times_unwinding:
+    "w \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> Ln(w * z) = Ln(w) + Ln(z) - (2*pi) * ii * unwinding(Ln w + Ln z)"
+
+
subsection{*Derivative of Ln away from the branch cut*}

lemma
@@ -944,6 +957,10 @@
lemma continuous_at_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z) Ln"

+lemma isCont_Ln' [simp]:
+   "\<lbrakk>isCont f z; Im(f z) = 0 \<Longrightarrow> 0 < Re(f z)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. Ln (f x)) z"
+  by (blast intro: isCont_o2 [OF _ continuous_at_Ln])
+
lemma continuous_within_Ln: "(Im(z) = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z within s) Ln"
using continuous_at_Ln continuous_at_imp_continuous_within by blast

@@ -956,7 +973,7 @@

subsection{*Relation to Real Logarithm*}

-lemma ln_of_real:
+lemma Ln_of_real:
assumes "0 < z"
shows "Ln(of_real z) = of_real(ln z)"
proof -
@@ -969,6 +986,9 @@
using assms by simp
qed

+corollary Ln_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> Re z > 0 \<Longrightarrow> Ln z \<in> \<real>"
+  by (auto simp: Ln_of_real elim: Reals_cases)
+

@@ -1091,7 +1111,7 @@
lemma Ln_1 [simp]: "Ln(1) = 0"
proof -
have "Ln (exp 0) = 0"
-    by (metis exp_zero ln_exp ln_of_real of_real_0 of_real_1 zero_less_one)
+    by (metis exp_zero ln_exp Ln_of_real of_real_0 of_real_1 zero_less_one)
then show ?thesis
by simp
qed
@@ -1262,6 +1282,10 @@
"(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z) csqrt"

+corollary isCont_csqrt' [simp]:
+   "\<lbrakk>isCont f z; Im(f z) = 0 \<Longrightarrow> 0 < Re(f z)\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. csqrt (f x)) z"
+  by (blast intro: isCont_o2 [OF _ continuous_at_csqrt])
+
lemma continuous_within_csqrt:
"(Im z = 0 \<Longrightarrow> 0 < Re(z)) \<Longrightarrow> continuous (at z within s) csqrt"
--- a/src/HOL/Transcendental.thy	Tue Mar 31 00:21:07 2015 +0200
+++ b/src/HOL/Transcendental.thy	Tue Mar 31 15:00:03 2015 +0100
@@ -1130,6 +1130,9 @@
done

+corollary exp_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> exp z \<in> \<real>"
+  by (metis Reals_cases Reals_of_real exp_of_real)
+
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
proof
have "exp x * exp (- x) = 1" by (simp add: exp_add_commuting[symmetric])
@@ -2432,6 +2435,9 @@
by blast
qed

+corollary sin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> sin z \<in> \<real>"
+  by (metis Reals_cases Reals_of_real sin_of_real)
+
lemma cos_of_real:
fixes x::real
shows "cos (of_real x) = of_real (cos x)"
@@ -2450,6 +2456,9 @@
by blast
qed

+corollary cos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> cos z \<in> \<real>"
+  by (metis Reals_cases Reals_of_real cos_of_real)
+
lemma diffs_sin_coeff: "diffs sin_coeff = cos_coeff"
by (simp add: diffs_def sin_coeff_Suc real_of_nat_def del: of_nat_Suc)

@@ -3661,6 +3670,16 @@
definition tan :: "'a \<Rightarrow> 'a::{real_normed_field,banach}"
where "tan = (\<lambda>x. sin x / cos x)"

+lemma tan_of_real:
+  fixes XXX :: "'a::{real_normed_field,banach,field_inverse_zero}"
+  shows  "of_real(tan x) = (tan(of_real x) :: 'a)"
+  by (simp add: tan_def sin_of_real cos_of_real)
+
+lemma tan_in_Reals [simp]:
+  fixes z :: "'a::{real_normed_field,banach,field_inverse_zero}"
+  shows "z \<in> \<real> \<Longrightarrow> tan z \<in> \<real>"