src/HOL/Analysis/Complex_Transcendental.thy
 author hoelzl Mon, 08 Aug 2016 14:13:14 +0200 changeset 63627 6ddb43c6b711 parent 63594 src/HOL/Multivariate_Analysis/Complex_Transcendental.thy@bd218a9320b5 child 63721 492bb53c3420 permissions -rw-r--r--
rename HOL-Multivariate_Analysis to HOL-Analysis.
```
section \<open>Complex Transcendental Functions\<close>

text\<open>By John Harrison et al.  Ported from HOL Light by L C Paulson (2015)\<close>

theory Complex_Transcendental
imports
Complex_Analysis_Basics
Summation_Tests
begin

(* TODO: Figure out what to do with MÃ¶bius transformations *)
definition "moebius a b c d = (\<lambda>z. (a*z+b) / (c*z+d :: 'a :: field))"

lemma moebius_inverse:
assumes "a * d \<noteq> b * c" "c * z + d \<noteq> 0"
shows   "moebius d (-b) (-c) a (moebius a b c d z) = z"
proof -
from assms have "(-c) * moebius a b c d z + a \<noteq> 0" unfolding moebius_def
with assms show ?thesis
qed

lemma moebius_inverse':
assumes "a * d \<noteq> b * c" "c * z - a \<noteq> 0"
shows   "moebius a b c d (moebius d (-b) (-c) a z) = z"
using assms moebius_inverse[of d a "-b" "-c" z]
by (auto simp: algebra_simps)

assumes "Im z \<noteq> 0" "r\<noteq>0"
shows "cmod (z + r) < cmod z + \<bar>r\<bar>"
proof (cases z)
case (Complex x y)
have "r * x / \<bar>r\<bar> < sqrt (x*x + y*y)"
apply (rule real_less_rsqrt)
using assms
using not_real_square_gt_zero by blast
then show ?thesis using assms Complex
apply (auto simp: cmod_def)
apply (rule power2_less_imp_less, auto)
done
qed

lemma cmod_diff_real_less: "Im z \<noteq> 0 \<Longrightarrow> x\<noteq>0 \<Longrightarrow> cmod (z - x) < cmod z + \<bar>x\<bar>"
by simp

lemma cmod_square_less_1_plus:
assumes "Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1"
shows "(cmod z)\<^sup>2 < 1 + cmod (1 - z\<^sup>2)"
using assms
apply (cases "Im z = 0 \<or> Re z = 0")
using abs_square_less_1
apply (force simp add: Re_power2 Im_power2 cmod_def)
using cmod_diff_real_less [of "1 - z\<^sup>2" "1"]
done

subsection\<open>The Exponential Function is Differentiable and Continuous\<close>

lemma field_differentiable_within_exp: "exp field_differentiable (at z within s)"
using DERIV_exp field_differentiable_at_within field_differentiable_def by blast

lemma continuous_within_exp:
fixes z::"'a::{real_normed_field,banach}"
shows "continuous (at z within s) exp"

lemma holomorphic_on_exp [holomorphic_intros]: "exp holomorphic_on s"

subsection\<open>Euler and de Moivre formulas.\<close>

text\<open>The sine series times @{term i}\<close>
lemma sin_ii_eq: "(\<lambda>n. (\<i> * sin_coeff n) * z^n) sums (\<i> * sin z)"
proof -
have "(\<lambda>n. \<i> * sin_coeff n *\<^sub>R z^n) sums (\<i> * sin z)"
using sin_converges sums_mult by blast
then show ?thesis
qed

theorem exp_Euler: "exp(\<i> * z) = cos(z) + \<i> * sin(z)"
proof -
have "(\<lambda>n. (cos_coeff n + \<i> * sin_coeff n) * z^n)
= (\<lambda>n. (\<i> * z) ^ n /\<^sub>R (fact n))"
proof
fix n
show "(cos_coeff n + \<i> * sin_coeff n) * z^n = (\<i> * z) ^ n /\<^sub>R (fact n)"
by (auto simp: cos_coeff_def sin_coeff_def scaleR_conv_of_real field_simps elim!: evenE oddE)
qed
also have "... sums (exp (\<i> * z))"
by (rule exp_converges)
finally have "(\<lambda>n. (cos_coeff n + \<i> * sin_coeff n) * z^n) sums (exp (\<i> * z))" .
moreover have "(\<lambda>n. (cos_coeff n + \<i> * sin_coeff n) * z^n) sums (cos z + \<i> * sin z)"
using sums_add [OF cos_converges [of z] sin_ii_eq [of z]]
ultimately show ?thesis
using sums_unique2 by blast
qed

corollary exp_minus_Euler: "exp(-(\<i> * z)) = cos(z) - \<i> * sin(z)"
using exp_Euler [of "-z"]
by simp

lemma sin_exp_eq: "sin z = (exp(\<i> * z) - exp(-(\<i> * z))) / (2*\<i>)"

lemma sin_exp_eq': "sin z = \<i> * (exp(-(\<i> * z)) - exp(\<i> * z)) / 2"

lemma cos_exp_eq:  "cos z = (exp(\<i> * z) + exp(-(\<i> * z))) / 2"

subsection\<open>Relationships between real and complex trig functions\<close>

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"

lemma DeMoivre: "(cos z + \<i> * sin z) ^ n = cos(n * z) + \<i> * sin(n * z)"
by (metis exp_of_nat_mult mult.left_commute)

lemma exp_cnj:
fixes z::complex
shows "cnj (exp z) = exp (cnj z)"
proof -
have "(\<lambda>n. cnj (z ^ n /\<^sub>R (fact n))) = (\<lambda>n. (cnj z)^n /\<^sub>R (fact n))"
by auto
also have "... sums (exp (cnj z))"
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)
ultimately show ?thesis
using sums_unique2
by blast
qed

lemma cnj_sin: "cnj(sin z) = sin(cnj z)"
by (simp add: sin_exp_eq exp_cnj field_simps)

lemma cnj_cos: "cnj(cos z) = cos(cnj z)"
by (simp add: cos_exp_eq exp_cnj field_simps)

lemma field_differentiable_at_sin: "sin field_differentiable at z"
using DERIV_sin field_differentiable_def by blast

lemma field_differentiable_within_sin: "sin field_differentiable (at z within s)"

lemma field_differentiable_at_cos: "cos field_differentiable at z"
using DERIV_cos field_differentiable_def by blast

lemma field_differentiable_within_cos: "cos field_differentiable (at z within s)"

lemma holomorphic_on_sin: "sin holomorphic_on s"

lemma holomorphic_on_cos: "cos holomorphic_on s"

subsection\<open>Get a nice real/imaginary separation in Euler's formula.\<close>

lemma Euler: "exp(z) = of_real(exp(Re z)) *
(of_real(cos(Im z)) + \<i> * of_real(sin(Im z)))"

lemma Re_sin: "Re(sin z) = sin(Re z) * (exp(Im z) + exp(-(Im z))) / 2"
by (simp add: sin_exp_eq field_simps Re_divide Im_exp)

lemma Im_sin: "Im(sin z) = cos(Re z) * (exp(Im z) - exp(-(Im z))) / 2"
by (simp add: sin_exp_eq field_simps Im_divide Re_exp)

lemma Re_cos: "Re(cos z) = cos(Re z) * (exp(Im z) + exp(-(Im z))) / 2"
by (simp add: cos_exp_eq field_simps Re_divide Re_exp)

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\<open>More on the Polar Representation of Complex Numbers\<close>

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
apply (metis exp_eq_one_iff norm_exp_eq_Re norm_one)
apply (metis Re_exp cos_one_2pi_int mult.commute mult.left_neutral norm_exp_eq_Re norm_one one_complex.simps(1))
by (metis Im_exp Re_exp complex_Re_Im_cancel_iff cos_one_2pi_int sin_double Re_complex_of_real complex_Re_numeral exp_zero mult.assoc mult.left_commute mult_eq_0_iff mult_numeral_1 numeral_One of_real_0 sin_zero_iff_int2)

lemma exp_eq: "exp w = exp z \<longleftrightarrow> (\<exists>n::int. w = z + (of_int (2 * n) * pi) * \<i>)"
(is "?lhs = ?rhs")
proof -
have "exp w = exp z \<longleftrightarrow> exp (w-z) = 1"
also have "... \<longleftrightarrow> (Re w = Re z \<and> (\<exists>n::int. Im w - Im z = of_int (2 * n) * pi))"
also have "... \<longleftrightarrow> ?rhs"
by (auto simp: algebra_simps intro!: complex_eqI)
finally show ?thesis .
qed

lemma exp_complex_eqI: "\<bar>Im w - Im z\<bar> < 2*pi \<Longrightarrow> exp w = exp z \<Longrightarrow> w = z"
by (auto simp: exp_eq abs_mult)

lemma exp_integer_2pi:
assumes "n \<in> \<int>"
shows "exp((2 * n * pi) * \<i>) = 1"
proof -
have "exp((2 * n * pi) * \<i>) = exp 0"
using assms
by (simp only: Ints_def exp_eq) auto
also have "... = 1"
by simp
finally show ?thesis .
qed

lemma sin_cos_eq_iff: "sin y = sin x \<and> cos y = cos x \<longleftrightarrow> (\<exists>n::int. y = x + 2 * n * pi)"
proof -
{ assume "sin y = sin x" "cos y = cos x"
then have "cos (y-x) = 1"
using cos_add [of y "-x"] by simp
then have "\<exists>n::int. y-x = n * 2 * pi"
using cos_one_2pi_int by blast }
then show ?thesis
done
qed

lemma exp_i_ne_1:
assumes "0 < x" "x < 2*pi"
shows "exp(\<i> * of_real x) \<noteq> 1"
proof
assume "exp (\<i> * of_real x) = 1"
then have "exp (\<i> * of_real x) = exp 0"
by simp
then obtain n where "\<i> * of_real x = (of_int (2 * n) * pi) * \<i>"
by (simp only: Ints_def exp_eq) auto
then have  "of_real x = (of_int (2 * n) * pi)"
by (metis complex_i_not_zero mult.commute mult_cancel_left of_real_eq_iff real_scaleR_def scaleR_conv_of_real)
then have  "x = (of_int (2 * n) * pi)"
by simp
then show False using assms
by (cases n) (auto simp: zero_less_mult_iff mult_less_0_iff)
qed

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:
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:
fixes z::complex
shows "cos z = 1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi))"
proof -
have "cos z = cos (2*(z/2))"
by simp
also have "... = 1 - 2 * sin (z/2) ^ 2"
by (simp only: cos_double_sin)
finally have [simp]: "cos z = 1 \<longleftrightarrow> sin (z/2) = 0"
by simp
show ?thesis
by (auto simp: sin_eq_0 of_real_numeral)
qed

lemma csin_eq_1:
fixes z::complex
shows "sin z = 1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi) + of_real pi/2)"
using cos_eq_1 [of "z - of_real pi/2"]

lemma csin_eq_minus1:
fixes z::complex
shows "sin z = -1 \<longleftrightarrow> (\<exists>n::int. z = of_real(2 * n * pi) + 3/2*pi)"
(is "_ = ?rhs")
proof -
have "sin z = -1 \<longleftrightarrow> sin (-z) = 1"
also have "...  \<longleftrightarrow> (\<exists>n::int. -z = of_real(2 * n * pi) + of_real pi/2)"
by (simp only: csin_eq_1)
also have "...  \<longleftrightarrow> (\<exists>n::int. z = - of_real(2 * n * pi) - of_real pi/2)"
apply (rule iff_exI)
by (metis (no_types)  is_num_normalize(8) minus_minus of_real_def real_vector.scale_minus_left uminus_add_conv_diff)
also have "... = ?rhs"
apply (auto simp: of_real_numeral)
apply (rule_tac [2] x="-(x+1)" in exI)
apply (rule_tac x="-(x+1)" in exI)
done
finally show ?thesis .
qed

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

lemma sin_eq_1: "sin x = 1 \<longleftrightarrow> (\<exists>n::int. x = (2 * n + 1 / 2) * pi)"
(is "_ = ?rhs")
proof -
have "sin x = 1 \<longleftrightarrow> sin (complex_of_real x) = 1"
by (metis of_real_1 one_complex.simps(1) real_sin_eq sin_of_real)
also have "...  \<longleftrightarrow> (\<exists>n::int. complex_of_real x = of_real(2 * n * pi) + of_real pi/2)"
by (simp only: csin_eq_1)
also have "...  \<longleftrightarrow> (\<exists>n::int. x = of_real(2 * n * pi) + of_real pi/2)"
apply (rule iff_exI)
apply (auto simp: algebra_simps of_real_numeral)
apply (rule injD [OF inj_of_real [where 'a = complex]])
apply (auto simp: of_real_numeral)
done
also have "... = ?rhs"
by (auto simp: algebra_simps)
finally show ?thesis .
qed

lemma sin_eq_minus1: "sin x = -1 \<longleftrightarrow> (\<exists>n::int. x = (2*n + 3/2) * pi)"  (is "_ = ?rhs")
proof -
have "sin x = -1 \<longleftrightarrow> sin (complex_of_real x) = -1"
by (metis Re_complex_of_real of_real_def scaleR_minus1_left sin_of_real)
also have "...  \<longleftrightarrow> (\<exists>n::int. complex_of_real x = of_real(2 * n * pi) + 3/2*pi)"
by (simp only: csin_eq_minus1)
also have "...  \<longleftrightarrow> (\<exists>n::int. x = of_real(2 * n * pi) + 3/2*pi)"
apply (rule iff_exI)
apply (auto simp: algebra_simps)
apply (rule injD [OF inj_of_real [where 'a = complex]], auto)
done
also have "... = ?rhs"
by (auto simp: algebra_simps)
finally show ?thesis .
qed

lemma cos_eq_minus1: "cos x = -1 \<longleftrightarrow> (\<exists>n::int. x = (2*n + 1) * pi)"
(is "_ = ?rhs")
proof -
have "cos x = -1 \<longleftrightarrow> cos (complex_of_real x) = -1"
by (metis Re_complex_of_real of_real_def scaleR_minus1_left cos_of_real)
also have "...  \<longleftrightarrow> (\<exists>n::int. complex_of_real x = of_real(2 * n * pi) + pi)"
by (simp only: ccos_eq_minus1)
also have "...  \<longleftrightarrow> (\<exists>n::int. x = of_real(2 * n * pi) + pi)"
apply (rule iff_exI)
apply (auto simp: algebra_simps)
apply (rule injD [OF inj_of_real [where 'a = complex]], auto)
done
also have "... = ?rhs"
by (auto simp: algebra_simps)
finally show ?thesis .
qed

lemma dist_exp_ii_1: "norm(exp(\<i> * of_real t) - 1) = 2 * \<bar>sin(t / 2)\<bar>"
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

lemma sinh_complex:
fixes z :: complex
shows "(exp z - inverse (exp z)) / 2 = -\<i> * sin(\<i> * z)"
by (simp add: sin_exp_eq divide_simps exp_minus of_real_numeral)

lemma sin_ii_times:
fixes z :: complex
shows "sin(\<i> * z) = \<i> * ((exp z - inverse (exp z)) / 2)"
using sinh_complex by auto

lemma sinh_real:
fixes x :: real
shows "of_real((exp x - inverse (exp x)) / 2) = -\<i> * sin(\<i> * of_real x)"
by (simp add: exp_of_real sin_ii_times of_real_numeral)

lemma cosh_complex:
fixes z :: complex
shows "(exp z + inverse (exp z)) / 2 = cos(\<i> * z)"
by (simp add: cos_exp_eq divide_simps exp_minus of_real_numeral exp_of_real)

lemma cosh_real:
fixes x :: real
shows "of_real((exp x + inverse (exp x)) / 2) = cos(\<i> * of_real x)"
by (simp add: cos_exp_eq divide_simps exp_minus of_real_numeral exp_of_real)

lemmas cos_ii_times = cosh_complex [symmetric]

lemma norm_cos_squared:
"norm(cos z) ^ 2 = cos(Re z) ^ 2 + (exp(Im z) - inverse(exp(Im z))) ^ 2 / 4"
apply (cases z)
apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide power_divide)
apply (simp only: left_diff_distrib [symmetric] power_mult_distrib)
apply (simp add: power2_eq_square algebra_simps divide_simps)
done

lemma norm_sin_squared:
"norm(sin z) ^ 2 = (exp(2 * Im z) + inverse(exp(2 * Im z)) - 2 * cos(2 * Re z)) / 4"
apply (cases z)
apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide power_divide)
apply (simp only: left_diff_distrib [symmetric] power_mult_distrib)
apply (simp add: power2_eq_square algebra_simps divide_simps)
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:
fixes z::complex
shows "norm(cos z) \<le> exp(norm z)"
proof -
have "Im z \<le> cmod z"
using abs_Im_le_cmod abs_le_D1 by auto
with exp_uminus_Im show ?thesis
apply (rule order_trans [OF norm_triangle_ineq], simp)
done
qed

lemma norm_cos_plus1_le:
fixes z::complex
shows "norm(1 + cos z) \<le> 2 * exp(norm z)"
proof -
have mono: "\<And>u w z::real. (1 \<le> w | 1 \<le> z) \<Longrightarrow> (w \<le> u & z \<le> u) \<Longrightarrow> 2 + w + z \<le> 4 * u"
by arith
have *: "Im z \<le> cmod z"
using abs_Im_le_cmod abs_le_D1 by auto
have triangle3: "\<And>x y z. norm(x + y + z) \<le> norm(x) + norm(y) + norm(z)"
have "norm(1 + cos z) = cmod (1 + (exp (\<i> * z) + exp (- (\<i> * z))) / 2)"
also have "... = cmod ((2 + exp (\<i> * z) + exp (- (\<i> * z))) / 2)"
also have "... = cmod (2 + exp (\<i> * z) + exp (- (\<i> * z))) / 2"
finally show ?thesis
apply (rule ssubst, simp)
apply (rule order_trans [OF triangle3], simp)
using exp_uminus_Im *
apply (auto intro: mono)
done
qed

subsection\<open>Taylor series for complex exponential, sine and cosine.\<close>

declare power_Suc [simp del]

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)"
by (rule convex_closed_segment [of 0 z])
next
fix k x
assume "x \<in> closed_segment 0 z" "k \<le> n"
show "(exp has_field_derivative exp x) (at x within closed_segment 0 z)"
using DERIV_exp DERIV_subset by blast
next
fix x
assume "x \<in> closed_segment 0 z"
then show "Re x \<le> \<bar>Re z\<bar>"
apply (auto simp: closed_segment_def scaleR_conv_of_real)
by (meson abs_ge_self abs_ge_zero linear mult_left_le_one_le mult_nonneg_nonpos order_trans)
next
show "0 \<in> closed_segment 0 z"
by (auto simp: closed_segment_def)
next
show "z \<in> closed_segment 0 z"
using of_real_1 zero_le_one by blast
qed

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>"
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"
by arith
show "cmod (sin (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>" using assms
apply (auto simp: scaleR_conv_of_real norm_mult norm_power sin_exp_eq norm_divide)
apply (rule order_trans [OF norm_triangle_ineq4])
apply (rule mono)
apply (auto simp: abs_if mult_left_le_one_le)
apply (meson mult_nonneg_nonneg neg_le_0_iff_le not_le order_trans)
apply (meson less_eq_real_def mult_nonneg_nonpos neg_0_le_iff_le order_trans)
done
show "cmod (cos (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>" using assms
apply (auto simp: scaleR_conv_of_real norm_mult norm_power cos_exp_eq norm_divide)
apply (rule order_trans [OF norm_triangle_ineq])
apply (rule mono)
apply (auto simp: abs_if mult_left_le_one_le)
apply (meson mult_nonneg_nonneg neg_le_0_iff_le not_le order_trans)
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))
\<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)"
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])
fix k x
show "((\<lambda>x. (- 1) ^ (k div 2) * (if even k then sin x else cos x)) has_field_derivative
(- 1) ^ (Suc k div 2) * (if odd k then sin x else cos x))
(at x within closed_segment 0 z)"
apply (auto simp: power_Suc)
apply (intro derivative_eq_intros | simp)+
done
next
fix x
assume "x \<in> closed_segment 0 z"
then show "cmod ((- 1) ^ (Suc n div 2) * (if odd n then sin x else cos x)) \<le> exp \<bar>Im z\<bar>"
by (auto simp: closed_segment_def norm_mult norm_power cmod_sin_le_exp cmod_cos_le_exp)
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)
show ?thesis
apply (rule order_trans [OF _ *])
done
qed

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)"
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])
fix k x
assume "x \<in> closed_segment 0 z" "k \<le> n"
show "((\<lambda>x. (- 1) ^ (Suc k div 2) * (if even k then cos x else sin x)) has_field_derivative
(- 1) ^ Suc (k div 2) * (if odd k then cos x else sin x))
(at x within closed_segment 0 z)"
apply (auto simp: power_Suc)
apply (intro derivative_eq_intros | simp)+
done
next
fix x
assume "x \<in> closed_segment 0 z"
then show "cmod ((- 1) ^ Suc (n div 2) * (if odd n then cos x else sin x)) \<le> exp \<bar>Im z\<bar>"
by (auto simp: closed_segment_def norm_mult norm_power cmod_sin_le_exp cmod_cos_le_exp)
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)
show ?thesis
apply (rule order_trans [OF _ *])
done
qed

declare power_Suc [simp]

text\<open>32-bit Approximation to e\<close>
lemma e_approx_32: "\<bar>exp(1) - 5837465777 / 2147483648\<bar> \<le> (inverse(2 ^ 32)::real)"
using Taylor_exp [of 1 14] exp_le
apply (simp add: setsum_left_distrib in_Reals_norm Re_exp atMost_nat_numeral fact_numeral)
apply (simp only: pos_le_divide_eq [symmetric], linarith)
done

lemma e_less_3: "exp 1 < (3::real)"
using e_approx_32
by (simp add: abs_if split: if_split_asm)

lemma ln3_gt_1: "ln 3 > (1::real)"
by (metis e_less_3 exp_less_cancel_iff exp_ln_iff less_trans ln_exp)

subsection\<open>The argument of a complex number\<close>

definition Arg :: "complex \<Rightarrow> real" where
"Arg z \<equiv> if z = 0 then 0
else THE t. 0 \<le> t \<and> t < 2*pi \<and>
z = of_real(norm z) * exp(\<i> * of_real t)"

lemma Arg_0 [simp]: "Arg(0) = 0"

lemma Arg_unique_lemma:
assumes z:  "z = of_real(norm z) * exp(\<i> * of_real t)"
and z': "z = of_real(norm z) * exp(\<i> * of_real t')"
and t:  "0 \<le> t"  "t < 2*pi"
and t': "0 \<le> t'" "t' < 2*pi"
and nz: "z \<noteq> 0"
shows "t' = t"
proof -
have [dest]: "\<And>x y z::real. x\<ge>0 \<Longrightarrow> x+y < z \<Longrightarrow> y<z"
by arith
have "of_real (cmod z) * exp (\<i> * of_real t') = of_real (cmod z) * exp (\<i> * of_real t)"
by (metis z z')
then have "exp (\<i> * of_real t') = exp (\<i> * of_real t)"
by (metis nz mult_left_cancel mult_zero_left z)
then have "sin t' = sin t \<and> cos t' = cos t"
apply (simp add: exp_Euler sin_of_real cos_of_real)
by (metis Complex_eq complex.sel)
then obtain n::int where n: "t' = t + 2 * n * pi"
by (auto simp: sin_cos_eq_iff)
then have "n=0"
apply (rule_tac z=n in int_cases)
using t t'
apply (auto simp: mult_less_0_iff algebra_simps)
done
then show "t' = t"
qed

lemma Arg: "0 \<le> Arg z & Arg z < 2*pi & z = of_real(norm z) * exp(\<i> * of_real(Arg z))"
proof (cases "z=0")
case True then show ?thesis
next
case False
obtain t where t: "0 \<le> t" "t < 2*pi"
and ReIm: "Re z / cmod z = cos t" "Im z / cmod z = sin t"
using sincos_total_2pi [OF complex_unit_circle [OF False]]
by blast
have z: "z = of_real(norm z) * exp(\<i> * of_real t)"
apply (rule complex_eqI)
using t False ReIm
apply (auto simp: exp_Euler sin_of_real cos_of_real divide_simps)
done
show ?thesis
apply (rule theI [where a=t])
using t z False
apply (auto intro: Arg_unique_lemma)
done
qed

corollary
shows Arg_ge_0: "0 \<le> Arg z"
and Arg_lt_2pi: "Arg z < 2*pi"
and Arg_eq: "z = of_real(norm z) * exp(\<i> * of_real(Arg z))"
using Arg by auto

lemma complex_norm_eq_1_exp: "norm z = 1 \<longleftrightarrow> (\<exists>t. z = exp(\<i> * of_real t))"
using Arg [of z] by auto

lemma Arg_unique: "\<lbrakk>of_real r * exp(\<i> * of_real a) = z; 0 < r; 0 \<le> a; a < 2*pi\<rbrakk> \<Longrightarrow> Arg z = a"
apply (rule Arg_unique_lemma [OF _ Arg_eq])
using Arg [of z]
apply (auto simp: norm_mult)
done

lemma Arg_minus: "z \<noteq> 0 \<Longrightarrow> Arg (-z) = (if Arg z < pi then Arg z + pi else Arg z - pi)"
apply (rule Arg_unique [of "norm z"])
apply (rule complex_eqI)
using Arg_ge_0 [of z] Arg_eq [of z] Arg_lt_2pi [of z] Arg_eq [of z]
apply auto
apply (auto simp: Re_exp Im_exp cos_diff sin_diff cis_conv_exp [symmetric])
apply (metis Re_rcis Im_rcis rcis_def)+
done

lemma Arg_times_of_real [simp]: "0 < r \<Longrightarrow> Arg (of_real r * z) = Arg z"
apply (cases "z=0", simp)
apply (rule Arg_unique [of "r * norm z"])
using Arg
apply auto
done

lemma Arg_times_of_real2 [simp]: "0 < r \<Longrightarrow> Arg (z * of_real r) = Arg z"
by (metis Arg_times_of_real mult.commute)

lemma Arg_divide_of_real [simp]: "0 < r \<Longrightarrow> Arg (z / of_real r) = Arg z"
by (metis Arg_times_of_real2 less_numeral_extra(3) nonzero_eq_divide_eq of_real_eq_0_iff)

lemma Arg_le_pi: "Arg z \<le> pi \<longleftrightarrow> 0 \<le> Im z"
proof (cases "z=0")
case True then show ?thesis
by simp
next
case False
have "0 \<le> Im z \<longleftrightarrow> 0 \<le> Im (of_real (cmod z) * exp (\<i> * complex_of_real (Arg z)))"
by (metis Arg_eq)
also have "... = (0 \<le> Im (exp (\<i> * complex_of_real (Arg z))))"
using False
also have "... \<longleftrightarrow> Arg z \<le> pi"
by (simp add: Im_exp) (metis Arg_ge_0 Arg_lt_2pi sin_lt_zero sin_ge_zero not_le)
finally show ?thesis
by blast
qed

lemma Arg_lt_pi: "0 < Arg z \<and> Arg z < pi \<longleftrightarrow> 0 < Im z"
proof (cases "z=0")
case True then show ?thesis
by simp
next
case False
have "0 < Im z \<longleftrightarrow> 0 < Im (of_real (cmod z) * exp (\<i> * complex_of_real (Arg z)))"
by (metis Arg_eq)
also have "... = (0 < Im (exp (\<i> * complex_of_real (Arg z))))"
using False
also have "... \<longleftrightarrow> 0 < Arg z \<and> Arg z < pi"
using Arg_ge_0  Arg_lt_2pi sin_le_zero sin_gt_zero
apply (auto simp: Im_exp)
using le_less apply fastforce
using not_le by blast
finally show ?thesis
by blast
qed

lemma Arg_eq_0: "Arg z = 0 \<longleftrightarrow> z \<in> \<real> \<and> 0 \<le> Re z"
proof (cases "z=0")
case True then show ?thesis
by simp
next
case False
have "z \<in> \<real> \<and> 0 \<le> Re z \<longleftrightarrow> z \<in> \<real> \<and> 0 \<le> Re (of_real (cmod z) * exp (\<i> * complex_of_real (Arg z)))"
by (metis Arg_eq)
also have "... \<longleftrightarrow> z \<in> \<real> \<and> 0 \<le> Re (exp (\<i> * complex_of_real (Arg z)))"
using False
also have "... \<longleftrightarrow> Arg z = 0"
apply (auto simp: Re_exp)
apply (metis Arg_lt_pi Arg_ge_0 Arg_le_pi cos_pi complex_is_Real_iff leD less_linear less_minus_one_simps(2) minus_minus neg_less_eq_nonneg order_refl)
using Arg_eq [of z]
apply (auto simp: Reals_def)
done
finally show ?thesis
by blast
qed

corollary Arg_gt_0:
assumes "z \<in> \<real> \<Longrightarrow> Re z < 0"
shows "Arg z > 0"
using Arg_eq_0 Arg_ge_0 assms dual_order.strict_iff_order by fastforce

lemma Arg_of_real: "Arg(of_real x) = 0 \<longleftrightarrow> 0 \<le> x"

lemma Arg_eq_pi: "Arg z = pi \<longleftrightarrow> z \<in> \<real> \<and> Re z < 0"
apply  (cases "z=0", simp)
using Arg_eq_0 [of "-z"]
apply (auto simp: complex_is_Real_iff Arg_minus)
apply (metis Arg_minus pi_gt_zero add.left_neutral minus_minus minus_zero)
done

lemma Arg_eq_0_pi: "Arg z = 0 \<or> Arg z = pi \<longleftrightarrow> z \<in> \<real>"
using Arg_eq_0 Arg_eq_pi not_le by auto

lemma Arg_inverse: "Arg(inverse z) = (if z \<in> \<real> \<and> 0 \<le> Re z then Arg z else 2*pi - Arg z)"
apply (cases "z=0", simp)
apply (rule Arg_unique [of "inverse (norm z)"])
using Arg_ge_0 [of z] Arg_lt_2pi [of z] Arg_eq [of z] Arg_eq_0 [of z] exp_two_pi_i
apply (auto simp: of_real_numeral algebra_simps exp_diff divide_simps)
done

lemma Arg_eq_iff:
assumes "w \<noteq> 0" "z \<noteq> 0"
shows "Arg w = Arg z \<longleftrightarrow> (\<exists>x. 0 < x & w = of_real x * z)"
using assms Arg_eq [of z] Arg_eq [of w]
apply auto
apply (rule_tac x="norm w / norm z" in exI)
by (metis mult.commute mult.left_commute)

lemma Arg_inverse_eq_0: "Arg(inverse z) = 0 \<longleftrightarrow> Arg z = 0"
using complex_is_Real_iff
apply (auto simp: divide_simps not_sum_power2_lt_zero)
done

lemma Arg_divide:
assumes "w \<noteq> 0" "z \<noteq> 0" "Arg w \<le> Arg z"
shows "Arg(z / w) = Arg z - Arg w"
apply (rule Arg_unique [of "norm(z / w)"])
using assms Arg_eq [of z] Arg_eq [of w] Arg_ge_0 [of w] Arg_lt_2pi [of z]
apply (auto simp: exp_diff norm_divide algebra_simps divide_simps)
done

lemma Arg_le_div_sum:
assumes "w \<noteq> 0" "z \<noteq> 0" "Arg w \<le> Arg z"
shows "Arg z = Arg w + Arg(z / w)"

lemma Arg_le_div_sum_eq:
assumes "w \<noteq> 0" "z \<noteq> 0"
shows "Arg w \<le> Arg z \<longleftrightarrow> Arg z = Arg w + Arg(z / w)"
using assms
by (auto simp: Arg_ge_0 intro: Arg_le_div_sum)

lemma Arg_diff:
assumes "w \<noteq> 0" "z \<noteq> 0"
shows "Arg w - Arg z = (if Arg z \<le> Arg w then Arg(w / z) else Arg(w/z) - 2*pi)"
using assms
apply (auto simp: Arg_ge_0 Arg_divide not_le)
using Arg_divide [of w z] Arg_inverse [of "w/z"]
apply auto
by (metis Arg_eq_0 less_irrefl minus_diff_eq right_minus_eq)

assumes "w \<noteq> 0" "z \<noteq> 0"
shows "Arg w + Arg z = (if Arg w + Arg z < 2*pi then Arg(w * z) else Arg(w * z) + 2*pi)"
using assms
using Arg_diff [of "w*z" z] Arg_le_div_sum_eq [of z "w*z"]
apply (auto simp: Arg_ge_0 Arg_divide not_le)
done

lemma Arg_times:
assumes "w \<noteq> 0" "z \<noteq> 0"
shows "Arg (w * z) = (if Arg w + Arg z < 2*pi then Arg w + Arg z
else (Arg w + Arg z) - 2*pi)"
by auto

lemma Arg_cnj: "Arg(cnj z) = (if z \<in> \<real> \<and> 0 \<le> Re z then Arg z else 2*pi - Arg z)"
apply (cases "z=0", simp)
apply (rule trans [of _ "Arg(inverse z)"])
apply (simp add: Arg_eq_iff divide_simps complex_norm_square [symmetric] mult.commute)
apply (metis norm_eq_zero of_real_power zero_less_power2)
apply (auto simp: of_real_numeral Arg_inverse)
done

lemma Arg_real: "z \<in> \<real> \<Longrightarrow> Arg z = (if 0 \<le> Re z then 0 else pi)"
using Arg_eq_0 Arg_eq_0_pi
by auto

lemma Arg_exp: "0 \<le> Im z \<Longrightarrow> Im z < 2*pi \<Longrightarrow> Arg(exp z) = Im z"
by (rule Arg_unique [of  "exp(Re z)"]) (auto simp: exp_eq_polar)

lemma complex_split_polar:
obtains r a::real where "z = complex_of_real r * (cos a + \<i> * sin a)" "0 \<le> r" "0 \<le> a" "a < 2*pi"
using Arg cis.ctr cis_conv_exp by fastforce

lemma Re_Im_le_cmod: "Im w * sin \<phi> + Re w * cos \<phi> \<le> cmod w"
proof (cases w rule: complex_split_polar)
case (1 r a) with sin_cos_le1 [of a \<phi>] show ?thesis
by (metis (no_types, hide_lams) abs_le_D1 distrib_left mult.commute mult.left_commute mult_left_le)
qed

subsection\<open>Analytic properties of tangent function\<close>

lemma cnj_tan: "cnj(tan z) = tan(cnj z)"
by (simp add: cnj_cos cnj_sin tan_def)

lemma field_differentiable_at_tan: "~(cos z = 0) \<Longrightarrow> tan field_differentiable at z"
unfolding field_differentiable_def
using DERIV_tan by blast

lemma field_differentiable_within_tan: "~(cos z = 0)
\<Longrightarrow> tan field_differentiable (at z within s)"
using field_differentiable_at_tan field_differentiable_at_within by blast

lemma continuous_within_tan: "~(cos z = 0) \<Longrightarrow> continuous (at z within s) tan"
using continuous_at_imp_continuous_within isCont_tan by blast

lemma continuous_on_tan [continuous_intros]: "(\<And>z. z \<in> s \<Longrightarrow> ~(cos z = 0)) \<Longrightarrow> continuous_on s tan"

lemma holomorphic_on_tan: "(\<And>z. z \<in> s \<Longrightarrow> ~(cos z = 0)) \<Longrightarrow> tan holomorphic_on s"

subsection\<open>Complex logarithms (the conventional principal value)\<close>

instantiation complex :: ln
begin

definition ln_complex :: "complex \<Rightarrow> complex"
where "ln_complex \<equiv> \<lambda>z. THE w. exp w = z & -pi < Im(w) & Im(w) \<le> pi"

lemma
assumes "z \<noteq> 0"
shows exp_Ln [simp]:  "exp(ln z) = z"
and mpi_less_Im_Ln: "-pi < Im(ln z)"
and Im_Ln_le_pi:    "Im(ln z) \<le> pi"
proof -
obtain \<psi> where z: "z / (cmod z) = Complex (cos \<psi>) (sin \<psi>)"
using complex_unimodular_polar [of "z / (norm z)"] assms
by (auto simp: norm_divide divide_simps)
obtain \<phi> where \<phi>: "- pi < \<phi>" "\<phi> \<le> pi" "sin \<phi> = sin \<psi>" "cos \<phi> = cos \<psi>"
using sincos_principal_value [of "\<psi>"] assms
by (auto simp: norm_divide divide_simps)
have "exp(ln z) = z & -pi < Im(ln z) & Im(ln z) \<le> pi" unfolding ln_complex_def
apply (rule theI [where a = "Complex (ln(norm z)) \<phi>"])
using z assms \<phi>
apply (auto simp: field_simps exp_complex_eqI exp_eq_polar cis.code)
done
then show "exp(ln z) = z" "-pi < Im(ln z)" "Im(ln z) \<le> pi"
by auto
qed

lemma Ln_exp [simp]:
assumes "-pi < Im(z)" "Im(z) \<le> pi"
shows "ln(exp z) = z"
apply (rule exp_complex_eqI)
using assms mpi_less_Im_Ln  [of "exp z"] Im_Ln_le_pi [of "exp z"]
apply auto
done

subsection\<open>Relation to Real Logarithm\<close>

lemma Ln_of_real:
assumes "0 < z"
shows "ln(of_real z::complex) = of_real(ln z)"
proof -
have "ln(of_real (exp (ln z))::complex) = ln (exp (of_real (ln z)))"
also have "... = of_real(ln z)"
using assms
by (subst Ln_exp) auto
finally show ?thesis
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)

corollary Im_Ln_of_real [simp]: "r > 0 \<Longrightarrow> Im (ln (of_real r)) = 0"

lemma cmod_Ln_Reals [simp]: "z \<in> \<real> \<Longrightarrow> 0 < Re z \<Longrightarrow> cmod (ln z) = norm (ln (Re z))"
using Ln_of_real by force

lemma Ln_1: "ln 1 = (0::complex)"
proof -
have "ln (exp 0) = (0::complex)"
by (metis (mono_tags, hide_lams) Ln_of_real exp_zero ln_one of_real_0 of_real_1 zero_less_one)
then show ?thesis
by simp
qed

instance
by intro_classes (rule ln_complex_def Ln_1)

end

abbreviation Ln :: "complex \<Rightarrow> complex"
where "Ln \<equiv> ln"

lemma Ln_eq_iff: "w \<noteq> 0 \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> (Ln w = Ln z \<longleftrightarrow> w = z)"
by (metis exp_Ln)

lemma Ln_unique: "exp(z) = w \<Longrightarrow> -pi < Im(z) \<Longrightarrow> Im(z) \<le> pi \<Longrightarrow> Ln w = z"
using Ln_exp by blast

lemma Re_Ln [simp]: "z \<noteq> 0 \<Longrightarrow> Re(Ln z) = ln(norm z)"
by (metis exp_Ln ln_exp norm_exp_eq_Re)

corollary ln_cmod_le:
assumes z: "z \<noteq> 0"
shows "ln (cmod z) \<le> cmod (Ln z)"
using norm_exp [of "Ln z", simplified exp_Ln [OF z]]
by (metis Re_Ln complex_Re_le_cmod z)

proposition exists_complex_root:
fixes z :: complex
assumes "n \<noteq> 0"  obtains w where "z = w ^ n"
apply (cases "z=0")
using assms apply (simp add: power_0_left)
apply (rule_tac w = "exp(Ln z / n)" in that)
apply (auto simp: assms exp_of_nat_mult [symmetric])
done

corollary exists_complex_root_nonzero:
fixes z::complex
assumes "z \<noteq> 0" "n \<noteq> 0"
obtains w where "w \<noteq> 0" "z = w ^ n"
by (metis exists_complex_root [of n z] assms power_0_left)

subsection\<open>The Unwinding Number and the Ln-product Formula\<close>

text\<open>Note that in this special case the unwinding number is -1, 0 or 1.\<close>

definition unwinding :: "complex \<Rightarrow> complex" where
"unwinding(z) = (z - Ln(exp z)) / (of_real(2*pi) * \<i>)"

lemma unwinding_2pi: "(2*pi) * \<i> * 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) * \<i> * unwinding(Ln w + Ln z)"

subsection\<open>Derivative of Ln away from the branch cut\<close>

lemma
assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0"
shows has_field_derivative_Ln: "(Ln has_field_derivative inverse(z)) (at z)"
and Im_Ln_less_pi:           "Im (Ln z) < pi"
proof -
have znz: "z \<noteq> 0"
using assms by auto
then have "Im (Ln z) \<noteq> pi"
by (metis (no_types) Im_exp Ln_in_Reals assms complex_nonpos_Reals_iff complex_is_Real_iff exp_Ln mult_zero_right not_less pi_neq_zero sin_pi znz)
then show *: "Im (Ln z) < pi" using assms Im_Ln_le_pi
have "(exp has_field_derivative z) (at (Ln z))"
by (metis znz DERIV_exp exp_Ln)
then show "(Ln has_field_derivative inverse(z)) (at z)"
apply (rule has_complex_derivative_inverse_strong_x
[where s = "{w. -pi < Im(w) \<and> Im(w) < pi}"])
using znz *
apply (auto simp: Transcendental.continuous_on_exp [OF continuous_on_id] open_Collect_conj open_halfspace_Im_gt open_halfspace_Im_lt mpi_less_Im_Ln)
done
qed

declare has_field_derivative_Ln [derivative_intros]
declare has_field_derivative_Ln [THEN DERIV_chain2, derivative_intros]

lemma field_differentiable_at_Ln: "z \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> Ln field_differentiable at z"
using field_differentiable_def has_field_derivative_Ln by blast

lemma field_differentiable_within_Ln: "z \<notin> \<real>\<^sub>\<le>\<^sub>0
\<Longrightarrow> Ln field_differentiable (at z within s)"
using field_differentiable_at_Ln field_differentiable_within_subset by blast

lemma continuous_at_Ln: "z \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> continuous (at z) Ln"

lemma isCont_Ln' [simp]:
"\<lbrakk>isCont f z; f z \<notin> \<real>\<^sub>\<le>\<^sub>0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. Ln (f x)) z"
by (blast intro: isCont_o2 [OF _ continuous_at_Ln])

lemma continuous_within_Ln: "z \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> continuous (at z within s) Ln"
using continuous_at_Ln continuous_at_imp_continuous_within by blast

lemma continuous_on_Ln [continuous_intros]: "(\<And>z. z \<in> s \<Longrightarrow> z \<notin> \<real>\<^sub>\<le>\<^sub>0) \<Longrightarrow> continuous_on s Ln"

lemma holomorphic_on_Ln: "(\<And>z. z \<in> s \<Longrightarrow> z \<notin> \<real>\<^sub>\<le>\<^sub>0) \<Longrightarrow> Ln holomorphic_on s"

lemma cos_lt_zero_pi: "pi/2 < x \<Longrightarrow> x < 3*pi/2 \<Longrightarrow> cos x < 0"
using cos_minus_pi cos_gt_zero_pi [of "x-pi"]
by simp

lemma Re_Ln_pos_lt:
assumes "z \<noteq> 0"
shows "\<bar>Im(Ln z)\<bar> < pi/2 \<longleftrightarrow> 0 < Re(z)"
proof -
{ fix w
assume "w = Ln z"
then have w: "Im w \<le> pi" "- pi < Im w"
using Im_Ln_le_pi [of z]  mpi_less_Im_Ln [of z]  assms
by auto
then have "\<bar>Im w\<bar> < pi/2 \<longleftrightarrow> 0 < Re(exp w)"
apply (auto simp: Re_exp zero_less_mult_iff cos_gt_zero_pi)
using cos_lt_zero_pi [of "-(Im w)"] cos_lt_zero_pi [of "(Im w)"]
apply (simp add: abs_if split: if_split_asm)
apply (metis (no_types) cos_minus cos_pi_half eq_divide_eq_numeral1(1) eq_numeral_simps(4)
less_numeral_extra(3) linorder_neqE_linordered_idom minus_mult_minus minus_mult_right
mult_numeral_1_right)
done
}
then show ?thesis using assms
by auto
qed

lemma Re_Ln_pos_le:
assumes "z \<noteq> 0"
shows "\<bar>Im(Ln z)\<bar> \<le> pi/2 \<longleftrightarrow> 0 \<le> Re(z)"
proof -
{ fix w
assume "w = Ln z"
then have w: "Im w \<le> pi" "- pi < Im w"
using Im_Ln_le_pi [of z]  mpi_less_Im_Ln [of z]  assms
by auto
then have "\<bar>Im w\<bar> \<le> pi/2 \<longleftrightarrow> 0 \<le> Re(exp w)"
apply (auto simp: Re_exp zero_le_mult_iff cos_ge_zero)
using cos_lt_zero_pi [of "- (Im w)"] cos_lt_zero_pi [of "(Im w)"] not_le
apply (auto simp: abs_if split: if_split_asm)
done
}
then show ?thesis using assms
by auto
qed

lemma Im_Ln_pos_lt:
assumes "z \<noteq> 0"
shows "0 < Im(Ln z) \<and> Im(Ln z) < pi \<longleftrightarrow> 0 < Im(z)"
proof -
{ fix w
assume "w = Ln z"
then have w: "Im w \<le> pi" "- pi < Im w"
using Im_Ln_le_pi [of z]  mpi_less_Im_Ln [of z]  assms
by auto
then have "0 < Im w \<and> Im w < pi \<longleftrightarrow> 0 < Im(exp w)"
using sin_gt_zero [of "- (Im w)"] sin_gt_zero [of "(Im w)"]
apply (auto simp: Im_exp zero_less_mult_iff)
using less_linear apply fastforce
using less_linear apply fastforce
done
}
then show ?thesis using assms
by auto
qed

lemma Im_Ln_pos_le:
assumes "z \<noteq> 0"
shows "0 \<le> Im(Ln z) \<and> Im(Ln z) \<le> pi \<longleftrightarrow> 0 \<le> Im(z)"
proof -
{ fix w
assume "w = Ln z"
then have w: "Im w \<le> pi" "- pi < Im w"
using Im_Ln_le_pi [of z]  mpi_less_Im_Ln [of z]  assms
by auto
then have "0 \<le> Im w \<and> Im w \<le> pi \<longleftrightarrow> 0 \<le> Im(exp w)"
using sin_ge_zero [of "- (Im w)"] sin_ge_zero [of "(Im w)"]
apply (auto simp: Im_exp zero_le_mult_iff sin_ge_zero)
apply (metis not_le not_less_iff_gr_or_eq pi_not_less_zero sin_eq_0_pi)
done }
then show ?thesis using assms
by auto
qed

lemma Re_Ln_pos_lt_imp: "0 < Re(z) \<Longrightarrow> \<bar>Im(Ln z)\<bar> < pi/2"
by (metis Re_Ln_pos_lt less_irrefl zero_complex.simps(1))

lemma Im_Ln_pos_lt_imp: "0 < Im(z) \<Longrightarrow> 0 < Im(Ln z) \<and> Im(Ln z) < pi"
by (metis Im_Ln_pos_lt not_le order_refl zero_complex.simps(2))

text\<open>A reference to the set of positive real numbers\<close>
lemma Im_Ln_eq_0: "z \<noteq> 0 \<Longrightarrow> (Im(Ln z) = 0 \<longleftrightarrow> 0 < Re(z) \<and> Im(z) = 0)"
by (metis Im_complex_of_real Im_exp Ln_in_Reals Re_Ln_pos_lt Re_Ln_pos_lt_imp
Re_complex_of_real complex_is_Real_iff exp_Ln exp_of_real pi_gt_zero)

lemma Im_Ln_eq_pi: "z \<noteq> 0 \<Longrightarrow> (Im(Ln z) = pi \<longleftrightarrow> Re(z) < 0 \<and> Im(z) = 0)"
by (metis Im_Ln_eq_0 Im_Ln_pos_le Im_Ln_pos_lt add.left_neutral complex_eq less_eq_real_def
mult_zero_right not_less_iff_gr_or_eq pi_ge_zero pi_neq_zero rcis_zero_arg rcis_zero_mod)

subsection\<open>More Properties of Ln\<close>

lemma cnj_Ln: "z \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> cnj(Ln z) = Ln(cnj z)"
apply (cases "z=0", auto)
apply (rule exp_complex_eqI)
apply (auto simp: abs_if split: if_split_asm)
using Im_Ln_less_pi Im_Ln_le_pi apply force
apply (metis complex_cnj_zero_iff diff_minus_eq_add diff_strict_mono minus_less_iff
mpi_less_Im_Ln mult.commute mult_2_right)
by (metis exp_Ln exp_cnj)

lemma Ln_inverse: "z \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> Ln(inverse z) = -(Ln z)"
apply (cases "z=0", auto)
apply (rule exp_complex_eqI)
using mpi_less_Im_Ln [of z] mpi_less_Im_Ln [of "inverse z"]
apply (auto simp: abs_if exp_minus split: if_split_asm)
done

lemma Ln_minus1 [simp]: "Ln(-1) = \<i> * pi"
apply (rule exp_complex_eqI)
using Im_Ln_le_pi [of "-1"] mpi_less_Im_Ln [of "-1"] cis_conv_exp cis_pi
apply (auto simp: abs_if)
done

lemma Ln_ii [simp]: "Ln \<i> = \<i> * of_real pi/2"
using Ln_exp [of "\<i> * (of_real pi/2)"]
unfolding exp_Euler
by simp

lemma Ln_minus_ii [simp]: "Ln(-\<i>) = - (\<i> * pi/2)"
proof -
have  "Ln(-\<i>) = Ln(inverse \<i>)"    by simp
also have "... = - (Ln \<i>)"         using Ln_inverse by blast
also have "... = - (\<i> * pi/2)"     by simp
finally show ?thesis .
qed

lemma Ln_times:
assumes "w \<noteq> 0" "z \<noteq> 0"
shows "Ln(w * z) =
(if Im(Ln w + Ln z) \<le> -pi then
(Ln(w) + Ln(z)) + \<i> * of_real(2*pi)
else if Im(Ln w + Ln z) > pi then
(Ln(w) + Ln(z)) - \<i> * of_real(2*pi)
else Ln(w) + Ln(z))"
using pi_ge_zero Im_Ln_le_pi [of w] Im_Ln_le_pi [of z]
using assms mpi_less_Im_Ln [of w] mpi_less_Im_Ln [of z]
by (auto simp: exp_add exp_diff sin_double cos_double exp_Euler intro!: Ln_unique)

corollary Ln_times_simple:
"\<lbrakk>w \<noteq> 0; z \<noteq> 0; -pi < Im(Ln w) + Im(Ln z); Im(Ln w) + Im(Ln z) \<le> pi\<rbrakk>
\<Longrightarrow> Ln(w * z) = Ln(w) + Ln(z)"

corollary Ln_times_of_real:
"\<lbrakk>r > 0; z \<noteq> 0\<rbrakk> \<Longrightarrow> Ln(of_real r * z) = ln r + Ln(z)"
using mpi_less_Im_Ln Im_Ln_le_pi
by (force simp: Ln_times)

corollary Ln_divide_of_real:
"\<lbrakk>r > 0; z \<noteq> 0\<rbrakk> \<Longrightarrow> Ln(z / of_real r) = Ln(z) - ln r"
using Ln_times_of_real [of "inverse r" z]
by (simp add: ln_inverse Ln_of_real mult.commute divide_inverse of_real_inverse [symmetric]
del: of_real_inverse)

lemma Ln_minus:
assumes "z \<noteq> 0"
shows "Ln(-z) = (if Im(z) \<le> 0 \<and> ~(Re(z) < 0 \<and> Im(z) = 0)
then Ln(z) + \<i> * pi
else Ln(z) - \<i> * pi)" (is "_ = ?rhs")
using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
Im_Ln_eq_pi [of z] Im_Ln_pos_lt [of z]
by (fastforce simp: exp_add exp_diff exp_Euler intro!: Ln_unique)

lemma Ln_inverse_if:
assumes "z \<noteq> 0"
shows "Ln (inverse z) = (if z \<in> \<real>\<^sub>\<le>\<^sub>0 then -(Ln z) + \<i> * 2 * complex_of_real pi else -(Ln z))"
proof (cases "z \<in> \<real>\<^sub>\<le>\<^sub>0")
case False then show ?thesis
next
case True
then have z: "Im z = 0" "Re z < 0"
using assms
apply (auto simp: complex_nonpos_Reals_iff)
by (metis complex_is_Real_iff le_imp_less_or_eq of_real_0 of_real_Re)
have "Ln(inverse z) = Ln(- (inverse (-z)))"
by simp
also have "... = Ln (inverse (-z)) + \<i> * complex_of_real pi"
using assms z
done
also have "... = - Ln (- z) + \<i> * complex_of_real pi"
apply (subst Ln_inverse)
using z by (auto simp add: complex_nonneg_Reals_iff)
also have "... = - (Ln z) + \<i> * 2 * complex_of_real pi"
apply (subst Ln_minus [OF assms])
using assms z
apply simp
done
finally show ?thesis by (simp add: True)
qed

lemma Ln_times_ii:
assumes "z \<noteq> 0"
shows  "Ln(\<i> * z) = (if 0 \<le> Re(z) | Im(z) < 0
then Ln(z) + \<i> * of_real pi/2
else Ln(z) - \<i> * of_real(3 * pi/2))"
using Im_Ln_le_pi [of z] mpi_less_Im_Ln [of z] assms
Im_Ln_eq_pi [of z] Im_Ln_pos_lt [of z] Re_Ln_pos_le [of z]
by (auto simp: Ln_times)

lemma Ln_of_nat: "0 < n \<Longrightarrow> Ln (of_nat n) = of_real (ln (of_nat n))"
by (subst of_real_of_nat_eq[symmetric], subst Ln_of_real[symmetric]) simp_all

lemma Ln_of_nat_over_of_nat:
assumes "m > 0" "n > 0"
shows   "Ln (of_nat m / of_nat n) = of_real (ln (of_nat m) - ln (of_nat n))"
proof -
have "of_nat m / of_nat n = (of_real (of_nat m / of_nat n) :: complex)" by simp
also from assms have "Ln ... = of_real (ln (of_nat m / of_nat n))"
also from assms have "... = of_real (ln (of_nat m) - ln (of_nat n))"
finally show ?thesis .
qed

subsection\<open>Relation between Ln and Arg, and hence continuity of Arg\<close>

lemma Arg_Ln:
assumes "0 < Arg z" shows "Arg z = Im(Ln(-z)) + pi"
proof (cases "z = 0")
case True
with assms show ?thesis
by simp
next
case False
then have "z / of_real(norm z) = exp(\<i> * of_real(Arg z))"
using Arg [of z]
by (metis abs_norm_cancel nonzero_mult_divide_cancel_left norm_of_real zero_less_norm_iff)
then have "- z / of_real(norm z) = exp (\<i> * (of_real (Arg z) - pi))"
using cis_conv_exp cis_pi
by (auto simp: exp_diff algebra_simps)
then have "ln (- z / of_real(norm z)) = ln (exp (\<i> * (of_real (Arg z) - pi)))"
by simp
also have "... = \<i> * (of_real(Arg z) - pi)"
using Arg [of z] assms pi_not_less_zero
by auto
finally have "Arg z =  Im (Ln (- z / of_real (cmod z))) + pi"
by simp
also have "... = Im (Ln (-z) - ln (cmod z)) + pi"
by (metis diff_0_right minus_diff_eq zero_less_norm_iff Ln_divide_of_real False)
also have "... = Im (Ln (-z)) + pi"
by simp
finally show ?thesis .
qed

lemma continuous_at_Arg:
assumes "z \<notin> \<real>\<^sub>\<ge>\<^sub>0"
shows "continuous (at z) Arg"
proof -
have *: "isCont (\<lambda>z. Im (Ln (- z)) + pi) z"
by (rule Complex.isCont_Im isCont_Ln' continuous_intros | simp add: assms complex_is_Real_iff)+
have [simp]: "\<And>x. \<lbrakk>Im x \<noteq> 0\<rbrakk> \<Longrightarrow> Im (Ln (- x)) + pi = Arg x"
using Arg_Ln Arg_gt_0 complex_is_Real_iff by auto
consider "Re z < 0" | "Im z \<noteq> 0" using assms
using complex_nonneg_Reals_iff not_le by blast
then have [simp]: "(\<lambda>z. Im (Ln (- z)) + pi) \<midarrow>z\<rightarrow> Arg z"
using "*"  by (simp add: isCont_def) (metis Arg_Ln Arg_gt_0 complex_is_Real_iff)
show ?thesis
apply (rule Lim_transform_within_open [where s= "-\<real>\<^sub>\<ge>\<^sub>0" and f = "\<lambda>z. Im(Ln(-z)) + pi"])
apply (auto simp add: not_le Arg_Ln [OF Arg_gt_0] complex_nonneg_Reals_iff closed_def [symmetric])
using assms apply (force simp add: complex_nonneg_Reals_iff)
done
qed

lemma Ln_series:
fixes z :: complex
assumes "norm z < 1"
shows   "(\<lambda>n. (-1)^Suc n / of_nat n * z^n) sums ln (1 + z)" (is "(\<lambda>n. ?f n * z^n) sums _")
proof -
let ?F = "\<lambda>z. \<Sum>n. ?f n * z^n" and ?F' = "\<lambda>z. \<Sum>n. diffs ?f n * z^n"
have r: "conv_radius ?f = 1"
(simp_all add: norm_divide LIMSEQ_Suc_n_over_n del: of_nat_Suc)

have "\<exists>c. \<forall>z\<in>ball 0 1. ln (1 + z) - ?F z = c"
proof (rule has_field_derivative_zero_constant)
fix z :: complex assume z': "z \<in> ball 0 1"
hence z: "norm z < 1" by (simp add: dist_0_norm)
define t :: complex where "t = of_real (1 + norm z) / 2"
from z have t: "norm z < norm t" "norm t < 1" unfolding t_def

have "Re (-z) \<le> norm (-z)" by (rule complex_Re_le_cmod)
also from z have "... < 1" by simp
finally have "((\<lambda>z. ln (1 + z)) has_field_derivative inverse (1+z)) (at z)"
by (auto intro!: derivative_eq_intros simp: complex_nonpos_Reals_iff)
moreover have "(?F has_field_derivative ?F' z) (at z)" using t r
by (intro termdiffs_strong[of _ t] summable_in_conv_radius) simp_all
ultimately have "((\<lambda>z. ln (1 + z) - ?F z) has_field_derivative (inverse (1 + z) - ?F' z))
(at z within ball 0 1)"
by (intro derivative_intros) (simp_all add: at_within_open[OF z'])
also have "(\<lambda>n. of_nat n * ?f n * z ^ (n - Suc 0)) sums ?F' z" using t r
by (intro diffs_equiv termdiff_converges[OF t(1)] summable_in_conv_radius) simp_all
from sums_split_initial_segment[OF this, of 1]
have "(\<lambda>i. (-z) ^ i) sums ?F' z" by (simp add: power_minus[of z] del: of_nat_Suc)
hence "?F' z = inverse (1 + z)" using z by (simp add: sums_iff suminf_geometric divide_inverse)
also have "inverse (1 + z) - inverse (1 + z) = 0" by simp
finally show "((\<lambda>z. ln (1 + z) - ?F z) has_field_derivative 0) (at z within ball 0 1)" .
qed simp_all
then obtain c where c: "\<And>z. z \<in> ball 0 1 \<Longrightarrow> ln (1 + z) - ?F z = c" by blast
from c[of 0] have "c = 0" by (simp only: powser_zero) simp
with c[of z] assms have "ln (1 + z) = ?F z" by (simp add: dist_0_norm)
moreover have "summable (\<lambda>n. ?f n * z^n)" using assms r
ultimately show ?thesis by (simp add: sums_iff)
qed

lemma Ln_approx_linear:
fixes z :: complex
assumes "norm z < 1"
shows   "norm (ln (1 + z) - z) \<le> norm z^2 / (1 - norm z)"
proof -
let ?f = "\<lambda>n. (-1)^Suc n / of_nat n"
from assms have "(\<lambda>n. ?f n * z^n) sums ln (1 + z)" using Ln_series by simp
moreover have "(\<lambda>n. (if n = 1 then 1 else 0) * z^n) sums z" using powser_sums_if[of 1] by simp
ultimately have "(\<lambda>n. (?f n - (if n = 1 then 1 else 0)) * z^n) sums (ln (1 + z) - z)"
by (subst left_diff_distrib, intro sums_diff) simp_all
from sums_split_initial_segment[OF this, of "Suc 1"]
have "(\<lambda>i. (-(z^2)) * inverse (2 + of_nat i) * (- z)^i) sums (Ln (1 + z) - z)"
by (simp add: power2_eq_square mult_ac power_minus[of z] divide_inverse)
hence "(Ln (1 + z) - z) = (\<Sum>i. (-(z^2)) * inverse (of_nat (i+2)) * (-z)^i)"
also have A: "summable (\<lambda>n. norm z^2 * (inverse (real_of_nat (Suc (Suc n))) * cmod z ^ n))"
by (rule summable_mult, rule summable_comparison_test_ev[OF _ summable_geometric[of "norm z"]])
(auto simp: assms field_simps intro!: always_eventually)
hence "norm (\<Sum>i. (-(z^2)) * inverse (of_nat (i+2)) * (-z)^i) \<le>
(\<Sum>i. norm (-(z^2) * inverse (of_nat (i+2)) * (-z)^i))"
by (intro summable_norm)
(auto simp: norm_power norm_inverse norm_mult mult_ac simp del: of_nat_add of_nat_Suc)
also have "norm ((-z)^2 * (-z)^i) * inverse (of_nat (i+2)) \<le> norm ((-z)^2 * (-z)^i) * 1" for i
by (intro mult_left_mono) (simp_all add: divide_simps)
hence "(\<Sum>i. norm (-(z^2) * inverse (of_nat (i+2)) * (-z)^i)) \<le>
(\<Sum>i. norm (-(z^2) * (-z)^i))" using A assms
apply (simp_all only: norm_power norm_inverse norm_divide norm_mult)
apply (intro suminf_le summable_mult summable_geometric)
apply (auto simp: norm_power field_simps simp del: of_nat_add of_nat_Suc)
done
also have "... = norm z^2 * (\<Sum>i. norm z^i)" using assms
by (subst suminf_mult [symmetric]) (auto intro!: summable_geometric simp: norm_mult norm_power)
also have "(\<Sum>i. norm z^i) = inverse (1 - norm z)" using assms
by (subst suminf_geometric) (simp_all add: divide_inverse)
also have "norm z^2 * ... = norm z^2 / (1 - norm z)" by (simp add: divide_inverse)
finally show ?thesis .
qed

text\<open>Relation between Arg and arctangent in upper halfplane\<close>
lemma Arg_arctan_upperhalf:
assumes "0 < Im z"
shows "Arg z = pi/2 - arctan(Re z / Im z)"
proof (cases "z = 0")
case True with assms show ?thesis
by simp
next
case False
show ?thesis
apply (rule Arg_unique [of "norm z"])
using False assms arctan [of "Re z / Im z"] pi_ge_two pi_half_less_two
apply (auto simp: exp_Euler cos_diff sin_diff)
using norm_complex_def [of z, symmetric]
apply (simp add: sin_of_real cos_of_real sin_arctan cos_arctan field_simps real_sqrt_divide)
apply (metis complex_eq mult.assoc ring_class.ring_distribs(2))
done
qed

lemma Arg_eq_Im_Ln:
assumes "0 \<le> Im z" "0 < Re z"
shows "Arg z = Im (Ln z)"
proof (cases "z = 0 \<or> Im z = 0")
case True then show ?thesis
using assms Arg_eq_0 complex_is_Real_iff
apply auto
by (metis Arg_eq_0_pi Arg_eq_pi Im_Ln_eq_0 Im_Ln_eq_pi less_numeral_extra(3) zero_complex.simps(1))
next
case False
then have "Arg z > 0"
using Arg_gt_0 complex_is_Real_iff by blast
then show ?thesis
using assms False
by (subst Arg_Ln) (auto simp: Ln_minus)
qed

lemma continuous_within_upperhalf_Arg:
assumes "z \<noteq> 0"
shows "continuous (at z within {z. 0 \<le> Im z}) Arg"
proof (cases "z \<in> \<real>\<^sub>\<ge>\<^sub>0")
case False then show ?thesis
using continuous_at_Arg continuous_at_imp_continuous_within by auto
next
case True
then have z: "z \<in> \<real>" "0 < Re z"
using assms  by (auto simp: complex_nonneg_Reals_iff complex_is_Real_iff complex_neq_0)
then have [simp]: "Arg z = 0" "Im (Ln z) = 0"
by (auto simp: Arg_eq_0 Im_Ln_eq_0 assms complex_is_Real_iff)
show ?thesis
proof (clarsimp simp add: continuous_within Lim_within dist_norm)
fix e::real
assume "0 < e"
moreover have "continuous (at z) (\<lambda>x. Im (Ln x))"
using z by (simp add: continuous_at_Ln complex_nonpos_Reals_iff)
ultimately
obtain d where d: "d>0" "\<And>x. x \<noteq> z \<Longrightarrow> cmod (x - z) < d \<Longrightarrow> \<bar>Im (Ln x)\<bar> < e"
by (auto simp: continuous_within Lim_within dist_norm)
{ fix x
assume "cmod (x - z) < Re z / 2"
then have "\<bar>Re x - Re z\<bar> < Re z / 2"
by (metis le_less_trans abs_Re_le_cmod minus_complex.simps(1))
then have "0 < Re x"
using z by linarith
}
then show "\<exists>d>0. \<forall>x. 0 \<le> Im x \<longrightarrow> x \<noteq> z \<and> cmod (x - z) < d \<longrightarrow> \<bar>Arg x\<bar> < e"
apply (rule_tac x="min d (Re z / 2)" in exI)
using z d
apply (auto simp: Arg_eq_Im_Ln)
done
qed
qed

lemma continuous_on_upperhalf_Arg: "continuous_on ({z. 0 \<le> Im z} - {0}) Arg"
apply (auto simp: continuous_on_eq_continuous_within)
by (metis Diff_subset continuous_within_subset continuous_within_upperhalf_Arg)

lemma open_Arg_less_Int:
assumes "0 \<le> s" "t \<le> 2*pi"
shows "open ({y. s < Arg y} \<inter> {y. Arg y < t})"
proof -
have 1: "continuous_on (UNIV - \<real>\<^sub>\<ge>\<^sub>0) Arg"
using continuous_at_Arg continuous_at_imp_continuous_within
by (auto simp: continuous_on_eq_continuous_within)
have 2: "open (UNIV - \<real>\<^sub>\<ge>\<^sub>0 :: complex set)"  by (simp add: open_Diff)
have "open ({z. s < z} \<inter> {z. z < t})"
using open_lessThan [of t] open_greaterThan [of s]
by (metis greaterThan_def lessThan_def open_Int)
moreover have "{y. s < Arg y} \<inter> {y. Arg y < t} \<subseteq> - \<real>\<^sub>\<ge>\<^sub>0"
using assms by (auto simp: Arg_real complex_nonneg_Reals_iff complex_is_Real_iff)
ultimately show ?thesis
using continuous_imp_open_vimage [OF 1 2, of  "{z. Re z > s} \<inter> {z. Re z < t}"]
by auto
qed

lemma open_Arg_gt: "open {z. t < Arg z}"
proof (cases "t < 0")
case True then have "{z. t < Arg z} = UNIV"
using Arg_ge_0 less_le_trans by auto
then show ?thesis
by simp
next
case False then show ?thesis
using open_Arg_less_Int [of t "2*pi"] Arg_lt_2pi
by auto
qed

lemma closed_Arg_le: "closed {z. Arg z \<le> t}"
using open_Arg_gt [of t]
by (simp add: closed_def Set.Collect_neg_eq [symmetric] not_le)

subsection\<open>Complex Powers\<close>

lemma powr_to_1 [simp]: "z powr 1 = (z::complex)"

lemma powr_nat:
fixes n::nat and z::complex shows "z powr n = (if z = 0 then 0 else z^n)"

fixes w::complex shows "w powr (z1 + z2) = w powr z1 * w powr z2"

lemma powr_minus_complex:
fixes w::complex shows  "w powr (-z) = inverse(w powr z)"

lemma powr_diff_complex:
fixes w::complex shows  "w powr (z1 - z2) = w powr z1 / w powr z2"
by (simp add: powr_def algebra_simps exp_diff)

lemma norm_powr_real: "w \<in> \<real> \<Longrightarrow> 0 < Re w \<Longrightarrow> norm(w powr z) = exp(Re z * ln(Re w))"
using Im_Ln_eq_0 complex_is_Real_iff norm_complex_def
by auto

lemma cnj_powr:
assumes "Im a = 0 \<Longrightarrow> Re a \<ge> 0"
shows   "cnj (a powr b) = cnj a powr cnj b"
proof (cases "a = 0")
case False
with assms have "a \<notin> \<real>\<^sub>\<le>\<^sub>0" by (auto simp: complex_eq_iff complex_nonpos_Reals_iff)
with False show ?thesis by (simp add: powr_def exp_cnj cnj_Ln)
qed simp

lemma powr_real_real:
"\<lbrakk>w \<in> \<real>; z \<in> \<real>; 0 < Re w\<rbrakk> \<Longrightarrow> w powr z = exp(Re z * ln(Re w))"
by (metis complex_eq complex_is_Real_iff diff_0 diff_0_right diff_minus_eq_add exp_ln exp_not_eq_zero
exp_of_real Ln_of_real mult_zero_right of_real_0 of_real_mult)

lemma powr_of_real:
fixes x::real and y::real
shows "0 \<le> x \<Longrightarrow> of_real x powr (of_real y::complex) = of_real (x powr y)"

lemma norm_powr_real_mono:
"\<lbrakk>w \<in> \<real>; 1 < Re w\<rbrakk>
\<Longrightarrow> cmod(w powr z1) \<le> cmod(w powr z2) \<longleftrightarrow> Re z1 \<le> Re z2"
by (auto simp: powr_def algebra_simps Reals_def Ln_of_real)

lemma powr_times_real:
"\<lbrakk>x \<in> \<real>; y \<in> \<real>; 0 \<le> Re x; 0 \<le> Re y\<rbrakk>
\<Longrightarrow> (x * y) powr z = x powr z * y powr z"
by (auto simp: Reals_def powr_def Ln_times exp_add algebra_simps less_eq_real_def Ln_of_real)

lemma powr_neg_real_complex:
shows   "(- of_real x) powr a = (-1) powr (of_real (sgn x) * a) * of_real x powr (a :: complex)"
proof (cases "x = 0")
assume x: "x \<noteq> 0"
hence "(-x) powr a = exp (a * ln (-of_real x))" by (simp add: powr_def)
also from x have "ln (-of_real x) = Ln (of_real x) + of_real (sgn x) * pi * \<i>"
also from x have "exp (a * ...) = cis pi powr (of_real (sgn x) * a) * of_real x powr a"
also note cis_pi
finally show ?thesis by simp
qed simp_all

lemma has_field_derivative_powr:
fixes z :: complex
shows "z \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> ((\<lambda>z. z powr s) has_field_derivative (s * z powr (s - 1))) (at z)"
apply (cases "z=0", auto)
apply (rule DERIV_transform_at [where d = "norm z" and f = "\<lambda>z. exp (s * Ln z)"])
apply (auto simp: dist_complex_def)
apply (intro derivative_eq_intros | simp)+
done

declare has_field_derivative_powr[THEN DERIV_chain2, derivative_intros]

lemma has_field_derivative_powr_right:
"w \<noteq> 0 \<Longrightarrow> ((\<lambda>z. w powr z) has_field_derivative Ln w * w powr z) (at z)"
apply (intro derivative_eq_intros | simp)+
done

lemma field_differentiable_powr_right:
fixes w::complex
shows
"w \<noteq> 0 \<Longrightarrow> (\<lambda>z. w powr z) field_differentiable (at z)"
using field_differentiable_def has_field_derivative_powr_right by blast

lemma holomorphic_on_powr_right:
"f holomorphic_on s \<Longrightarrow> w \<noteq> 0 \<Longrightarrow> (\<lambda>z. w powr (f z)) holomorphic_on s"
unfolding holomorphic_on_def field_differentiable_def
by (metis (full_types) DERIV_chain' has_field_derivative_powr_right)

lemma norm_powr_real_powr:
"w \<in> \<real> \<Longrightarrow> 0 \<le> Re w \<Longrightarrow> cmod (w powr z) = Re w powr Re z"
by (cases "w = 0") (auto simp add: norm_powr_real powr_def Im_Ln_eq_0
complex_is_Real_iff in_Reals_norm complex_eq_iff)

lemma tendsto_ln_complex [tendsto_intros]:
assumes "(f \<longlongrightarrow> a) F" "a \<notin> \<real>\<^sub>\<le>\<^sub>0"
shows   "((\<lambda>z. ln (f z :: complex)) \<longlongrightarrow> ln a) F"
using tendsto_compose[OF continuous_at_Ln[of a, unfolded isCont_def] assms(1)] assms(2) by simp

lemma tendsto_powr_complex:
fixes f g :: "_ \<Rightarrow> complex"
assumes a: "a \<notin> \<real>\<^sub>\<le>\<^sub>0"
assumes f: "(f \<longlongrightarrow> a) F" and g: "(g \<longlongrightarrow> b) F"
shows   "((\<lambda>z. f z powr g z) \<longlongrightarrow> a powr b) F"
proof -
from a have [simp]: "a \<noteq> 0" by auto
from f g a have "((\<lambda>z. exp (g z * ln (f z))) \<longlongrightarrow> a powr b) F" (is ?P)
by (auto intro!: tendsto_intros simp: powr_def)
also {
have "eventually (\<lambda>z. z \<noteq> 0) (nhds a)"
by (intro t1_space_nhds) simp_all
with f have "eventually (\<lambda>z. f z \<noteq> 0) F" using filterlim_iff by blast
}
hence "?P \<longleftrightarrow> ((\<lambda>z. f z powr g z) \<longlongrightarrow> a powr b) F"
by (intro tendsto_cong refl) (simp_all add: powr_def mult_ac)
finally show ?thesis .
qed

lemma tendsto_powr_complex_0:
fixes f g :: "'a \<Rightarrow> complex"
assumes f: "(f \<longlongrightarrow> 0) F" and g: "(g \<longlongrightarrow> b) F" and b: "Re b > 0"
shows   "((\<lambda>z. f z powr g z) \<longlongrightarrow> 0) F"
proof (rule tendsto_norm_zero_cancel)
define h where
"h = (\<lambda>z. if f z = 0 then 0 else exp (Re (g z) * ln (cmod (f z)) + abs (Im (g z)) * pi))"
{
fix z :: 'a assume z: "f z \<noteq> 0"
define c where "c = abs (Im (g z)) * pi"
from mpi_less_Im_Ln[OF z] Im_Ln_le_pi[OF z]
have "abs (Im (Ln (f z))) \<le> pi" by simp
from mult_left_mono[OF this, of "abs (Im (g z))"]
have "abs (Im (g z) * Im (ln (f z))) \<le> c" by (simp add: abs_mult c_def)
hence "-Im (g z) * Im (ln (f z)) \<le> c" by simp
hence "norm (f z powr g z) \<le> h z" by (simp add: powr_def field_simps h_def c_def)
}
hence le: "norm (f z powr g z) \<le> h z" for z by (cases "f x = 0") (simp_all add: h_def)

have g': "(g \<longlongrightarrow> b) (inf F (principal {z. f z \<noteq> 0}))"
by (rule tendsto_mono[OF _ g]) simp_all
have "((\<lambda>x. norm (f x)) \<longlongrightarrow> 0) (inf F (principal {z. f z \<noteq> 0}))"
by (subst tendsto_norm_zero_iff, rule tendsto_mono[OF _ f]) simp_all
moreover {
have "filterlim (\<lambda>x. norm (f x)) (principal {0<..}) (principal {z. f z \<noteq> 0})"
by (auto simp: filterlim_def)
hence "filterlim (\<lambda>x. norm (f x)) (principal {0<..})
(inf F (principal {z. f z \<noteq> 0}))"
by (rule filterlim_mono) simp_all
}
ultimately have norm: "filterlim (\<lambda>x. norm (f x)) (at_right 0) (inf F (principal {z. f z \<noteq> 0}))"

have A: "LIM x inf F (principal {z. f z \<noteq> 0}). Re (g x) * -ln (cmod (f x)) :> at_top"
by (rule filterlim_tendsto_pos_mult_at_top tendsto_intros g' b
filterlim_compose[OF filterlim_uminus_at_top_at_bot] filterlim_compose[OF ln_at_0] norm)+
have B: "LIM x inf F (principal {z. f z \<noteq> 0}).
-\<bar>Im (g x)\<bar> * pi + -(Re (g x) * ln (cmod (f x))) :> at_top"
by (rule filterlim_tendsto_add_at_top tendsto_intros g')+ (insert A, simp_all)
have C: "(h \<longlongrightarrow> 0) F" unfolding h_def
by (intro filterlim_If tendsto_const filterlim_compose[OF exp_at_bot])
(insert B, auto simp: filterlim_uminus_at_bot algebra_simps)
show "((\<lambda>x. norm (f x powr g x)) \<longlongrightarrow> 0) F"
by (rule Lim_null_comparison[OF always_eventually C]) (insert le, auto)
qed

lemma tendsto_powr_complex' [tendsto_intros]:
fixes f g :: "_ \<Rightarrow> complex"
assumes fz: "a \<notin> \<real>\<^sub>\<le>\<^sub>0 \<or> (a = 0 \<and> Re b > 0)"
assumes fg: "(f \<longlongrightarrow> a) F" "(g \<longlongrightarrow> b) F"
shows   "((\<lambda>z. f z powr g z) \<longlongrightarrow> a powr b) F"
proof (cases "a = 0")
case True
with assms show ?thesis by (auto intro!: tendsto_powr_complex_0)
next
case False
with assms show ?thesis by (auto intro!: tendsto_powr_complex elim!: nonpos_Reals_cases)
qed

lemma continuous_powr_complex:
assumes "f (netlimit F) \<notin> \<real>\<^sub>\<le>\<^sub>0" "continuous F f" "continuous F g"
shows   "continuous F (\<lambda>z. f z powr g z :: complex)"
using assms unfolding continuous_def by (intro tendsto_powr_complex) simp_all

lemma isCont_powr_complex [continuous_intros]:
assumes "f z \<notin> \<real>\<^sub>\<le>\<^sub>0" "isCont f z" "isCont g z"
shows   "isCont (\<lambda>z. f z powr g z :: complex) z"
using assms unfolding isCont_def by (intro tendsto_powr_complex) simp_all

lemma continuous_on_powr_complex [continuous_intros]:
assumes "A \<subseteq> {z. Re (f z) \<ge> 0 \<or> Im (f z) \<noteq> 0}"
assumes "\<And>z. z \<in> A \<Longrightarrow> f z = 0 \<Longrightarrow> Re (g z) > 0"
assumes "continuous_on A f" "continuous_on A g"
shows   "continuous_on A (\<lambda>z. f z powr g z)"
unfolding continuous_on_def
proof
fix z assume z: "z \<in> A"
show "((\<lambda>z. f z powr g z) \<longlongrightarrow> f z powr g z) (at z within A)"
proof (cases "f z = 0")
case False
from assms(1,2) z have "Re (f z) \<ge> 0 \<or> Im (f z) \<noteq> 0" "f z = 0 \<longrightarrow> Re (g z) > 0" by auto
with assms(3,4) z show ?thesis
by (intro tendsto_powr_complex')
(auto elim!: nonpos_Reals_cases simp: complex_eq_iff continuous_on_def)
next
case True
with assms z show ?thesis
by (auto intro!: tendsto_powr_complex_0 simp: continuous_on_def)
qed
qed

subsection\<open>Some Limits involving Logarithms\<close>

lemma lim_Ln_over_power:
fixes s::complex
assumes "0 < Re s"
shows "((\<lambda>n. Ln n / (n powr s)) \<longlongrightarrow> 0) sequentially"
proof (simp add: lim_sequentially dist_norm, clarify)
fix e::real
assume e: "0 < e"
have "\<exists>xo>0. \<forall>x\<ge>xo. 0 < e * 2 + (e * Re s * 2 - 2) * x + e * (Re s)\<^sup>2 * x\<^sup>2"
proof (rule_tac x="2/(e * (Re s)\<^sup>2)" in exI, safe)
show "0 < 2 / (e * (Re s)\<^sup>2)"
using e assms by (simp add: field_simps)
next
fix x::real
assume x: "2 / (e * (Re s)\<^sup>2) \<le> x"
then have "x>0"
using e assms
by (metis less_le_trans mult_eq_0_iff mult_pos_pos pos_less_divide_eq power2_eq_square
zero_less_numeral)
then show "0 < e * 2 + (e * Re s * 2 - 2) * x + e * (Re s)\<^sup>2 * x\<^sup>2"
using e assms x
apply (auto simp: field_simps)
apply (rule_tac y = "e * (x\<^sup>2 * (Re s)\<^sup>2)" in le_less_trans)
apply (auto simp: power2_eq_square field_simps add_pos_pos)
done
qed
then have "\<exists>xo>0. \<forall>x\<ge>xo. x / e < 1 + (Re s * x) + (1/2) * (Re s * x)^2"
using e  by (simp add: field_simps)
then have "\<exists>xo>0. \<forall>x\<ge>xo. x / e < exp (Re s * x)"
using assms
by (force intro: less_le_trans [OF _ exp_lower_taylor_quadratic])
then have "\<exists>xo>0. \<forall>x\<ge>xo. x < e * exp (Re s * x)"
using e   by (auto simp: field_simps)
with e show "\<exists>no. \<forall>n\<ge>no. norm (Ln (of_nat n) / of_nat n powr s) < e"
apply (auto simp: norm_divide norm_powr_real divide_simps)
apply (rule_tac x="nat \<lceil>exp xo\<rceil>" in exI)
apply clarify
apply (drule_tac x="ln n" in spec)
apply (auto simp: exp_less_mono nat_ceiling_le_eq not_le)
apply (metis exp_less_mono exp_ln not_le of_nat_0_less_iff)
done
qed

lemma lim_Ln_over_n: "((\<lambda>n. Ln(of_nat n) / of_nat n) \<longlongrightarrow> 0) sequentially"
using lim_Ln_over_power [of 1]
by simp

lemma Ln_Reals_eq: "x \<in> \<real> \<Longrightarrow> Re x > 0 \<Longrightarrow> Ln x = of_real (ln (Re x))"
using Ln_of_real by force

lemma powr_Reals_eq: "x \<in> \<real> \<Longrightarrow> Re x > 0 \<Longrightarrow> x powr complex_of_real y = of_real (x powr y)"

lemma lim_ln_over_power:
fixes s :: real
assumes "0 < s"
shows "((\<lambda>n. ln n / (n powr s)) \<longlongrightarrow> 0) sequentially"
using lim_Ln_over_power [of "of_real s", THEN filterlim_sequentially_Suc [THEN iffD2]] assms
apply (subst filterlim_sequentially_Suc [symmetric])
Ln_Reals_eq norm_powr_real_powr norm_divide)
done

lemma lim_ln_over_n: "((\<lambda>n. ln(real_of_nat n) / of_nat n) \<longlongrightarrow> 0) sequentially"
using lim_ln_over_power [of 1, THEN filterlim_sequentially_Suc [THEN iffD2]]
apply (subst filterlim_sequentially_Suc [symmetric])
done

lemma lim_1_over_complex_power:
assumes "0 < Re s"
shows "((\<lambda>n. 1 / (of_nat n powr s)) \<longlongrightarrow> 0) sequentially"
proof -
have "\<forall>n>0. 3 \<le> n \<longrightarrow> 1 \<le> ln (real_of_nat n)"
using ln3_gt_1
by (force intro: order_trans [of _ "ln 3"] ln3_gt_1)
moreover have "(\<lambda>n. cmod (Ln (of_nat n) / of_nat n powr s)) \<longlonglongrightarrow> 0"
using lim_Ln_over_power [OF assms]
by (metis tendsto_norm_zero_iff)
ultimately show ?thesis
apply (auto intro!: Lim_null_comparison [where g = "\<lambda>n. norm (Ln(of_nat n) / of_nat n powr s)"])
apply (auto simp: norm_divide divide_simps eventually_sequentially)
done
qed

lemma lim_1_over_real_power:
fixes s :: real
assumes "0 < s"
shows "((\<lambda>n. 1 / (of_nat n powr s)) \<longlongrightarrow> 0) sequentially"
using lim_1_over_complex_power [of "of_real s", THEN filterlim_sequentially_Suc [THEN iffD2]] assms
apply (subst filterlim_sequentially_Suc [symmetric])
apply (simp add: Ln_Reals_eq norm_powr_real_powr norm_divide)
done

lemma lim_1_over_Ln: "((\<lambda>n. 1 / Ln(of_nat n)) \<longlongrightarrow> 0) sequentially"
proof (clarsimp simp add: lim_sequentially dist_norm norm_divide divide_simps)
fix r::real
assume "0 < r"
have ir: "inverse (exp (inverse r)) > 0"
by simp
obtain n where n: "1 < of_nat n * inverse (exp (inverse r))"
using ex_less_of_nat_mult [of _ 1, OF ir]
by auto
then have "exp (inverse r) < of_nat n"
then have "ln (exp (inverse r)) < ln (of_nat n)"
by (metis exp_gt_zero less_trans ln_exp ln_less_cancel_iff)
with \<open>0 < r\<close> have "1 < r * ln (real_of_nat n)"
moreover have "n > 0" using n
using neq0_conv by fastforce
ultimately show "\<exists>no. \<forall>n. Ln (of_nat n) \<noteq> 0 \<longrightarrow> no \<le> n \<longrightarrow> 1 < r * cmod (Ln (of_nat n))"
using n \<open>0 < r\<close>
apply (rule_tac x=n in exI)
apply (auto simp: divide_simps)
apply (erule less_le_trans, auto)
done
qed

lemma lim_1_over_ln: "((\<lambda>n. 1 / ln(real_of_nat n)) \<longlongrightarrow> 0) sequentially"
using lim_1_over_Ln [THEN filterlim_sequentially_Suc [THEN iffD2]]
apply (subst filterlim_sequentially_Suc [symmetric])
apply (simp add: Ln_Reals_eq norm_powr_real_powr norm_divide)
done

subsection\<open>Relation between Square Root and exp/ln, hence its derivative\<close>

lemma csqrt_exp_Ln:
assumes "z \<noteq> 0"
shows "csqrt z = exp(Ln(z) / 2)"
proof -
have "(exp (Ln z / 2))\<^sup>2 = (exp (Ln z))"
by (metis exp_double nonzero_mult_divide_cancel_left times_divide_eq_right zero_neq_numeral)
also have "... = z"
using assms exp_Ln by blast
finally have "csqrt z = csqrt ((exp (Ln z / 2))\<^sup>2)"
by simp
also have "... = exp (Ln z / 2)"
apply (subst csqrt_square)
using cos_gt_zero_pi [of "(Im (Ln z) / 2)"] Im_Ln_le_pi mpi_less_Im_Ln assms
apply (auto simp: Re_exp Im_exp zero_less_mult_iff zero_le_mult_iff, fastforce+)
done
finally show ?thesis using assms csqrt_square
by simp
qed

lemma csqrt_inverse:
assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0"
shows "csqrt (inverse z) = inverse (csqrt z)"
proof (cases "z=0", simp)
assume "z \<noteq> 0"
then show ?thesis
using assms csqrt_exp_Ln Ln_inverse exp_minus
by (simp add: csqrt_exp_Ln Ln_inverse exp_minus)
qed

lemma cnj_csqrt:
assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0"
shows "cnj(csqrt z) = csqrt(cnj z)"
proof (cases "z=0", simp)
assume "z \<noteq> 0"
then show ?thesis
by (simp add: assms cnj_Ln csqrt_exp_Ln exp_cnj)
qed

lemma has_field_derivative_csqrt:
assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0"
shows "(csqrt has_field_derivative inverse(2 * csqrt z)) (at z)"
proof -
have z: "z \<noteq> 0"
using assms by auto
then have *: "inverse z = inverse (2*z) * 2"
have [simp]: "exp (Ln z / 2) * inverse z = inverse (csqrt z)"
by (simp add: z field_simps csqrt_exp_Ln [symmetric]) (metis power2_csqrt power2_eq_square)
have "Im z = 0 \<Longrightarrow> 0 < Re z"
using assms complex_nonpos_Reals_iff not_less by blast
with z have "((\<lambda>z. exp (Ln z / 2)) has_field_derivative inverse (2 * csqrt z)) (at z)"
by (force intro: derivative_eq_intros * simp add: assms)
then show ?thesis
apply (rule DERIV_transform_at[where d = "norm z"])
apply (intro z derivative_eq_intros | simp add: assms)+
using z
apply (metis csqrt_exp_Ln dist_0_norm less_irrefl)
done
qed

lemma field_differentiable_at_csqrt:
"z \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> csqrt field_differentiable at z"
using field_differentiable_def has_field_derivative_csqrt by blast

lemma field_differentiable_within_csqrt:
"z \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> csqrt field_differentiable (at z within s)"
using field_differentiable_at_csqrt field_differentiable_within_subset by blast

lemma continuous_at_csqrt:
"z \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> continuous (at z) csqrt"

corollary isCont_csqrt' [simp]:
"\<lbrakk>isCont f z; f z \<notin> \<real>\<^sub>\<le>\<^sub>0\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. csqrt (f x)) z"
by (blast intro: isCont_o2 [OF _ continuous_at_csqrt])

lemma continuous_within_csqrt:
"z \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> continuous (at z within s) csqrt"

lemma continuous_on_csqrt [continuous_intros]:
"(\<And>z. z \<in> s \<Longrightarrow> z \<notin> \<real>\<^sub>\<le>\<^sub>0) \<Longrightarrow> continuous_on s csqrt"

lemma holomorphic_on_csqrt:
"(\<And>z. z \<in> s \<Longrightarrow> z \<notin> \<real>\<^sub>\<le>\<^sub>0) \<Longrightarrow> csqrt holomorphic_on s"

lemma continuous_within_closed_nontrivial:
"closed s \<Longrightarrow> a \<notin> s ==> continuous (at a within s) f"
using open_Compl
by (force simp add: continuous_def eventually_at_topological filterlim_iff open_Collect_neg)

lemma continuous_within_csqrt_posreal:
"continuous (at z within (\<real> \<inter> {w. 0 \<le> Re(w)})) csqrt"
proof (cases "z \<in> \<real>\<^sub>\<le>\<^sub>0")
case True
then have "Im z = 0" "Re z < 0 \<or> z = 0"
using cnj.code complex_cnj_zero_iff  by (auto simp: complex_nonpos_Reals_iff) fastforce
then show ?thesis
apply (auto simp: continuous_within_closed_nontrivial [OF closed_Real_halfspace_Re_ge])
apply (auto simp: continuous_within_eps_delta norm_conv_dist [symmetric])
apply (rule_tac x="e^2" in exI)
apply (auto simp: Reals_def)
by (metis linear not_less real_sqrt_less_iff real_sqrt_pow2_iff real_sqrt_power)
next
case False
then show ?thesis   by (blast intro: continuous_within_csqrt)
qed

subsection\<open>Complex arctangent\<close>

text\<open>The branch cut gives standard bounds in the real case.\<close>

definition Arctan :: "complex \<Rightarrow> complex" where
"Arctan \<equiv> \<lambda>z. (\<i>/2) * Ln((1 - \<i>*z) / (1 + \<i>*z))"

lemma Arctan_def_moebius: "Arctan z = \<i>/2 * Ln (moebius (-\<i>) 1 \<i> 1 z)"

lemma Ln_conv_Arctan:
assumes "z \<noteq> -1"
shows   "Ln z = -2*\<i> * Arctan (moebius 1 (- 1) (- \<i>) (- \<i>) z)"
proof -
have "Arctan (moebius 1 (- 1) (- \<i>) (- \<i>) z) =
\<i>/2 * Ln (moebius (- \<i>) 1 \<i> 1 (moebius 1 (- 1) (- \<i>) (- \<i>) z))"
also from assms have "\<i> * z \<noteq> \<i> * (-1)" by (subst mult_left_cancel) simp
hence "\<i> * z - -\<i> \<noteq> 0" by (simp add: eq_neg_iff_add_eq_0)
from moebius_inverse'[OF _ this, of 1 1]
have "moebius (- \<i>) 1 \<i> 1 (moebius 1 (- 1) (- \<i>) (- \<i>) z) = z" by simp
finally show ?thesis by (simp add: field_simps)
qed

lemma Arctan_0 [simp]: "Arctan 0 = 0"

lemma Im_complex_div_lemma: "Im((1 - \<i>*z) / (1 + \<i>*z)) = 0 \<longleftrightarrow> Re z = 0"
by (auto simp: Im_complex_div_eq_0 algebra_simps)

lemma Re_complex_div_lemma: "0 < Re((1 - \<i>*z) / (1 + \<i>*z)) \<longleftrightarrow> norm z < 1"
by (simp add: Re_complex_div_gt_0 algebra_simps cmod_def power2_eq_square)

lemma tan_Arctan:
assumes "z\<^sup>2 \<noteq> -1"
shows [simp]:"tan(Arctan z) = z"
proof -
have "1 + \<i>*z \<noteq> 0"
by (metis assms complex_i_mult_minus i_squared minus_unique power2_eq_square power2_minus)
moreover
have "1 - \<i>*z \<noteq> 0"
by (metis assms complex_i_mult_minus i_squared power2_eq_square power2_minus right_minus_eq)
ultimately
show ?thesis
by (simp add: Arctan_def tan_def sin_exp_eq cos_exp_eq exp_minus csqrt_exp_Ln [symmetric]
divide_simps power2_eq_square [symmetric])
qed

lemma Arctan_tan [simp]:
assumes "\<bar>Re z\<bar> < pi/2"
shows "Arctan(tan z) = z"
proof -
have ge_pi2: "\<And>n::int. \<bar>of_int (2*n + 1) * pi/2\<bar> \<ge> pi/2"
by (case_tac n rule: int_cases) (auto simp: abs_mult)
have "exp (\<i>*z)*exp (\<i>*z) = -1 \<longleftrightarrow> exp (2*\<i>*z) = -1"
also have "... \<longleftrightarrow> exp (2*\<i>*z) = exp (\<i>*pi)"
using cis_conv_exp cis_pi by auto
also have "... \<longleftrightarrow> exp (2*\<i>*z - \<i>*pi) = 1"
also have "... \<longleftrightarrow> Re(\<i>*2*z - \<i>*pi) = 0 \<and> (\<exists>n::int. Im(\<i>*2*z - \<i>*pi) = of_int (2 * n) * pi)"
also have "... \<longleftrightarrow> Im z = 0 \<and> (\<exists>n::int. 2 * Re z = of_int (2*n + 1) * pi)"
also have "... \<longleftrightarrow> False"
using assms ge_pi2
apply (auto simp: algebra_simps)
by (metis abs_mult_pos not_less of_nat_less_0_iff of_nat_numeral)
finally have *: "exp (\<i>*z)*exp (\<i>*z) + 1 \<noteq> 0"
show ?thesis
using assms *
apply (simp add: Arctan_def tan_def sin_exp_eq cos_exp_eq exp_minus divide_simps
ii_times_eq_iff power2_eq_square [symmetric])
apply (rule Ln_unique)
apply (auto simp: divide_simps exp_minus)
apply (simp add: algebra_simps exp_double [symmetric])
done
qed

lemma
assumes "Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1"
shows Re_Arctan_bounds: "\<bar>Re(Arctan z)\<bar> < pi/2"
and has_field_derivative_Arctan: "(Arctan has_field_derivative inverse(1 + z\<^sup>2)) (at z)"
proof -
have nz0: "1 + \<i>*z \<noteq> 0"
using assms
by (metis abs_one complex_i_mult_minus diff_0_right diff_minus_eq_add ii.simps(1) ii.simps(2)
less_irrefl minus_diff_eq mult.right_neutral right_minus_eq)
have "z \<noteq> -\<i>" using assms
by auto
then have zz: "1 + z * z \<noteq> 0"
by (metis abs_one assms i_squared ii.simps less_irrefl minus_unique square_eq_iff)
have nz1: "1 - \<i>*z \<noteq> 0"
using assms by (force simp add: ii_times_eq_iff)
have nz2: "inverse (1 + \<i>*z) \<noteq> 0"
using assms
by (metis Im_complex_div_lemma Re_complex_div_lemma cmod_eq_Im divide_complex_def
less_irrefl mult_zero_right zero_complex.simps(1) zero_complex.simps(2))
have nzi: "((1 - \<i>*z) * inverse (1 + \<i>*z)) \<noteq> 0"
using nz1 nz2 by auto
have "Im ((1 - \<i>*z) / (1 + \<i>*z)) = 0 \<Longrightarrow> 0 < Re ((1 - \<i>*z) / (1 + \<i>*z))"
apply (simp add: divide_simps split: if_split_asm)
using assms
apply (auto simp: algebra_simps abs_square_less_1 [unfolded power2_eq_square])
done
then have *: "((1 - \<i>*z) / (1 + \<i>*z)) \<notin> \<real>\<^sub>\<le>\<^sub>0"
show "\<bar>Re(Arctan z)\<bar> < pi/2"
unfolding Arctan_def divide_complex_def
using mpi_less_Im_Ln [OF nzi]
apply (auto simp: abs_if intro!: Im_Ln_less_pi * [unfolded divide_complex_def])
done
show "(Arctan has_field_derivative inverse(1 + z\<^sup>2)) (at z)"
unfolding Arctan_def scaleR_conv_of_real
apply (rule DERIV_cong)
apply (intro derivative_eq_intros | simp add: nz0 *)+
using nz0 nz1 zz
apply (auto simp: algebra_simps)
done
qed

lemma field_differentiable_at_Arctan: "(Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1) \<Longrightarrow> Arctan field_differentiable at z"
using has_field_derivative_Arctan
by (auto simp: field_differentiable_def)

lemma field_differentiable_within_Arctan:
"(Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1) \<Longrightarrow> Arctan field_differentiable (at z within s)"
using field_differentiable_at_Arctan field_differentiable_at_within by blast

declare has_field_derivative_Arctan [derivative_intros]
declare has_field_derivative_Arctan [THEN DERIV_chain2, derivative_intros]

lemma continuous_at_Arctan:
"(Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1) \<Longrightarrow> continuous (at z) Arctan"

lemma continuous_within_Arctan:
"(Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1) \<Longrightarrow> continuous (at z within s) Arctan"
using continuous_at_Arctan continuous_at_imp_continuous_within by blast

lemma continuous_on_Arctan [continuous_intros]:
"(\<And>z. z \<in> s \<Longrightarrow> Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1) \<Longrightarrow> continuous_on s Arctan"
by (auto simp: continuous_at_imp_continuous_on continuous_within_Arctan)

lemma holomorphic_on_Arctan:
"(\<And>z. z \<in> s \<Longrightarrow> Re z = 0 \<Longrightarrow> \<bar>Im z\<bar> < 1) \<Longrightarrow> Arctan holomorphic_on s"

lemma Arctan_series:
assumes z: "norm (z :: complex) < 1"
defines "g \<equiv> \<lambda>n. if odd n then -\<i>*\<i>^n / n else 0"
defines "h \<equiv> \<lambda>z n. (-1)^n / of_nat (2*n+1) * (z::complex)^(2*n+1)"
shows   "(\<lambda>n. g n * z^n) sums Arctan z"
and     "h z sums Arctan z"
proof -
define G where [abs_def]: "G z = (\<Sum>n. g n * z^n)" for z
have summable: "summable (\<lambda>n. g n * u^n)" if "norm u < 1" for u
proof (cases "u = 0")
assume u: "u \<noteq> 0"
have "(\<lambda>n. ereal (norm (h u n) / norm (h u (Suc n)))) = (\<lambda>n. ereal (inverse (norm u)^2) *
ereal ((2 + inverse (real (Suc n))) / (2 - inverse (real (Suc n)))))"
proof
fix n
have "ereal (norm (h u n) / norm (h u (Suc n))) =
ereal (inverse (norm u)^2) * ereal ((of_nat (2*Suc n+1) / of_nat (Suc n)) /
(of_nat (2*Suc n-1) / of_nat (Suc n)))"
by (simp add: h_def norm_mult norm_power norm_divide divide_simps
also have "of_nat (2*Suc n+1) / of_nat (Suc n) = (2::real) + inverse (real (Suc n))"
by (auto simp: divide_simps simp del: of_nat_Suc) simp_all?
also have "of_nat (2*Suc n-1) / of_nat (Suc n) = (2::real) - inverse (real (Suc n))"
by (auto simp: divide_simps simp del: of_nat_Suc) simp_all?
finally show "ereal (norm (h u n) / norm (h u (Suc n))) = ereal (inverse (norm u)^2) *
ereal ((2 + inverse (real (Suc n))) / (2 - inverse (real (Suc n))))" .
qed
also have "\<dots> \<longlonglongrightarrow> ereal (inverse (norm u)^2) * ereal ((2 + 0) / (2 - 0))"
by (intro tendsto_intros LIMSEQ_inverse_real_of_nat) simp_all
finally have "liminf (\<lambda>n. ereal (cmod (h u n) / cmod (h u (Suc n)))) = inverse (norm u)^2"
by (intro lim_imp_Liminf) simp_all
moreover from power_strict_mono[OF that, of 2] u have "inverse (norm u)^2 > 1"
ultimately have A: "liminf (\<lambda>n. ereal (cmod (h u n) / cmod (h u (Suc n)))) > 1" by simp
from u have "summable (h u)"
by (intro summable_norm_cancel[OF ratio_test_convergence[OF _ A]])
(auto simp: h_def norm_divide norm_mult norm_power simp del: of_nat_Suc
intro!: mult_pos_pos divide_pos_pos always_eventually)
thus "summable (\<lambda>n. g n * u^n)"
by (subst summable_mono_reindex[of "\<lambda>n. 2*n+1", symmetric])
(auto simp: power_mult subseq_def g_def h_def elim!: oddE)

have "\<exists>c. \<forall>u\<in>ball 0 1. Arctan u - G u = c"
proof (rule has_field_derivative_zero_constant)
fix u :: complex assume "u \<in> ball 0 1"
hence u: "norm u < 1" by (simp add: dist_0_norm)
define K where "K = (norm u + 1) / 2"
from u and abs_Im_le_cmod[of u] have Im_u: "\<bar>Im u\<bar> < 1" by linarith
from u have K: "0 \<le> K" "norm u < K" "K < 1" by (simp_all add: K_def)
hence "(G has_field_derivative (\<Sum>n. diffs g n * u ^ n)) (at u)" unfolding G_def
by (intro termdiffs_strong[of _ "of_real K"] summable) simp_all
also have "(\<lambda>n. diffs g n * u^n) = (\<lambda>n. if even n then (\<i>*u)^n else 0)"
by (intro ext) (simp_all del: of_nat_Suc add: g_def diffs_def power_mult_distrib)
also have "suminf \<dots> = (\<Sum>n. (-(u^2))^n)"
by (subst suminf_mono_reindex[of "\<lambda>n. 2*n", symmetric])
(auto elim!: evenE simp: subseq_def power_mult power_mult_distrib)
also from u have "norm u^2 < 1^2" by (intro power_strict_mono) simp_all
hence "(\<Sum>n. (-(u^2))^n) = inverse (1 + u^2)"
by (subst suminf_geometric) (simp_all add: norm_power inverse_eq_divide)
finally have "(G has_field_derivative inverse (1 + u\<^sup>2)) (at u)" .
from DERIV_diff[OF has_field_derivative_Arctan this] Im_u u
show "((\<lambda>u. Arctan u - G u) has_field_derivative 0) (at u within ball 0 1)"
by (simp_all add: dist_0_norm at_within_open[OF _ open_ball])
qed simp_all
then obtain c where c: "\<And>u. norm u < 1 \<Longrightarrow> Arctan u - G u = c" by (auto simp: dist_0_norm)
from this[of 0] have "c = 0" by (simp add: G_def g_def powser_zero)
with c z have "Arctan z = G z" by simp
with summable[OF z] show "(\<lambda>n. g n * z^n) sums Arctan z" unfolding G_def by (simp add: sums_iff)
thus "h z sums Arctan z" by (subst (asm) sums_mono_reindex[of "\<lambda>n. 2*n+1", symmetric])
(auto elim!: oddE simp: subseq_def power_mult g_def h_def)
qed

text \<open>A quickly-converging series for the logarithm, based on the arctangent.\<close>
assumes x: "x > (0::real)"
shows "(\<lambda>n. (2*((x - 1) / (x + 1)) ^ (2*n+1) / of_nat (2*n+1))) sums ln x"
proof -
define y :: complex where "y = of_real ((x-1)/(x+1))"
from x have x': "complex_of_real x \<noteq> of_real (-1)"  by (subst of_real_eq_iff) auto
from x have "\<bar>x - 1\<bar> < \<bar>x + 1\<bar>" by linarith
hence "norm (complex_of_real (x - 1) / complex_of_real (x + 1)) < 1"
hence "norm (\<i> * y) < 1" unfolding y_def by (subst norm_mult) simp
hence "(\<lambda>n. (-2*\<i>) * ((-1)^n / of_nat (2*n+1) * (\<i>*y)^(2*n+1))) sums ((-2*\<i>) * Arctan (\<i>*y))"
by (intro Arctan_series sums_mult) simp_all
also have "(\<lambda>n. (-2*\<i>) * ((-1)^n / of_nat (2*n+1) * (\<i>*y)^(2*n+1))) =
(\<lambda>n. (-2*\<i>) * ((-1)^n * (\<i>*y*(-y\<^sup>2)^n)/of_nat (2*n+1)))"
by (intro ext) (simp_all add: power_mult power_mult_distrib)
also have "\<dots> = (\<lambda>n. 2*y* ((-1) * (-y\<^sup>2))^n/of_nat (2*n+1))"
by (intro ext, subst power_mult_distrib) (simp add: algebra_simps power_mult)
also have "\<dots> = (\<lambda>n. 2*y^(2*n+1) / of_nat (2*n+1))"
also have "\<dots> = (\<lambda>n. of_real (2*((x-1)/(x+1))^(2*n+1) / of_nat (2*n+1)))"
by (intro ext) (simp add: y_def)
also have "\<i> * y = (of_real x - 1) / (-\<i> * (of_real x + 1))"
by (subst divide_divide_eq_left [symmetric]) (simp add: y_def)
also have "\<dots> = moebius 1 (-1) (-\<i>) (-\<i>) (of_real x)" by (simp add: moebius_def algebra_simps)
also from x' have "-2*\<i>*Arctan \<dots> = Ln (of_real x)" by (intro Ln_conv_Arctan [symmetric]) simp_all
also from x have "\<dots> = ln x" by (rule Ln_of_real)
finally show ?thesis by (subst (asm) sums_of_real_iff)
qed

subsection \<open>Real arctangent\<close>

lemma norm_exp_ii_times [simp]: "norm (exp(\<i> * of_real y)) = 1"
by simp

lemma norm_exp_imaginary: "norm(exp z) = 1 \<Longrightarrow> Re z = 0"

lemma Im_Arctan_of_real [simp]: "Im (Arctan (of_real x)) = 0"
unfolding Arctan_def divide_complex_def
apply (rule norm_exp_imaginary)
apply (subst exp_Ln, auto)
apply (auto simp: divide_simps)
apply (metis power_one sum_power2_eq_zero_iff zero_neq_one, algebra)
done

lemma arctan_eq_Re_Arctan: "arctan x = Re (Arctan (of_real x))"
proof (rule arctan_unique)
show "- (pi / 2) < Re (Arctan (complex_of_real x))"
apply (rule Im_Ln_less_pi)
apply (auto simp: Im_complex_div_lemma complex_nonpos_Reals_iff)
done
next
have *: " (1 - \<i>*x) / (1 + \<i>*x) \<noteq> 0"
show "Re (Arctan (complex_of_real x)) < pi / 2"
using mpi_less_Im_Ln [OF *]
next
have "tan (Re (Arctan (of_real x))) = Re (tan (Arctan (of_real x)))"
apply (auto simp: tan_def Complex.Re_divide Re_sin Re_cos Im_sin Im_cos)
also have "... = x"
apply (subst tan_Arctan, auto)
by (metis diff_0_right minus_diff_eq mult_zero_left not_le of_real_1 of_real_eq_iff of_real_minus of_real_power power2_eq_square real_minus_mult_self_le zero_less_one)
finally show "tan (Re (Arctan (complex_of_real x))) = x" .
qed

lemma Arctan_of_real: "Arctan (of_real x) = of_real (arctan x)"
unfolding arctan_eq_Re_Arctan divide_complex_def

lemma Arctan_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> Arctan z \<in> \<real>"
by (metis Reals_cases Reals_of_real Arctan_of_real)

declare arctan_one [simp]

lemma arctan_less_pi4_pos: "x < 1 \<Longrightarrow> arctan x < pi/4"
by (metis arctan_less_iff arctan_one)

lemma arctan_less_pi4_neg: "-1 < x \<Longrightarrow> -(pi/4) < arctan x"
by (metis arctan_less_iff arctan_minus arctan_one)

lemma arctan_less_pi4: "\<bar>x\<bar> < 1 \<Longrightarrow> \<bar>arctan x\<bar> < pi/4"
by (metis abs_less_iff arctan_less_pi4_pos arctan_minus)

lemma arctan_le_pi4: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> \<bar>arctan x\<bar> \<le> pi/4"
by (metis abs_le_iff arctan_le_iff arctan_minus arctan_one)

lemma abs_arctan: "\<bar>arctan x\<bar> = arctan \<bar>x\<bar>"

assumes "\<bar>arctan x + arctan y\<bar> < pi/2"
shows "arctan x + arctan y = arctan((x + y) / (1 - x * y))"
proof (rule arctan_unique [symmetric])
show 12: "- (pi / 2) < arctan x + arctan y" "arctan x + arctan y < pi / 2"
using assms by linarith+
show "tan (arctan x + arctan y) = (x + y) / (1 - x * y)"
using cos_gt_zero_pi [OF 12]
qed

lemma arctan_inverse:
assumes "0 < x"
shows "arctan(inverse x) = pi/2 - arctan x"
proof -
have "arctan(inverse x) = arctan(inverse(tan(arctan x)))"
also have "... = arctan (tan (pi / 2 - arctan x))"
also have "... = pi/2 - arctan x"
proof -
have "0 < pi - arctan x"
using arctan_ubound [of x] pi_gt_zero by linarith
with assms show ?thesis
qed
finally show ?thesis .
qed

assumes "\<bar>x * y\<bar> < 1"
shows "(arctan x + arctan y = arctan((x + y) / (1 - x * y)))"
proof (cases "x = 0 \<or> y = 0")
case True then show ?thesis
by auto
next
case False
then have *: "\<bar>arctan x\<bar> < pi / 2 - \<bar>arctan y\<bar>" using assms
apply (auto simp add: abs_arctan arctan_inverse [symmetric] arctan_less_iff)
done
show ?thesis
using * by linarith
qed

lemma abs_arctan_le:
fixes x::real shows "\<bar>arctan x\<bar> \<le> \<bar>x\<bar>"
proof -
{ fix w::complex and z::complex
assume *: "w \<in> \<real>" "z \<in> \<real>"
have "cmod (Arctan w - Arctan z) \<le> 1 * cmod (w-z)"
apply (rule field_differentiable_bound [OF convex_Reals, of Arctan _ 1])
apply (rule has_field_derivative_at_within [OF has_field_derivative_Arctan])
apply (simp add: norm_divide divide_simps in_Reals_norm complex_is_Real_iff power2_eq_square)
using * by auto
}
then have "cmod (Arctan (of_real x) - Arctan 0) \<le> 1 * cmod (of_real x -0)"
using Reals_0 Reals_of_real by blast
then show ?thesis
qed

lemma arctan_le_self: "0 \<le> x \<Longrightarrow> arctan x \<le> x"
by (metis abs_arctan_le abs_of_nonneg zero_le_arctan_iff)

lemma abs_tan_ge: "\<bar>x\<bar> < pi/2 \<Longrightarrow> \<bar>x\<bar> \<le> \<bar>tan x\<bar>"
by (metis abs_arctan_le abs_less_iff arctan_tan minus_less_iff)

lemma arctan_bounds:
assumes "0 \<le> x" "x < 1"
shows arctan_lower_bound:
"(\<Sum>k<2 * n. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1))) \<le> arctan x"
(is "(\<Sum>k<_. (- 1)^ k * ?a k) \<le> _")
and arctan_upper_bound:
"arctan x \<le> (\<Sum>k<2 * n + 1. (- 1) ^ k * (1 / real (k * 2 + 1) * x ^ (k * 2 + 1)))"
proof -
have tendsto_zero: "?a \<longlonglongrightarrow> 0"
using assms
apply -
apply (rule tendsto_eq_rhs[where x="0 * 0"])
subgoal by (intro tendsto_mult real_tendsto_divide_at_top)
(auto simp: filterlim_real_sequentially filterlim_sequentially_iff_filterlim_real
intro!: real_tendsto_divide_at_top tendsto_power_zero filterlim_real_sequentially
subgoal by simp
done
have nonneg: "0 \<le> ?a n" for n
by (force intro!: divide_nonneg_nonneg mult_nonneg_nonneg zero_le_power assms)
have le: "?a (Suc n) \<le> ?a n" for n
by (rule mult_mono[OF _ power_decreasing]) (auto simp: divide_simps assms less_imp_le)
from summable_Leibniz'(4)[of ?a, OF tendsto_zero nonneg le, of n]
summable_Leibniz'(2)[of ?a, OF tendsto_zero nonneg le, of n]
assms
show "(\<Sum>k<2*n. (- 1)^ k * ?a k) \<le> arctan x" "arctan x \<le> (\<Sum>k<2 * n + 1. (- 1)^ k * ?a k)"
by (auto simp: arctan_series)
qed

subsection \<open>Bounds on pi using real arctangent\<close>

lemma pi_machin: "pi = 16 * arctan (1 / 5) - 4 * arctan (1 / 239)"
using machin
by simp

lemma pi_approx: "3.141592653588 \<le> pi" "pi \<le> 3.1415926535899"
unfolding pi_machin
using arctan_bounds[of "1/5"   4]
arctan_bounds[of "1/239" 4]

subsection\<open>Inverse Sine\<close>

definition Arcsin :: "complex \<Rightarrow> complex" where
"Arcsin \<equiv> \<lambda>z. -\<i> * Ln(\<i> * z + csqrt(1 - z\<^sup>2))"

lemma Arcsin_body_lemma: "\<i> * z + csqrt(1 - z\<^sup>2) \<noteq> 0"
using power2_csqrt [of "1 - z\<^sup>2"]
apply auto
by (metis complex_i_mult_minus diff_add_cancel diff_minus_eq_add diff_self mult.assoc mult.left_commute numeral_One power2_csqrt power2_eq_square zero_neq_numeral)

lemma Arcsin_range_lemma: "\<bar>Re z\<bar> < 1 \<Longrightarrow> 0 < Re(\<i> * z + csqrt(1 - z\<^sup>2))"
using Complex.cmod_power2 [of z, symmetric]
by (simp add: real_less_rsqrt algebra_simps Re_power2 cmod_square_less_1_plus)

lemma Re_Arcsin: "Re(Arcsin z) = Im (Ln (\<i> * z + csqrt(1 - z\<^sup>2)))"

lemma Im_Arcsin: "Im(Arcsin z) = - ln (cmod (\<i> * z + csqrt (1 - z\<^sup>2)))"

lemma one_minus_z2_notin_nonpos_Reals:
assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
shows "1 - z\<^sup>2 \<notin> \<real>\<^sub>\<le>\<^sub>0"
using assms
apply (auto simp: complex_nonpos_Reals_iff Re_power2 Im_power2)
using power2_less_0 [of "Im z"] apply force
using abs_square_less_1 not_le by blast

lemma isCont_Arcsin_lemma:
assumes le0: "Re (\<i> * z + csqrt (1 - z\<^sup>2)) \<le> 0" and "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
shows False
proof (cases "Im z = 0")
case True
then show ?thesis
using assms by (fastforce simp: cmod_def abs_square_less_1 [symmetric])
next
case False
have neq: "(cmod z)\<^sup>2 \<noteq> 1 + cmod (1 - z\<^sup>2)"
assume "(Re z)\<^sup>2 + (Im z)\<^sup>2 = 1 + sqrt ((1 - Re (z\<^sup>2))\<^sup>2 + (Im (z\<^sup>2))\<^sup>2)"
then have "((Re z)\<^sup>2 + (Im z)\<^sup>2 - 1)\<^sup>2 = ((1 - Re (z\<^sup>2))\<^sup>2 + (Im (z\<^sup>2))\<^sup>2)"
by simp
then show False using False
qed
moreover have 2: "(Im z)\<^sup>2 = (1 + ((Im z)\<^sup>2 + cmod (1 - z\<^sup>2)) - (Re z)\<^sup>2) / 2"
using le0
apply simp
apply (drule sqrt_le_D)
using cmod_power2 [of z] norm_triangle_ineq2 [of "z^2" 1]
apply (simp add: norm_power Re_power2 norm_minus_commute [of 1])
done
ultimately show False
by (simp add: Re_power2 Im_power2 cmod_power2)
qed

lemma isCont_Arcsin:
assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
shows "isCont Arcsin z"
proof -
have *: "\<i> * z + csqrt (1 - z\<^sup>2) \<notin> \<real>\<^sub>\<le>\<^sub>0"
by (metis isCont_Arcsin_lemma assms complex_nonpos_Reals_iff)
show ?thesis
using assms
apply (rule isCont_Ln' isCont_csqrt' continuous_intros)+
apply (rule *)
done
qed

lemma isCont_Arcsin' [simp]:
shows "isCont f z \<Longrightarrow> (Im (f z) = 0 \<Longrightarrow> \<bar>Re (f z)\<bar> < 1) \<Longrightarrow> isCont (\<lambda>x. Arcsin (f x)) z"
by (blast intro: isCont_o2 [OF _ isCont_Arcsin])

lemma sin_Arcsin [simp]: "sin(Arcsin z) = z"
proof -
have "\<i>*z*2 + csqrt (1 - z\<^sup>2)*2 = 0 \<longleftrightarrow> (\<i>*z)*2 + csqrt (1 - z\<^sup>2)*2 = 0"
by (simp add: algebra_simps)  \<comment>\<open>Cancelling a factor of 2\<close>
moreover have "... \<longleftrightarrow> (\<i>*z) + csqrt (1 - z\<^sup>2) = 0"
by (metis Arcsin_body_lemma distrib_right no_zero_divisors zero_neq_numeral)
ultimately show ?thesis
apply (simp add: sin_exp_eq Arcsin_def Arcsin_body_lemma exp_minus divide_simps)
apply (simp add: power2_eq_square [symmetric] algebra_simps)
done
qed

lemma Re_eq_pihalf_lemma:
"\<bar>Re z\<bar> = pi/2 \<Longrightarrow> Im z = 0 \<Longrightarrow>
Re ((exp (\<i>*z) + inverse (exp (\<i>*z))) / 2) = 0 \<and> 0 \<le> Im ((exp (\<i>*z) + inverse (exp (\<i>*z))) / 2)"
apply (simp add: cos_ii_times [symmetric] Re_cos Im_cos abs_if del: eq_divide_eq_numeral1)
by (metis cos_minus cos_pi_half)

lemma Re_less_pihalf_lemma:
assumes "\<bar>Re z\<bar> < pi / 2"
shows "0 < Re ((exp (\<i>*z) + inverse (exp (\<i>*z))) / 2)"
proof -
have "0 < cos (Re z)" using assms
using cos_gt_zero_pi by auto
then show ?thesis
qed

lemma Arcsin_sin:
assumes "\<bar>Re z\<bar> < pi/2 \<or> (\<bar>Re z\<bar> = pi/2 \<and> Im z = 0)"
shows "Arcsin(sin z) = z"
proof -
have "Arcsin(sin z) = - (\<i> * Ln (csqrt (1 - (\<i> * (exp (\<i>*z) - inverse (exp (\<i>*z))))\<^sup>2 / 4) - (inverse (exp (\<i>*z)) - exp (\<i>*z)) / 2))"
by (simp add: sin_exp_eq Arcsin_def exp_minus power_divide)
also have "... = - (\<i> * Ln (csqrt (((exp (\<i>*z) + inverse (exp (\<i>*z)))/2)\<^sup>2) - (inverse (exp (\<i>*z)) - exp (\<i>*z)) / 2))"
also have "... = - (\<i> * Ln (((exp (\<i>*z) + inverse (exp (\<i>*z)))/2) - (inverse (exp (\<i>*z)) - exp (\<i>*z)) / 2))"
apply (subst csqrt_square)
using assms Re_eq_pihalf_lemma Re_less_pihalf_lemma
apply auto
done
also have "... =  - (\<i> * Ln (exp (\<i>*z)))"
also have "... = z"
apply (subst Complex_Transcendental.Ln_exp)
using assms
apply (auto simp: abs_if simp del: eq_divide_eq_numeral1 split: if_split_asm)
done
finally show ?thesis .
qed

lemma Arcsin_unique:
"\<lbrakk>sin z = w; \<bar>Re z\<bar> < pi/2 \<or> (\<bar>Re z\<bar> = pi/2 \<and> Im z = 0)\<rbrakk> \<Longrightarrow> Arcsin w = z"
by (metis Arcsin_sin)

lemma Arcsin_0 [simp]: "Arcsin 0 = 0"
by (metis Arcsin_sin norm_zero pi_half_gt_zero real_norm_def sin_zero zero_complex.simps(1))

lemma Arcsin_1 [simp]: "Arcsin 1 = pi/2"
by (metis Arcsin_sin Im_complex_of_real Re_complex_of_real numeral_One of_real_numeral pi_half_ge_zero real_sqrt_abs real_sqrt_pow2 real_sqrt_power sin_of_real sin_pi_half)

lemma Arcsin_minus_1 [simp]: "Arcsin(-1) = - (pi/2)"
by (metis Arcsin_1 Arcsin_sin Im_complex_of_real Re_complex_of_real abs_of_nonneg of_real_minus pi_half_ge_zero power2_minus real_sqrt_abs sin_Arcsin sin_minus)

lemma has_field_derivative_Arcsin:
assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
shows "(Arcsin has_field_derivative inverse(cos(Arcsin z))) (at z)"
proof -
have "(sin (Arcsin z))\<^sup>2 \<noteq> 1"
using assms
apply atomize
apply (auto simp: complex_eq_iff Re_power2 Im_power2 abs_square_eq_1)
apply (metis abs_minus_cancel abs_one abs_power2 numeral_One numeral_neq_neg_one)
by (metis abs_minus_cancel abs_one abs_power2 one_neq_neg_one)
then have "cos (Arcsin z) \<noteq> 0"
by (metis diff_0_right power_zero_numeral sin_squared_eq)
then show ?thesis
apply (rule has_complex_derivative_inverse_basic [OF DERIV_sin _ _ open_ball [of z 1]])
apply (auto intro: isCont_Arcsin assms)
done
qed

declare has_field_derivative_Arcsin [derivative_intros]
declare has_field_derivative_Arcsin [THEN DERIV_chain2, derivative_intros]

lemma field_differentiable_at_Arcsin:
"(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arcsin field_differentiable at z"
using field_differentiable_def has_field_derivative_Arcsin by blast

lemma field_differentiable_within_Arcsin:
"(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arcsin field_differentiable (at z within s)"
using field_differentiable_at_Arcsin field_differentiable_within_subset by blast

lemma continuous_within_Arcsin:
"(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous (at z within s) Arcsin"
using continuous_at_imp_continuous_within isCont_Arcsin by blast

lemma continuous_on_Arcsin [continuous_intros]:
"(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous_on s Arcsin"

lemma holomorphic_on_Arcsin: "(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arcsin holomorphic_on s"

subsection\<open>Inverse Cosine\<close>

definition Arccos :: "complex \<Rightarrow> complex" where
"Arccos \<equiv> \<lambda>z. -\<i> * Ln(z + \<i> * csqrt(1 - z\<^sup>2))"

lemma Arccos_range_lemma: "\<bar>Re z\<bar> < 1 \<Longrightarrow> 0 < Im(z + \<i> * csqrt(1 - z\<^sup>2))"
using Arcsin_range_lemma [of "-z"]
by simp

lemma Arccos_body_lemma: "z + \<i> * csqrt(1 - z\<^sup>2) \<noteq> 0"
using Arcsin_body_lemma [of z]
by (metis complex_i_mult_minus diff_add_cancel minus_diff_eq minus_unique mult.assoc mult.left_commute
power2_csqrt power2_eq_square zero_neq_one)

lemma Re_Arccos: "Re(Arccos z) = Im (Ln (z + \<i> * csqrt(1 - z\<^sup>2)))"

lemma Im_Arccos: "Im(Arccos z) = - ln (cmod (z + \<i> * csqrt (1 - z\<^sup>2)))"

text\<open>A very tricky argument to find!\<close>
lemma isCont_Arccos_lemma:
assumes eq0: "Im (z + \<i> * csqrt (1 - z\<^sup>2)) = 0" and "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
shows False
proof (cases "Im z = 0")
case True
then show ?thesis
using assms by (fastforce simp add: cmod_def abs_square_less_1 [symmetric])
next
case False
have Imz: "Im z = - sqrt ((1 + ((Im z)\<^sup>2 + cmod (1 - z\<^sup>2)) - (Re z)\<^sup>2) / 2)"
using eq0 abs_Re_le_cmod [of "1-z\<^sup>2"]
have "(cmod z)\<^sup>2 - 1 \<noteq> cmod (1 - z\<^sup>2)"
assume "(Re z)\<^sup>2 + (Im z)\<^sup>2 - 1 = sqrt ((1 - Re (z\<^sup>2))\<^sup>2 + (Im (z\<^sup>2))\<^sup>2)"
then have "((Re z)\<^sup>2 + (Im z)\<^sup>2 - 1)\<^sup>2 = ((1 - Re (z\<^sup>2))\<^sup>2 + (Im (z\<^sup>2))\<^sup>2)"
by simp
then show False using False
qed
moreover have "(Im z)\<^sup>2 = ((1 + ((Im z)\<^sup>2 + cmod (1 - z\<^sup>2)) - (Re z)\<^sup>2) / 2)"
apply (subst Imz)
using abs_Re_le_cmod [of "1-z\<^sup>2"]
done
ultimately show False
qed

lemma isCont_Arccos:
assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
shows "isCont Arccos z"
proof -
have "z + \<i> * csqrt (1 - z\<^sup>2) \<notin> \<real>\<^sub>\<le>\<^sub>0"
by (metis complex_nonpos_Reals_iff isCont_Arccos_lemma assms)
with assms show ?thesis
apply (rule isCont_Ln' isCont_csqrt' continuous_intros)+
done
qed

lemma isCont_Arccos' [simp]:
shows "isCont f z \<Longrightarrow> (Im (f z) = 0 \<Longrightarrow> \<bar>Re (f z)\<bar> < 1) \<Longrightarrow> isCont (\<lambda>x. Arccos (f x)) z"
by (blast intro: isCont_o2 [OF _ isCont_Arccos])

lemma cos_Arccos [simp]: "cos(Arccos z) = z"
proof -
have "z*2 + \<i> * (2 * csqrt (1 - z\<^sup>2)) = 0 \<longleftrightarrow> z*2 + \<i> * csqrt (1 - z\<^sup>2)*2 = 0"
by (simp add: algebra_simps)  \<comment>\<open>Cancelling a factor of 2\<close>
moreover have "... \<longleftrightarrow> z + \<i> * csqrt (1 - z\<^sup>2) = 0"
by (metis distrib_right mult_eq_0_iff zero_neq_numeral)
ultimately show ?thesis
apply (simp add: cos_exp_eq Arccos_def Arccos_body_lemma exp_minus field_simps)
done
qed

lemma Arccos_cos:
assumes "0 < Re z & Re z < pi \<or>
Re z = 0 & 0 \<le> Im z \<or>
Re z = pi & Im z \<le> 0"
shows "Arccos(cos z) = z"
proof -
have *: "((\<i> - (exp (\<i> * z))\<^sup>2 * \<i>) / (2 * exp (\<i> * z))) = sin z"
by (simp add: sin_exp_eq exp_minus field_simps power2_eq_square)
have "1 - (exp (\<i> * z) + inverse (exp (\<i> * z)))\<^sup>2 / 4 = ((\<i> - (exp (\<i> * z))\<^sup>2 * \<i>) / (2 * exp (\<i> * z)))\<^sup>2"
then have "Arccos(cos z) = - (\<i> * Ln ((exp (\<i> * z) + inverse (exp (\<i> * z))) / 2 +
\<i> * csqrt (((\<i> - (exp (\<i> * z))\<^sup>2 * \<i>) / (2 * exp (\<i> * z)))\<^sup>2)))"
by (simp add: cos_exp_eq Arccos_def exp_minus power_divide)
also have "... = - (\<i> * Ln ((exp (\<i> * z) + inverse (exp (\<i> * z))) / 2 +
\<i> * ((\<i> - (exp (\<i> * z))\<^sup>2 * \<i>) / (2 * exp (\<i> * z)))))"
apply (subst csqrt_square)
using assms Re_sin_pos [of z] Im_sin_nonneg [of z] Im_sin_nonneg2 [of z]
apply (auto simp: * Re_sin Im_sin)
done
also have "... =  - (\<i> * Ln (exp (\<i>*z)))"
also have "... = z"
using assms
apply (subst Complex_Transcendental.Ln_exp, auto)
done
finally show ?thesis .
qed

lemma Arccos_unique:
"\<lbrakk>cos z = w;
0 < Re z \<and> Re z < pi \<or>
Re z = 0 \<and> 0 \<le> Im z \<or>
Re z = pi \<and> Im z \<le> 0\<rbrakk> \<Longrightarrow> Arccos w = z"
using Arccos_cos by blast

lemma Arccos_0 [simp]: "Arccos 0 = pi/2"
by (rule Arccos_unique) (auto simp: of_real_numeral)

lemma Arccos_1 [simp]: "Arccos 1 = 0"
by (rule Arccos_unique) auto

lemma Arccos_minus1: "Arccos(-1) = pi"
by (rule Arccos_unique) auto

lemma has_field_derivative_Arccos:
assumes "(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1)"
shows "(Arccos has_field_derivative - inverse(sin(Arccos z))) (at z)"
proof -
have "(cos (Arccos z))\<^sup>2 \<noteq> 1"
using assms
apply atomize
apply (auto simp: complex_eq_iff Re_power2 Im_power2 abs_square_eq_1)
apply (metis abs_minus_cancel abs_one abs_power2 numeral_One numeral_neq_neg_one)
apply (metis left_minus less_irrefl power_one sum_power2_gt_zero_iff zero_neq_neg_one)
done
then have "- sin (Arccos z) \<noteq> 0"
by (metis cos_squared_eq diff_0_right mult_zero_left neg_0_equal_iff_equal power2_eq_square)
then have "(Arccos has_field_derivative inverse(- sin(Arccos z))) (at z)"
apply (rule has_complex_derivative_inverse_basic [OF DERIV_cos _ _ open_ball [of z 1]])
apply (auto intro: isCont_Arccos assms)
done
then show ?thesis
by simp
qed

declare has_field_derivative_Arcsin [derivative_intros]
declare has_field_derivative_Arcsin [THEN DERIV_chain2, derivative_intros]

lemma field_differentiable_at_Arccos:
"(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arccos field_differentiable at z"
using field_differentiable_def has_field_derivative_Arccos by blast

lemma field_differentiable_within_Arccos:
"(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arccos field_differentiable (at z within s)"
using field_differentiable_at_Arccos field_differentiable_within_subset by blast

lemma continuous_within_Arccos:
"(Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous (at z within s) Arccos"
using continuous_at_imp_continuous_within isCont_Arccos by blast

lemma continuous_on_Arccos [continuous_intros]:
"(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> continuous_on s Arccos"

lemma holomorphic_on_Arccos: "(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arccos holomorphic_on s"

subsection\<open>Upper and Lower Bounds for Inverse Sine and Cosine\<close>

lemma Arcsin_bounds: "\<bar>Re z\<bar> < 1 \<Longrightarrow> \<bar>Re(Arcsin z)\<bar> < pi/2"
unfolding Re_Arcsin
by (blast intro: Re_Ln_pos_lt_imp Arcsin_range_lemma)

lemma Arccos_bounds: "\<bar>Re z\<bar> < 1 \<Longrightarrow> 0 < Re(Arccos z) \<and> Re(Arccos z) < pi"
unfolding Re_Arccos
by (blast intro!: Im_Ln_pos_lt_imp Arccos_range_lemma)

lemma Re_Arccos_bounds: "-pi < Re(Arccos z) \<and> Re(Arccos z) \<le> pi"
unfolding Re_Arccos
by (blast intro!: mpi_less_Im_Ln Im_Ln_le_pi Arccos_body_lemma)

lemma Re_Arccos_bound: "\<bar>Re(Arccos z)\<bar> \<le> pi"
by (meson Re_Arccos_bounds abs_le_iff less_eq_real_def minus_less_iff)

lemma Re_Arcsin_bounds: "-pi < Re(Arcsin z) & Re(Arcsin z) \<le> pi"
unfolding Re_Arcsin
by (blast intro!: mpi_less_Im_Ln Im_Ln_le_pi Arcsin_body_lemma)

lemma Re_Arcsin_bound: "\<bar>Re(Arcsin z)\<bar> \<le> pi"
by (meson Re_Arcsin_bounds abs_le_iff less_eq_real_def minus_less_iff)

subsection\<open>Interrelations between Arcsin and Arccos\<close>

lemma cos_Arcsin_nonzero:
assumes "z\<^sup>2 \<noteq> 1" shows "cos(Arcsin z) \<noteq> 0"
proof -
have eq: "(\<i> * z * (csqrt (1 - z\<^sup>2)))\<^sup>2 = z\<^sup>2 * (z\<^sup>2 - 1)"
have "\<i> * z * (csqrt (1 - z\<^sup>2)) \<noteq> z\<^sup>2 - 1"
proof
assume "\<i> * z * (csqrt (1 - z\<^sup>2)) = z\<^sup>2 - 1"
then have "(\<i> * z * (csqrt (1 - z\<^sup>2)))\<^sup>2 = (z\<^sup>2 - 1)\<^sup>2"
by simp
then have "z\<^sup>2 * (z\<^sup>2 - 1) = (z\<^sup>2 - 1)*(z\<^sup>2 - 1)"
using eq power2_eq_square by auto
then show False
using assms by simp
qed
then have "1 + \<i> * z * (csqrt (1 - z * z)) \<noteq> z\<^sup>2"
then have "2*(1 + \<i> * z * (csqrt (1 - z * z))) \<noteq> 2*z\<^sup>2"  (*FIXME cancel_numeral_factor*)
by (metis mult_cancel_left zero_neq_numeral)
then have "(\<i> * z + csqrt (1 - z\<^sup>2))\<^sup>2 \<noteq> -1"
using assms
apply (auto simp: power2_sum)
done
then show ?thesis
apply (simp add: cos_exp_eq Arcsin_def exp_minus)
done
qed

lemma sin_Arccos_nonzero:
assumes "z\<^sup>2 \<noteq> 1" shows "sin(Arccos z) \<noteq> 0"
proof -
have eq: "(\<i> * z * (csqrt (1 - z\<^sup>2)))\<^sup>2 = -(z\<^sup>2) * (1 - z\<^sup>2)"
have "\<i> * z * (csqrt (1 - z\<^sup>2)) \<noteq> 1 - z\<^sup>2"
proof
assume "\<i> * z * (csqrt (1 - z\<^sup>2)) = 1 - z\<^sup>2"
then have "(\<i> * z * (csqrt (1 - z\<^sup>2)))\<^sup>2 = (1 - z\<^sup>2)\<^sup>2"
by simp
then have "-(z\<^sup>2) * (1 - z\<^sup>2) = (1 - z\<^sup>2)*(1 - z\<^sup>2)"
using eq power2_eq_square by auto
then have "-(z\<^sup>2) = (1 - z\<^sup>2)"
using assms
then show False
using assms by simp
qed
then have "z\<^sup>2 + \<i> * z * (csqrt (1 - z\<^sup>2)) \<noteq> 1"
then have "2*(z\<^sup>2 + \<i> * z * (csqrt (1 - z\<^sup>2))) \<noteq> 2*1"
by (metis mult_cancel_left2 zero_neq_numeral)  (*FIXME cancel_numeral_factor*)
then have "(z + \<i> * csqrt (1 - z\<^sup>2))\<^sup>2 \<noteq> 1"
using assms
apply (auto simp: Power.comm_semiring_1_class.power2_sum power_mult_distrib)
done
then show ?thesis
apply (simp add: sin_exp_eq Arccos_def exp_minus)
done
qed

lemma cos_sin_csqrt:
assumes "0 < cos(Re z)  \<or>  cos(Re z) = 0 \<and> Im z * sin(Re z) \<le> 0"
shows "cos z = csqrt(1 - (sin z)\<^sup>2)"
apply (rule csqrt_unique [THEN sym])
using assms
apply (auto simp: Re_cos Im_cos add_pos_pos mult_le_0_iff zero_le_mult_iff)
done

lemma sin_cos_csqrt:
assumes "0 < sin(Re z)  \<or>  sin(Re z) = 0 \<and> 0 \<le> Im z * cos(Re z)"
shows "sin z = csqrt(1 - (cos z)\<^sup>2)"
apply (rule csqrt_unique [THEN sym])
using assms
apply (auto simp: Re_sin Im_sin add_pos_pos mult_le_0_iff zero_le_mult_iff)
done

lemma Arcsin_Arccos_csqrt_pos:
"(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> Arcsin z = Arccos(csqrt(1 - z\<^sup>2))"

lemma Arccos_Arcsin_csqrt_pos:
"(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> Arccos z = Arcsin(csqrt(1 - z\<^sup>2))"

lemma sin_Arccos:
"0 < Re z | Re z = 0 & 0 \<le> Im z \<Longrightarrow> sin(Arccos z) = csqrt(1 - z\<^sup>2)"

lemma cos_Arcsin:
"0 < Re z | Re z = 0 & 0 \<le> Im z \<Longrightarrow> cos(Arcsin z) = csqrt(1 - z\<^sup>2)"

subsection\<open>Relationship with Arcsin on the Real Numbers\<close>

lemma Im_Arcsin_of_real:
assumes "\<bar>x\<bar> \<le> 1"
shows "Im (Arcsin (of_real x)) = 0"
proof -
have "csqrt (1 - (of_real x)\<^sup>2) = (if x^2 \<le> 1 then sqrt (1 - x^2) else \<i> * sqrt (x^2 - 1))"
by (simp add: of_real_sqrt del: csqrt_of_real_nonneg)
then have "cmod (\<i> * of_real x + csqrt (1 - (of_real x)\<^sup>2))^2 = 1"
using assms abs_square_le_1
then have "cmod (\<i> * of_real x + csqrt (1 - (of_real x)\<^sup>2)) = 1"
then show ?thesis
qed

corollary Arcsin_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> \<bar>Re z\<bar> \<le> 1 \<Longrightarrow> Arcsin z \<in> \<real>"
by (metis Im_Arcsin_of_real Re_complex_of_real Reals_cases complex_is_Real_iff)

lemma arcsin_eq_Re_Arcsin:
assumes "\<bar>x\<bar> \<le> 1"
shows "arcsin x = Re (Arcsin (of_real x))"
unfolding arcsin_def
proof (rule the_equality, safe)
show "- (pi / 2) \<le> Re (Arcsin (complex_of_real x))"
using Re_Ln_pos_le [OF Arcsin_body_lemma, of "of_real x"]
by (auto simp: Complex.in_Reals_norm Re_Arcsin)
next
show "Re (Arcsin (complex_of_real x)) \<le> pi / 2"
using Re_Ln_pos_le [OF Arcsin_body_lemma, of "of_real x"]
by (auto simp: Complex.in_Reals_norm Re_Arcsin)
next
show "sin (Re (Arcsin (complex_of_real x))) = x"
using Re_sin [of "Arcsin (of_real x)"] Arcsin_body_lemma [of "of_real x"]
next
fix x'
assume "- (pi / 2) \<le> x'" "x' \<le> pi / 2" "x = sin x'"
then show "x' = Re (Arcsin (complex_of_real (sin x')))"
apply (subst Arcsin_sin)
apply (auto simp: )
done
qed

lemma of_real_arcsin: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> of_real(arcsin x) = Arcsin(of_real x)"
by (metis Im_Arcsin_of_real add.right_neutral arcsin_eq_Re_Arcsin complex_eq mult_zero_right of_real_0)

subsection\<open>Relationship with Arccos on the Real Numbers\<close>

lemma Im_Arccos_of_real:
assumes "\<bar>x\<bar> \<le> 1"
shows "Im (Arccos (of_real x)) = 0"
proof -
have "csqrt (1 - (of_real x)\<^sup>2) = (if x^2 \<le> 1 then sqrt (1 - x^2) else \<i> * sqrt (x^2 - 1))"
by (simp add: of_real_sqrt del: csqrt_of_real_nonneg)
then have "cmod (of_real x + \<i> * csqrt (1 - (of_real x)\<^sup>2))^2 = 1"
using assms abs_square_le_1
then have "cmod (of_real x + \<i> * csqrt (1 - (of_real x)\<^sup>2)) = 1"
then show ?thesis
qed

corollary Arccos_in_Reals [simp]: "z \<in> \<real> \<Longrightarrow> \<bar>Re z\<bar> \<le> 1 \<Longrightarrow> Arccos z \<in> \<real>"
by (metis Im_Arccos_of_real Re_complex_of_real Reals_cases complex_is_Real_iff)

lemma arccos_eq_Re_Arccos:
assumes "\<bar>x\<bar> \<le> 1"
shows "arccos x = Re (Arccos (of_real x))"
unfolding arccos_def
proof (rule the_equality, safe)
show "0 \<le> Re (Arccos (complex_of_real x))"
using Im_Ln_pos_le [OF Arccos_body_lemma, of "of_real x"]
by (auto simp: Complex.in_Reals_norm Re_Arccos)
next
show "Re (Arccos (complex_of_real x)) \<le> pi"
using Im_Ln_pos_le [OF Arccos_body_lemma, of "of_real x"]
by (auto simp: Complex.in_Reals_norm Re_Arccos)
next
show "cos (Re (Arccos (complex_of_real x))) = x"
using Re_cos [of "Arccos (of_real x)"] Arccos_body_lemma [of "of_real x"]
next
fix x'
assume "0 \<le> x'" "x' \<le> pi" "x = cos x'"
then show "x' = Re (Arccos (complex_of_real (cos x')))"
apply (subst Arccos_cos)
apply (auto simp: )
done
qed

lemma of_real_arccos: "\<bar>x\<bar> \<le> 1 \<Longrightarrow> of_real(arccos x) = Arccos(of_real x)"
by (metis Im_Arccos_of_real add.right_neutral arccos_eq_Re_Arccos complex_eq mult_zero_right of_real_0)

subsection\<open>Some interrelationships among the real inverse trig functions.\<close>

lemma arccos_arctan:
assumes "-1 < x" "x < 1"
shows "arccos x = pi/2 - arctan(x / sqrt(1 - x\<^sup>2))"
proof -
have "arctan(x / sqrt(1 - x\<^sup>2)) - (pi/2 - arccos x) = 0"
proof (rule sin_eq_0_pi)
show "- pi < arctan (x / sqrt (1 - x\<^sup>2)) - (pi / 2 - arccos x)"
using arctan_lbound [of "x / sqrt(1 - x\<^sup>2)"]  arccos_bounded [of x] assms
next
show "arctan (x / sqrt (1 - x\<^sup>2)) - (pi / 2 - arccos x) < pi"
using arctan_ubound [of "x / sqrt(1 - x\<^sup>2)"]  arccos_bounded [of x] assms
next
show "sin (arctan (x / sqrt (1 - x\<^sup>2)) - (pi / 2 - arccos x)) = 0"
using assms
power2_eq_square square_eq_1_iff)
qed
then show ?thesis
by simp
qed

lemma arcsin_plus_arccos:
assumes "-1 \<le> x" "x \<le> 1"
shows "arcsin x + arccos x = pi/2"
proof -
have "arcsin x = pi/2 - arccos x"
apply (rule sin_inj_pi)
using assms arcsin [OF assms] arccos [OF assms]
apply (auto simp: algebra_simps sin_diff)
done
then show ?thesis
qed

lemma arcsin_arccos_eq: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin x = pi/2 - arccos x"
using arcsin_plus_arccos by force

lemma arccos_arcsin_eq: "-1 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos x = pi/2 - arcsin x"
using arcsin_plus_arccos by force

lemma arcsin_arctan: "-1 < x \<Longrightarrow> x < 1 \<Longrightarrow> arcsin x = arctan(x / sqrt(1 - x\<^sup>2))"

lemma csqrt_1_diff_eq: "csqrt (1 - (of_real x)\<^sup>2) = (if x^2 \<le> 1 then sqrt (1 - x^2) else \<i> * sqrt (x^2 - 1))"
by ( simp add: of_real_sqrt del: csqrt_of_real_nonneg)

lemma arcsin_arccos_sqrt_pos: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arcsin x = arccos(sqrt(1 - x\<^sup>2))"
apply (simp add: abs_square_le_1 arcsin_eq_Re_Arcsin arccos_eq_Re_Arccos)
apply (subst Arcsin_Arccos_csqrt_pos)
apply (auto simp: power_le_one csqrt_1_diff_eq)
done

lemma arcsin_arccos_sqrt_neg: "-1 \<le> x \<Longrightarrow> x \<le> 0 \<Longrightarrow> arcsin x = -arccos(sqrt(1 - x\<^sup>2))"
using arcsin_arccos_sqrt_pos [of "-x"]

lemma arccos_arcsin_sqrt_pos: "0 \<le> x \<Longrightarrow> x \<le> 1 \<Longrightarrow> arccos x = arcsin(sqrt(1 - x\<^sup>2))"
apply (simp add: abs_square_le_1 arcsin_eq_Re_Arcsin arccos_eq_Re_Arccos)
apply (subst Arccos_Arcsin_csqrt_pos)
apply (auto simp: power_le_one csqrt_1_diff_eq)
done

lemma arccos_arcsin_sqrt_neg: "-1 \<le> x \<Longrightarrow> x \<le> 0 \<Longrightarrow> arccos x = pi - arcsin(sqrt(1 - x\<^sup>2))"
using arccos_arcsin_sqrt_pos [of "-x"]

subsection\<open>continuity results for arcsin and arccos.\<close>

lemma continuous_on_Arcsin_real [continuous_intros]:
"continuous_on {w \<in> \<real>. \<bar>Re w\<bar> \<le> 1} Arcsin"
proof -
have "continuous_on {w \<in> \<real>. \<bar>Re w\<bar> \<le> 1} (\<lambda>x. complex_of_real (arcsin (Re x))) =
continuous_on {w \<in> \<real>. \<bar>Re w\<bar> \<le> 1} (\<lambda>x. complex_of_real (Re (Arcsin (of_real (Re x)))))"
by (rule continuous_on_cong [OF refl]) (simp add: arcsin_eq_Re_Arcsin)
also have "... = ?thesis"
by (rule continuous_on_cong [OF refl]) simp
finally show ?thesis
using continuous_on_arcsin [OF continuous_on_Re [OF continuous_on_id], of "{w \<in> \<real>. \<bar>Re w\<bar> \<le> 1}"]
continuous_on_of_real
by fastforce
qed

lemma continuous_within_Arcsin_real:
"continuous (at z within {w \<in> \<real>. \<bar>Re w\<bar> \<le> 1}) Arcsin"
proof (cases "z \<in> {w \<in> \<real>. \<bar>Re w\<bar> \<le> 1}")
case True then show ?thesis
using continuous_on_Arcsin_real continuous_on_eq_continuous_within
by blast
next
case False
with closed_real_abs_le [of 1] show ?thesis
by (rule continuous_within_closed_nontrivial)
qed

lemma continuous_on_Arccos_real:
"continuous_on {w \<in> \<real>. \<bar>Re w\<bar> \<le> 1} Arccos"
proof -
have "continuous_on {w \<in> \<real>. \<bar>Re w\<bar> \<le> 1} (\<lambda>x. complex_of_real (arccos (Re x))) =
continuous_on {w \<in> \<real>. \<bar>Re w\<bar> \<le> 1} (\<lambda>x. complex_of_real (Re (Arccos (of_real (Re x)))))"
by (rule continuous_on_cong [OF refl]) (simp add: arccos_eq_Re_Arccos)
also have "... = ?thesis"
by (rule continuous_on_cong [OF refl]) simp
finally show ?thesis
using continuous_on_arccos [OF continuous_on_Re [OF continuous_on_id], of "{w \<in> \<real>. \<bar>Re w\<bar> \<le> 1}"]
continuous_on_of_real
by fastforce
qed

lemma continuous_within_Arccos_real:
"continuous (at z within {w \<in> \<real>. \<bar>Re w\<bar> \<le> 1}) Arccos"
proof (cases "z \<in> {w \<in> \<real>. \<bar>Re w\<bar> \<le> 1}")
case True then show ?thesis
using continuous_on_Arccos_real continuous_on_eq_continuous_within
by blast
next
case False
with closed_real_abs_le [of 1] show ?thesis
by (rule continuous_within_closed_nontrivial)
qed

subsection\<open>Roots of unity\<close>

lemma complex_root_unity:
fixes j::nat
assumes "n \<noteq> 0"
shows "exp(2 * of_real pi * \<i> * of_nat j / of_nat n)^n = 1"
proof -
have *: "of_nat j * (complex_of_real pi * 2) = complex_of_real (2 * real j * pi)"
then show ?thesis
apply (simp add: exp_of_nat_mult [symmetric] mult_ac exp_Euler)
apply (simp only: * cos_of_real sin_of_real)
done
qed

lemma complex_root_unity_eq:
fixes j::nat and k::nat
assumes "1 \<le> n"
shows "(exp(2 * of_real pi * \<i> * of_nat j / of_nat n) = exp(2 * of_real pi * \<i> * of_nat k / of_nat n)
\<longleftrightarrow> j mod n = k mod n)"
proof -
have "(\<exists>z::int. \<i> * (of_nat j * (of_real pi * 2)) =
\<i> * (of_nat k * (of_real pi * 2)) + \<i> * (of_int z * (of_nat n * (of_real pi * 2)))) \<longleftrightarrow>
(\<exists>z::int. of_nat j * (\<i> * (of_real pi * 2)) =
(of_nat k + of_nat n * of_int z) * (\<i> * (of_real pi * 2)))"
also have "... \<longleftrightarrow> (\<exists>z::int. of_nat j = of_nat k + of_nat n * (of_int z :: complex))"
by simp
also have "... \<longleftrightarrow> (\<exists>z::int. of_nat j = of_nat k + of_nat n * z)"
apply (rule HOL.iff_exI)
apply (auto simp: )
using of_int_eq_iff apply fastforce
also have "... \<longleftrightarrow> int j mod int n = int k mod int n"
by (auto simp: zmod_eq_dvd_iff dvd_def algebra_simps)
also have "... \<longleftrightarrow> j mod n = k mod n"
by (metis of_nat_eq_iff zmod_int)
finally have "(\<exists>z. \<i> * (of_nat j * (of_real pi * 2)) =
\<i> * (of_nat k * (of_real pi * 2)) + \<i> * (of_int z * (of_nat n * (of_real pi * 2)))) \<longleftrightarrow> j mod n = k mod n" .
note * = this
show ?thesis
using assms
by (simp add: exp_eq divide_simps mult_ac of_real_numeral *)
qed

corollary bij_betw_roots_unity:
"bij_betw (\<lambda>j. exp(2 * of_real pi * \<i> * of_nat j / of_nat n))
{..<n}  {exp(2 * of_real pi * \<i> * of_nat j / of_nat n) | j. j < n}"
by (auto simp: bij_betw_def inj_on_def complex_root_unity_eq)

lemma complex_root_unity_eq_1:
fixes j::nat and k::nat
assumes "1 \<le> n"
shows "exp(2 * of_real pi * \<i> * of_nat j / of_nat n) = 1 \<longleftrightarrow> n dvd j"
proof -
have "1 = exp(2 * of_real pi * \<i> * (of_nat n / of_nat n))"
using assms by simp
then have "exp(2 * of_real pi * \<i> * (of_nat j / of_nat n)) = 1 \<longleftrightarrow> j mod n = n mod n"
using complex_root_unity_eq [of n j n] assms
by simp
then show ?thesis
by auto
qed

lemma finite_complex_roots_unity_explicit:
"finite {exp(2 * of_real pi * \<i> * of_nat j / of_nat n) | j::nat. j < n}"
by simp

lemma card_complex_roots_unity_explicit:
"card {exp(2 * of_real pi * \<i> * of_nat j / of_nat n) | j::nat. j < n} = n"
by (simp add:  Finite_Set.bij_betw_same_card [OF bij_betw_roots_unity, symmetric])

lemma complex_roots_unity:
assumes "1 \<le> n"
shows "{z::complex. z^n = 1} = {exp(2 * of_real pi * \<i> * of_nat j / of_nat n) | j::nat. j < n}"
apply (rule Finite_Set.card_seteq [symmetric])
using assms
apply (auto simp: card_complex_roots_unity_explicit finite_roots_unity complex_root_unity card_roots_unity)
done

lemma card_complex_roots_unity: "1 \<le> n \<Longrightarrow> card {z::complex. z^n = 1} = n"