diff --git a/src/HOL/Analysis/Complex_Transcendental.thy b/src/HOL/Analysis/Complex_Transcendental.thy
--- a/src/HOL/Analysis/Complex_Transcendental.thy
+++ b/src/HOL/Analysis/Complex_Transcendental.thy
@@ -1,4063 +1,4061 @@
 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 "HOL-Library.Periodic_Fun"
 begin
 
 subsection\<open>Möbius transformations\<close>
 
 (* TODO: Figure out what to do with Möbius transformations *)
 definition\<^marker>\<open>tag important\<close> "moebius a b c d \<equiv> (\<lambda>z. (a*z+b) / (c*z+d :: 'a :: field))"
 
 theorem 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
     by (simp add: field_simps)
   with assms show ?thesis
     unfolding moebius_def by (simp add: moebius_def divide_simps) (simp add: algebra_simps)?
 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)
 
 lemma cmod_add_real_less:
   assumes "Im z \<noteq> 0" "r\<noteq>0"
     shows "cmod (z + r) < cmod z + \<bar>r\<bar>"
 proof (cases z)
   case (Complex x y)
   then have "0 < y * y"
     using assms mult_neg_neg by force
   with assms have "r * x / \<bar>r\<bar> < sqrt (x*x + y*y)"
     by (simp add: real_less_rsqrt power2_eq_square)
   then show ?thesis using assms Complex
     apply (simp add: cmod_def)
     apply (rule power2_less_imp_less, auto)
     apply (simp add: power2_eq_square field_simps)
     done
 qed
 
 lemma cmod_diff_real_less: "Im z \<noteq> 0 \<Longrightarrow> x\<noteq>0 \<Longrightarrow> cmod (z - x) < cmod z + \<bar>x\<bar>"
   using cmod_add_real_less [of z "-x"]
   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)"
 proof (cases "Im z = 0 \<or> Re z = 0")
   case True
   with assms abs_square_less_1 show ?thesis
     by (force simp add: Re_power2 Im_power2 cmod_def)
 next
   case False
   with cmod_diff_real_less [of "1 - z\<^sup>2" "1"] show ?thesis
     by (simp add: norm_power Im_power2)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>The Exponential Function\<close>
 
 lemma norm_exp_i_times [simp]: "norm (exp(\<i> * of_real y)) = 1"
   by simp
 
 lemma norm_exp_imaginary: "norm(exp z) = 1 \<Longrightarrow> Re z = 0"
   by simp
 
 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"
 by (simp add: continuous_at_imp_continuous_within)
 
 lemma holomorphic_on_exp [holomorphic_intros]: "exp holomorphic_on s"
   by (simp add: field_differentiable_within_exp holomorphic_on_def)
 
 lemma holomorphic_on_exp' [holomorphic_intros]:
   "f holomorphic_on s \<Longrightarrow> (\<lambda>x. exp (f x)) holomorphic_on s"
   using holomorphic_on_compose[OF _ holomorphic_on_exp] by (simp add: o_def)
 
 subsection\<open>Euler and de Moivre formulas\<close>
 
 text\<open>The sine series times \<^term>\<open>i\<close>\<close>
 lemma sin_i_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
     by (simp add: scaleR_conv_of_real field_simps)
 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_i_eq [of z]]
     by (simp add: field_simps scaleR_conv_of_real)
   ultimately show ?thesis
     using sums_unique2 by blast
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> 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>)"
   by (simp add: exp_Euler exp_minus_Euler)
 
 lemma sin_exp_eq': "sin z = \<i> * (exp(-(\<i> * z)) - exp(\<i> * z)) / 2"
   by (simp add: exp_Euler exp_minus_Euler)
 
 lemma cos_exp_eq:  "cos z = (exp(\<i> * z) + exp(-(\<i> * z))) / 2"
   by (simp add: exp_Euler exp_minus_Euler)
 
 theorem Euler: "exp(z) = of_real(exp(Re z)) *
               (of_real(cos(Im z)) + \<i> * of_real(sin(Im z)))"
 by (cases z) (simp add: exp_add exp_Euler cos_of_real exp_of_real sin_of_real Complex_eq)
 
 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\<^marker>\<open>tag unimportant\<close>\<open>Relationships between real and complex trigonometric and hyperbolic functions\<close>
 
 lemma real_sin_eq [simp]: "Re(sin(of_real x)) = sin x"
   by (simp add: sin_of_real)
 
 lemma real_cos_eq [simp]: "Re(cos(of_real x)) = cos x"
   by (simp add: cos_of_real)
 
 lemma DeMoivre: "(cos z + \<i> * sin z) ^ n = cos(n * z) + \<i> * sin(n * z)"
   by (metis exp_Euler [symmetric] exp_of_nat_mult mult.left_commute)
 
 lemma exp_cnj: "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)"
   by (simp add: field_differentiable_at_sin field_differentiable_at_within)
 
 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)"
   by (simp add: field_differentiable_at_cos field_differentiable_at_within)
 
 lemma holomorphic_on_sin: "sin holomorphic_on S"
   by (simp add: field_differentiable_within_sin holomorphic_on_def)
 
 lemma holomorphic_on_cos: "cos holomorphic_on S"
   by (simp add: field_differentiable_within_cos holomorphic_on_def)
 
 lemma holomorphic_on_sin' [holomorphic_intros]:
   assumes "f holomorphic_on A"
   shows   "(\<lambda>x. sin (f x)) holomorphic_on A"
   using holomorphic_on_compose[OF assms holomorphic_on_sin] by (simp add: o_def)
 
 lemma holomorphic_on_cos' [holomorphic_intros]:
   assumes "f holomorphic_on A"
   shows   "(\<lambda>x. cos (f x)) holomorphic_on A"
   using holomorphic_on_compose[OF assms holomorphic_on_cos] by (simp add: o_def)
 
 subsection\<^marker>\<open>tag unimportant\<close>\<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)"
   by (simp add: Complex_eq exp_add exp_Euler exp_of_real sin_of_real cos_of_real)
 
 lemma exp_eq_1: "exp z = 1 \<longleftrightarrow> Re(z) = 0 \<and> (\<exists>n::int. Im(z) = of_int (2 * n) * pi)"
                  (is "?lhs = ?rhs")
 proof
   assume "exp z = 1"
   then have "Re z = 0"
     by (metis exp_eq_one_iff norm_exp_eq_Re norm_one)
   with \<open>?lhs\<close> show ?rhs
     by (metis Re_exp complex_Re_of_int cos_one_2pi_int exp_zero mult.commute mult_numeral_1 numeral_One of_int_mult of_int_numeral)
 next
   assume ?rhs then show ?lhs
     using Im_exp Re_exp complex_eq_iff
     by (simp add: cos_one_2pi_int cos_one_sin_zero mult.commute)
 qed
 
 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"
     by (simp add: exp_diff)
   also have "... \<longleftrightarrow> (Re w = Re z \<and> (\<exists>n::int. Im w - Im z = of_int (2 * n) * pi))"
     by (simp add: exp_eq_1)
   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 unfolding Ints_def exp_eq by auto
   also have "... = 1"
     by simp
   finally show ?thesis .
 qed
 
 lemma exp_plus_2pin [simp]: "exp (z + \<i> * (of_int n * (of_real pi * 2))) = exp z"
   by (simp add: exp_eq)
 
 lemma exp_integer_2pi_plus1:
   assumes "n \<in> \<int>"
   shows "exp(((2 * n + 1) * pi) * \<i>) = - 1"
 proof -
   from assms obtain n' where [simp]: "n = of_int n'"
     by (auto simp: Ints_def)
   have "exp(((2 * n + 1) * pi) * \<i>) = exp (pi * \<i>)"
     using assms by (subst exp_eq) (auto intro!: exI[of _ n'] simp: algebra_simps)
   also have "... = - 1"
     by simp
   finally show ?thesis .
 qed
 
 lemma inj_on_exp_pi:
   fixes z::complex shows "inj_on exp (ball z pi)"
 proof (clarsimp simp: inj_on_def exp_eq)
   fix y n
   assume "dist z (y + 2 * of_int n * of_real pi * \<i>) < pi"
          "dist z y < pi"
   then have "dist y (y + 2 * of_int n * of_real pi * \<i>) < pi+pi"
     using dist_commute_lessI dist_triangle_less_add by blast
   then have "norm (2 * of_int n * of_real pi * \<i>) < 2*pi"
     by (simp add: dist_norm)
   then show "n = 0"
     by (auto simp: norm_mult)
 qed
 
 lemma cmod_add_squared:
   fixes r1 r2::real
   assumes "r1 \<ge> 0" "r2 \<ge> 0"
   shows "(cmod (r1 * exp (\<i> * \<theta>1) + r2 * exp (\<i> * \<theta>2)))\<^sup>2 = r1\<^sup>2 + r2\<^sup>2 + 2 * r1 * r2 * cos (\<theta>1 - \<theta>2)" (is "(cmod (?z1 + ?z2))\<^sup>2 = ?rhs")
 proof -
   have "(cmod (?z1 + ?z2))\<^sup>2 = (?z1 + ?z2) * cnj (?z1 + ?z2)"
     by (rule complex_norm_square)
   also have "\<dots> = (?z1 * cnj ?z1 + ?z2 * cnj ?z2) + (?z1 * cnj ?z2 + cnj ?z1 * ?z2)"
     by (simp add: algebra_simps)
   also have "\<dots> = (norm ?z1)\<^sup>2 + (norm ?z2)\<^sup>2 + 2 * Re (?z1 * cnj ?z2)"
     unfolding complex_norm_square [symmetric] cnj_add_mult_eq_Re by simp
   also have "\<dots> = ?rhs"
     by (simp add: norm_mult) (simp add: exp_Euler complex_is_Real_iff [THEN iffD1] cos_diff algebra_simps)
   finally show ?thesis
     using of_real_eq_iff by blast
 qed
 
 lemma cmod_diff_squared:
   fixes r1 r2::real
   assumes "r1 \<ge> 0" "r2 \<ge> 0"
   shows "(cmod (r1 * exp (\<i> * \<theta>1) - r2 * exp (\<i> * \<theta>2)))\<^sup>2 = r1\<^sup>2 + r2\<^sup>2 - 2*r1*r2*cos (\<theta>1 - \<theta>2)" (is "(cmod (?z1 - ?z2))\<^sup>2 = ?rhs")
 proof -
   have "exp (\<i> * (\<theta>2 + pi)) = - exp (\<i> * \<theta>2)"
     by (simp add: exp_Euler cos_plus_pi sin_plus_pi)
   then have "(cmod (?z1 - ?z2))\<^sup>2 = cmod (?z1 + r2 * exp (\<i> * (\<theta>2 + pi))) ^2"
     by simp
   also have "\<dots> = r1\<^sup>2 + r2\<^sup>2 + 2*r1*r2*cos (\<theta>1 - (\<theta>2 + pi))"
     using assms cmod_add_squared by blast
   also have "\<dots> = ?rhs"
     by (simp add: add.commute diff_add_eq_diff_diff_swap)
   finally show ?thesis .
 qed
 
 lemma polar_convergence:
   fixes R::real
   assumes "\<And>j. r j > 0" "R > 0"
   shows "((\<lambda>j. r j * exp (\<i> * \<theta> j)) \<longlonglongrightarrow> (R * exp (\<i> * \<Theta>))) \<longleftrightarrow>
          (r \<longlonglongrightarrow> R) \<and> (\<exists>k. (\<lambda>j. \<theta> j - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> \<Theta>)"    (is "(?z \<longlonglongrightarrow> ?Z) = ?rhs")
 proof
   assume L: "?z \<longlonglongrightarrow> ?Z"
   have rR: "r \<longlonglongrightarrow> R"
     using tendsto_norm [OF L] assms by (auto simp: norm_mult abs_of_pos)
   moreover obtain k where "(\<lambda>j. \<theta> j - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> \<Theta>"
   proof -
     have "cos (\<theta> j - \<Theta>) = ((r j)\<^sup>2 + R\<^sup>2 - (norm(?z j - ?Z))\<^sup>2) / (2 * R * r j)" for j
       apply (subst cmod_diff_squared)
       using assms by (auto simp: field_split_simps less_le)
     moreover have "(\<lambda>j. ((r j)\<^sup>2 + R\<^sup>2 - (norm(?z j - ?Z))\<^sup>2) / (2 * R * r j)) \<longlonglongrightarrow> ((R\<^sup>2 + R\<^sup>2 - (norm(?Z - ?Z))\<^sup>2) / (2 * R * R))"
       by (intro L rR tendsto_intros) (use \<open>R > 0\<close> in force)
     moreover have "((R\<^sup>2 + R\<^sup>2 - (norm(?Z - ?Z))\<^sup>2) / (2 * R * R)) = 1"
       using \<open>R > 0\<close> by (simp add: power2_eq_square field_split_simps)
     ultimately have "(\<lambda>j. cos (\<theta> j - \<Theta>)) \<longlonglongrightarrow> 1"
       by auto
     then show ?thesis
       using that cos_diff_limit_1 by blast
   qed
   ultimately show ?rhs
     by metis
 next
   assume R: ?rhs
   show "?z \<longlonglongrightarrow> ?Z"
   proof (rule tendsto_mult)
     show "(\<lambda>x. complex_of_real (r x)) \<longlonglongrightarrow> of_real R"
       using R by (auto simp: tendsto_of_real_iff)
     obtain k where "(\<lambda>j. \<theta> j - of_int (k j) * (2 * pi)) \<longlonglongrightarrow> \<Theta>"
       using R by metis
     then have "(\<lambda>j. complex_of_real (\<theta> j - of_int (k j) * (2 * pi))) \<longlonglongrightarrow> of_real \<Theta>"
       using tendsto_of_real_iff by force
     then have "(\<lambda>j.  exp (\<i> * of_real (\<theta> j - of_int (k j) * (2 * pi)))) \<longlonglongrightarrow> exp (\<i> * \<Theta>)"
       using tendsto_mult [OF tendsto_const] isCont_exp isCont_tendsto_compose by blast
     moreover have "exp (\<i> * of_real (\<theta> j - of_int (k j) * (2 * pi))) = exp (\<i> * \<theta> j)" for j
       unfolding exp_eq
       by (rule_tac x="- k j" in exI) (auto simp: algebra_simps)
     ultimately show "(\<lambda>j. exp (\<i> * \<theta> j)) \<longlonglongrightarrow> exp (\<i> * \<Theta>)"
       by auto
   qed
 qed
 
 lemma sin_cos_eq_iff: "sin y = sin x \<and> cos y = cos x \<longleftrightarrow> (\<exists>n::int. y = x + 2 * pi * n)"
 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 = 2 * pi * n"
       using cos_one_2pi_int by auto }
   then show ?thesis
   apply (auto simp: sin_add cos_add)
   apply (metis add.commute diff_add_cancel)
   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)
 
 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"]
   by (simp add: sin_diff algebra_simps)
 
 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)
 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"]
   by (simp add: cos_diff algebra_simps)
 
 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"
     by (simp add: equation_minus_iff)
   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)"
     by (rule iff_exI) (metis add.inverse_inverse add_uminus_conv_diff minus_add_distrib)
   also have "... = ?rhs"
     apply safe
     apply (rule_tac [2] x="-(x+1)" in exI)
     apply (rule_tac x="-(x+1)" in exI)
     apply (simp_all add: algebra_simps)
     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"]
   by (simp add: sin_diff algebra_simps equation_minus_iff)
 
 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)"
     by (rule iff_exI) (auto simp: algebra_simps intro: injD [OF inj_of_real [where 'a = complex]])
   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)"
     by (rule iff_exI) (auto simp: algebra_simps intro: injD [OF inj_of_real [where 'a = complex]])
   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)"
     by (rule iff_exI) (auto simp: algebra_simps intro: injD [OF inj_of_real [where 'a = complex]])
   also have "... = ?rhs"
     by (auto simp: algebra_simps)
   finally show ?thesis .
 qed
 
 lemma dist_exp_i_1: "norm(exp(\<i> * of_real t) - 1) = 2 * \<bar>sin(t / 2)\<bar>"
 proof -
   have "sqrt (2 - cos t * 2) = 2 * \<bar>sin (t / 2)\<bar>"
     using cos_double_sin [of "t/2"] by (simp add: real_sqrt_mult)
   then show ?thesis
     by (simp add: exp_Euler cmod_def power2_diff sin_of_real cos_of_real algebra_simps)
 qed
 
 lemma sin_cx_2pi [simp]: "\<lbrakk>z = of_int m; even m\<rbrakk> \<Longrightarrow> sin (z * complex_of_real pi) = 0"
   by (simp add: sin_eq_0)
 
 lemma cos_cx_2pi [simp]: "\<lbrakk>z = of_int m; even m\<rbrakk> \<Longrightarrow> cos (z * complex_of_real pi) = 1"
   using cos_eq_1 by auto
 
 lemma complex_sin_eq:
   fixes w :: complex
   shows "sin w = sin z \<longleftrightarrow> (\<exists>n \<in> \<int>. w = z + of_real(2*n*pi) \<or> w = -z + of_real((2*n + 1)*pi))"
         (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then have "sin w - sin z = 0"
     by (auto simp: algebra_simps)
   then have "sin ((w - z) / 2)*cos ((w + z) / 2) = 0"
     by (auto simp: sin_diff_sin)
   then consider "sin ((w - z) / 2) = 0" | "cos ((w + z) / 2) = 0"
     using mult_eq_0_iff by blast
   then show ?rhs
   proof cases
     case 1
     then show ?thesis
       by (simp add: sin_eq_0 algebra_simps) (metis Ints_of_int of_real_of_int_eq)
   next
     case 2
     then show ?thesis
       by (simp add: cos_eq_0 algebra_simps) (metis Ints_of_int of_real_of_int_eq)
   qed
 next
   assume ?rhs
   then consider n::int where "w = z + of_real (2 * of_int n * pi)" 
               | n::int where  " w = -z + of_real ((2 * of_int n + 1) * pi)"
     using Ints_cases by blast
   then show ?lhs
   proof cases
     case 1
     then show ?thesis
       using Periodic_Fun.sin.plus_of_int [of z n]
       by (auto simp: algebra_simps)
   next
     case 2
     then show ?thesis 
       using Periodic_Fun.sin.plus_of_int [of "-z" "n"]
       apply (simp add: algebra_simps)
       by (metis add.commute add.inverse_inverse add_diff_cancel_left diff_add_cancel sin_plus_pi)
   qed
 qed
 
 lemma complex_cos_eq:
   fixes w :: complex
   shows "cos w = cos z \<longleftrightarrow> (\<exists>n \<in> \<int>. w = z + of_real(2*n*pi) \<or> w = -z + of_real(2*n*pi))"
         (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then have "cos w - cos z = 0"
     by (auto simp: algebra_simps)
   then have "sin ((w + z) / 2) * sin ((z - w) / 2) = 0"
     by (auto simp: cos_diff_cos)
   then consider "sin ((w + z) / 2) = 0" | "sin ((z - w) / 2) = 0"
     using mult_eq_0_iff by blast
   then show ?rhs
   proof cases
     case 1
     then obtain n where "w + z = of_int n * (complex_of_real pi * 2)"
       by (auto simp: sin_eq_0 algebra_simps)
     then have "w = -z + of_real(2 * of_int n * pi)"
       by (auto simp: algebra_simps)
     then show ?thesis
       using Ints_of_int by blast
   next
     case 2
     then obtain n where "z = w + of_int n * (complex_of_real pi * 2)"
       by (auto simp: sin_eq_0 algebra_simps)
     then have "w = z + complex_of_real (2 * of_int(-n) * pi)"
       by (auto simp: algebra_simps)
     then show ?thesis
       using Ints_of_int by blast
   qed
 next
   assume ?rhs
   then obtain n::int where w: "w = z + of_real (2* of_int n*pi) \<or>
                                w = -z + of_real(2*n*pi)"
     using Ints_cases  by (metis of_int_mult of_int_numeral)
   then show ?lhs
     using Periodic_Fun.cos.plus_of_int [of z n]
     apply (simp add: algebra_simps)
     by (metis cos.plus_of_int cos_minus minus_add_cancel mult.commute)
 qed
 
 lemma sin_eq:
    "sin x = sin y \<longleftrightarrow> (\<exists>n \<in> \<int>. x = y + 2*n*pi \<or> x = -y + (2*n + 1)*pi)"
   using complex_sin_eq [of x y]
   by (simp only: sin_of_real Re_complex_of_real of_real_add [symmetric] of_real_minus [symmetric] of_real_mult [symmetric] of_real_eq_iff)
 
 lemma cos_eq:
    "cos x = cos y \<longleftrightarrow> (\<exists>n \<in> \<int>. x = y + 2*n*pi \<or> x = -y + 2*n*pi)"
   using complex_cos_eq [of x y]
   by (simp only: cos_of_real Re_complex_of_real of_real_add [symmetric] of_real_minus [symmetric] of_real_mult [symmetric] of_real_eq_iff)
 
 lemma sinh_complex:
   fixes z :: complex
   shows "(exp z - inverse (exp z)) / 2 = -\<i> * sin(\<i> * z)"
   by (simp add: sin_exp_eq field_split_simps exp_minus)
 
 lemma sin_i_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_i_times)
 
 lemma cosh_complex:
   fixes z :: complex
   shows "(exp z + inverse (exp z)) / 2 = cos(\<i> * z)"
   by (simp add: cos_exp_eq field_split_simps exp_minus 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 field_split_simps exp_minus exp_of_real)
 
 lemmas cos_i_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"
 proof (cases z)
   case (Complex x1 x2)
   then show ?thesis
     apply (simp only: cos_add cmod_power2 cos_of_real sin_of_real Complex_eq)
     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 sin_squared_eq)
     apply (simp add: power2_eq_square field_split_simps)
     done
 qed
 
 lemma norm_sin_squared:
   "norm(sin z) ^ 2 = (exp(2 * Im z) + inverse(exp(2 * Im z)) - 2 * cos(2 * Re z)) / 4"
 proof (cases z)
   case (Complex x1 x2)
   then show ?thesis
     apply (simp only: sin_add cmod_power2 cos_of_real sin_of_real cos_double_cos exp_double Complex_eq)
     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 cos_squared_eq)
     apply (simp add: power2_eq_square field_split_simps)
     done
 qed
 
 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
   then have "exp (- Im z) + exp (Im z) \<le> exp (cmod z) * 2"
     by (metis exp_uminus_Im add_mono exp_le_cancel_iff mult_2_right)
   then show ?thesis
     by (force simp add: cos_exp_eq norm_divide intro: order_trans [OF norm_triangle_ineq])
 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)"
     by (simp add: norm_add_rule_thm)
   have "norm(1 + cos z) = cmod (1 + (exp (\<i> * z) + exp (- (\<i> * z))) / 2)"
     by (simp add: cos_exp_eq)
   also have "... = cmod ((2 + exp (\<i> * z) + exp (- (\<i> * z))) / 2)"
     by (simp add: field_simps)
   also have "... = cmod (2 + exp (\<i> * z) + exp (- (\<i> * z))) / 2"
     by (simp add: norm_divide)
   finally show ?thesis
     by (metis exp_eq_one_iff exp_le_cancel_iff mult_2 norm_cos_le norm_ge_zero norm_one norm_triangle_mono)
 qed
 
 lemma sinh_conv_sin: "sinh z = -\<i> * sin (\<i>*z)"
   by (simp add: sinh_field_def sin_i_times exp_minus)
 
 lemma cosh_conv_cos: "cosh z = cos (\<i>*z)"
   by (simp add: cosh_field_def cos_i_times exp_minus)
 
 lemma tanh_conv_tan: "tanh z = -\<i> * tan (\<i>*z)"
   by (simp add: tanh_def sinh_conv_sin cosh_conv_cos tan_def)
 
 lemma sin_conv_sinh: "sin z = -\<i> * sinh (\<i>*z)"
   by (simp add: sinh_conv_sin)
 
 lemma cos_conv_cosh: "cos z = cosh (\<i>*z)"
   by (simp add: cosh_conv_cos)
 
 lemma tan_conv_tanh: "tan z = -\<i> * tanh (\<i>*z)"
   by (simp add: tan_def sin_conv_sinh cos_conv_cosh tanh_def)
 
 lemma sinh_complex_eq_iff:
   "sinh (z :: complex) = sinh w \<longleftrightarrow>
       (\<exists>n\<in>\<int>. z = w - 2 * \<i> * of_real n * of_real pi \<or>
               z = -(2 * complex_of_real n + 1) * \<i> * complex_of_real pi - w)" (is "_ = ?rhs")
 proof -
   have "sinh z = sinh w \<longleftrightarrow> sin (\<i> * z) = sin (\<i> * w)"
     by (simp add: sinh_conv_sin)
   also have "\<dots> \<longleftrightarrow> ?rhs"
     by (subst complex_sin_eq) (force simp: field_simps complex_eq_iff)
   finally show ?thesis .
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Taylor series for complex exponential, sine and cosine\<close>
 
 declare power_Suc [simp del]
 
 lemma Taylor_exp_field:
   fixes z::"'a::{banach,real_normed_field}"
   shows "norm (exp z - (\<Sum>i\<le>n. z ^ i / fact i)) \<le> exp (norm z) * (norm z ^ Suc n) / fact n"
 proof (rule field_Taylor[of _ n "\<lambda>k. exp" "exp (norm z)" 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: "x \<in> closed_segment 0 z"
   have "norm (exp x) \<le> exp (norm x)"
     by (rule norm_exp)
   also have "norm x \<le> norm z"
     using x by (auto simp: closed_segment_def intro!: mult_left_le_one_le)
   finally show "norm (exp x) \<le> exp (norm z)"
     by simp
 qed auto
 
 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 obtain u where u: "x = complex_of_real u * z" "0 \<le> u" "u \<le> 1"
     by (auto simp: closed_segment_def scaleR_conv_of_real)
   then have "u * Re z \<le> \<bar>Re z\<bar>"
     by (metis abs_ge_self abs_ge_zero mult.commute mult.right_neutral mult_mono)
   then show "Re x \<le> \<bar>Re z\<bar>"
     by (simp add: u)
 qed auto
 
 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)/2 \<le> u"
     by simp
   have *: "(cmod (exp (\<i> * (u * z))) + cmod (exp (- (\<i> * (u * z)))) ) / 2 \<le> exp \<bar>Im z\<bar>"
   proof (rule mono)
     show "cmod (exp (\<i> * (u * z))) \<le> exp \<bar>Im z\<bar>"
       using assms
       by (auto simp: abs_if mult_left_le_one_le not_less intro: order_trans [of _ 0])
     show "cmod (exp (- (\<i> * (u * z)))) \<le> exp \<bar>Im z\<bar>"
       using assms
       by (auto simp: abs_if mult_left_le_one_le mult_nonneg_nonpos intro: order_trans [of _ 0])
   qed
   have "cmod (sin (u *\<^sub>R z)) = cmod (exp (\<i> * (u * z)) - exp (- (\<i> * (u * z)))) / 2"
     by (auto simp: scaleR_conv_of_real norm_mult norm_power sin_exp_eq norm_divide)
   also have "... \<le> (cmod (exp (\<i> * (u * z))) + cmod (exp (- (\<i> * (u * z)))) ) / 2"
     by (intro divide_right_mono norm_triangle_ineq4) simp
   also have "... \<le> exp \<bar>Im z\<bar>"
     by (rule *)
   finally show "cmod (sin (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>" .
   have "cmod (cos (u *\<^sub>R z)) = cmod (exp (\<i> * (u * z)) + exp (- (\<i> * (u * z)))) / 2"
     by (auto simp: scaleR_conv_of_real norm_mult norm_power cos_exp_eq norm_divide)
   also have "... \<le> (cmod (exp (\<i> * (u * z))) + cmod (exp (- (\<i> * (u * z)))) ) / 2"
     by (intro divide_right_mono norm_triangle_ineq) simp
   also have "... \<le> exp \<bar>Im z\<bar>"
     by (rule *)
   finally show "cmod (cos (u *\<^sub>R z)) \<le> exp \<bar>Im z\<bar>" .
 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
     by (simp add: ** order_trans [OF _ *])
 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
     by (simp add: ** order_trans [OF _ *])
 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: sum_distrib_right in_Reals_norm Re_exp atMost_nat_numeral fact_numeral)
   apply (simp only: pos_le_divide_eq [symmetric])
   done
 
 lemma e_less_272: "exp 1 < (272/100::real)"
   using e_approx_32
   by (simp add: abs_if split: if_split_asm)
 
 lemma ln_272_gt_1: "ln (272/100) > (1::real)"
   by (metis e_less_272 exp_less_cancel_iff exp_ln_iff less_trans ln_exp)
 
 text\<open>Apparently redundant. But many arguments involve integers.\<close>
 lemma ln3_gt_1: "ln 3 > (1::real)"
   by (simp add: less_trans [OF ln_272_gt_1])
 
 subsection\<open>The argument of a complex number (HOL Light version)\<close>
 
 definition\<^marker>\<open>tag important\<close> is_Arg :: "[complex,real] \<Rightarrow> bool"
   where "is_Arg z r \<equiv> z = of_real(norm z) * exp(\<i> * of_real r)"
 
 definition\<^marker>\<open>tag important\<close> Arg2pi :: "complex \<Rightarrow> real"
   where "Arg2pi z \<equiv> if z = 0 then 0 else THE t. 0 \<le> t \<and> t < 2*pi \<and> is_Arg z t"
 
 lemma is_Arg_2pi_iff: "is_Arg z (r + of_int k * (2 * pi)) \<longleftrightarrow> is_Arg z r"
   by (simp add: algebra_simps is_Arg_def)
 
 lemma is_Arg_eqI:
   assumes r: "is_Arg z r" and s: "is_Arg z s" and rs: "abs (r-s) < 2*pi" and "z \<noteq> 0"
   shows "r = s"
 proof -
   have zr: "z = (cmod z) * exp (\<i> * r)" and zs: "z = (cmod z) * exp (\<i> * s)"
     using r s by (auto simp: is_Arg_def)
   with \<open>z \<noteq> 0\<close> have eq: "exp (\<i> * r) = exp (\<i> * s)"
     by (metis mult_eq_0_iff mult_left_cancel)
   have  "\<i> * r = \<i> * s"
     by (rule exp_complex_eqI) (use rs in \<open>auto simp: eq exp_complex_eqI\<close>)
   then show ?thesis
     by simp
 qed
 
 text\<open>This function returns the angle of a complex number from its representation in polar coordinates.
 Due to periodicity, its range is arbitrary. \<^term>\<open>Arg2pi\<close> follows HOL Light in adopting the interval \<open>[0,2\<pi>)\<close>.
 But we have the same periodicity issue with logarithms, and it is usual to adopt the same interval
 for the complex logarithm and argument functions. Further on down, we shall define both functions for the interval \<open>(-\<pi>,\<pi>]\<close>.
 The present version is provided for compatibility.\<close>
 
 lemma Arg2pi_0 [simp]: "Arg2pi(0) = 0"
   by (simp add: Arg2pi_def)
 
 lemma Arg2pi_unique_lemma:
   assumes z:  "is_Arg z t"
       and z': "is_Arg z 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' is_Arg_def)
   then have "exp (\<i> * of_real t') = exp (\<i> * of_real t)"
     by (metis nz mult_left_cancel mult_zero_left z is_Arg_def)
   then have "sin t' = sin t \<and> cos t' = cos t"
     by (metis cis.simps cis_conv_exp)
   then obtain n::int where n: "t' = t + 2 * n * pi"
     by (auto simp: sin_cos_eq_iff)
   then have "n=0"
     by (cases n) (use t t' in \<open>auto simp: mult_less_0_iff algebra_simps\<close>)
   then show "t' = t"
     by (simp add: n)
 qed
 
 lemma Arg2pi: "0 \<le> Arg2pi z \<and> Arg2pi z < 2*pi \<and> is_Arg z (Arg2pi z)"
 proof (cases "z=0")
   case True then show ?thesis
     by (simp add: Arg2pi_def is_Arg_def)
 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: "is_Arg z t"
     unfolding is_Arg_def
     using t False ReIm
     by (intro complex_eqI) (auto simp: exp_Euler sin_of_real cos_of_real field_split_simps)
   show ?thesis
     apply (simp add: Arg2pi_def False)
     apply (rule theI [where a=t])
     using t z False
     apply (auto intro: Arg2pi_unique_lemma)
     done
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close>
   shows Arg2pi_ge_0: "0 \<le> Arg2pi z"
     and Arg2pi_lt_2pi: "Arg2pi z < 2*pi"
     and Arg2pi_eq: "z = of_real(norm z) * exp(\<i> * of_real(Arg2pi z))"
   using Arg2pi is_Arg_def by auto
 
 lemma complex_norm_eq_1_exp: "norm z = 1 \<longleftrightarrow> exp(\<i> * of_real (Arg2pi z)) = z"
   by (metis Arg2pi_eq cis_conv_exp mult.left_neutral norm_cis of_real_1)
 
 lemma Arg2pi_unique: "\<lbrakk>of_real r * exp(\<i> * of_real a) = z; 0 < r; 0 \<le> a; a < 2*pi\<rbrakk> \<Longrightarrow> Arg2pi z = a"
   by (rule Arg2pi_unique_lemma [unfolded is_Arg_def, OF _ Arg2pi_eq]) (use Arg2pi [of z] in \<open>auto simp: norm_mult\<close>)
 
 lemma cos_Arg2pi: "cmod z * cos (Arg2pi z) = Re z" and sin_Arg2pi: "cmod z * sin (Arg2pi z) = Im z"
   using Arg2pi_eq [of z] cis_conv_exp Re_rcis Im_rcis unfolding rcis_def by metis+
 
 lemma Arg2pi_minus:
   assumes "z \<noteq> 0" shows "Arg2pi (-z) = (if Arg2pi z < pi then Arg2pi z + pi else Arg2pi z - pi)"
   apply (rule Arg2pi_unique [of "norm z", OF complex_eqI])
   using cos_Arg2pi sin_Arg2pi Arg2pi_ge_0 Arg2pi_lt_2pi [of z] assms
   by (auto simp: Re_exp Im_exp)
 
 lemma Arg2pi_times_of_real [simp]:
   assumes "0 < r" shows "Arg2pi (of_real r * z) = Arg2pi z"
 proof (cases "z=0")
   case False
   show ?thesis
     by (rule Arg2pi_unique [of "r * norm z"]) (use Arg2pi False assms is_Arg_def in auto)
 qed auto
 
 lemma Arg2pi_times_of_real2 [simp]: "0 < r \<Longrightarrow> Arg2pi (z * of_real r) = Arg2pi z"
   by (metis Arg2pi_times_of_real mult.commute)
 
 lemma Arg2pi_divide_of_real [simp]: "0 < r \<Longrightarrow> Arg2pi (z / of_real r) = Arg2pi z"
   by (metis Arg2pi_times_of_real2 less_numeral_extra(3) nonzero_eq_divide_eq of_real_eq_0_iff)
 
 lemma Arg2pi_le_pi: "Arg2pi z \<le> pi \<longleftrightarrow> 0 \<le> Im z"
 proof (cases "z=0")
   case False
   have "0 \<le> Im z \<longleftrightarrow> 0 \<le> Im (of_real (cmod z) * exp (\<i> * complex_of_real (Arg2pi z)))"
     by (metis Arg2pi_eq)
   also have "... = (0 \<le> Im (exp (\<i> * complex_of_real (Arg2pi z))))"
     using False  by (simp add: zero_le_mult_iff)
   also have "... \<longleftrightarrow> Arg2pi z \<le> pi"
     by (simp add: Im_exp) (metis Arg2pi_ge_0 Arg2pi_lt_2pi sin_lt_zero sin_ge_zero not_le)
   finally show ?thesis
     by blast
 qed auto
 
 lemma Arg2pi_lt_pi: "0 < Arg2pi z \<and> Arg2pi z < pi \<longleftrightarrow> 0 < Im z"
 proof (cases "z=0")
   case False
   have "0 < Im z \<longleftrightarrow> 0 < Im (of_real (cmod z) * exp (\<i> * complex_of_real (Arg2pi z)))"
     by (metis Arg2pi_eq)
   also have "... = (0 < Im (exp (\<i> * complex_of_real (Arg2pi z))))"
     using False by (simp add: zero_less_mult_iff)
   also have "... \<longleftrightarrow> 0 < Arg2pi z \<and> Arg2pi z < pi" (is "_ = ?rhs")
   proof -
     have "0 < sin (Arg2pi z) \<Longrightarrow> ?rhs"
       by (meson Arg2pi_ge_0 Arg2pi_lt_2pi less_le_trans not_le sin_le_zero sin_x_le_x)
     then show ?thesis
       by (auto simp: Im_exp sin_gt_zero)
   qed
   finally show ?thesis
     by blast
 qed auto
 
 lemma Arg2pi_eq_0: "Arg2pi z = 0 \<longleftrightarrow> z \<in> \<real> \<and> 0 \<le> Re z"
 proof (cases "z=0")
   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 (Arg2pi z)))"
     by (metis Arg2pi_eq)
   also have "... \<longleftrightarrow> z \<in> \<real> \<and> 0 \<le> Re (exp (\<i> * complex_of_real (Arg2pi z)))"
     using False  by (simp add: zero_le_mult_iff)
   also have "... \<longleftrightarrow> Arg2pi z = 0"
   proof -
     have [simp]: "Arg2pi z = 0 \<Longrightarrow> z \<in> \<real>"
       using Arg2pi_eq [of z] by (auto simp: Reals_def)
     moreover have "\<lbrakk>z \<in> \<real>; 0 \<le> cos (Arg2pi z)\<rbrakk> \<Longrightarrow> Arg2pi z = 0"
       by (metis Arg2pi_lt_pi Arg2pi_ge_0 Arg2pi_le_pi cos_pi complex_is_Real_iff leD less_linear less_minus_one_simps(2) minus_minus neg_less_eq_nonneg order_refl)
     ultimately show ?thesis
       by (auto simp: Re_exp)
   qed
   finally show ?thesis
     by blast
 qed auto
 
 corollary\<^marker>\<open>tag unimportant\<close> Arg2pi_gt_0:
   assumes "z \<notin> \<real>\<^sub>\<ge>\<^sub>0"
     shows "Arg2pi z > 0"
   using Arg2pi_eq_0 Arg2pi_ge_0 assms dual_order.strict_iff_order
   unfolding nonneg_Reals_def by fastforce
 
 lemma Arg2pi_eq_pi: "Arg2pi z = pi \<longleftrightarrow> z \<in> \<real> \<and> Re z < 0"
     using Arg2pi_le_pi [of z] Arg2pi_lt_pi [of z] Arg2pi_eq_0 [of z] Arg2pi_ge_0 [of z] 
     by (fastforce simp: complex_is_Real_iff)
 
 lemma Arg2pi_eq_0_pi: "Arg2pi z = 0 \<or> Arg2pi z = pi \<longleftrightarrow> z \<in> \<real>"
   using Arg2pi_eq_0 Arg2pi_eq_pi not_le by auto
 
 lemma Arg2pi_of_real: "Arg2pi (of_real r) = (if r<0 then pi else 0)"
   using Arg2pi_eq_0_pi Arg2pi_eq_pi by fastforce
 
 lemma Arg2pi_real: "z \<in> \<real> \<Longrightarrow> Arg2pi z = (if 0 \<le> Re z then 0 else pi)"
   using Arg2pi_eq_0 Arg2pi_eq_0_pi by auto
 
 lemma Arg2pi_inverse: "Arg2pi(inverse z) = (if z \<in> \<real> then Arg2pi z else 2*pi - Arg2pi z)"
 proof (cases "z=0")
   case False
   show ?thesis
     apply (rule Arg2pi_unique [of "inverse (norm z)"])
     using Arg2pi_eq False Arg2pi_ge_0 [of z] Arg2pi_lt_2pi [of z] Arg2pi_eq_0 [of z]
     by (auto simp: Arg2pi_real in_Reals_norm exp_diff field_simps)
 qed auto
 
 lemma Arg2pi_eq_iff:
   assumes "w \<noteq> 0" "z \<noteq> 0"
      shows "Arg2pi w = Arg2pi z \<longleftrightarrow> (\<exists>x. 0 < x & w = of_real x * z)"
   using assms Arg2pi_eq [of z] Arg2pi_eq [of w]
   apply auto
   apply (rule_tac x="norm w / norm z" in exI)
   apply (simp add: field_split_simps)
   by (metis mult.commute mult.left_commute)
 
 lemma Arg2pi_inverse_eq_0: "Arg2pi(inverse z) = 0 \<longleftrightarrow> Arg2pi z = 0"
   by (metis Arg2pi_eq_0 Arg2pi_inverse inverse_inverse_eq)
 
 lemma Arg2pi_divide:
   assumes "w \<noteq> 0" "z \<noteq> 0" "Arg2pi w \<le> Arg2pi z"
     shows "Arg2pi(z / w) = Arg2pi z - Arg2pi w"
   apply (rule Arg2pi_unique [of "norm(z / w)"])
   using assms Arg2pi_eq Arg2pi_ge_0 [of w] Arg2pi_lt_2pi [of z]
   apply (auto simp: exp_diff norm_divide field_simps)
   done
 
 lemma Arg2pi_le_div_sum:
   assumes "w \<noteq> 0" "z \<noteq> 0" "Arg2pi w \<le> Arg2pi z"
     shows "Arg2pi z = Arg2pi w + Arg2pi(z / w)"
   by (simp add: Arg2pi_divide assms)
 
 lemma Arg2pi_le_div_sum_eq:
   assumes "w \<noteq> 0" "z \<noteq> 0"
     shows "Arg2pi w \<le> Arg2pi z \<longleftrightarrow> Arg2pi z = Arg2pi w + Arg2pi(z / w)"
   using assms by (auto simp: Arg2pi_ge_0 intro: Arg2pi_le_div_sum)
 
 lemma Arg2pi_diff:
   assumes "w \<noteq> 0" "z \<noteq> 0"
     shows "Arg2pi w - Arg2pi z = (if Arg2pi z \<le> Arg2pi w then Arg2pi(w / z) else Arg2pi(w/z) - 2*pi)"
   using assms Arg2pi_divide Arg2pi_inverse [of "w/z"] Arg2pi_eq_0_pi
   by (force simp add: Arg2pi_ge_0 Arg2pi_divide not_le split: if_split_asm)
 
 lemma Arg2pi_add:
   assumes "w \<noteq> 0" "z \<noteq> 0"
     shows "Arg2pi w + Arg2pi z = (if Arg2pi w + Arg2pi z < 2*pi then Arg2pi(w * z) else Arg2pi(w * z) + 2*pi)"
   using assms Arg2pi_diff [of "w*z" z] Arg2pi_le_div_sum_eq [of z "w*z"]
   apply (auto simp: Arg2pi_ge_0 Arg2pi_divide not_le)
   apply (metis Arg2pi_lt_2pi add.commute)
   apply (metis (no_types) Arg2pi add.commute diff_0 diff_add_cancel diff_less_eq diff_minus_eq_add not_less)
   done
 
 lemma Arg2pi_times:
   assumes "w \<noteq> 0" "z \<noteq> 0"
     shows "Arg2pi (w * z) = (if Arg2pi w + Arg2pi z < 2*pi then Arg2pi w + Arg2pi z
                             else (Arg2pi w + Arg2pi z) - 2*pi)"
   using Arg2pi_add [OF assms]
   by auto
 
 lemma Arg2pi_cnj_eq_inverse: "z\<noteq>0 \<Longrightarrow> Arg2pi (cnj z) = Arg2pi (inverse z)"
   apply (simp add: Arg2pi_eq_iff field_split_simps complex_norm_square [symmetric])
   by (metis of_real_power zero_less_norm_iff zero_less_power)
 
 lemma Arg2pi_cnj: "Arg2pi(cnj z) = (if z \<in> \<real> then Arg2pi z else 2*pi - Arg2pi z)"
 proof (cases "z=0")
   case False
   then show ?thesis
     by (simp add: Arg2pi_cnj_eq_inverse Arg2pi_inverse)
 qed auto
 
 lemma Arg2pi_exp: "0 \<le> Im z \<Longrightarrow> Im z < 2*pi \<Longrightarrow> Arg2pi(exp z) = Im z"
   by (rule Arg2pi_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 Arg2pi cis.ctr cis_conv_exp unfolding Complex_eq is_Arg_def 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
     apply (simp add: norm_mult cmod_unit_one)
     by (metis (no_types, hide_lams) abs_le_D1 distrib_left mult.commute mult.left_commute mult_left_le)
 qed
 
 subsection\<^marker>\<open>tag unimportant\<close>\<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 \<noteq> 0 \<Longrightarrow> tan field_differentiable at z"
   unfolding field_differentiable_def
   using DERIV_tan by blast
 
 lemma field_differentiable_within_tan: "cos z \<noteq> 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 \<noteq> 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 \<noteq> 0) \<Longrightarrow> continuous_on s tan"
   by (simp add: continuous_at_imp_continuous_on)
 
 lemma holomorphic_on_tan: "(\<And>z. z \<in> s \<Longrightarrow> cos z \<noteq> 0) \<Longrightarrow> tan holomorphic_on s"
   by (simp add: field_differentiable_within_tan holomorphic_on_def)
 
 
 subsection\<open>The principal branch of the Complex logarithm\<close>
 
 instantiation complex :: ln
 begin
 
 definition\<^marker>\<open>tag important\<close> ln_complex :: "complex \<Rightarrow> complex"
   where "ln_complex \<equiv> \<lambda>z. THE w. exp w = z & -pi < Im(w) & Im(w) \<le> pi"
 
 text\<open>NOTE: within this scope, the constant Ln is not yet available!\<close>
 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 field_split_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 field_split_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"
 proof (rule exp_complex_eqI)
   show "\<bar>Im (ln (exp z)) - Im z\<bar> < 2 * pi"
     using assms mpi_less_Im_Ln  [of "exp z"] Im_Ln_le_pi [of "exp z"] by auto
 qed auto
 
 subsection\<^marker>\<open>tag unimportant\<close>\<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)))"
     by (simp add: exp_of_real)
   also have "... = of_real(ln z)"
     using assms by (subst Ln_exp) auto
   finally show ?thesis
     using assms by simp
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> 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\<^marker>\<open>tag unimportant\<close> Im_Ln_of_real [simp]: "r > 0 \<Longrightarrow> Im (ln (of_real r)) = 0"
   by (simp add: Ln_of_real)
 
 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_Reals_eq: "\<lbrakk>x \<in> \<real>; Re x > 0\<rbrakk> \<Longrightarrow> ln x = of_real (ln (Re x))"
   using Ln_of_real by force
 
 lemma Ln_1 [simp]: "ln 1 = (0::complex)"
 proof -
   have "ln (exp 0) = (0::complex)"
     by (simp add: del: exp_zero)
   then show ?thesis
     by simp
 qed
 
 
 lemma Ln_eq_zero_iff [simp]: "x \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> ln x = 0 \<longleftrightarrow> x = 1" for x::complex
   by auto (metis exp_Ln exp_zero nonpos_Reals_zero_I)
 
 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\<^marker>\<open>tag unimportant\<close> 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\<^marker>\<open>tag unimportant\<close> exists_complex_root:
   fixes z :: complex
   assumes "n \<noteq> 0"  obtains w where "z = w ^ n"
 proof (cases "z=0")
   case False
   then show ?thesis
     by (rule_tac w = "exp(Ln z / n)" in that) (simp add: assms exp_of_nat_mult [symmetric])
 qed (use assms in auto)
 
 corollary\<^marker>\<open>tag unimportant\<close> 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\<^marker>\<open>tag unimportant\<close>\<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 [simp]: "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
     by (simp add: le_neq_trans)
   let ?U = "{w. -pi < Im(w) \<and> Im(w) < pi}"
   have 1: "open ?U"
     by (simp add: open_Collect_conj open_halfspace_Im_gt open_halfspace_Im_lt)
   have 2: "\<And>x. x \<in> ?U \<Longrightarrow> (exp has_derivative blinfun_apply(Blinfun ((*) (exp x)))) (at x)"
     by (simp add: bounded_linear_Blinfun_apply bounded_linear_mult_right has_field_derivative_imp_has_derivative)   
   have 3: "continuous_on ?U (\<lambda>x. Blinfun ((*) (exp x)))"
     unfolding blinfun_mult_right.abs_eq [symmetric] by (intro continuous_intros)
   have 4: "Ln z \<in> ?U"
     by (auto simp: mpi_less_Im_Ln *)
   have 5: "Blinfun ((*) (inverse z)) o\<^sub>L Blinfun ((*) (exp (Ln z))) = id_blinfun"
     by (rule blinfun_eqI) (simp add: bounded_linear_mult_right bounded_linear_Blinfun_apply)
   obtain U' V g g' where "open U'" and sub: "U' \<subseteq> ?U"
     and "Ln z \<in> U'" "open V" "z \<in> V"
     and hom: "homeomorphism U' V exp g"
     and g: "\<And>y. y \<in> V \<Longrightarrow> (g has_derivative (g' y)) (at y)"
     and g': "\<And>y. y \<in> V \<Longrightarrow> g' y = inv ((*) (exp (g y)))"
     and bij: "\<And>y. y \<in> V \<Longrightarrow> bij ((*) (exp (g y)))"
     using inverse_function_theorem [OF 1 2 3 4 5]
     by (simp add: bounded_linear_Blinfun_apply bounded_linear_mult_right) blast
   show "(Ln has_field_derivative inverse(z)) (at z)"
     unfolding has_field_derivative_def
   proof (rule has_derivative_transform_within_open)
     show g_eq_Ln: "g y = Ln y" if "y \<in> V" for y
     proof -
       obtain x where "y = exp x" "x \<in> U'"
         using hom homeomorphism_image1 that \<open>y \<in> V\<close> by blast
       then show ?thesis
         using sub hom homeomorphism_apply1 by fastforce
     qed
     have "0 \<notin> V"
       by (meson exp_not_eq_zero hom homeomorphism_def)
     then have "\<And>y. y \<in> V \<Longrightarrow> g' y = inv ((*) y)"
       by (metis exp_Ln g' g_eq_Ln)
     then have g': "g' z = (\<lambda>x. x/z)"
       by (metis (no_types, hide_lams) bij \<open>z \<in> V\<close> bij_inv_eq_iff exp_Ln g_eq_Ln nonzero_mult_div_cancel_left znz)
     show "(g has_derivative (*) (inverse z)) (at z)"
       using g [OF \<open>z \<in> V\<close>] g'
       by (simp add: \<open>z \<in> V\<close> field_class.field_divide_inverse has_derivative_imp_has_field_derivative has_field_derivative_imp_has_derivative)
   qed (auto simp: \<open>z \<in> V\<close> \<open>open V\<close>)
 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"
   by (simp add: field_differentiable_imp_continuous_at field_differentiable_within_Ln)
 
 lemma isCont_Ln' [simp,continuous_intros]:
    "\<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 [continuous_intros]: "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"
   by (simp add: continuous_at_imp_continuous_on continuous_within_Ln)
 
 lemma continuous_on_Ln' [continuous_intros]:
   "continuous_on S f \<Longrightarrow> (\<And>z. z \<in> S \<Longrightarrow> f z \<notin> \<real>\<^sub>\<le>\<^sub>0) \<Longrightarrow> continuous_on S (\<lambda>x. Ln (f x))"
   by (rule continuous_on_compose2[OF continuous_on_Ln, of "UNIV - nonpos_Reals" S f]) auto
 
 lemma holomorphic_on_Ln [holomorphic_intros]: "(\<And>z. z \<in> S \<Longrightarrow> z \<notin> \<real>\<^sub>\<le>\<^sub>0) \<Longrightarrow> Ln holomorphic_on S"
   by (simp add: field_differentiable_within_Ln holomorphic_on_def)
 
 lemma holomorphic_on_Ln' [holomorphic_intros]:
   "(\<And>z. z \<in> A \<Longrightarrow> f z \<notin> \<real>\<^sub>\<le>\<^sub>0) \<Longrightarrow> f holomorphic_on A \<Longrightarrow> (\<lambda>z. Ln (f z)) holomorphic_on A"
   using holomorphic_on_compose_gen[OF _ holomorphic_on_Ln, of f A "- \<real>\<^sub>\<le>\<^sub>0"]
   by (auto simp: o_def)
 
 lemma tendsto_Ln [tendsto_intros]:
   fixes L F
   assumes "(f \<longlongrightarrow> L) F" "L \<notin> \<real>\<^sub>\<le>\<^sub>0"
   shows   "((\<lambda>x. Ln (f x)) \<longlongrightarrow> Ln L) F"
 proof -
   have "nhds L \<ge> filtermap f F"
     using assms(1) by (simp add: filterlim_def)
   moreover have "\<forall>\<^sub>F y in nhds L. y \<in> - \<real>\<^sub>\<le>\<^sub>0"
     using eventually_nhds_in_open[of "- \<real>\<^sub>\<le>\<^sub>0" L] assms by (auto simp: open_Compl)
   ultimately have "\<forall>\<^sub>F y in filtermap f F. y \<in> - \<real>\<^sub>\<le>\<^sub>0" by (rule filter_leD)
   moreover have "continuous_on (-\<real>\<^sub>\<le>\<^sub>0) Ln" by (rule continuous_on_Ln) auto
   ultimately show ?thesis using continuous_on_tendsto_compose[of "- \<real>\<^sub>\<le>\<^sub>0" Ln f L F] assms
     by (simp add: eventually_filtermap)
 qed
 
 lemma divide_ln_mono:
   fixes x y::real
   assumes "3 \<le> x" "x \<le> y"
   shows "x / ln x \<le> y / ln y"
 proof (rule exE [OF complex_mvt_line [of x y "\<lambda>z. z / Ln z" "\<lambda>z. 1/(Ln z) - 1/(Ln z)^2"]];
     clarsimp simp add: closed_segment_Reals closed_segment_eq_real_ivl assms)
   show "\<And>u. \<lbrakk>x \<le> u; u \<le> y\<rbrakk> \<Longrightarrow> ((\<lambda>z. z / Ln z) has_field_derivative 1 / Ln u - 1 / (Ln u)\<^sup>2) (at u)"
     using \<open>3 \<le> x\<close> by (force intro!: derivative_eq_intros simp: field_simps power_eq_if)
   show "x / ln x \<le> y / ln y"
     if "Re (y / Ln y) - Re (x / Ln x) = (Re (1 / Ln u) - Re (1 / (Ln u)\<^sup>2)) * (y - x)"
     and x: "x \<le> u" "u \<le> y" for u
   proof -
     have eq: "y / ln y = (1 / ln u - 1 / (ln u)\<^sup>2) * (y - x) + x / ln x"
       using that \<open>3 \<le> x\<close> by (auto simp: Ln_Reals_eq in_Reals_norm group_add_class.diff_eq_eq)
     show ?thesis
       using exp_le \<open>3 \<le> x\<close> x by (simp add: eq) (simp add: power_eq_if divide_simps ln_ge_iff)
   qed
 qed
 
 theorem 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"
     by (intro conv_radius_ratio_limit_nonzero[of _ 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
     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
       by (simp_all add: field_simps norm_divide del: of_real_add)
 
     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
   moreover have "summable (\<lambda>n. ?f n * z^n)" using assms r
     by (intro summable_in_conv_radius) simp_all
   ultimately show ?thesis by (simp add: sums_iff)
 qed
 
 lemma Ln_series': "cmod z < 1 \<Longrightarrow> (\<lambda>n. - ((-z)^n) / of_nat n) sums ln (1 + z)"
   by (drule Ln_series) (simp add: power_minus')
 
 lemma ln_series':
   assumes "abs (x::real) < 1"
   shows   "(\<lambda>n. - ((-x)^n) / of_nat n) sums ln (1 + x)"
 proof -
   from assms have "(\<lambda>n. - ((-of_real x)^n) / of_nat n) sums ln (1 + complex_of_real x)"
     by (intro Ln_series') simp_all
   also have "(\<lambda>n. - ((-of_real x)^n) / of_nat n) = (\<lambda>n. complex_of_real (- ((-x)^n) / of_nat n))"
     by (rule ext) simp
   also from assms have "ln (1 + complex_of_real x) = of_real (ln (1 + x))"
     by (subst Ln_of_real [symmetric]) simp_all
   finally show ?thesis by (subst (asm) sums_of_real_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)"
     by (simp add: sums_iff)
   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: field_split_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
     unfolding 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
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Quadrant-type results for Ln\<close>
 
 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
     have "\<bar>Im w\<bar> < pi/2 \<longleftrightarrow> 0 < Re(exp w)"
     proof
       assume "\<bar>Im w\<bar> < pi/2" then show "0 < Re(exp w)" 
         by (auto simp: Re_exp cos_gt_zero_pi split: if_split_asm)
     next
       assume R: "0 < Re(exp w)" then 
       have "\<bar>Im w\<bar> \<noteq> pi/2"
         by (metis cos_minus cos_pi_half mult_eq_0_iff Re_exp abs_if order_less_irrefl)
       then show "\<bar>Im w\<bar> < pi/2"
         using cos_lt_zero_pi [of "-(Im w)"] cos_lt_zero_pi [of "(Im w)"] w R
         by (force simp: Re_exp zero_less_mult_iff abs_if not_less_iff_gr_or_eq)
     qed
   }
   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)"
       using cos_lt_zero_pi [of "- (Im w)"] cos_lt_zero_pi [of "(Im w)"] not_le
       by (auto simp: Re_exp zero_le_mult_iff abs_if intro: cos_ge_zero)
   }
   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)"] less_linear
       by (fastforce simp add: Im_exp zero_less_mult_iff)
   }
   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 "abs(Im w)"] sin_zero_pi_iff [of "Im w"]
       by (force simp: Im_exp zero_le_mult_iff sin_ge_zero) }
   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\<^marker>\<open>tag unimportant\<close>\<open>More Properties of Ln\<close>
 
 lemma cnj_Ln: assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "cnj(Ln z) = Ln(cnj z)"
 proof (cases "z=0")
   case False
   show ?thesis
   proof (rule exp_complex_eqI)
     have "\<bar>Im (cnj (Ln z)) - Im (Ln (cnj z))\<bar> \<le> \<bar>Im (cnj (Ln z))\<bar> + \<bar>Im (Ln (cnj z))\<bar>"
       by (rule abs_triangle_ineq4)
     also have "... < pi + pi"
     proof -
       have "\<bar>Im (cnj (Ln z))\<bar> < pi"
         by (simp add: False Im_Ln_less_pi abs_if assms minus_less_iff mpi_less_Im_Ln)
       moreover have "\<bar>Im (Ln (cnj z))\<bar> \<le> pi"
         by (meson abs_le_iff complex_cnj_zero_iff less_eq_real_def minus_less_iff  False Im_Ln_le_pi mpi_less_Im_Ln)
       ultimately show ?thesis
         by simp
     qed
     finally show "\<bar>Im (cnj (Ln z)) - Im (Ln (cnj z))\<bar> < 2 * pi"
       by simp
     show "exp (cnj (Ln z)) = exp (Ln (cnj z))"
       by (metis False complex_cnj_zero_iff exp_Ln exp_cnj)
   qed
 qed (use assms in auto)
 
 
 lemma Ln_inverse: assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "Ln(inverse z) = -(Ln z)"
 proof (cases "z=0")
   case False
   show ?thesis
   proof (rule exp_complex_eqI)
     have "\<bar>Im (Ln (inverse z)) - Im (- Ln z)\<bar> \<le> \<bar>Im (Ln (inverse z))\<bar> + \<bar>Im (- Ln z)\<bar>"
       by (rule abs_triangle_ineq4)
     also have "... < pi + pi"
     proof -
       have "\<bar>Im (Ln (inverse z))\<bar> < pi"
         by (simp add: False Im_Ln_less_pi abs_if assms minus_less_iff mpi_less_Im_Ln)
       moreover have "\<bar>Im (- Ln z)\<bar> \<le> pi"
         using False Im_Ln_le_pi mpi_less_Im_Ln by fastforce
       ultimately show ?thesis
         by simp
     qed
     finally show "\<bar>Im (Ln (inverse z)) - Im (- Ln z)\<bar> < 2 * pi"
       by simp
     show "exp (Ln (inverse z)) = exp (- Ln z)"
       by (simp add: False exp_minus)
   qed
 qed (use assms in auto)
 
 lemma Ln_minus1 [simp]: "Ln(-1) = \<i> * pi"
 proof (rule exp_complex_eqI)
   show "\<bar>Im (Ln (- 1)) - Im (\<i> * pi)\<bar> < 2 * pi"
     using Im_Ln_le_pi [of "-1"] mpi_less_Im_Ln [of "-1"] by auto
 qed auto
 
 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\<^marker>\<open>tag unimportant\<close> 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)"
   by (simp add: Ln_times)
 
 corollary\<^marker>\<open>tag unimportant\<close> 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\<^marker>\<open>tag unimportant\<close> Ln_times_Reals:
     "\<lbrakk>r \<in> Reals; Re r > 0; z \<noteq> 0\<rbrakk> \<Longrightarrow> Ln(r * z) = ln (Re r) + Ln(z)"
   using Ln_Reals_eq Ln_times_of_real by fastforce
 
 corollary\<^marker>\<open>tag unimportant\<close> 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)
 
 corollary\<^marker>\<open>tag unimportant\<close> Ln_prod:
   fixes f :: "'a \<Rightarrow> complex"
   assumes "finite A" "\<And>x. x \<in> A \<Longrightarrow> f x \<noteq> 0"
   shows "\<exists>n. Ln (prod f A) = (\<Sum>x \<in> A. Ln (f x) + (of_int (n x) * (2 * pi)) * \<i>)"
   using assms
 proof (induction A)
   case (insert x A)
   then obtain n where n: "Ln (prod f A) = (\<Sum>x\<in>A. Ln (f x) + of_real (of_int (n x) * (2 * pi)) * \<i>)"
     by auto
   define D where "D \<equiv> Im (Ln (f x)) + Im (Ln (prod f A))"
   define q::int where "q \<equiv> (if D \<le> -pi then 1 else if D > pi then -1 else 0)"
   have "prod f A \<noteq> 0" "f x \<noteq> 0"
     by (auto simp: insert.hyps insert.prems)
   with insert.hyps pi_ge_zero show ?case
     by (rule_tac x="n(x:=q)" in exI) (force simp: Ln_times q_def D_def n intro!: sum.cong)
 qed auto
 
 lemma Ln_minus:
   assumes "z \<noteq> 0"
     shows "Ln(-z) = (if Im(z) \<le> 0 \<and> \<not>(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
     by (simp add: Ln_inverse)
 next
   case True
   then have z: "Im z = 0" "Re z < 0" "- z \<notin> \<real>\<^sub>\<le>\<^sub>0"
     using assms complex_eq_iff complex_nonpos_Reals_iff by auto
   have "Ln(inverse z) = Ln(- (inverse (-z)))"
     by simp
   also have "... = Ln (inverse (-z)) + \<i> * complex_of_real pi"
     using assms z by (simp add: Ln_minus divide_less_0_iff)
   also have "... = - Ln (- z) + \<i> * complex_of_real pi"
     using z Ln_inverse by presburger
   also have "... = - (Ln z) + \<i> * 2 * complex_of_real pi"
     using Ln_minus assms z by auto
   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 (simp add: Ln_times) auto
 
 lemma Ln_of_nat [simp]: "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))"
     by (simp add: Ln_of_real[symmetric])
   also from assms have "... = of_real (ln (of_nat m) - ln (of_nat n))"
     by (simp add: ln_div)
   finally show ?thesis .
 qed
 
 subsection\<open>The Argument of a Complex Number\<close>
 
 text\<open>Unlike in HOL Light, it's defined for the same interval as the complex logarithm: \<open>(-\<pi>,\<pi>]\<close>.\<close>
 
 lemma Arg_eq_Im_Ln:
-  assumes "z \<noteq> 0" shows "arg z = Im (Ln z)"
+  assumes "z \<noteq> 0" shows "Arg z = Im (Ln z)"
 proof (rule arg_unique)
   show "sgn z = cis (Im (Ln z))"
     by (metis assms exp_Ln exp_eq_polar nonzero_mult_div_cancel_left norm_eq_zero 
               norm_exp_eq_Re of_real_eq_0_iff sgn_eq)
   show "- pi < Im (Ln z)"
     by (simp add: assms mpi_less_Im_Ln)
   show "Im (Ln z) \<le> pi"
     by (simp add: Im_Ln_le_pi assms)
 qed
 
 text \<open>The 1990s definition of argument coincides with the more recent one\<close>
-lemma Arg_definition_equivalence:
-  shows "arg z = (if z = 0 then 0 else Im (Ln z))"
-  by (simp add: Arg_eq_Im_Ln arg_zero)
-
-definition\<^marker>\<open>tag important\<close> Arg :: "complex \<Rightarrow> real" where "Arg z \<equiv> (if z = 0 then 0 else Im (Ln z))"
+lemma\<^marker>\<open>tag important\<close> Arg_def:
+  shows "Arg z = (if z = 0 then 0 else Im (Ln z))"
+  by (simp add: Arg_eq_Im_Ln Arg_zero)
 
 lemma Arg_of_real: "Arg (of_real r) = (if r<0 then pi else 0)"
   by (simp add: Im_Ln_eq_pi Arg_def)
 
 lemma mpi_less_Arg: "-pi < Arg z"
     and Arg_le_pi: "Arg z \<le> pi"
   by (auto simp: Arg_def mpi_less_Im_Ln Im_Ln_le_pi)
 
 lemma
   assumes "z \<noteq> 0"
   shows Arg_eq: "z = of_real(norm z) * exp(\<i> * Arg z)"
   using assms exp_Ln exp_eq_polar
   by (auto simp:  Arg_def)
 
 lemma is_Arg_Arg: "z \<noteq> 0 \<Longrightarrow> is_Arg z (Arg z)"
   by (simp add: Arg_eq is_Arg_def)
 
 lemma Argument_exists:
   assumes "z \<noteq> 0" and R: "R = {r-pi<..r+pi}"
   obtains s where "is_Arg z s" "s\<in>R"
 proof -
   let ?rp = "r - Arg z + pi"
   define k where "k \<equiv> \<lfloor>?rp / (2 * pi)\<rfloor>"
   have "(Arg z + of_int k * (2 * pi)) \<in> R"
     using floor_divide_lower [of "2*pi" ?rp] floor_divide_upper [of "2*pi" ?rp]
     by (auto simp: k_def algebra_simps R)
   then show ?thesis
     using Arg_eq \<open>z \<noteq> 0\<close> is_Arg_2pi_iff is_Arg_def that by blast
 qed
 
 lemma Argument_exists_unique:
   assumes "z \<noteq> 0" and R: "R = {r-pi<..r+pi}"
   obtains s where "is_Arg z s" "s\<in>R" "\<And>t. \<lbrakk>is_Arg z t; t\<in>R\<rbrakk> \<Longrightarrow> s=t"
 proof -
   obtain s where s: "is_Arg z s" "s\<in>R"
     using Argument_exists [OF assms] .
   moreover have "\<And>t. \<lbrakk>is_Arg z t; t\<in>R\<rbrakk> \<Longrightarrow> s=t"
     using assms s  by (auto simp: is_Arg_eqI)
   ultimately show thesis
     using that by blast
 qed
 
 lemma Argument_Ex1:
   assumes "z \<noteq> 0" and R: "R = {r-pi<..r+pi}"
   shows "\<exists>!s. is_Arg z s \<and> s \<in> R"
   using Argument_exists_unique [OF assms]  by metis
 
 lemma Arg_divide:
   assumes "w \<noteq> 0" "z \<noteq> 0"
   shows "is_Arg (z / w) (Arg z - Arg w)"
   using Arg_eq [of z] Arg_eq [of w] Arg_eq [of "norm(z / w)"] assms
   by (auto simp: is_Arg_def norm_divide field_simps exp_diff Arg_of_real)
 
 lemma Arg_unique_lemma:
   assumes z:  "is_Arg z t"
       and z': "is_Arg z t'"
       and t:  "- pi < t"  "t \<le> pi"
       and t': "- pi < t'" "t' \<le> pi"
       and nz: "z \<noteq> 0"
     shows "t' = t"
   using Arg2pi_unique_lemma [of "- (inverse z)"]
 proof -
   have "pi - t' = pi - t"
   proof (rule Arg2pi_unique_lemma [of "- (inverse z)"])
     have "- (inverse z) = - (inverse (of_real(norm z) * exp(\<i> * t)))"
       by (metis is_Arg_def z)
     also have "... = (cmod (- inverse z)) * exp (\<i> * (pi - t))"
       by (auto simp: field_simps exp_diff norm_divide)
     finally show "is_Arg (- inverse z) (pi - t)"
       unfolding is_Arg_def .
     have "- (inverse z) = - (inverse (of_real(norm z) * exp(\<i> * t')))"
       by (metis is_Arg_def z')
     also have "... = (cmod (- inverse z)) * exp (\<i> * (pi - t'))"
       by (auto simp: field_simps exp_diff norm_divide)
     finally show "is_Arg (- inverse z) (pi - t')"
       unfolding is_Arg_def .
   qed (use assms in auto)
   then show ?thesis
     by simp
 qed
 
 lemma complex_norm_eq_1_exp_eq: "norm z = 1 \<longleftrightarrow> exp(\<i> * (Arg z)) = z"
   by (metis Arg_eq exp_not_eq_zero exp_zero mult.left_neutral norm_zero of_real_1 norm_exp_i_times)
 
 lemma Arg_unique: "\<lbrakk>of_real r * exp(\<i> * a) = z; 0 < r; -pi < a; a \<le> pi\<rbrakk> \<Longrightarrow> Arg z = a"
   by (rule Arg_unique_lemma [unfolded is_Arg_def, OF _ Arg_eq])
      (use mpi_less_Arg Arg_le_pi in \<open>auto simp: norm_mult\<close>)
 
 lemma Arg_minus:
   assumes "z \<noteq> 0"
   shows "Arg (-z) = (if Arg z \<le> 0 then Arg z + pi else Arg z - pi)"
 proof -
   have [simp]: "cmod z * cos (Arg z) = Re z"
     using assms Arg_eq [of z] by (metis Re_exp exp_Ln norm_exp_eq_Re Arg_def)
   have [simp]: "cmod z * sin (Arg z) = Im z"
     using assms Arg_eq [of z] by (metis Im_exp exp_Ln norm_exp_eq_Re Arg_def)
   show ?thesis
     apply (rule Arg_unique [of "norm z", OF complex_eqI])
     using mpi_less_Arg [of z] Arg_le_pi [of z] assms
     by (auto simp: Re_exp Im_exp)
 qed
 
 lemma Arg_times_of_real [simp]:
   assumes "0 < r" shows "Arg (of_real r * z) = Arg z"
 proof (cases "z=0")
   case True
   then show ?thesis
     by (simp add: Arg_def)
 next
   case False
   with Arg_eq assms show ?thesis
   by (auto simp: mpi_less_Arg Arg_le_pi intro!:  Arg_unique [of "r * norm z"])
 qed
 
 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_less_0: "0 \<le> Arg z \<longleftrightarrow> 0 \<le> Im z"
   using Im_Ln_le_pi Im_Ln_pos_le
   by (simp add: Arg_def)
 
 lemma Arg_eq_pi: "Arg z = pi \<longleftrightarrow> Re z < 0 \<and> Im z = 0"
   by (auto simp: Arg_def Im_Ln_eq_pi)
 
 lemma Arg_lt_pi: "0 < Arg z \<and> Arg z < pi \<longleftrightarrow> 0 < Im z"
   using Arg_less_0 [of z] Im_Ln_pos_lt
   by (auto simp: order.order_iff_strict Arg_def)
 
 lemma Arg_eq_0: "Arg z = 0 \<longleftrightarrow> z \<in> \<real> \<and> 0 \<le> Re z"
   using complex_is_Real_iff
   by (simp add: Arg_def Im_Ln_eq_0) (metis less_eq_real_def of_real_Re of_real_def scale_zero_left)
 
 corollary\<^marker>\<open>tag unimportant\<close> Arg_ne_0: assumes "z \<notin> \<real>\<^sub>\<ge>\<^sub>0" shows "Arg z \<noteq> 0"
   using assms by (auto simp: nonneg_Reals_def Arg_eq_0)
 
 lemma Arg_eq_pi_iff: "Arg z = pi \<longleftrightarrow> z \<in> \<real> \<and> Re z < 0"
 proof (cases "z=0")
   case False
   then show ?thesis
     using Arg_eq_0 [of "-z"] Arg_eq_pi complex_is_Real_iff by blast
 qed (simp add: Arg_def)
 
 lemma Arg_eq_0_pi: "Arg z = 0 \<or> Arg z = pi \<longleftrightarrow> z \<in> \<real>"
   using Arg_eq_pi_iff Arg_eq_0 by force
 
 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_inverse: "Arg(inverse z) = (if z \<in> \<real> then Arg z else - Arg z)"
 proof (cases "z \<in> \<real>")
   case True
   then show ?thesis
     by simp (metis Arg2pi_inverse Arg2pi_real Arg_real Reals_inverse)
 next
   case False
   then have z: "Arg z < pi" "z \<noteq> 0"
     using Arg_eq_0_pi Arg_le_pi by (auto simp: less_eq_real_def)
   show ?thesis
     apply (rule Arg_unique [of "inverse (norm z)"])
     using False z mpi_less_Arg [of z] Arg_eq [of z]
     by (auto simp: exp_minus field_simps)
 qed
 
 lemma Arg_eq_iff:
   assumes "w \<noteq> 0" "z \<noteq> 0"
   shows "Arg w = Arg z \<longleftrightarrow> (\<exists>x. 0 < x \<and> w = of_real x * z)" (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then have "w = complex_of_real (cmod w / cmod z) * z"
     by (metis Arg_eq assms divide_divide_eq_right eq_divide_eq exp_not_eq_zero of_real_divide)
   then show ?rhs
     using assms divide_pos_pos zero_less_norm_iff by blast
 qed auto
 
 lemma Arg_inverse_eq_0: "Arg(inverse z) = 0 \<longleftrightarrow> Arg z = 0"
   by (metis Arg_eq_0 Arg_inverse inverse_inverse_eq)
 
 lemma Arg_cnj_eq_inverse: "z\<noteq>0 \<Longrightarrow> Arg (cnj z) = Arg (inverse z)"
   using Arg2pi_cnj_eq_inverse Arg2pi_eq_iff Arg_eq_iff by auto
 
 lemma Arg_cnj: "Arg(cnj z) = (if z \<in> \<real> then Arg z else - Arg z)"
   by (metis Arg_cnj_eq_inverse Arg_inverse Reals_0 complex_cnj_zero)
 
 lemma Arg_exp: "-pi < Im z \<Longrightarrow> Im z \<le> pi \<Longrightarrow> Arg(exp z) = Im z"
   by (rule Arg_unique [of "exp(Re z)"]) (auto simp: exp_eq_polar)
 
 lemma Ln_Arg: "z\<noteq>0 \<Longrightarrow> Ln(z) = ln(norm z) + \<i> * Arg(z)"
   by (metis Arg_def Re_Ln complex_eq)
 
 lemma continuous_at_Arg:
   assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0"
     shows "continuous (at z) Arg"
 proof -
   have [simp]: "(\<lambda>z. Im (Ln z)) \<midarrow>z\<rightarrow> Arg z"
     using Arg_def assms continuous_at by fastforce
   show ?thesis
     unfolding continuous_at
   proof (rule Lim_transform_within_open)
     show "\<And>w. \<lbrakk>w \<in> - \<real>\<^sub>\<le>\<^sub>0; w \<noteq> z\<rbrakk> \<Longrightarrow> Im (Ln w) = Arg w"
       by (metis Arg_def Compl_iff nonpos_Reals_zero_I)
   qed (use assms in auto)
 qed
 
 lemma continuous_within_Arg: "z \<notin> \<real>\<^sub>\<le>\<^sub>0 \<Longrightarrow> continuous (at z within S) Arg"
   using continuous_at_Arg continuous_at_imp_continuous_within by blast
 
 
 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. But it's always an integer.\<close>
 
 lemma is_Arg_exp_Im: "is_Arg (exp z) (Im z)"
   using exp_eq_polar is_Arg_def norm_exp_eq_Re by auto
 
 lemma is_Arg_exp_diff_2pi:
   assumes "is_Arg (exp z) \<theta>"
   shows "\<exists>k. Im z - of_int k * (2 * pi) = \<theta>"
 proof (intro exI is_Arg_eqI)
   let ?k = "\<lfloor>(Im z - \<theta>) / (2 * pi)\<rfloor>"
   show "is_Arg (exp z) (Im z - real_of_int ?k * (2 * pi))"
     by (metis diff_add_cancel is_Arg_2pi_iff is_Arg_exp_Im)
   show "\<bar>Im z - real_of_int ?k * (2 * pi) - \<theta>\<bar> < 2 * pi"
     using floor_divide_upper [of "2*pi" "Im z - \<theta>"] floor_divide_lower [of "2*pi" "Im z - \<theta>"]
     by (auto simp: algebra_simps abs_if)
 qed (auto simp: is_Arg_exp_Im assms)
 
 lemma Arg_exp_diff_2pi: "\<exists>k. Im z - of_int k * (2 * pi) = Arg (exp z)"
   using is_Arg_exp_diff_2pi [OF is_Arg_Arg] by auto
 
 lemma unwinding_in_Ints: "(z - Ln(exp z)) / (of_real(2*pi) * \<i>) \<in> \<int>"
   using Arg_exp_diff_2pi [of z]
   by (force simp: Ints_def image_def field_simps Arg_def intro!: complex_eqI)
 
 definition\<^marker>\<open>tag important\<close> unwinding :: "complex \<Rightarrow> int" where
    "unwinding z \<equiv> THE k. of_int k = (z - Ln(exp z)) / (of_real(2*pi) * \<i>)"
 
 lemma unwinding: "of_int (unwinding z) = (z - Ln(exp z)) / (of_real(2*pi) * \<i>)"
   using unwinding_in_Ints [of z]
   unfolding unwinding_def Ints_def by force
 
 lemma unwinding_2pi: "(2*pi) * \<i> * unwinding(z) = z - Ln(exp z)"
   by (simp add: unwinding)
 
 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)"
   using unwinding_2pi by (simp add: exp_add)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Relation between Ln and Arg2pi, and hence continuity of Arg2pi\<close>
 
 lemma Arg2pi_Ln:
   assumes "0 < Arg2pi z" shows "Arg2pi 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(Arg2pi z))"
     using Arg2pi [of z]
     by (metis is_Arg_def abs_norm_cancel nonzero_mult_div_cancel_left norm_of_real zero_less_norm_iff)
   then have "- z / of_real(norm z) = exp (\<i> * (of_real (Arg2pi 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 (Arg2pi z) - pi)))"
     by simp
   also have "... = \<i> * (of_real(Arg2pi z) - pi)"
     using Arg2pi [of z] assms pi_not_less_zero
     by auto
   finally have "Arg2pi 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_Arg2pi:
   assumes "z \<notin> \<real>\<^sub>\<ge>\<^sub>0"
     shows "continuous (at z) Arg2pi"
 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]: "Im x \<noteq> 0 \<Longrightarrow> Im (Ln (- x)) + pi = Arg2pi x" for x
     using Arg2pi_Ln  by (simp add: Arg2pi_gt_0 complex_nonneg_Reals_iff)
   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> Arg2pi z"
     using "*" by (simp add: Arg2pi_Ln Arg2pi_gt_0 assms continuous_within) 
   show ?thesis
     unfolding continuous_at
   proof (rule Lim_transform_within_open)
     show "\<And>x. \<lbrakk>x \<in> - \<real>\<^sub>\<ge>\<^sub>0; x \<noteq> z\<rbrakk> \<Longrightarrow> Im (Ln (- x)) + pi = Arg2pi x"
       by (auto simp add: Arg2pi_Ln [OF Arg2pi_gt_0] complex_nonneg_Reals_iff)
   qed (use assms in auto)
 qed
 
 
 text\<open>Relation between Arg2pi and arctangent in upper halfplane\<close>
 lemma Arg2pi_arctan_upperhalf:
   assumes "0 < Im z"
     shows "Arg2pi z = pi/2 - arctan(Re z / Im z)"
 proof (cases "z = 0")
   case False
   show ?thesis
   proof (rule Arg2pi_unique [of "norm z"])
     show "(cmod z) * exp (\<i> * (pi / 2 - arctan (Re z / Im z))) = z"
       apply (rule complex_eqI)
       using assms norm_complex_def [of z, symmetric]
       unfolding exp_Euler cos_diff sin_diff sin_of_real cos_of_real
       by (simp_all add: field_simps real_sqrt_divide sin_arctan cos_arctan)
   qed (use False arctan [of "Re z / Im z"] in auto)
 qed (use assms in auto)
 
 lemma Arg2pi_eq_Im_Ln:
   assumes "0 \<le> Im z" "0 < Re z"
     shows "Arg2pi z = Im (Ln z)"
 proof (cases "Im z = 0")
   case True then show ?thesis
     using Arg2pi_eq_0 Ln_in_Reals assms(2) complex_is_Real_iff by auto
 next
   case False
   then have *: "Arg2pi z > 0"
     using Arg2pi_gt_0 complex_is_Real_iff by blast
   then have "z \<noteq> 0"
     by auto
   with * assms False show ?thesis
     by (subst Arg2pi_Ln) (auto simp: Ln_minus)
 qed
 
 lemma continuous_within_upperhalf_Arg2pi:
   assumes "z \<noteq> 0"
     shows "continuous (at z within {z. 0 \<le> Im z}) Arg2pi"
 proof (cases "z \<in> \<real>\<^sub>\<ge>\<^sub>0")
   case False then show ?thesis
     using continuous_at_Arg2pi 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]: "Arg2pi z = 0" "Im (Ln z) = 0"
     by (auto simp: Arg2pi_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>Arg2pi x\<bar> < e"
       apply (rule_tac x="min d (Re z / 2)" in exI)
       using z d by (auto simp: Arg2pi_eq_Im_Ln)
   qed
 qed
 
 lemma continuous_on_upperhalf_Arg2pi: "continuous_on ({z. 0 \<le> Im z} - {0}) Arg2pi"
   unfolding continuous_on_eq_continuous_within
   by (metis DiffE Diff_subset continuous_within_subset continuous_within_upperhalf_Arg2pi insertCI)
 
 lemma open_Arg2pi2pi_less_Int:
   assumes "0 \<le> s" "t \<le> 2*pi"
     shows "open ({y. s < Arg2pi y} \<inter> {y. Arg2pi y < t})"
 proof -
   have 1: "continuous_on (UNIV - \<real>\<^sub>\<ge>\<^sub>0) Arg2pi"
     using continuous_at_Arg2pi 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 < Arg2pi y} \<inter> {y. Arg2pi y < t} \<subseteq> - \<real>\<^sub>\<ge>\<^sub>0"
     using assms by (auto simp: Arg2pi_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_Arg2pi2pi_gt: "open {z. t < Arg2pi z}"
 proof (cases "t < 0")
   case True then have "{z. t < Arg2pi z} = UNIV"
     using Arg2pi_ge_0 less_le_trans by auto
   then show ?thesis
     by simp
 next
   case False then show ?thesis
     using open_Arg2pi2pi_less_Int [of t "2*pi"] Arg2pi_lt_2pi
     by auto
 qed
 
 lemma closed_Arg2pi2pi_le: "closed {z. Arg2pi z \<le> t}"
   using open_Arg2pi2pi_gt [of t]
   by (simp add: closed_def Set.Collect_neg_eq [symmetric] not_le)
 
 subsection\<^marker>\<open>tag unimportant\<close>\<open>Complex Powers\<close>
 
 lemma powr_to_1 [simp]: "z powr 1 = (z::complex)"
   by (simp add: powr_def)
 
 lemma powr_nat:
   fixes n::nat and z::complex shows "z powr n = (if z = 0 then 0 else z^n)"
   by (simp add: exp_of_nat_mult powr_def)
 
 lemma norm_powr_real: "w \<in> \<real> \<Longrightarrow> 0 < Re w \<Longrightarrow> norm(w powr z) = exp(Re z * ln(Re w))"
   using Ln_Reals_eq norm_exp_eq_Re by (auto simp: Im_Ln_eq_0 powr_def norm_complex_def)
 
 lemma powr_complexpow [simp]:
   fixes x::complex shows "x \<noteq> 0 \<Longrightarrow> x powr (of_nat n) = x^n"
   by (induct n) (auto simp: ac_simps powr_add)
 
 lemma powr_complexnumeral [simp]:
   fixes x::complex shows "x \<noteq> 0 \<Longrightarrow> x powr (numeral n) = x ^ (numeral n)"
   by (metis of_nat_numeral powr_complexpow)
 
 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:
   assumes "w \<in> \<real>" "z \<in> \<real>" "0 < Re w"
   shows "w powr z = exp(Re z * ln(Re w))"
 proof -
   have "w \<noteq> 0"
     using assms by auto
   with assms show ?thesis
     by (simp add: powr_def Ln_Reals_eq of_real_exp)
 qed
 
 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)"
   by (simp_all add: powr_def exp_eq_polar)
 
 lemma powr_of_int:
   fixes z::complex and n::int
   assumes "z\<noteq>(0::complex)"
   shows "z powr of_int n = (if n\<ge>0 then z^nat n else inverse (z^nat (-n)))"
   by (metis assms not_le of_int_of_nat powr_complexpow powr_minus)
 
 lemma powr_Reals_eq: "\<lbrakk>x \<in> \<real>; y \<in> \<real>; Re x \<ge> 0\<rbrakk> \<Longrightarrow> x powr y = of_real (Re x powr Re y)"
   by (metis of_real_Re powr_of_real)
 
 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 Re_powr_le: "r \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> Re (r powr z) \<le> Re r powr Re z"
   by (auto simp: powr_def nonneg_Reals_def order_trans [OF complex_Re_le_cmod])
 
 lemma
   fixes w::complex
   shows Reals_powr [simp]: "\<lbrakk>w \<in> \<real>\<^sub>\<ge>\<^sub>0; z \<in> \<real>\<rbrakk> \<Longrightarrow> w powr z \<in> \<real>"
   and nonneg_Reals_powr [simp]: "\<lbrakk>w \<in> \<real>\<^sub>\<ge>\<^sub>0; z \<in> \<real>\<rbrakk> \<Longrightarrow> w powr z \<in> \<real>\<^sub>\<ge>\<^sub>0"
   by (auto simp: nonneg_Reals_def Reals_def powr_of_real)
 
 lemma powr_neg_real_complex:
   "(- 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>"
     by (simp add: Ln_minus Ln_of_real)
   also from x have "exp (a * ...) = cis pi powr (of_real (sgn x) * a) * of_real x powr a"
     by (simp add: powr_def exp_add algebra_simps Ln_of_real cis_conv_exp)
   also note cis_pi
   finally show ?thesis by simp
 qed simp_all
 
 lemma has_field_derivative_powr:
   fixes z :: complex
   assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0"
   shows "((\<lambda>z. z powr s) has_field_derivative (s * z powr (s - 1))) (at z)"
 proof (cases "z=0")
   case False
   then have \<section>: "exp (s * Ln z) * inverse z = exp ((s - 1) * Ln z)"
     by (simp add: divide_complex_def exp_diff left_diff_distrib')
   show ?thesis
     unfolding powr_def
   proof (rule has_field_derivative_transform_within)
     show "((\<lambda>z. exp (s * Ln z)) has_field_derivative s * (if z = 0 then 0 else exp ((s - 1) * Ln z)))
            (at z)"
       by (intro derivative_eq_intros | simp add: assms False \<section>)+
   qed (use False in auto)
 qed (use assms in auto)
 
 declare has_field_derivative_powr[THEN DERIV_chain2, derivative_intros]
 
 lemma has_field_derivative_powr_of_int:
   fixes z :: complex
   assumes gderiv:"(g has_field_derivative gd) (at z within S)" and "g z\<noteq>0"
   shows "((\<lambda>z. g z powr of_int n) has_field_derivative (n * g z powr (of_int n - 1) * gd)) (at z within S)"
 proof -
   define dd where "dd = of_int n * g z powr (of_int (n - 1)) * gd"
   obtain e where "e>0" and e_dist:"\<forall>y\<in>S. dist z y < e \<longrightarrow> g y \<noteq> 0"
     using DERIV_continuous[OF gderiv,THEN continuous_within_avoid] \<open>g z\<noteq>0\<close> by auto
   have ?thesis when "n\<ge>0"
   proof -
     define dd' where "dd' = of_int n * g z ^ (nat n - 1) * gd"
     have "dd=dd'"
     proof (cases "n=0")
       case False
       then have "n-1 \<ge>0" using \<open>n\<ge>0\<close> by auto
       then have "g z powr (of_int (n - 1)) = g z ^ (nat n - 1)"
         using powr_of_int[OF \<open>g z\<noteq>0\<close>,of "n-1"] by (simp add: nat_diff_distrib')
       then show ?thesis unfolding dd_def dd'_def by simp
     qed (simp add:dd_def dd'_def)
     then have "((\<lambda>z. g z powr of_int n) has_field_derivative dd) (at z within S)
                 \<longleftrightarrow> ((\<lambda>z. g z powr of_int n) has_field_derivative dd') (at z within S)"
       by simp
     also have "... \<longleftrightarrow> ((\<lambda>z. g z ^ nat n) has_field_derivative dd') (at z within S)"
     proof (rule has_field_derivative_cong_eventually)
       show "\<forall>\<^sub>F x in at z within S. g x powr of_int n = g x ^ nat n"
         unfolding eventually_at
         apply (rule exI[where x=e])
         using powr_of_int that \<open>e>0\<close> e_dist by (simp add: dist_commute)
     qed (use powr_of_int \<open>g z\<noteq>0\<close> that in simp)
     also have "..." unfolding dd'_def using gderiv that
       by (auto intro!: derivative_eq_intros)
     finally have "((\<lambda>z. g z powr of_int n) has_field_derivative dd) (at z within S)" .
     then show ?thesis unfolding dd_def by simp
   qed
   moreover have ?thesis when "n<0"
   proof -
     define dd' where "dd' = of_int n / g z ^ (nat (1 - n)) * gd"
     have "dd=dd'"
     proof -
       have "g z powr of_int (n - 1) = inverse (g z ^ nat (1-n))"
         using powr_of_int[OF \<open>g z\<noteq>0\<close>,of "n-1"] that by auto
       then show ?thesis
         unfolding dd_def dd'_def by (simp add: divide_inverse)
     qed
     then have "((\<lambda>z. g z powr of_int n) has_field_derivative dd) (at z within S)
                 \<longleftrightarrow> ((\<lambda>z. g z powr of_int n) has_field_derivative dd') (at z within S)"
       by simp
     also have "... \<longleftrightarrow> ((\<lambda>z. inverse (g z ^ nat (-n))) has_field_derivative dd') (at z within S)"
     proof (rule has_field_derivative_cong_eventually)
       show "\<forall>\<^sub>F x in at z within S. g x powr of_int n = inverse (g x ^ nat (- n))"
          unfolding eventually_at
         apply (rule exI[where x=e])
         using powr_of_int that \<open>e>0\<close> e_dist by (simp add: dist_commute)
     qed (use powr_of_int \<open>g z\<noteq>0\<close> that in simp)
     also have "..."
     proof -
       have "nat (- n) + nat (1 - n) - Suc 0 = nat (- n) + nat (- n)"
         by auto
       then show ?thesis
         unfolding dd'_def using gderiv that \<open>g z\<noteq>0\<close>
         by (auto intro!: derivative_eq_intros simp add:field_split_simps power_add[symmetric])
     qed
     finally have "((\<lambda>z. g z powr of_int n) has_field_derivative dd) (at z within S)" .
     then show ?thesis unfolding dd_def by simp
   qed
   ultimately show ?thesis by force
 qed
 
 lemma field_differentiable_powr_of_int:
   fixes z :: complex
   assumes gderiv: "g field_differentiable (at z within S)" and "g z \<noteq> 0"
   shows "(\<lambda>z. g z powr of_int n) field_differentiable (at z within S)"
 using has_field_derivative_powr_of_int assms(2) field_differentiable_def gderiv by blast
 
 lemma holomorphic_on_powr_of_int [holomorphic_intros]:
   assumes holf: "f holomorphic_on S" and 0: "\<And>z. z\<in>S \<Longrightarrow> f z \<noteq> 0"
   shows "(\<lambda>z. (f z) powr of_int n) holomorphic_on S"
 proof (cases "n\<ge>0")
   case True
   then have "?thesis \<longleftrightarrow> (\<lambda>z. (f z) ^ nat n) holomorphic_on S"
     by (metis (no_types, lifting) 0 holomorphic_cong powr_of_int)
   moreover have "(\<lambda>z. (f z) ^ nat n) holomorphic_on S"
     using holf by (auto intro: holomorphic_intros)
   ultimately show ?thesis by auto
 next
   case False
   then have "?thesis \<longleftrightarrow> (\<lambda>z. inverse (f z) ^ nat (-n)) holomorphic_on S"
     by (metis (no_types, lifting) "0" holomorphic_cong power_inverse powr_of_int)
   moreover have "(\<lambda>z. inverse (f z) ^ nat (-n)) holomorphic_on S"
     using assms by (auto intro!:holomorphic_intros)
   ultimately show ?thesis by auto
 qed
 
 lemma has_field_derivative_powr_right [derivative_intros]:
     "w \<noteq> 0 \<Longrightarrow> ((\<lambda>z. w powr z) has_field_derivative Ln w * w powr z) (at z)"
   unfolding powr_def by (intro derivative_eq_intros | simp)+
 
 lemma field_differentiable_powr_right [derivative_intros]:
   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 [holomorphic_intros]:
   assumes "f holomorphic_on s"
   shows "(\<lambda>z. w powr (f z)) holomorphic_on s"
 proof (cases "w = 0")
   case False
   with assms show ?thesis
     unfolding holomorphic_on_def field_differentiable_def
     by (metis (full_types) DERIV_chain' has_field_derivative_powr_right)
 qed simp
 
 lemma holomorphic_on_divide_gen [holomorphic_intros]:
   assumes f: "f holomorphic_on s" and g: "g holomorphic_on s" and 0: "\<And>z z'. \<lbrakk>z \<in> s; z' \<in> s\<rbrakk> \<Longrightarrow> g z = 0 \<longleftrightarrow> g z' = 0"
 shows "(\<lambda>z. f z / g z) holomorphic_on s"
 proof (cases "\<exists>z\<in>s. g z = 0")
   case True
   with 0 have "g z = 0" if "z \<in> s" for z
     using that by blast
   then show ?thesis
     using g holomorphic_transform by auto
 next
   case False
   with 0 have "g z \<noteq> 0" if "z \<in> s" for z
     using that by blast
   with holomorphic_on_divide show ?thesis
     using f g by blast
 qed
 
 lemma norm_powr_real_powr:
   "w \<in> \<real> \<Longrightarrow> 0 \<le> Re w \<Longrightarrow> cmod (w powr z) = Re w powr Re z"
   by (metis dual_order.order_iff_strict norm_powr_real norm_zero of_real_0 of_real_Re powr_def)
 
 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}))"
     by (simp add: filterlim_inf at_within_def)
 
   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 "a \<notin> \<real>\<^sub>\<le>\<^sub>0 \<or> (a = 0 \<and> Re b > 0)" and "(f \<longlongrightarrow> a) F" "(g \<longlongrightarrow> b) F"
   shows   "((\<lambda>z. f z powr g z) \<longlongrightarrow> a powr b) F"
   using assms tendsto_powr_complex tendsto_powr_complex_0 by fastforce
 
 lemma tendsto_neg_powr_complex_of_real:
   assumes "filterlim f at_top F" and "Re s < 0"
   shows   "((\<lambda>x. complex_of_real (f x) powr s) \<longlongrightarrow> 0) F"
 proof -
   have "((\<lambda>x. norm (complex_of_real (f x) powr s)) \<longlongrightarrow> 0) F"
   proof (rule Lim_transform_eventually)
     from assms(1) have "eventually (\<lambda>x. f x \<ge> 0) F"
       by (auto simp: filterlim_at_top)
     thus "eventually (\<lambda>x. f x powr Re s = norm (of_real (f x) powr s)) F"
       by eventually_elim (simp add: norm_powr_real_powr)
     from assms show "((\<lambda>x. f x powr Re s) \<longlongrightarrow> 0) F"
       by (intro tendsto_neg_powr)
   qed
   thus ?thesis by (simp add: tendsto_norm_zero_iff)
 qed
 
 lemma tendsto_neg_powr_complex_of_nat:
   assumes "filterlim f at_top F" and "Re s < 0"
   shows   "((\<lambda>x. of_nat (f x) powr s) \<longlongrightarrow> 0) F"
 proof -
   have "((\<lambda>x. of_real (real (f x)) powr s) \<longlongrightarrow> 0) F" using assms(2)
     by (intro filterlim_compose[OF _ tendsto_neg_powr_complex_of_real]
               filterlim_compose[OF _ assms(1)] filterlim_real_sequentially filterlim_ident) auto
   thus ?thesis by simp
 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\<^marker>\<open>tag unimportant\<close>\<open>Some Limits involving Logarithms\<close>
 
 lemma lim_Ln_over_power:
   fixes s::complex
   assumes "0 < Re s"
     shows "(\<lambda>n. Ln (of_nat n) / of_nat n powr s) \<longlonglongrightarrow> 0"
 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"
     have "2 / (e * (Re s)\<^sup>2) > 0"
       using e assms by simp
     with x have "x > 0"
       by linarith
     then have "x * 2 \<le> e * (x\<^sup>2 * (Re s)\<^sup>2)"
       using e assms x by (auto simp: power2_eq_square field_simps)
     also have "... < e * (2 + (x * (Re s * 2) + x\<^sup>2 * (Re s)\<^sup>2))"
       using e assms \<open>x > 0\<close>
       by (auto simp: power2_eq_square field_simps add_pos_pos)
     finally show "0 < e * 2 + (e * Re s * 2 - 2) * x + e * (Re s)\<^sup>2 * x\<^sup>2"
       by (auto simp: algebra_simps)
   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 obtain xo where "xo > 0" and xo: "\<And>x. x \<ge> xo \<Longrightarrow> x < e * exp (Re s * x)"
     using e  by (auto simp: field_simps)
   have "norm (Ln (of_nat n) / of_nat n powr s) < e" if "n \<ge> nat \<lceil>exp xo\<rceil>" for n
   proof -
     have "ln (real n) \<ge> xo"
       using that exp_gt_zero ln_ge_iff [of n] nat_ceiling_le_eq by fastforce
     then show ?thesis
       using e xo [of "ln n"]  by (auto simp: norm_divide norm_powr_real field_split_simps)
   qed
   then show "\<exists>no. \<forall>n\<ge>no. norm (Ln (of_nat n) / of_nat n powr s) < e"
     by blast
 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 lim_ln_over_power:
   fixes s :: real
   assumes "0 < s"
   shows "((\<lambda>n. ln n / (n powr s)) \<longlongrightarrow> 0) sequentially"
 proof -
   have "(\<lambda>n. ln (Suc n) / (Suc n) powr s) \<longlonglongrightarrow> 0"
     using lim_Ln_over_power [of "of_real s", THEN filterlim_sequentially_Suc [THEN iffD2]] assms
     by (simp add: lim_sequentially dist_norm Ln_Reals_eq norm_powr_real_powr norm_divide)
   then show ?thesis
     using filterlim_sequentially_Suc[of "\<lambda>n::nat. ln n / n powr s"] by auto
 qed
 
 lemma lim_ln_over_n [tendsto_intros]: "((\<lambda>n. ln(real_of_nat n) / of_nat n) \<longlongrightarrow> 0) sequentially"
   using lim_ln_over_power [of 1] by auto
 
 lemma lim_log_over_n [tendsto_intros]:
   "(\<lambda>n. log k n/n) \<longlonglongrightarrow> 0"
 proof -
   have *: "log k n/n = (1/ln k) * (ln n / n)" for n
     unfolding log_def by auto
   have "(\<lambda>n. (1/ln k) * (ln n / n)) \<longlonglongrightarrow> (1/ln k) * 0"
     by (intro tendsto_intros)
   then show ?thesis
     unfolding * by auto
 qed
 
 lemma lim_1_over_complex_power:
   assumes "0 < Re s"
   shows "(\<lambda>n. 1 / of_nat n powr s) \<longlonglongrightarrow> 0"
 proof (rule Lim_null_comparison)
   have "\<forall>n>0. 3 \<le> n \<longrightarrow> 1 \<le> ln (real_of_nat n)"
     using ln_272_gt_1
     by (force intro: order_trans [of _ "ln (272/100)"])
   then show "\<forall>\<^sub>F x in sequentially. cmod (1 / of_nat x powr s) \<le> cmod (Ln (of_nat x) / of_nat x powr s)"
     by (auto simp: norm_divide field_split_simps eventually_sequentially)
   show "(\<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)
 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])
   by (simp add: lim_sequentially dist_norm Ln_Reals_eq norm_powr_real_powr norm_divide)
 
 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 field_split_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"
     by (simp add: field_split_simps)
   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)"
     by (simp add: field_simps)
   moreover have "n > 0" using n
     using neq0_conv by fastforce
   ultimately show "\<exists>no. \<forall>k. Ln (of_nat k) \<noteq> 0 \<longrightarrow> no \<le> k \<longrightarrow> 1 < r * cmod (Ln (of_nat k))"
     using n \<open>0 < r\<close>
     by (rule_tac x=n in exI) (force simp: field_split_simps intro: less_le_trans)
 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])
   by (simp add: lim_sequentially dist_norm Ln_Reals_eq norm_powr_real_powr norm_divide)
 
 lemma lim_ln1_over_ln: "(\<lambda>n. ln(Suc n) / ln n) \<longlonglongrightarrow> 1"
 proof (rule Lim_transform_eventually)
   have "(\<lambda>n. ln(1 + 1/n) / ln n) \<longlonglongrightarrow> 0"
   proof (rule Lim_transform_bound)
     show "(inverse o real) \<longlonglongrightarrow> 0"
       by (metis comp_def lim_inverse_n lim_explicit)
     show "\<forall>\<^sub>F n in sequentially. norm (ln (1 + 1 / n) / ln n) \<le> norm ((inverse \<circ> real) n)"
     proof
       fix n::nat
       assume n: "3 \<le> n"
       then have "ln 3 \<le> ln n" and ln0: "0 \<le> ln n"
         by auto
       with ln3_gt_1 have "1/ ln n \<le> 1"
         by (simp add: field_split_simps)
       moreover have "ln (1 + 1 / real n) \<le> 1/n"
         by (simp add: ln_add_one_self_le_self)
       ultimately have "ln (1 + 1 / real n) * (1 / ln n) \<le> (1/n) * 1"
         by (intro mult_mono) (use n in auto)
       then show "norm (ln (1 + 1 / n) / ln n) \<le> norm ((inverse \<circ> real) n)"
         by (simp add: field_simps ln0)
       qed
   qed
   then show "(\<lambda>n. 1 + ln(1 + 1/n) / ln n) \<longlonglongrightarrow> 1"
     by (metis (full_types) add.right_neutral tendsto_add_const_iff)
   show "\<forall>\<^sub>F k in sequentially. 1 + ln (1 + 1 / k) / ln k = ln(Suc k) / ln k"
     by (simp add: field_split_simps ln_div eventually_sequentiallyI [of 2])
 qed
 
 lemma lim_ln_over_ln1: "(\<lambda>n. ln n / ln(Suc n)) \<longlonglongrightarrow> 1"
 proof -
   have "(\<lambda>n. inverse (ln(Suc n) / ln n)) \<longlonglongrightarrow> inverse 1"
     by (rule tendsto_inverse [OF lim_ln1_over_ln]) auto
   then show ?thesis
     by simp
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<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_div_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 (rule csqrt_square)
     using cos_gt_zero_pi [of "(Im (Ln z) / 2)"] Im_Ln_le_pi mpi_less_Im_Ln assms
     by (fastforce simp: Re_exp Im_exp )
   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")
   case False
   then show ?thesis
     using assms csqrt_exp_Ln Ln_inverse exp_minus
     by (simp add: csqrt_exp_Ln Ln_inverse exp_minus)
 qed auto
 
 lemma cnj_csqrt:
   assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0"
     shows "cnj(csqrt z) = csqrt(cnj z)"
 proof (cases "z=0")
   case False
   then show ?thesis
      by (simp add: assms cnj_Ln csqrt_exp_Ln exp_cnj)
 qed auto
 
 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"
     by (simp add: field_split_simps)
   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
   proof (rule has_field_derivative_transform_within)
     show "\<And>x. dist x z < cmod z \<Longrightarrow> exp (Ln x / 2) = csqrt x"
       by (metis csqrt_exp_Ln dist_0_norm less_irrefl)
   qed (use z in auto)
 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"
   by (simp add: field_differentiable_within_csqrt field_differentiable_imp_continuous_at)
 
 corollary\<^marker>\<open>tag unimportant\<close> 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"
   by (simp add: field_differentiable_imp_continuous_at field_differentiable_within_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"
   by (simp add: continuous_at_imp_continuous_on continuous_within_csqrt)
 
 lemma holomorphic_on_csqrt:
     "(\<And>z. z \<in> s \<Longrightarrow> z \<notin> \<real>\<^sub>\<le>\<^sub>0) \<Longrightarrow> csqrt holomorphic_on s"
   by (simp add: field_differentiable_within_csqrt holomorphic_on_def)
 
 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
   have [simp]: "Im z = 0" and 0: "Re z < 0 \<or> z = 0"
     using True cnj.code complex_cnj_zero_iff  by (auto simp: Complex_eq complex_nonpos_Reals_iff) fastforce
   show ?thesis
     using 0
   proof
     assume "Re z < 0"
     then show ?thesis
       by (auto simp: continuous_within_closed_nontrivial [OF closed_Real_halfspace_Re_ge])
   next
     assume "z = 0"
     moreover
     have "\<And>e. 0 < e
          \<Longrightarrow> \<forall>x'\<in>\<real> \<inter> {w. 0 \<le> Re w}. cmod x' < e^2 \<longrightarrow> cmod (csqrt x') < e"
       by (auto simp: Reals_def real_less_lsqrt)
     ultimately show ?thesis
       using zero_less_power by (fastforce simp: continuous_within_eps_delta)
   qed
 qed (blast intro: continuous_within_csqrt)
 
 subsection\<open>Complex arctangent\<close>
 
 text\<open>The branch cut gives standard bounds in the real case.\<close>
 
 definition\<^marker>\<open>tag important\<close> 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)"
   by (simp add: Arctan_def moebius_def add_ac)
 
 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))"
     by (simp add: Arctan_def_moebius)
   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"
   by (simp add: Arctan_def)
 
 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 "Ln ((1 - \<i> * tan z) / (1 + \<i> * tan z)) = 2 * z / \<i>"
   proof (rule Ln_unique)
     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"
       by (metis distrib_right exp_add mult_2)
     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"
       by (metis (no_types) diff_add_cancel diff_minus_eq_add exp_add exp_minus_inverse mult.commute)
     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)"
       by (simp add: exp_eq_1)
     also have "... \<longleftrightarrow> Im z = 0 \<and> (\<exists>n::int. 2 * Re z = of_int (2*n + 1) * pi)"
       by (simp add: algebra_simps)
     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"
       by (auto simp: add.commute minus_unique)
     then show "exp (2 * z / \<i>) = (1 - \<i> * tan z) / (1 + \<i> * tan z)"
       apply (simp add: tan_def sin_exp_eq cos_exp_eq exp_minus divide_simps)
       by (simp add: algebra_simps flip: power2_eq_square exp_double)
   qed (use assms in auto)
   then show ?thesis
     by (auto simp: Arctan_def)
 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 add_diff_cancel_left' complex_i_mult_minus diff_0 i_squared imaginary_unit.simps
                 less_asym neg_equal_iff_equal)
   have "z \<noteq> -\<i>" using assms
     by auto
   then have zz: "1 + z * z \<noteq> 0"
     by (metis abs_one assms i_squared imaginary_unit.simps less_irrefl minus_unique square_eq_iff)
   have nz1: "1 - \<i>*z \<noteq> 0"
     using assms by (force simp add: i_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_complex_def)
     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"
     by (auto simp add: complex_nonpos_Reals_iff)
   show "\<bar>Re(Arctan z)\<bar> < pi/2"
     unfolding Arctan_def divide_complex_def
     using mpi_less_Im_Ln [OF nzi]
     by (auto simp: abs_if intro!: Im_Ln_less_pi * [unfolded divide_complex_def])
   show "(Arctan has_field_derivative inverse(1 + z\<^sup>2)) (at z)"
     unfolding Arctan_def scaleR_conv_of_real
     apply (intro derivative_eq_intros | simp add: nz0 *)+
     using nz1 zz
     apply (simp add: field_split_simps power2_eq_square)
     apply algebra
     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"
   by (simp add: field_differentiable_imp_continuous_at field_differentiable_within_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"
   by (simp add: field_differentiable_within_Arctan holomorphic_on_def)
 
 theorem 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 (((2*Suc n+1) / (Suc n)) /
                  ((2*Suc n-1) / (Suc n)))"
       by (simp add: h_def norm_mult norm_power norm_divide field_split_simps
                     power2_eq_square eval_nat_numeral del: of_nat_add of_nat_Suc)
       also have "of_nat (2*Suc n+1) / of_nat (Suc n) = (2::real) + inverse (real (Suc n))"
         by (auto simp: field_split_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: field_split_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"
       by (simp add: field_split_simps)
     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 strict_mono_def g_def h_def elim!: oddE)
   qed (simp add: h_def)
 
   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)
     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: strict_mono_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: 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
   from this[of 0] have "c = 0" by (simp add: G_def g_def)
   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: strict_mono_def power_mult g_def h_def)
 qed
 
 text \<open>A quickly-converging series for the logarithm, based on the arctangent.\<close>
 theorem ln_series_quadratic:
   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"
     by (simp add: norm_divide del: of_real_add of_real_diff)
   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))"
     by (subst power_add, subst power_mult) (simp add: mult_ac)
   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\<^marker>\<open>tag unimportant\<close> \<open>Real arctangent\<close>
 
 lemma Im_Arctan_of_real [simp]: "Im (Arctan (of_real x)) = 0"
 proof -
   have ne: "1 + x\<^sup>2 \<noteq> 0"
     by (metis power_one sum_power2_eq_zero_iff zero_neq_one)
   have ne1: "1 + \<i> * complex_of_real x \<noteq> 0"
     using Complex_eq complex_eq_cancel_iff2 by fastforce 
   have "Re (Ln ((1 - \<i> * x) * inverse (1 + \<i> * x))) = 0"
     apply (rule norm_exp_imaginary)
     using ne
     apply (simp add: ne1 cmod_def)
     apply (auto simp: field_split_simps)
     apply algebra
     done
   then show ?thesis
     unfolding Arctan_def divide_complex_def by (simp add: complex_eq_iff)
 qed
 
 lemma arctan_eq_Re_Arctan: "arctan x = Re (Arctan (of_real x))"
 proof (rule arctan_unique)
   have "(1 - \<i> * x) / (1 + \<i> * x) \<notin> \<real>\<^sub>\<le>\<^sub>0"
     by (auto simp: Im_complex_div_lemma complex_nonpos_Reals_iff)
   then show "- (pi / 2) < Re (Arctan (complex_of_real x))"
     by (simp add: Arctan_def Im_Ln_less_pi)
 next
   have *: " (1 - \<i>*x) / (1 + \<i>*x) \<noteq> 0"
     by (simp add: field_split_simps) ( simp add: complex_eq_iff)
   show "Re (Arctan (complex_of_real x)) < pi / 2"
     using mpi_less_Im_Ln [OF *]
     by (simp add: Arctan_def)
 next
   have "tan (Re (Arctan (of_real x))) = Re (tan (Arctan (of_real x)))"
     by (auto simp: tan_def Complex.Re_divide Re_sin Re_cos Im_sin Im_cos field_simps power2_eq_square)
   also have "... = x"
   proof -
     have "(complex_of_real x)\<^sup>2 \<noteq> - 1"
       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)
     then show ?thesis
       by simp
   qed
   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
   by (simp add: complex_eq_iff)
 
 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>"
   by (simp add: abs_if arctan_minus)
 
 lemma arctan_add_raw:
   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]
     by (simp add: arctan tan_add)
 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)))"
     by (simp add: arctan)
   also have "... = arctan (tan (pi / 2 - arctan x))"
     by (simp add: tan_cot)
   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
       by (simp add: Transcendental.arctan_tan)
   qed
   finally show ?thesis .
 qed
 
 lemma arctan_add_small:
   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 False
   with assms have "\<bar>x\<bar> < inverse \<bar>y\<bar>"
     by (simp add: field_split_simps abs_mult)
   with False have "\<bar>arctan x\<bar> < pi / 2 - \<bar>arctan y\<bar>" using assms
     by (auto simp add: abs_arctan arctan_inverse [symmetric] arctan_less_iff)
   then show ?thesis
     by (intro arctan_add_raw) linarith
 qed auto
 
 lemma abs_arctan_le:
   fixes x::real shows "\<bar>arctan x\<bar> \<le> \<bar>x\<bar>"
 proof -
   have 1: "\<And>x. x \<in> \<real> \<Longrightarrow> cmod (inverse (1 + x\<^sup>2)) \<le> 1"
     by (simp add: norm_divide divide_simps in_Reals_norm complex_is_Real_iff power2_eq_square)
   have "cmod (Arctan w - Arctan z) \<le> 1 * cmod (w-z)" if "w \<in> \<real>" "z \<in> \<real>" for 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])
     using 1 that by (auto simp: Reals_def)
   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
     by (simp add: Arctan_of_real)
 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"
   proof (rule tendsto_eq_rhs)
     show "(\<lambda>k. 1 / real (k * 2 + 1) * x ^ (k * 2 + 1)) \<longlonglongrightarrow> 0 * 0"
       using assms
       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
           tendsto_eq_intros filterlim_at_top_mult_tendsto_pos filterlim_tendsto_add_at_top)
   qed simp
   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: field_split_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\<^marker>\<open>tag unimportant\<close> \<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]
   by (simp_all add: eval_nat_numeral)
 
 lemma pi_gt3: "pi > 3"
   using pi_approx by simp
 
 
 subsection\<open>Inverse Sine\<close>
 
 definition\<^marker>\<open>tag important\<close> 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"]
   by (metis add.inverse_inverse complex_i_mult_minus diff_0 diff_add_cancel diff_minus_eq_add mult.assoc mult.commute numeral_One 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)))"
   by (simp add: Arcsin_def)
 
 lemma Im_Arcsin: "Im(Arcsin z) = - ln (cmod (\<i> * z + csqrt (1 - z\<^sup>2)))"
   by (simp add: Arcsin_def Arcsin_body_lemma)
 
 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"
 proof (cases "Im z = 0")
   case True
   with assms show ?thesis
     by (simp add: complex_nonpos_Reals_iff flip: abs_square_less_1)
 next
   case False
   have "\<not> (Im z)\<^sup>2 \<le> - 1"
     using False power2_less_eq_zero_iff by fastforce
   with False show ?thesis
     by (auto simp add: complex_nonpos_Reals_iff Re_power2 Im_power2)
 qed
 
 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 leim: "(cmod (1 - z\<^sup>2) + (1 - Re (z\<^sup>2))) / 2 \<le> (Im z)\<^sup>2"
     using le0 sqrt_le_D by fastforce
   have neq: "(cmod z)\<^sup>2 \<noteq> 1 + cmod (1 - z\<^sup>2)"
   proof (clarsimp simp add: cmod_def)
     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
       by (simp add: power2_eq_square algebra_simps)
   qed
   moreover have 2: "(Im z)\<^sup>2 = (1 + ((Im z)\<^sup>2 + cmod (1 - z\<^sup>2)) - (Re z)\<^sup>2) / 2"
     using leim cmod_power2 [of z] norm_triangle_ineq2 [of "z^2" 1]
     by (simp add: norm_power Re_power2 norm_minus_commute [of 1])
   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 1: "\<i> * z + csqrt (1 - z\<^sup>2) \<notin> \<real>\<^sub>\<le>\<^sub>0"
     by (metis isCont_Arcsin_lemma assms complex_nonpos_Reals_iff)
   have 2: "1 - z\<^sup>2 \<notin> \<real>\<^sub>\<le>\<^sub>0"
     by (simp add: one_minus_z2_notin_nonpos_Reals assms)
   show ?thesis
     using assms unfolding Arcsin_def by (intro isCont_Ln' isCont_csqrt' continuous_intros 1 2)
 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: algebra_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_i_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
     by (simp add: cos_i_times [symmetric] Re_cos Im_cos add_pos_pos)
 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))"
     by (simp add: field_simps power2_eq_square)
   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 by auto
   also have "... =  - (\<i> * Ln (exp (\<i>*z)))"
     by (simp add: field_simps power2_eq_square)
   also have "... = z"
     using assms by (auto simp: abs_if simp del: eq_divide_eq_numeral1 split: if_split_asm)
   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 one_minus_z2_notin_nonpos_Reals by force
   then have "cos (Arcsin z) \<noteq> 0"
     by (metis diff_0_right power_zero_numeral sin_squared_eq)
   then show ?thesis
     by (rule has_field_derivative_inverse_basic [OF DERIV_sin _ _ open_ball [of z 1]]) (auto intro: isCont_Arcsin assms)
 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"
   by (simp add: continuous_at_imp_continuous_on)
 
 lemma holomorphic_on_Arcsin: "(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arcsin holomorphic_on s"
   by (simp add: field_differentiable_within_Arcsin holomorphic_on_def)
 
 
 subsection\<open>Inverse Cosine\<close>
 
 definition\<^marker>\<open>tag important\<close> 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 Arcsin_body_lemma complex_i_mult_minus diff_minus_eq_add power2_minus right_minus_eq)
 
 lemma Re_Arccos: "Re(Arccos z) = Im (Ln (z + \<i> * csqrt(1 - z\<^sup>2)))"
   by (simp add: Arccos_def)
 
 lemma Im_Arccos: "Im(Arccos z) = - ln (cmod (z + \<i> * csqrt (1 - z\<^sup>2)))"
   by (simp add: Arccos_def Arccos_body_lemma)
 
 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"]
     by (simp add: Re_power2 algebra_simps)
   have "(cmod z)\<^sup>2 - 1 \<noteq> cmod (1 - z\<^sup>2)"
   proof (clarsimp simp add: cmod_def)
     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
       by (simp add: power2_eq_square algebra_simps)
   qed
   moreover have "(Im z)\<^sup>2 = (1 + ((Im z)\<^sup>2 + cmod (1 - z\<^sup>2)) - (Re z)\<^sup>2) / 2"
     using abs_Re_le_cmod [of "1-z\<^sup>2"] by (subst Imz) (simp add: Re_power2)
   ultimately show False
     by (simp add: cmod_power2)
 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
     unfolding Arccos_def
     by (simp_all add: one_minus_z2_notin_nonpos_Reals assms)
 qed
 
 lemma isCont_Arccos' [simp]:
   "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
     by (simp add: cos_exp_eq Arccos_def Arccos_body_lemma exp_minus field_simps flip: power2_eq_square)
 qed
 
 lemma Arccos_cos:
     assumes "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"
       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"
     by (simp add: field_simps power2_eq_square)
   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]
     by (auto simp: * Re_sin Im_sin)
   also have "... =  - (\<i> * Ln (exp (\<i>*z)))"
     by (simp add: field_simps power2_eq_square)
   also have "... = z"
     using assms
     by (subst Complex_Transcendental.Ln_exp, auto)
   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
 
 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 "x\<^sup>2 \<noteq> -1" for x::real
     by (sos "((R<1 + (([~1] * A=0) + (R<1 * (R<1 * [x__]^2)))))")
   with assms have "(cos (Arccos z))\<^sup>2 \<noteq> 1"
     by (auto simp: complex_eq_iff Re_power2 Im_power2 abs_square_eq_1)
   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)"
     by (rule has_field_derivative_inverse_basic [OF DERIV_cos _ _ open_ball [of z 1]])
        (auto intro: isCont_Arccos assms)
   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"
   by (simp add: continuous_at_imp_continuous_on)
 
 lemma holomorphic_on_Arccos: "(\<And>z. z \<in> s \<Longrightarrow> Im z = 0 \<Longrightarrow> \<bar>Re z\<bar> < 1) \<Longrightarrow> Arccos holomorphic_on s"
   by (simp add: field_differentiable_within_Arccos holomorphic_on_def)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<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 Im_Arccos_bound: "\<bar>Im (Arccos w)\<bar> \<le> cmod w"
 proof -
   have "(Im (Arccos w))\<^sup>2 \<le> (cmod (cos (Arccos w)))\<^sup>2 - (cos (Re (Arccos w)))\<^sup>2"
     using norm_cos_squared [of "Arccos w"] real_le_abs_sinh [of "Im (Arccos w)"]
     by (simp only: abs_le_square_iff) (simp add: field_split_simps)
   also have "... \<le> (cmod w)\<^sup>2"
     by (auto simp: cmod_power2)
   finally show ?thesis
     using abs_le_square_iff by force
 qed
 
 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)
 
 lemma norm_Arccos_bounded:
   fixes w :: complex
   shows "norm (Arccos w) \<le> pi + norm w"
 proof -
   have Re: "(Re (Arccos w))\<^sup>2 \<le> pi\<^sup>2" "(Im (Arccos w))\<^sup>2 \<le> (cmod w)\<^sup>2"
     using Re_Arccos_bound [of w] Im_Arccos_bound [of w] abs_le_square_iff by force+
   have "Arccos w \<bullet> Arccos w \<le> pi\<^sup>2 + (cmod w)\<^sup>2"
     using Re by (simp add: dot_square_norm cmod_power2 [of "Arccos w"])
   then have "cmod (Arccos w) \<le> pi + cmod (cos (Arccos w))"
     apply (simp add: norm_le_square)
     by (metis dot_square_norm norm_ge_zero norm_le_square pi_ge_zero triangle_lemma)
   then show "cmod (Arccos w) \<le> pi + cmod w"
     by auto
 qed
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<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)"
     by (simp add: algebra_simps)
   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"
     by (metis add_minus_cancel power2_eq_square uminus_add_conv_diff)
   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 (simp add: power2_sum)
     apply (simp add: power2_eq_square algebra_simps)
     done
   then show ?thesis
     apply (simp add: cos_exp_eq Arcsin_def exp_minus)
     apply (simp add: divide_simps Arcsin_body_lemma)
     apply (metis add.commute minus_unique power2_eq_square)
     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)"
     by (simp add: algebra_simps)
   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
       by (metis add.commute add.right_neutral diff_add_cancel mult_right_cancel)
     then show False
       using assms by simp
   qed
   then have "z\<^sup>2 + \<i> * z * (csqrt (1 - z\<^sup>2)) \<noteq> 1"
     by (simp add: algebra_simps)
   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
     by (metis Arccos_def add.commute add.left_neutral cancel_comm_monoid_add_class.diff_cancel cos_Arccos csqrt_0 mult_zero_right)
   then show ?thesis
     apply (simp add: sin_exp_eq Arccos_def exp_minus)
     apply (simp add: divide_simps Arccos_body_lemma)
     apply (simp add: power2_eq_square)
     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)"
 proof (rule csqrt_unique [THEN sym])
   show "(cos z)\<^sup>2 = 1 - (sin z)\<^sup>2"
     by (simp add: cos_squared_eq)
 qed (use assms in \<open>auto simp: Re_cos Im_cos add_pos_pos mult_le_0_iff zero_le_mult_iff\<close>)
 
 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)"
 proof (rule csqrt_unique [THEN sym])
   show "(sin z)\<^sup>2 = 1 - (cos z)\<^sup>2"
     by (simp add: sin_squared_eq)
 qed (use assms in \<open>auto simp: Re_sin Im_sin add_pos_pos mult_le_0_iff zero_le_mult_iff\<close>)
 
 lemma Arcsin_Arccos_csqrt_pos:
     "(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> Arcsin z = Arccos(csqrt(1 - z\<^sup>2))"
   by (simp add: Arcsin_def Arccos_def Complex.csqrt_square add.commute)
 
 lemma Arccos_Arcsin_csqrt_pos:
     "(0 < Re z | Re z = 0 & 0 \<le> Im z) \<Longrightarrow> Arccos z = Arcsin(csqrt(1 - z\<^sup>2))"
   by (simp add: Arcsin_def Arccos_def Complex.csqrt_square add.commute)
 
 lemma sin_Arccos:
     "0 < Re z | Re z = 0 & 0 \<le> Im z \<Longrightarrow> sin(Arccos z) = csqrt(1 - z\<^sup>2)"
   by (simp add: Arccos_Arcsin_csqrt_pos)
 
 lemma cos_Arcsin:
     "0 < Re z | Re z = 0 & 0 \<le> Im z \<Longrightarrow> cos(Arcsin z) = csqrt(1 - z\<^sup>2)"
   by (simp add: Arcsin_Arccos_csqrt_pos)
 
 
 subsection\<^marker>\<open>tag unimportant\<close>\<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
     by (force simp add: Complex.cmod_power2)
   then have "cmod (\<i> * of_real x + csqrt (1 - (of_real x)\<^sup>2)) = 1"
     by (simp add: norm_complex_def)
   then show ?thesis
     by (simp add: Im_Arcsin exp_minus)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> 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"]
     by (simp add: Im_Arcsin_of_real assms)
 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')))"
     unfolding sin_of_real [symmetric] 
     by (subst Arcsin_sin) auto
 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\<^marker>\<open>tag unimportant\<close>\<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
     by (force simp add: Complex.cmod_power2)
   then have "cmod (of_real x + \<i> * csqrt (1 - (of_real x)\<^sup>2)) = 1"
     by (simp add: norm_complex_def)
   then show ?thesis
     by (simp add: Im_Arccos exp_minus)
 qed
 
 corollary\<^marker>\<open>tag unimportant\<close> 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"]
     by (simp add: Im_Arccos_of_real assms)
 next
   fix x'
   assume "0 \<le> x'" "x' \<le> pi" "x = cos x'"
   then show "x' = Re (Arccos (complex_of_real (cos x')))"
     unfolding cos_of_real [symmetric]
     by (subst Arccos_cos) auto
 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\<^marker>\<open>tag unimportant\<close>\<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
       by (simp add: algebra_simps)
   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
       by (simp add: algebra_simps)
   next
     show "sin (arctan (x / sqrt (1 - x\<^sup>2)) - (pi / 2 - arccos x)) = 0"
       using assms
       by (simp add: algebra_simps sin_diff cos_add sin_arccos sin_arctan cos_arctan
                     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]
     by (auto simp: algebra_simps sin_diff)
   then show ?thesis
     by (simp add: algebra_simps)
 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))"
   by (simp add: arccos_arctan arcsin_arccos_eq)
 
 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"]
   by (simp add: arcsin_minus)
 
 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"]
   by (simp add: arccos_minus)
 
 subsection\<^marker>\<open>tag unimportant\<close>\<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
 
 lemma sinh_ln_complex: "x \<noteq> 0 \<Longrightarrow> sinh (ln x :: complex) = (x - inverse x) / 2"
   by (simp add: sinh_def exp_minus scaleR_conv_of_real exp_of_real)
 
 lemma cosh_ln_complex: "x \<noteq> 0 \<Longrightarrow> cosh (ln x :: complex) = (x + inverse x) / 2"
   by (simp add: cosh_def exp_minus scaleR_conv_of_real)
 
 lemma tanh_ln_complex: "x \<noteq> 0 \<Longrightarrow> tanh (ln x :: complex) = (x ^ 2 - 1) / (x ^ 2 + 1)"
   by (simp add: tanh_def sinh_ln_complex cosh_ln_complex divide_simps power2_eq_square)
 
 
 subsection\<open>Roots of unity\<close>
 
 theorem 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)"
     by (simp)
   then show ?thesis
     apply (simp add: exp_of_nat_mult [symmetric] mult_ac exp_Euler)
     apply (simp only: * cos_of_real sin_of_real)
     apply simp
     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)))"
       by (simp add: algebra_simps)
     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)"
       by (metis (mono_tags, hide_lams) of_int_add of_int_eq_iff of_int_mult of_int_of_nat_eq)
     also have "... \<longleftrightarrow> int j mod int n = int k mod int n"
       by (auto simp: mod_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 field_split_simps *)
 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. 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"
   by (simp add: card_complex_roots_unity_explicit complex_roots_unity)
 
 lemma complex_not_root_unity:
     "1 \<le> n \<Longrightarrow> \<exists>u::complex. norm u = 1 \<and> u^n \<noteq> 1"
   apply (rule_tac x="exp (of_real pi * \<i> * of_real (1 / n))" in exI)
   apply (auto simp: Re_complex_div_eq_0 exp_of_nat_mult [symmetric] mult_ac exp_Euler)
   done
 
 end
diff --git a/src/HOL/Complex.thy b/src/HOL/Complex.thy
--- a/src/HOL/Complex.thy
+++ b/src/HOL/Complex.thy
@@ -1,1324 +1,1327 @@
 (*  Title:       HOL/Complex.thy
     Author:      Jacques D. Fleuriot, 2001 University of Edinburgh
     Author:      Lawrence C Paulson, 2003/4
 *)
 
 section \<open>Complex Numbers: Rectangular and Polar Representations\<close>
 
 theory Complex
 imports Transcendental
 begin
 
 text \<open>
   We use the \<^theory_text>\<open>codatatype\<close> command to define the type of complex numbers. This
   allows us to use \<^theory_text>\<open>primcorec\<close> to define complex functions by defining their
   real and imaginary result separately.
 \<close>
 
 codatatype complex = Complex (Re: real) (Im: real)
 
 lemma complex_surj: "Complex (Re z) (Im z) = z"
   by (rule complex.collapse)
 
 lemma complex_eqI [intro?]: "Re x = Re y \<Longrightarrow> Im x = Im y \<Longrightarrow> x = y"
   by (rule complex.expand) simp
 
 lemma complex_eq_iff: "x = y \<longleftrightarrow> Re x = Re y \<and> Im x = Im y"
   by (auto intro: complex.expand)
 
 
 subsection \<open>Addition and Subtraction\<close>
 
 instantiation complex :: ab_group_add
 begin
 
 primcorec zero_complex
   where
     "Re 0 = 0"
   | "Im 0 = 0"
 
 primcorec plus_complex
   where
     "Re (x + y) = Re x + Re y"
   | "Im (x + y) = Im x + Im y"
 
 primcorec uminus_complex
   where
     "Re (- x) = - Re x"
   | "Im (- x) = - Im x"
 
 primcorec minus_complex
   where
     "Re (x - y) = Re x - Re y"
   | "Im (x - y) = Im x - Im y"
 
 instance
   by standard (simp_all add: complex_eq_iff)
 
 end
 
 
 subsection \<open>Multiplication and Division\<close>
 
 instantiation complex :: field
 begin
 
 primcorec one_complex
   where
     "Re 1 = 1"
   | "Im 1 = 0"
 
 primcorec times_complex
   where
     "Re (x * y) = Re x * Re y - Im x * Im y"
   | "Im (x * y) = Re x * Im y + Im x * Re y"
 
 primcorec inverse_complex
   where
     "Re (inverse x) = Re x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
   | "Im (inverse x) = - Im x / ((Re x)\<^sup>2 + (Im x)\<^sup>2)"
 
 definition "x div y = x * inverse y" for x y :: complex
 
 instance
   by standard
      (simp_all add: complex_eq_iff divide_complex_def
       distrib_left distrib_right right_diff_distrib left_diff_distrib
       power2_eq_square add_divide_distrib [symmetric])
 
 end
 
 lemma Re_divide: "Re (x / y) = (Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
   by (simp add: divide_complex_def add_divide_distrib)
 
 lemma Im_divide: "Im (x / y) = (Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2)"
   by (simp add: divide_complex_def diff_divide_distrib)
 
 lemma Complex_divide:
     "(x / y) = Complex ((Re x * Re y + Im x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2))
                        ((Im x * Re y - Re x * Im y) / ((Re y)\<^sup>2 + (Im y)\<^sup>2))"
   by (metis Im_divide Re_divide complex_surj)
 
 lemma Re_power2: "Re (x ^ 2) = (Re x)^2 - (Im x)^2"
   by (simp add: power2_eq_square)
 
 lemma Im_power2: "Im (x ^ 2) = 2 * Re x * Im x"
   by (simp add: power2_eq_square)
 
 lemma Re_power_real [simp]: "Im x = 0 \<Longrightarrow> Re (x ^ n) = Re x ^ n "
   by (induct n) simp_all
 
 lemma Im_power_real [simp]: "Im x = 0 \<Longrightarrow> Im (x ^ n) = 0"
   by (induct n) simp_all
 
 
 subsection \<open>Scalar Multiplication\<close>
 
 instantiation complex :: real_field
 begin
 
 primcorec scaleR_complex
   where
     "Re (scaleR r x) = r * Re x"
   | "Im (scaleR r x) = r * Im x"
 
 instance
 proof
   fix a b :: real and x y :: complex
   show "scaleR a (x + y) = scaleR a x + scaleR a y"
     by (simp add: complex_eq_iff distrib_left)
   show "scaleR (a + b) x = scaleR a x + scaleR b x"
     by (simp add: complex_eq_iff distrib_right)
   show "scaleR a (scaleR b x) = scaleR (a * b) x"
     by (simp add: complex_eq_iff mult.assoc)
   show "scaleR 1 x = x"
     by (simp add: complex_eq_iff)
   show "scaleR a x * y = scaleR a (x * y)"
     by (simp add: complex_eq_iff algebra_simps)
   show "x * scaleR a y = scaleR a (x * y)"
     by (simp add: complex_eq_iff algebra_simps)
 qed
 
 end
 
 
 subsection \<open>Numerals, Arithmetic, and Embedding from R\<close>
 
 abbreviation complex_of_real :: "real \<Rightarrow> complex"
   where "complex_of_real \<equiv> of_real"
 
 declare [[coercion "of_real :: real \<Rightarrow> complex"]]
 declare [[coercion "of_rat :: rat \<Rightarrow> complex"]]
 declare [[coercion "of_int :: int \<Rightarrow> complex"]]
 declare [[coercion "of_nat :: nat \<Rightarrow> complex"]]
 
 lemma complex_Re_of_nat [simp]: "Re (of_nat n) = of_nat n"
   by (induct n) simp_all
 
 lemma complex_Im_of_nat [simp]: "Im (of_nat n) = 0"
   by (induct n) simp_all
 
 lemma complex_Re_of_int [simp]: "Re (of_int z) = of_int z"
   by (cases z rule: int_diff_cases) simp
 
 lemma complex_Im_of_int [simp]: "Im (of_int z) = 0"
   by (cases z rule: int_diff_cases) simp
 
 lemma complex_Re_numeral [simp]: "Re (numeral v) = numeral v"
   using complex_Re_of_int [of "numeral v"] by simp
 
 lemma complex_Im_numeral [simp]: "Im (numeral v) = 0"
   using complex_Im_of_int [of "numeral v"] by simp
 
 lemma Re_complex_of_real [simp]: "Re (complex_of_real z) = z"
   by (simp add: of_real_def)
 
 lemma Im_complex_of_real [simp]: "Im (complex_of_real z) = 0"
   by (simp add: of_real_def)
 
 lemma Re_divide_numeral [simp]: "Re (z / numeral w) = Re z / numeral w"
   by (simp add: Re_divide sqr_conv_mult)
 
 lemma Im_divide_numeral [simp]: "Im (z / numeral w) = Im z / numeral w"
   by (simp add: Im_divide sqr_conv_mult)
 
 lemma Re_divide_of_nat [simp]: "Re (z / of_nat n) = Re z / of_nat n"
   by (cases n) (simp_all add: Re_divide field_split_simps power2_eq_square del: of_nat_Suc)
 
 lemma Im_divide_of_nat [simp]: "Im (z / of_nat n) = Im z / of_nat n"
   by (cases n) (simp_all add: Im_divide field_split_simps power2_eq_square del: of_nat_Suc)
 
 lemma of_real_Re [simp]: "z \<in> \<real> \<Longrightarrow> of_real (Re z) = z"
   by (auto simp: Reals_def)
 
 lemma complex_Re_fact [simp]: "Re (fact n) = fact n"
 proof -
   have "(fact n :: complex) = of_real (fact n)"
     by simp
   also have "Re \<dots> = fact n"
     by (subst Re_complex_of_real) simp_all
   finally show ?thesis .
 qed
 
 lemma complex_Im_fact [simp]: "Im (fact n) = 0"
   by (subst of_nat_fact [symmetric]) (simp only: complex_Im_of_nat)
 
 lemma Re_prod_Reals: "(\<And>x. x \<in> A \<Longrightarrow> f x \<in> \<real>) \<Longrightarrow> Re (prod f A) = prod (\<lambda>x. Re (f x)) A"
 proof (induction A rule: infinite_finite_induct)
   case (insert x A)
   hence "Re (prod f (insert x A)) = Re (f x) * Re (prod f A) - Im (f x) * Im (prod f A)"
     by simp
   also from insert.prems have "f x \<in> \<real>" by simp
   hence "Im (f x) = 0" by (auto elim!: Reals_cases)
   also have "Re (prod f A) = (\<Prod>x\<in>A. Re (f x))"
     by (intro insert.IH insert.prems) auto
   finally show ?case using insert.hyps by simp
 qed auto
 
 
 subsection \<open>The Complex Number $i$\<close>
 
 primcorec imaginary_unit :: complex  ("\<i>")
   where
     "Re \<i> = 0"
   | "Im \<i> = 1"
 
 lemma Complex_eq: "Complex a b = a + \<i> * b"
   by (simp add: complex_eq_iff)
 
 lemma complex_eq: "a = Re a + \<i> * Im a"
   by (simp add: complex_eq_iff)
 
 lemma fun_complex_eq: "f = (\<lambda>x. Re (f x) + \<i> * Im (f x))"
   by (simp add: fun_eq_iff complex_eq)
 
 lemma i_squared [simp]: "\<i> * \<i> = -1"
   by (simp add: complex_eq_iff)
 
 lemma power2_i [simp]: "\<i>\<^sup>2 = -1"
   by (simp add: power2_eq_square)
 
 lemma inverse_i [simp]: "inverse \<i> = - \<i>"
   by (rule inverse_unique) simp
 
 lemma divide_i [simp]: "x / \<i> = - \<i> * x"
   by (simp add: divide_complex_def)
 
 lemma complex_i_mult_minus [simp]: "\<i> * (\<i> * x) = - x"
   by (simp add: mult.assoc [symmetric])
 
 lemma complex_i_not_zero [simp]: "\<i> \<noteq> 0"
   by (simp add: complex_eq_iff)
 
 lemma complex_i_not_one [simp]: "\<i> \<noteq> 1"
   by (simp add: complex_eq_iff)
 
 lemma complex_i_not_numeral [simp]: "\<i> \<noteq> numeral w"
   by (simp add: complex_eq_iff)
 
 lemma complex_i_not_neg_numeral [simp]: "\<i> \<noteq> - numeral w"
   by (simp add: complex_eq_iff)
 
 lemma complex_split_polar: "\<exists>r a. z = complex_of_real r * (cos a + \<i> * sin a)"
   by (simp add: complex_eq_iff polar_Ex)
 
 lemma i_even_power [simp]: "\<i> ^ (n * 2) = (-1) ^ n"
   by (metis mult.commute power2_i power_mult)
 
 lemma Re_i_times [simp]: "Re (\<i> * z) = - Im z"
   by simp
 
 lemma Im_i_times [simp]: "Im (\<i> * z) = Re z"
   by simp
 
 lemma i_times_eq_iff: "\<i> * w = z \<longleftrightarrow> w = - (\<i> * z)"
   by auto
 
 lemma divide_numeral_i [simp]: "z / (numeral n * \<i>) = - (\<i> * z) / numeral n"
   by (metis divide_divide_eq_left divide_i mult.commute mult_minus_right)
 
 lemma imaginary_eq_real_iff [simp]:
   assumes "y \<in> Reals" "x \<in> Reals"
   shows "\<i> * y = x \<longleftrightarrow> x=0 \<and> y=0"
     using assms
     unfolding Reals_def
     apply clarify
       apply (rule iffI)
     apply (metis Im_complex_of_real Im_i_times Re_complex_of_real mult_eq_0_iff of_real_0)
     by simp
 
 lemma real_eq_imaginary_iff [simp]:
   assumes "y \<in> Reals" "x \<in> Reals"
   shows "x = \<i> * y  \<longleftrightarrow> x=0 \<and> y=0"
     using assms imaginary_eq_real_iff by fastforce
 
 subsection \<open>Vector Norm\<close>
 
 instantiation complex :: real_normed_field
 begin
 
 definition "norm z = sqrt ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
 
 abbreviation cmod :: "complex \<Rightarrow> real"
   where "cmod \<equiv> norm"
 
 definition complex_sgn_def: "sgn x = x /\<^sub>R cmod x"
 
 definition dist_complex_def: "dist x y = cmod (x - y)"
 
 definition uniformity_complex_def [code del]:
   "(uniformity :: (complex \<times> complex) filter) = (INF e\<in>{0 <..}. principal {(x, y). dist x y < e})"
 
 definition open_complex_def [code del]:
   "open (U :: complex set) \<longleftrightarrow> (\<forall>x\<in>U. eventually (\<lambda>(x', y). x' = x \<longrightarrow> y \<in> U) uniformity)"
 
 instance
 proof
   fix r :: real and x y :: complex and S :: "complex set"
   show "(norm x = 0) = (x = 0)"
     by (simp add: norm_complex_def complex_eq_iff)
   show "norm (x + y) \<le> norm x + norm y"
     by (simp add: norm_complex_def complex_eq_iff real_sqrt_sum_squares_triangle_ineq)
   show "norm (scaleR r x) = \<bar>r\<bar> * norm x"
     by (simp add: norm_complex_def complex_eq_iff power_mult_distrib distrib_left [symmetric]
         real_sqrt_mult)
   show "norm (x * y) = norm x * norm y"
     by (simp add: norm_complex_def complex_eq_iff real_sqrt_mult [symmetric]
         power2_eq_square algebra_simps)
 qed (rule complex_sgn_def dist_complex_def open_complex_def uniformity_complex_def)+
 
 end
 
 declare uniformity_Abort[where 'a = complex, code]
 
 lemma norm_ii [simp]: "norm \<i> = 1"
   by (simp add: norm_complex_def)
 
 lemma cmod_unit_one: "cmod (cos a + \<i> * sin a) = 1"
   by (simp add: norm_complex_def)
 
 lemma cmod_complex_polar: "cmod (r * (cos a + \<i> * sin a)) = \<bar>r\<bar>"
   by (simp add: norm_mult cmod_unit_one)
 
 lemma complex_Re_le_cmod: "Re x \<le> cmod x"
   unfolding norm_complex_def by (rule real_sqrt_sum_squares_ge1)
 
 lemma complex_mod_minus_le_complex_mod: "- cmod x \<le> cmod x"
   by (rule order_trans [OF _ norm_ge_zero]) simp
 
 lemma complex_mod_triangle_ineq2: "cmod (b + a) - cmod b \<le> cmod a"
   by (rule ord_le_eq_trans [OF norm_triangle_ineq2]) simp
 
 lemma abs_Re_le_cmod: "\<bar>Re x\<bar> \<le> cmod x"
   by (simp add: norm_complex_def)
 
 lemma abs_Im_le_cmod: "\<bar>Im x\<bar> \<le> cmod x"
   by (simp add: norm_complex_def)
 
 lemma cmod_le: "cmod z \<le> \<bar>Re z\<bar> + \<bar>Im z\<bar>"
   apply (subst complex_eq)
   apply (rule order_trans)
    apply (rule norm_triangle_ineq)
   apply (simp add: norm_mult)
   done
 
 lemma cmod_eq_Re: "Im z = 0 \<Longrightarrow> cmod z = \<bar>Re z\<bar>"
   by (simp add: norm_complex_def)
 
 lemma cmod_eq_Im: "Re z = 0 \<Longrightarrow> cmod z = \<bar>Im z\<bar>"
   by (simp add: norm_complex_def)
 
 lemma cmod_power2: "(cmod z)\<^sup>2 = (Re z)\<^sup>2 + (Im z)\<^sup>2"
   by (simp add: norm_complex_def)
 
 lemma cmod_plus_Re_le_0_iff: "cmod z + Re z \<le> 0 \<longleftrightarrow> Re z = - cmod z"
   using abs_Re_le_cmod[of z] by auto
 
 lemma cmod_Re_le_iff: "Im x = Im y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> \<bar>Re x\<bar> \<le> \<bar>Re y\<bar>"
   by (metis add.commute add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff)
 
 lemma cmod_Im_le_iff: "Re x = Re y \<Longrightarrow> cmod x \<le> cmod y \<longleftrightarrow> \<bar>Im x\<bar> \<le> \<bar>Im y\<bar>"
   by (metis add_le_cancel_left norm_complex_def real_sqrt_abs real_sqrt_le_iff)
 
 lemma Im_eq_0: "\<bar>Re z\<bar> = cmod z \<Longrightarrow> Im z = 0"
   by (subst (asm) power_eq_iff_eq_base[symmetric, where n=2]) (auto simp add: norm_complex_def)
 
 lemma abs_sqrt_wlog: "(\<And>x. x \<ge> 0 \<Longrightarrow> P x (x\<^sup>2)) \<Longrightarrow> P \<bar>x\<bar> (x\<^sup>2)"
   for x::"'a::linordered_idom"
   by (metis abs_ge_zero power2_abs)
 
 lemma complex_abs_le_norm: "\<bar>Re z\<bar> + \<bar>Im z\<bar> \<le> sqrt 2 * norm z"
   unfolding norm_complex_def
   apply (rule abs_sqrt_wlog [where x="Re z"])
   apply (rule abs_sqrt_wlog [where x="Im z"])
   apply (rule power2_le_imp_le)
    apply (simp_all add: power2_sum add.commute sum_squares_bound real_sqrt_mult [symmetric])
   done
 
 lemma complex_unit_circle: "z \<noteq> 0 \<Longrightarrow> (Re z / cmod z)\<^sup>2 + (Im z / cmod z)\<^sup>2 = 1"
   by (simp add: norm_complex_def complex_eq_iff power2_eq_square add_divide_distrib [symmetric])
 
 
 text \<open>Properties of complex signum.\<close>
 
 lemma sgn_eq: "sgn z = z / complex_of_real (cmod z)"
   by (simp add: sgn_div_norm divide_inverse scaleR_conv_of_real mult.commute)
 
 lemma Re_sgn [simp]: "Re(sgn z) = Re(z)/cmod z"
   by (simp add: complex_sgn_def divide_inverse)
 
 lemma Im_sgn [simp]: "Im(sgn z) = Im(z)/cmod z"
   by (simp add: complex_sgn_def divide_inverse)
 
 
 subsection \<open>Absolute value\<close>
 
 instantiation complex :: field_abs_sgn
 begin
 
 definition abs_complex :: "complex \<Rightarrow> complex"
   where "abs_complex = of_real \<circ> norm"
 
 instance
   apply standard
          apply (auto simp add: abs_complex_def complex_sgn_def norm_mult)
   apply (auto simp add: scaleR_conv_of_real field_simps)
   done
 
 end
 
 
 subsection \<open>Completeness of the Complexes\<close>
 
 lemma bounded_linear_Re: "bounded_linear Re"
   by (rule bounded_linear_intro [where K=1]) (simp_all add: norm_complex_def)
 
 lemma bounded_linear_Im: "bounded_linear Im"
   by (rule bounded_linear_intro [where K=1]) (simp_all add: norm_complex_def)
 
 lemmas Cauchy_Re = bounded_linear.Cauchy [OF bounded_linear_Re]
 lemmas Cauchy_Im = bounded_linear.Cauchy [OF bounded_linear_Im]
 lemmas tendsto_Re [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Re]
 lemmas tendsto_Im [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_Im]
 lemmas isCont_Re [simp] = bounded_linear.isCont [OF bounded_linear_Re]
 lemmas isCont_Im [simp] = bounded_linear.isCont [OF bounded_linear_Im]
 lemmas continuous_Re [simp] = bounded_linear.continuous [OF bounded_linear_Re]
 lemmas continuous_Im [simp] = bounded_linear.continuous [OF bounded_linear_Im]
 lemmas continuous_on_Re [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Re]
 lemmas continuous_on_Im [continuous_intros] = bounded_linear.continuous_on[OF bounded_linear_Im]
 lemmas has_derivative_Re [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Re]
 lemmas has_derivative_Im [derivative_intros] = bounded_linear.has_derivative[OF bounded_linear_Im]
 lemmas sums_Re = bounded_linear.sums [OF bounded_linear_Re]
 lemmas sums_Im = bounded_linear.sums [OF bounded_linear_Im]
 
 lemma tendsto_Complex [tendsto_intros]:
   "(f \<longlongrightarrow> a) F \<Longrightarrow> (g \<longlongrightarrow> b) F \<Longrightarrow> ((\<lambda>x. Complex (f x) (g x)) \<longlongrightarrow> Complex a b) F"
   unfolding Complex_eq by (auto intro!: tendsto_intros)
 
 lemma tendsto_complex_iff:
   "(f \<longlongrightarrow> x) F \<longleftrightarrow> (((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F \<and> ((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F)"
 proof safe
   assume "((\<lambda>x. Re (f x)) \<longlongrightarrow> Re x) F" "((\<lambda>x. Im (f x)) \<longlongrightarrow> Im x) F"
   from tendsto_Complex[OF this] show "(f \<longlongrightarrow> x) F"
     unfolding complex.collapse .
 qed (auto intro: tendsto_intros)
 
 lemma continuous_complex_iff:
   "continuous F f \<longleftrightarrow> continuous F (\<lambda>x. Re (f x)) \<and> continuous F (\<lambda>x. Im (f x))"
   by (simp only: continuous_def tendsto_complex_iff)
 
 lemma continuous_on_of_real_o_iff [simp]:
      "continuous_on S (\<lambda>x. complex_of_real (g x)) = continuous_on S g"
   using continuous_on_Re continuous_on_of_real  by fastforce
 
 lemma continuous_on_of_real_id [simp]:
      "continuous_on S (of_real :: real \<Rightarrow> 'a::real_normed_algebra_1)"
   by (rule continuous_on_of_real [OF continuous_on_id])
 
 lemma has_vector_derivative_complex_iff: "(f has_vector_derivative x) F \<longleftrightarrow>
     ((\<lambda>x. Re (f x)) has_field_derivative (Re x)) F \<and>
     ((\<lambda>x. Im (f x)) has_field_derivative (Im x)) F"
   by (simp add: has_vector_derivative_def has_field_derivative_def has_derivative_def
       tendsto_complex_iff algebra_simps bounded_linear_scaleR_left bounded_linear_mult_right)
 
 lemma has_field_derivative_Re[derivative_intros]:
   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Re (f x)) has_field_derivative (Re D)) F"
   unfolding has_vector_derivative_complex_iff by safe
 
 lemma has_field_derivative_Im[derivative_intros]:
   "(f has_vector_derivative D) F \<Longrightarrow> ((\<lambda>x. Im (f x)) has_field_derivative (Im D)) F"
   unfolding has_vector_derivative_complex_iff by safe
 
 instance complex :: banach
 proof
   fix X :: "nat \<Rightarrow> complex"
   assume X: "Cauchy X"
   then have "(\<lambda>n. Complex (Re (X n)) (Im (X n))) \<longlonglongrightarrow>
     Complex (lim (\<lambda>n. Re (X n))) (lim (\<lambda>n. Im (X n)))"
     by (intro tendsto_Complex convergent_LIMSEQ_iff[THEN iffD1]
         Cauchy_convergent_iff[THEN iffD1] Cauchy_Re Cauchy_Im)
   then show "convergent X"
     unfolding complex.collapse by (rule convergentI)
 qed
 
 declare DERIV_power[where 'a=complex, unfolded of_nat_def[symmetric], derivative_intros]
 
 
 subsection \<open>Complex Conjugation\<close>
 
 primcorec cnj :: "complex \<Rightarrow> complex"
   where
     "Re (cnj z) = Re z"
   | "Im (cnj z) = - Im z"
 
 lemma complex_cnj_cancel_iff [simp]: "cnj x = cnj y \<longleftrightarrow> x = y"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_cnj [simp]: "cnj (cnj z) = z"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_zero [simp]: "cnj 0 = 0"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_zero_iff [iff]: "cnj z = 0 \<longleftrightarrow> z = 0"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_one_iff [simp]: "cnj z = 1 \<longleftrightarrow> z = 1"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_add [simp]: "cnj (x + y) = cnj x + cnj y"
   by (simp add: complex_eq_iff)
 
 lemma cnj_sum [simp]: "cnj (sum f s) = (\<Sum>x\<in>s. cnj (f x))"
   by (induct s rule: infinite_finite_induct) auto
 
 lemma complex_cnj_diff [simp]: "cnj (x - y) = cnj x - cnj y"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_minus [simp]: "cnj (- x) = - cnj x"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_one [simp]: "cnj 1 = 1"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_mult [simp]: "cnj (x * y) = cnj x * cnj y"
   by (simp add: complex_eq_iff)
 
 lemma cnj_prod [simp]: "cnj (prod f s) = (\<Prod>x\<in>s. cnj (f x))"
   by (induct s rule: infinite_finite_induct) auto
 
 lemma complex_cnj_inverse [simp]: "cnj (inverse x) = inverse (cnj x)"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_divide [simp]: "cnj (x / y) = cnj x / cnj y"
   by (simp add: divide_complex_def)
 
 lemma complex_cnj_power [simp]: "cnj (x ^ n) = cnj x ^ n"
   by (induct n) simp_all
 
 lemma complex_cnj_of_nat [simp]: "cnj (of_nat n) = of_nat n"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_of_int [simp]: "cnj (of_int z) = of_int z"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_numeral [simp]: "cnj (numeral w) = numeral w"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_neg_numeral [simp]: "cnj (- numeral w) = - numeral w"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_scaleR [simp]: "cnj (scaleR r x) = scaleR r (cnj x)"
   by (simp add: complex_eq_iff)
 
 lemma complex_mod_cnj [simp]: "cmod (cnj z) = cmod z"
   by (simp add: norm_complex_def)
 
 lemma complex_cnj_complex_of_real [simp]: "cnj (of_real x) = of_real x"
   by (simp add: complex_eq_iff)
 
 lemma complex_cnj_i [simp]: "cnj \<i> = - \<i>"
   by (simp add: complex_eq_iff)
 
 lemma complex_add_cnj: "z + cnj z = complex_of_real (2 * Re z)"
   by (simp add: complex_eq_iff)
 
 lemma complex_diff_cnj: "z - cnj z = complex_of_real (2 * Im z) * \<i>"
   by (simp add: complex_eq_iff)
 
 lemma Ints_cnj [intro]: "x \<in> \<int> \<Longrightarrow> cnj x \<in> \<int>"
   by (auto elim!: Ints_cases)
 
 lemma cnj_in_Ints_iff [simp]: "cnj x \<in> \<int> \<longleftrightarrow> x \<in> \<int>"
   using Ints_cnj[of x] Ints_cnj[of "cnj x"] by auto
 
 lemma complex_mult_cnj: "z * cnj z = complex_of_real ((Re z)\<^sup>2 + (Im z)\<^sup>2)"
   by (simp add: complex_eq_iff power2_eq_square)
 
 lemma cnj_add_mult_eq_Re: "z * cnj w + cnj z * w = 2 * Re (z * cnj w)"
   by (rule complex_eqI) auto
 
 lemma complex_mod_mult_cnj: "cmod (z * cnj z) = (cmod z)\<^sup>2"
   by (simp add: norm_mult power2_eq_square)
 
 lemma complex_mod_sqrt_Re_mult_cnj: "cmod z = sqrt (Re (z * cnj z))"
   by (simp add: norm_complex_def power2_eq_square)
 
 lemma complex_In_mult_cnj_zero [simp]: "Im (z * cnj z) = 0"
   by simp
 
 lemma complex_cnj_fact [simp]: "cnj (fact n) = fact n"
   by (subst of_nat_fact [symmetric], subst complex_cnj_of_nat) simp
 
 lemma complex_cnj_pochhammer [simp]: "cnj (pochhammer z n) = pochhammer (cnj z) n"
   by (induct n arbitrary: z) (simp_all add: pochhammer_rec)
 
 lemma bounded_linear_cnj: "bounded_linear cnj"
   using complex_cnj_add complex_cnj_scaleR by (rule bounded_linear_intro [where K=1]) simp
 
 lemma linear_cnj: "linear cnj"
   using bounded_linear.linear[OF bounded_linear_cnj] .
 
 lemmas tendsto_cnj [tendsto_intros] = bounded_linear.tendsto [OF bounded_linear_cnj]
   and isCont_cnj [simp] = bounded_linear.isCont [OF bounded_linear_cnj]
   and continuous_cnj [simp, continuous_intros] = bounded_linear.continuous [OF bounded_linear_cnj]
   and continuous_on_cnj [simp, continuous_intros] = bounded_linear.continuous_on [OF bounded_linear_cnj]
   and has_derivative_cnj [simp, derivative_intros] = bounded_linear.has_derivative [OF bounded_linear_cnj]
 
 lemma lim_cnj: "((\<lambda>x. cnj(f x)) \<longlongrightarrow> cnj l) F \<longleftrightarrow> (f \<longlongrightarrow> l) F"
   by (simp add: tendsto_iff dist_complex_def complex_cnj_diff [symmetric] del: complex_cnj_diff)
 
 lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
   by (simp add: sums_def lim_cnj cnj_sum [symmetric] del: cnj_sum)
 
 lemma differentiable_cnj_iff:
   "(\<lambda>z. cnj (f z)) differentiable at x within A \<longleftrightarrow> f differentiable at x within A"
 proof
   assume "(\<lambda>z. cnj (f z)) differentiable at x within A"
   then obtain D where "((\<lambda>z. cnj (f z)) has_derivative D) (at x within A)"
     by (auto simp: differentiable_def)
   from has_derivative_cnj[OF this] show "f differentiable at x within A"
     by (auto simp: differentiable_def)
 next
   assume "f differentiable at x within A"
   then obtain D where "(f has_derivative D) (at x within A)"
     by (auto simp: differentiable_def)
   from has_derivative_cnj[OF this] show "(\<lambda>z. cnj (f z)) differentiable at x within A"
     by (auto simp: differentiable_def)
 qed
 
 lemma has_vector_derivative_cnj [derivative_intros]:
   assumes "(f has_vector_derivative f') (at z within A)"
   shows   "((\<lambda>z. cnj (f z)) has_vector_derivative cnj f') (at z within A)"
   using assms by (auto simp: has_vector_derivative_complex_iff intro: derivative_intros)
 
 
 subsection \<open>Basic Lemmas\<close>
 
 lemma complex_eq_0: "z=0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 = 0"
   by (metis zero_complex.sel complex_eqI sum_power2_eq_zero_iff)
 
 lemma complex_neq_0: "z\<noteq>0 \<longleftrightarrow> (Re z)\<^sup>2 + (Im z)\<^sup>2 > 0"
   by (metis complex_eq_0 less_numeral_extra(3) sum_power2_gt_zero_iff)
 
 lemma complex_norm_square: "of_real ((norm z)\<^sup>2) = z * cnj z"
   by (cases z)
     (auto simp: complex_eq_iff norm_complex_def power2_eq_square[symmetric] of_real_power[symmetric]
       simp del: of_real_power)
 
 lemma complex_div_cnj: "a / b = (a * cnj b) / (norm b)\<^sup>2"
   using complex_norm_square by auto
 
 lemma Re_complex_div_eq_0: "Re (a / b) = 0 \<longleftrightarrow> Re (a * cnj b) = 0"
   by (auto simp add: Re_divide)
 
 lemma Im_complex_div_eq_0: "Im (a / b) = 0 \<longleftrightarrow> Im (a * cnj b) = 0"
   by (auto simp add: Im_divide)
 
 lemma complex_div_gt_0: "(Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0) \<and> (Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0)"
 proof (cases "b = 0")
   case True
   then show ?thesis by auto
 next
   case False
   then have "0 < (Re b)\<^sup>2 + (Im b)\<^sup>2"
     by (simp add: complex_eq_iff sum_power2_gt_zero_iff)
   then show ?thesis
     by (simp add: Re_divide Im_divide zero_less_divide_iff)
 qed
 
 lemma Re_complex_div_gt_0: "Re (a / b) > 0 \<longleftrightarrow> Re (a * cnj b) > 0"
   and Im_complex_div_gt_0: "Im (a / b) > 0 \<longleftrightarrow> Im (a * cnj b) > 0"
   using complex_div_gt_0 by auto
 
 lemma Re_complex_div_ge_0: "Re (a / b) \<ge> 0 \<longleftrightarrow> Re (a * cnj b) \<ge> 0"
   by (metis le_less Re_complex_div_eq_0 Re_complex_div_gt_0)
 
 lemma Im_complex_div_ge_0: "Im (a / b) \<ge> 0 \<longleftrightarrow> Im (a * cnj b) \<ge> 0"
   by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 le_less)
 
 lemma Re_complex_div_lt_0: "Re (a / b) < 0 \<longleftrightarrow> Re (a * cnj b) < 0"
   by (metis less_asym neq_iff Re_complex_div_eq_0 Re_complex_div_gt_0)
 
 lemma Im_complex_div_lt_0: "Im (a / b) < 0 \<longleftrightarrow> Im (a * cnj b) < 0"
   by (metis Im_complex_div_eq_0 Im_complex_div_gt_0 less_asym neq_iff)
 
 lemma Re_complex_div_le_0: "Re (a / b) \<le> 0 \<longleftrightarrow> Re (a * cnj b) \<le> 0"
   by (metis not_le Re_complex_div_gt_0)
 
 lemma Im_complex_div_le_0: "Im (a / b) \<le> 0 \<longleftrightarrow> Im (a * cnj b) \<le> 0"
   by (metis Im_complex_div_gt_0 not_le)
 
 lemma Re_divide_of_real [simp]: "Re (z / of_real r) = Re z / r"
   by (simp add: Re_divide power2_eq_square)
 
 lemma Im_divide_of_real [simp]: "Im (z / of_real r) = Im z / r"
   by (simp add: Im_divide power2_eq_square)
 
 lemma Re_divide_Reals [simp]: "r \<in> \<real> \<Longrightarrow> Re (z / r) = Re z / Re r"
   by (metis Re_divide_of_real of_real_Re)
 
 lemma Im_divide_Reals [simp]: "r \<in> \<real> \<Longrightarrow> Im (z / r) = Im z / Re r"
   by (metis Im_divide_of_real of_real_Re)
 
 lemma Re_sum[simp]: "Re (sum f s) = (\<Sum>x\<in>s. Re (f x))"
   by (induct s rule: infinite_finite_induct) auto
 
 lemma Im_sum[simp]: "Im (sum f s) = (\<Sum>x\<in>s. Im(f x))"
   by (induct s rule: infinite_finite_induct) auto
 
 lemma sums_complex_iff: "f sums x \<longleftrightarrow> ((\<lambda>x. Re (f x)) sums Re x) \<and> ((\<lambda>x. Im (f x)) sums Im x)"
   unfolding sums_def tendsto_complex_iff Im_sum Re_sum ..
 
 lemma summable_complex_iff: "summable f \<longleftrightarrow> summable (\<lambda>x. Re (f x)) \<and>  summable (\<lambda>x. Im (f x))"
   unfolding summable_def sums_complex_iff[abs_def] by (metis complex.sel)
 
 lemma summable_complex_of_real [simp]: "summable (\<lambda>n. complex_of_real (f n)) \<longleftrightarrow> summable f"
   unfolding summable_complex_iff by simp
 
 lemma summable_Re: "summable f \<Longrightarrow> summable (\<lambda>x. Re (f x))"
   unfolding summable_complex_iff by blast
 
 lemma summable_Im: "summable f \<Longrightarrow> summable (\<lambda>x. Im (f x))"
   unfolding summable_complex_iff by blast
 
 lemma complex_is_Nat_iff: "z \<in> \<nat> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_nat i)"
   by (auto simp: Nats_def complex_eq_iff)
 
 lemma complex_is_Int_iff: "z \<in> \<int> \<longleftrightarrow> Im z = 0 \<and> (\<exists>i. Re z = of_int i)"
   by (auto simp: Ints_def complex_eq_iff)
 
 lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
   by (auto simp: Reals_def complex_eq_iff)
 
 lemma Reals_cnj_iff: "z \<in> \<real> \<longleftrightarrow> cnj z = z"
   by (auto simp: complex_is_Real_iff complex_eq_iff)
 
 lemma in_Reals_norm: "z \<in> \<real> \<Longrightarrow> norm z = \<bar>Re z\<bar>"
   by (simp add: complex_is_Real_iff norm_complex_def)
 
 lemma Re_Reals_divide: "r \<in> \<real> \<Longrightarrow> Re (r / z) = Re r * Re z / (norm z)\<^sup>2"
   by (simp add: Re_divide complex_is_Real_iff cmod_power2)
 
 lemma Im_Reals_divide: "r \<in> \<real> \<Longrightarrow> Im (r / z) = -Re r * Im z / (norm z)\<^sup>2"
   by (simp add: Im_divide complex_is_Real_iff cmod_power2)
 
 lemma series_comparison_complex:
   fixes f:: "nat \<Rightarrow> 'a::banach"
   assumes sg: "summable g"
     and "\<And>n. g n \<in> \<real>" "\<And>n. Re (g n) \<ge> 0"
     and fg: "\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> norm(g n)"
   shows "summable f"
 proof -
   have g: "\<And>n. cmod (g n) = Re (g n)"
     using assms by (metis abs_of_nonneg in_Reals_norm)
   show ?thesis
     apply (rule summable_comparison_test' [where g = "\<lambda>n. norm (g n)" and N=N])
     using sg
      apply (auto simp: summable_def)
      apply (rule_tac x = "Re s" in exI)
      apply (auto simp: g sums_Re)
     apply (metis fg g)
     done
 qed
 
 
 subsection \<open>Polar Form for Complex Numbers\<close>
 
 lemma complex_unimodular_polar:
   assumes "norm z = 1"
   obtains t where "0 \<le> t" "t < 2 * pi" "z = Complex (cos t) (sin t)"
   by (metis cmod_power2 one_power2 complex_surj sincos_total_2pi [of "Re z" "Im z"] assms)
 
 
 subsubsection \<open>$\cos \theta + i \sin \theta$\<close>
 
 primcorec cis :: "real \<Rightarrow> complex"
   where
     "Re (cis a) = cos a"
   | "Im (cis a) = sin a"
 
 lemma cis_zero [simp]: "cis 0 = 1"
   by (simp add: complex_eq_iff)
 
 lemma norm_cis [simp]: "norm (cis a) = 1"
   by (simp add: norm_complex_def)
 
 lemma sgn_cis [simp]: "sgn (cis a) = cis a"
   by (simp add: sgn_div_norm)
 
 lemma cis_2pi [simp]: "cis (2 * pi) = 1"
   by (simp add: cis.ctr complex_eq_iff)
 
 lemma cis_neq_zero [simp]: "cis a \<noteq> 0"
   by (metis norm_cis norm_zero zero_neq_one)
 
 lemma cis_cnj: "cnj (cis t) = cis (-t)"
   by (simp add: complex_eq_iff)
 
 lemma cis_mult: "cis a * cis b = cis (a + b)"
   by (simp add: complex_eq_iff cos_add sin_add)
 
 lemma DeMoivre: "(cis a) ^ n = cis (real n * a)"
   by (induct n) (simp_all add: algebra_simps cis_mult)
 
 lemma cis_inverse [simp]: "inverse (cis a) = cis (- a)"
   by (simp add: complex_eq_iff)
 
 lemma cis_divide: "cis a / cis b = cis (a - b)"
   by (simp add: divide_complex_def cis_mult)
 
 lemma cos_n_Re_cis_pow_n: "cos (real n * a) = Re (cis a ^ n)"
   by (auto simp add: DeMoivre)
 
 lemma sin_n_Im_cis_pow_n: "sin (real n * a) = Im (cis a ^ n)"
   by (auto simp add: DeMoivre)
 
 lemma cis_pi [simp]: "cis pi = -1"
   by (simp add: complex_eq_iff)
 
 lemma cis_pi_half[simp]: "cis (pi / 2) = \<i>"
   by (simp add: cis.ctr complex_eq_iff)
 
 lemma cis_minus_pi_half[simp]: "cis (-(pi / 2)) = -\<i>"
   by (simp add: cis.ctr complex_eq_iff)
 
 lemma cis_multiple_2pi[simp]: "n \<in> \<int> \<Longrightarrow> cis (2 * pi * n) = 1"
   by (auto elim!: Ints_cases simp: cis.ctr one_complex.ctr)
 
 
 subsubsection \<open>$r(\cos \theta + i \sin \theta)$\<close>
 
 definition rcis :: "real \<Rightarrow> real \<Rightarrow> complex"
   where "rcis r a = complex_of_real r * cis a"
 
 lemma Re_rcis [simp]: "Re(rcis r a) = r * cos a"
   by (simp add: rcis_def)
 
 lemma Im_rcis [simp]: "Im(rcis r a) = r * sin a"
   by (simp add: rcis_def)
 
 lemma rcis_Ex: "\<exists>r a. z = rcis r a"
   by (simp add: complex_eq_iff polar_Ex)
 
 lemma complex_mod_rcis [simp]: "cmod (rcis r a) = \<bar>r\<bar>"
   by (simp add: rcis_def norm_mult)
 
 lemma cis_rcis_eq: "cis a = rcis 1 a"
   by (simp add: rcis_def)
 
 lemma rcis_mult: "rcis r1 a * rcis r2 b = rcis (r1 * r2) (a + b)"
   by (simp add: rcis_def cis_mult)
 
 lemma rcis_zero_mod [simp]: "rcis 0 a = 0"
   by (simp add: rcis_def)
 
 lemma rcis_zero_arg [simp]: "rcis r 0 = complex_of_real r"
   by (simp add: rcis_def)
 
 lemma rcis_eq_zero_iff [simp]: "rcis r a = 0 \<longleftrightarrow> r = 0"
   by (simp add: rcis_def)
 
 lemma DeMoivre2: "(rcis r a) ^ n = rcis (r ^ n) (real n * a)"
   by (simp add: rcis_def power_mult_distrib DeMoivre)
 
 lemma rcis_inverse: "inverse(rcis r a) = rcis (1 / r) (- a)"
   by (simp add: divide_inverse rcis_def)
 
 lemma rcis_divide: "rcis r1 a / rcis r2 b = rcis (r1 / r2) (a - b)"
   by (simp add: rcis_def cis_divide [symmetric])
 
-
 subsubsection \<open>Complex exponential\<close>
 
 lemma exp_Reals_eq:
   assumes "z \<in> \<real>"
   shows   "exp z = of_real (exp (Re z))"
   using assms by (auto elim!: Reals_cases simp: exp_of_real)
 
 lemma cis_conv_exp: "cis b = exp (\<i> * b)"
 proof -
   have "(\<i> * complex_of_real b) ^ n /\<^sub>R fact n =
       of_real (cos_coeff n * b^n) + \<i> * of_real (sin_coeff n * b^n)"
     for n :: nat
   proof -
     have "\<i> ^ n = fact n *\<^sub>R (cos_coeff n + \<i> * sin_coeff n)"
       by (induct n)
         (simp_all add: sin_coeff_Suc cos_coeff_Suc complex_eq_iff Re_divide Im_divide field_simps
           power2_eq_square add_nonneg_eq_0_iff)
     then show ?thesis
       by (simp add: field_simps)
   qed
   then show ?thesis
     using sin_converges [of b] cos_converges [of b]
     by (auto simp add: Complex_eq cis.ctr exp_def simp del: of_real_mult
         intro!: sums_unique sums_add sums_mult sums_of_real)
 qed
 
 lemma exp_eq_polar: "exp z = exp (Re z) * cis (Im z)"
   unfolding cis_conv_exp exp_of_real [symmetric] mult_exp_exp
   by (cases z) (simp add: Complex_eq)
 
 lemma Re_exp: "Re (exp z) = exp (Re z) * cos (Im z)"
   unfolding exp_eq_polar by simp
 
 lemma Im_exp: "Im (exp z) = exp (Re z) * sin (Im z)"
   unfolding exp_eq_polar by simp
 
 lemma norm_cos_sin [simp]: "norm (Complex (cos t) (sin t)) = 1"
   by (simp add: norm_complex_def)
 
 lemma norm_exp_eq_Re [simp]: "norm (exp z) = exp (Re z)"
   by (simp add: cis.code cmod_complex_polar exp_eq_polar Complex_eq)
 
 lemma complex_exp_exists: "\<exists>a r. z = complex_of_real r * exp a"
   apply (insert rcis_Ex [of z])
   apply (auto simp add: exp_eq_polar rcis_def mult.assoc [symmetric])
   apply (rule_tac x = "\<i> * complex_of_real a" in exI)
   apply auto
   done
 
 lemma exp_pi_i [simp]: "exp (of_real pi * \<i>) = -1"
   by (metis cis_conv_exp cis_pi mult.commute)
 
 lemma exp_pi_i' [simp]: "exp (\<i> * of_real pi) = -1"
   using cis_conv_exp cis_pi by auto
 
 lemma exp_two_pi_i [simp]: "exp (2 * of_real pi * \<i>) = 1"
   by (simp add: exp_eq_polar complex_eq_iff)
 
 lemma exp_two_pi_i' [simp]: "exp (\<i> * (of_real pi * 2)) = 1"
   by (metis exp_two_pi_i mult.commute)
 
 lemma continuous_on_cis [continuous_intros]:
   "continuous_on A f \<Longrightarrow> continuous_on A (\<lambda>x. cis (f x))"
   by (auto simp: cis_conv_exp intro!: continuous_intros)
 
 
 subsubsection \<open>Complex argument\<close>
 
-definition arg :: "complex \<Rightarrow> real"
-  where "arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> - pi < a \<and> a \<le> pi))"
+definition Arg :: "complex \<Rightarrow> real"
+  where "Arg z = (if z = 0 then 0 else (SOME a. sgn z = cis a \<and> - pi < a \<and> a \<le> pi))"
 
-lemma arg_zero: "arg 0 = 0"
-  by (simp add: arg_def)
+lemma Arg_zero: "Arg 0 = 0"
+  by (simp add: Arg_def)
 
 lemma arg_unique:
   assumes "sgn z = cis x" and "-pi < x" and "x \<le> pi"
-  shows "arg z = x"
+  shows "Arg z = x"
 proof -
   from assms have "z \<noteq> 0" by auto
   have "(SOME a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi) = x"
   proof
     fix a
     define d where "d = a - x"
     assume a: "sgn z = cis a \<and> - pi < a \<and> a \<le> pi"
     from a assms have "- (2*pi) < d \<and> d < 2*pi"
       unfolding d_def by simp
     moreover
     from a assms have "cos a = cos x" and "sin a = sin x"
       by (simp_all add: complex_eq_iff)
     then have cos: "cos d = 1"
       by (simp add: d_def cos_diff)
     moreover from cos have "sin d = 0"
       by (rule cos_one_sin_zero)
     ultimately have "d = 0"
       by (auto simp: sin_zero_iff elim!: evenE dest!: less_2_cases)
     then show "a = x"
       by (simp add: d_def)
   qed (simp add: assms del: Re_sgn Im_sgn)
-  with \<open>z \<noteq> 0\<close> show "arg z = x"
-    by (simp add: arg_def)
+  with \<open>z \<noteq> 0\<close> show "Arg z = x"
+    by (simp add: Arg_def)
 qed
 
 lemma arg_correct:
   assumes "z \<noteq> 0"
-  shows "sgn z = cis (arg z) \<and> -pi < arg z \<and> arg z \<le> pi"
-proof (simp add: arg_def assms, rule someI_ex)
+  shows "sgn z = cis (Arg z) \<and> -pi < Arg z \<and> Arg z \<le> pi"
+proof (simp add: Arg_def assms, rule someI_ex)
   obtain r a where z: "z = rcis r a"
     using rcis_Ex by fast
   with assms have "r \<noteq> 0" by auto
   define b where "b = (if 0 < r then a else a + pi)"
   have b: "sgn z = cis b"
     using \<open>r \<noteq> 0\<close> by (simp add: z b_def rcis_def of_real_def sgn_scaleR sgn_if complex_eq_iff)
   have cis_2pi_nat: "cis (2 * pi * real_of_nat n) = 1" for n
     by (induct n) (simp_all add: distrib_left cis_mult [symmetric] complex_eq_iff)
   have cis_2pi_int: "cis (2 * pi * real_of_int x) = 1" for x
     by (cases x rule: int_diff_cases)
       (simp add: right_diff_distrib cis_divide [symmetric] cis_2pi_nat)
   define c where "c = b - 2 * pi * of_int \<lceil>(b - pi) / (2 * pi)\<rceil>"
   have "sgn z = cis c"
     by (simp add: b c_def cis_divide [symmetric] cis_2pi_int)
   moreover have "- pi < c \<and> c \<le> pi"
     using ceiling_correct [of "(b - pi) / (2*pi)"]
     by (simp add: c_def less_divide_eq divide_le_eq algebra_simps del: le_of_int_ceiling)
   ultimately show "\<exists>a. sgn z = cis a \<and> -pi < a \<and> a \<le> pi"
     by fast
 qed
 
-lemma arg_bounded: "- pi < arg z \<and> arg z \<le> pi"
-  by (cases "z = 0") (simp_all add: arg_zero arg_correct)
+lemma Arg_bounded: "- pi < Arg z \<and> Arg z \<le> pi"
+  by (cases "z = 0") (simp_all add: Arg_zero arg_correct)
 
-lemma cis_arg: "z \<noteq> 0 \<Longrightarrow> cis (arg z) = sgn z"
+lemma cis_Arg: "z \<noteq> 0 \<Longrightarrow> cis (Arg z) = sgn z"
   by (simp add: arg_correct)
 
-lemma rcis_cmod_arg: "rcis (cmod z) (arg z) = z"
-  by (cases "z = 0") (simp_all add: rcis_def cis_arg sgn_div_norm of_real_def)
+lemma rcis_cmod_Arg: "rcis (cmod z) (Arg z) = z"
+  by (cases "z = 0") (simp_all add: rcis_def cis_Arg sgn_div_norm of_real_def)
 
-lemma cos_arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (arg y) = 0"
-  using cis_arg [of y] by (simp add: complex_eq_iff)
+lemma rcis_cnj:
+  shows "cnj a = rcis (cmod a) (- Arg a)"
+  by (metis cis_cnj complex_cnj_complex_of_real complex_cnj_mult rcis_cmod_Arg rcis_def)
 
-lemma arg_ii [simp]: "arg \<i> = pi/2"
+lemma cos_Arg_i_mult_zero [simp]: "y \<noteq> 0 \<Longrightarrow> Re y = 0 \<Longrightarrow> cos (Arg y) = 0"
+  using cis_Arg [of y] by (simp add: complex_eq_iff)
+
+lemma Arg_ii [simp]: "Arg \<i> = pi/2"
   by (rule arg_unique; simp add: sgn_eq)
 
-lemma arg_minus_ii [simp]: "arg (-\<i>) = -pi/2"
+lemma Arg_minus_ii [simp]: "Arg (-\<i>) = -pi/2"
 proof (rule arg_unique)
   show "sgn (- \<i>) = cis (- pi / 2)"
     by (simp add: sgn_eq)
   show "- pi / 2 \<le> pi"
     using pi_not_less_zero by linarith
 qed auto
 
 subsection \<open>Complex n-th roots\<close>
 
 lemma bij_betw_roots_unity:
   assumes "n > 0"
   shows   "bij_betw (\<lambda>k. cis (2 * pi * real k / real n)) {..<n} {z. z ^ n = 1}"
     (is "bij_betw ?f _ _")
   unfolding bij_betw_def
 proof (intro conjI)
   show inj: "inj_on ?f {..<n}" unfolding inj_on_def
   proof (safe, goal_cases)
     case (1 k l)
     hence kl: "k < n" "l < n" by simp_all
     from 1 have "1 = ?f k / ?f l" by simp
     also have "\<dots> = cis (2*pi*(real k - real l)/n)"
       using assms by (simp add: field_simps cis_divide)
     finally have "cos (2*pi*(real k - real l) / n) = 1"
       by (simp add: complex_eq_iff)
     then obtain m :: int where "2 * pi * (real k - real l) / real n = real_of_int m * 2 * pi"
       by (subst (asm) cos_one_2pi_int) blast
     hence "real_of_int (int k - int l) = real_of_int (m * int n)"
       unfolding of_int_diff of_int_mult using assms
       by (simp add: nonzero_divide_eq_eq)
     also note of_int_eq_iff
     finally have *: "abs m * n = abs (int k - int l)" by (simp add: abs_mult)
     also have "\<dots> < int n" using kl by linarith
     finally have "m = 0" using assms by simp
     with * show "k = l" by simp
   qed
 
   have subset: "?f ` {..<n} \<subseteq> {z. z ^ n = 1}"
   proof safe
     fix k :: nat
     have "cis (2 * pi * real k / real n) ^ n = cis (2 * pi) ^ k"
       using assms by (simp add: DeMoivre mult_ac)
     also have "cis (2 * pi) = 1" by (simp add: complex_eq_iff)
     finally show "?f k ^ n = 1" by simp
   qed
 
   have "n = card {..<n}" by simp
   also from assms and subset have "\<dots> \<le> card {z::complex. z ^ n = 1}"
     by (intro card_inj_on_le[OF inj]) (auto simp: finite_roots_unity)
   finally have card: "card {z::complex. z ^ n = 1} = n"
     using assms by (intro antisym card_roots_unity) auto
 
   have "card (?f ` {..<n}) = card {z::complex. z ^ n = 1}"
     using card inj by (subst card_image) auto
   with subset and assms show "?f ` {..<n} = {z::complex. z ^ n = 1}"
     by (intro card_subset_eq finite_roots_unity) auto
 qed
 
 lemma card_roots_unity_eq:
   assumes "n > 0"
   shows   "card {z::complex. z ^ n = 1} = n"
   using bij_betw_same_card [OF bij_betw_roots_unity [OF assms]] by simp
 
 lemma bij_betw_nth_root_unity:
   fixes c :: complex and n :: nat
   assumes c: "c \<noteq> 0" and n: "n > 0"
-  defines "c' \<equiv> root n (norm c) * cis (arg c / n)"
+  defines "c' \<equiv> root n (norm c) * cis (Arg c / n)"
   shows "bij_betw (\<lambda>z. c' * z) {z. z ^ n = 1} {z. z ^ n = c}"
 proof -
-  have "c' ^ n = of_real (root n (norm c) ^ n) * cis (arg c)"
+  have "c' ^ n = of_real (root n (norm c) ^ n) * cis (Arg c)"
     unfolding of_real_power using n by (simp add: c'_def power_mult_distrib DeMoivre)
   also from n have "root n (norm c) ^ n = norm c" by simp
-  also from c have "of_real \<dots> * cis (arg c) = c" by (simp add: cis_arg Complex.sgn_eq)
+  also from c have "of_real \<dots> * cis (Arg c) = c" by (simp add: cis_Arg Complex.sgn_eq)
   finally have [simp]: "c' ^ n = c" .
 
   show ?thesis unfolding bij_betw_def inj_on_def
   proof safe
     fix z :: complex assume "z ^ n = 1"
     hence "(c' * z) ^ n = c' ^ n" by (simp add: power_mult_distrib)
-    also have "c' ^ n = of_real (root n (norm c) ^ n) * cis (arg c)"
+    also have "c' ^ n = of_real (root n (norm c) ^ n) * cis (Arg c)"
       unfolding of_real_power using n by (simp add: c'_def power_mult_distrib DeMoivre)
     also from n have "root n (norm c) ^ n = norm c" by simp
-    also from c have "\<dots> * cis (arg c) = c" by (simp add: cis_arg Complex.sgn_eq)
+    also from c have "\<dots> * cis (Arg c) = c" by (simp add: cis_Arg Complex.sgn_eq)
     finally show "(c' * z) ^ n = c" .
   next
     fix z assume z: "c = z ^ n"
     define z' where "z' = z / c'"
     from c and n have "c' \<noteq> 0" by (auto simp: c'_def)
     with n c have "z = c' * z'" and "z' ^ n = 1"
       by (auto simp: z'_def power_divide z)
     thus "z \<in> (\<lambda>z. c' * z) ` {z. z ^ n = 1}" by blast
   qed (insert c n, auto simp: c'_def)
 qed
 
 lemma finite_nth_roots [intro]:
   assumes "n > 0"
   shows   "finite {z::complex. z ^ n = c}"
 proof (cases "c = 0")
   case True
   with assms have "{z::complex. z ^ n = c} = {0}" by auto
   thus ?thesis by simp
 next
   case False
   from assms have "finite {z::complex. z ^ n = 1}" by (intro finite_roots_unity) simp_all
   also have "?this \<longleftrightarrow> ?thesis"
     by (rule bij_betw_finite, rule bij_betw_nth_root_unity) fact+
   finally show ?thesis .
 qed
 
 lemma card_nth_roots:
   assumes "c \<noteq> 0" "n > 0"
   shows   "card {z::complex. z ^ n = c} = n"
 proof -
   have "card {z. z ^ n = c} = card {z::complex. z ^ n = 1}"
     by (rule sym, rule bij_betw_same_card, rule bij_betw_nth_root_unity) fact+
   also have "\<dots> = n" by (rule card_roots_unity_eq) fact+
   finally show ?thesis .
 qed
 
 lemma sum_roots_unity:
   assumes "n > 1"
   shows   "\<Sum>{z::complex. z ^ n = 1} = 0"
 proof -
   define \<omega> where "\<omega> = cis (2 * pi / real n)"
   have [simp]: "\<omega> \<noteq> 1"
   proof
     assume "\<omega> = 1"
     with assms obtain k :: int where "2 * pi / real n = 2 * pi * of_int k"
       by (auto simp: \<omega>_def complex_eq_iff cos_one_2pi_int)
     with assms have "real n * of_int k = 1" by (simp add: field_simps)
     also have "real n * of_int k = of_int (int n * k)" by simp
     also have "1 = (of_int 1 :: real)" by simp
     also note of_int_eq_iff
     finally show False using assms by (auto simp: zmult_eq_1_iff)
   qed
 
   have "(\<Sum>z | z ^ n = 1. z :: complex) = (\<Sum>k<n. cis (2 * pi * real k / real n))"
     using assms by (intro sum.reindex_bij_betw [symmetric] bij_betw_roots_unity) auto
   also have "\<dots> = (\<Sum>k<n. \<omega> ^ k)"
     by (intro sum.cong refl) (auto simp: \<omega>_def DeMoivre mult_ac)
   also have "\<dots> = (\<omega> ^ n - 1) / (\<omega> - 1)"
     by (subst geometric_sum) auto
   also have "\<omega> ^ n - 1 = cis (2 * pi) - 1" using assms by (auto simp: \<omega>_def DeMoivre)
   also have "\<dots> = 0" by (simp add: complex_eq_iff)
   finally show ?thesis by simp
 qed
 
 lemma sum_nth_roots:
   assumes "n > 1"
   shows   "\<Sum>{z::complex. z ^ n = c} = 0"
 proof (cases "c = 0")
   case True
   with assms have "{z::complex. z ^ n = c} = {0}" by auto
   also have "\<Sum>\<dots> = 0" by simp
   finally show ?thesis .
 next
   case False
-  define c' where "c' = root n (norm c) * cis (arg c / n)"
+  define c' where "c' = root n (norm c) * cis (Arg c / n)"
   from False and assms have "(\<Sum>{z. z ^ n = c}) = (\<Sum>z | z ^ n = 1. c' * z)"
     by (subst sum.reindex_bij_betw [OF bij_betw_nth_root_unity, symmetric])
        (auto simp: sum_distrib_left finite_roots_unity c'_def)
   also from assms have "\<dots> = 0"
     by (simp add: sum_distrib_left [symmetric] sum_roots_unity)
   finally show ?thesis .
 qed
 
 subsection \<open>Square root of complex numbers\<close>
 
 primcorec csqrt :: "complex \<Rightarrow> complex"
   where
     "Re (csqrt z) = sqrt ((cmod z + Re z) / 2)"
   | "Im (csqrt z) = (if Im z = 0 then 1 else sgn (Im z)) * sqrt ((cmod z - Re z) / 2)"
 
 lemma csqrt_of_real_nonneg [simp]: "Im x = 0 \<Longrightarrow> Re x \<ge> 0 \<Longrightarrow> csqrt x = sqrt (Re x)"
   by (simp add: complex_eq_iff norm_complex_def)
 
 lemma csqrt_of_real_nonpos [simp]: "Im x = 0 \<Longrightarrow> Re x \<le> 0 \<Longrightarrow> csqrt x = \<i> * sqrt \<bar>Re x\<bar>"
   by (simp add: complex_eq_iff norm_complex_def)
 
 lemma of_real_sqrt: "x \<ge> 0 \<Longrightarrow> of_real (sqrt x) = csqrt (of_real x)"
   by (simp add: complex_eq_iff norm_complex_def)
 
 lemma csqrt_0 [simp]: "csqrt 0 = 0"
   by simp
 
 lemma csqrt_1 [simp]: "csqrt 1 = 1"
   by simp
 
 lemma csqrt_ii [simp]: "csqrt \<i> = (1 + \<i>) / sqrt 2"
   by (simp add: complex_eq_iff Re_divide Im_divide real_sqrt_divide real_div_sqrt)
 
 lemma power2_csqrt[simp,algebra]: "(csqrt z)\<^sup>2 = z"
 proof (cases "Im z = 0")
   case True
   then show ?thesis
     using real_sqrt_pow2[of "Re z"] real_sqrt_pow2[of "- Re z"]
     by (cases "0::real" "Re z" rule: linorder_cases)
       (simp_all add: complex_eq_iff Re_power2 Im_power2 power2_eq_square cmod_eq_Re)
 next
   case False
   moreover have "cmod z * cmod z - Re z * Re z = Im z * Im z"
     by (simp add: norm_complex_def power2_eq_square)
   moreover have "\<bar>Re z\<bar> \<le> cmod z"
     by (simp add: norm_complex_def)
   ultimately show ?thesis
     by (simp add: Re_power2 Im_power2 complex_eq_iff real_sgn_eq
         field_simps real_sqrt_mult[symmetric] real_sqrt_divide)
 qed
 
 lemma csqrt_eq_0 [simp]: "csqrt z = 0 \<longleftrightarrow> z = 0"
   by auto (metis power2_csqrt power_eq_0_iff)
 
 lemma csqrt_eq_1 [simp]: "csqrt z = 1 \<longleftrightarrow> z = 1"
   by auto (metis power2_csqrt power2_eq_1_iff)
 
 lemma csqrt_principal: "0 < Re (csqrt z) \<or> Re (csqrt z) = 0 \<and> 0 \<le> Im (csqrt z)"
   by (auto simp add: not_less cmod_plus_Re_le_0_iff Im_eq_0)
 
 lemma Re_csqrt: "0 \<le> Re (csqrt z)"
   by (metis csqrt_principal le_less)
 
 lemma csqrt_square:
   assumes "0 < Re b \<or> (Re b = 0 \<and> 0 \<le> Im b)"
   shows "csqrt (b^2) = b"
 proof -
   have "csqrt (b^2) = b \<or> csqrt (b^2) = - b"
     by (simp add: power2_eq_iff[symmetric])
   moreover have "csqrt (b^2) \<noteq> -b \<or> b = 0"
     using csqrt_principal[of "b ^ 2"] assms
     by (intro disjCI notI) (auto simp: complex_eq_iff)
   ultimately show ?thesis
     by auto
 qed
 
 lemma csqrt_unique: "w\<^sup>2 = z \<Longrightarrow> 0 < Re w \<or> Re w = 0 \<and> 0 \<le> Im w \<Longrightarrow> csqrt z = w"
   by (auto simp: csqrt_square)
 
 lemma csqrt_minus [simp]:
   assumes "Im x < 0 \<or> (Im x = 0 \<and> 0 \<le> Re x)"
   shows "csqrt (- x) = \<i> * csqrt x"
 proof -
   have "csqrt ((\<i> * csqrt x)^2) = \<i> * csqrt x"
   proof (rule csqrt_square)
     have "Im (csqrt x) \<le> 0"
       using assms by (auto simp add: cmod_eq_Re mult_le_0_iff field_simps complex_Re_le_cmod)
     then show "0 < Re (\<i> * csqrt x) \<or> Re (\<i> * csqrt x) = 0 \<and> 0 \<le> Im (\<i> * csqrt x)"
       by (auto simp add: Re_csqrt simp del: csqrt.simps)
   qed
   also have "(\<i> * csqrt x)^2 = - x"
     by (simp add: power_mult_distrib)
   finally show ?thesis .
 qed
 
 
 text \<open>Legacy theorem names\<close>
 
 lemmas cmod_def = norm_complex_def
 
 lemma legacy_Complex_simps:
   shows Complex_eq_0: "Complex a b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
     and complex_add: "Complex a b + Complex c d = Complex (a + c) (b + d)"
     and complex_minus: "- (Complex a b) = Complex (- a) (- b)"
     and complex_diff: "Complex a b - Complex c d = Complex (a - c) (b - d)"
     and Complex_eq_1: "Complex a b = 1 \<longleftrightarrow> a = 1 \<and> b = 0"
     and Complex_eq_neg_1: "Complex a b = - 1 \<longleftrightarrow> a = - 1 \<and> b = 0"
     and complex_mult: "Complex a b * Complex c d = Complex (a * c - b * d) (a * d + b * c)"
     and complex_inverse: "inverse (Complex a b) = Complex (a / (a\<^sup>2 + b\<^sup>2)) (- b / (a\<^sup>2 + b\<^sup>2))"
     and Complex_eq_numeral: "Complex a b = numeral w \<longleftrightarrow> a = numeral w \<and> b = 0"
     and Complex_eq_neg_numeral: "Complex a b = - numeral w \<longleftrightarrow> a = - numeral w \<and> b = 0"
     and complex_scaleR: "scaleR r (Complex a b) = Complex (r * a) (r * b)"
     and Complex_eq_i: "Complex x y = \<i> \<longleftrightarrow> x = 0 \<and> y = 1"
     and i_mult_Complex: "\<i> * Complex a b = Complex (- b) a"
     and Complex_mult_i: "Complex a b * \<i> = Complex (- b) a"
     and i_complex_of_real: "\<i> * complex_of_real r = Complex 0 r"
     and complex_of_real_i: "complex_of_real r * \<i> = Complex 0 r"
     and Complex_add_complex_of_real: "Complex x y + complex_of_real r = Complex (x+r) y"
     and complex_of_real_add_Complex: "complex_of_real r + Complex x y = Complex (r+x) y"
     and Complex_mult_complex_of_real: "Complex x y * complex_of_real r = Complex (x*r) (y*r)"
     and complex_of_real_mult_Complex: "complex_of_real r * Complex x y = Complex (r*x) (r*y)"
     and complex_eq_cancel_iff2: "(Complex x y = complex_of_real xa) = (x = xa \<and> y = 0)"
     and complex_cnj: "cnj (Complex a b) = Complex a (- b)"
     and Complex_sum': "sum (\<lambda>x. Complex (f x) 0) s = Complex (sum f s) 0"
     and Complex_sum: "Complex (sum f s) 0 = sum (\<lambda>x. Complex (f x) 0) s"
     and complex_of_real_def: "complex_of_real r = Complex r 0"
     and complex_norm: "cmod (Complex x y) = sqrt (x\<^sup>2 + y\<^sup>2)"
   by (simp_all add: norm_complex_def field_simps complex_eq_iff Re_divide Im_divide)
 
 lemma Complex_in_Reals: "Complex x 0 \<in> \<real>"
   by (metis Reals_of_real complex_of_real_def)
 
 end
diff --git a/src/HOL/Complex_Analysis/Complex_Residues.thy b/src/HOL/Complex_Analysis/Complex_Residues.thy
--- a/src/HOL/Complex_Analysis/Complex_Residues.thy
+++ b/src/HOL/Complex_Analysis/Complex_Residues.thy
@@ -1,545 +1,540 @@
 theory Complex_Residues
   imports Complex_Singularities
 begin
 
 subsection \<open>Definition of residues\<close>
 
 text\<open>Wenda Li and LC Paulson (2016). A Formal Proof of Cauchy's Residue Theorem.
     Interactive Theorem Proving\<close>
 
 definition\<^marker>\<open>tag important\<close> residue :: "(complex \<Rightarrow> complex) \<Rightarrow> complex \<Rightarrow> complex" where
   "residue f z = (SOME int. \<exists>e>0. \<forall>\<epsilon>>0. \<epsilon><e
     \<longrightarrow> (f has_contour_integral 2*pi* \<i> *int) (circlepath z \<epsilon>))"
 
-lemma Eps_cong:
-  assumes "\<And>x. P x = Q x"
-  shows   "Eps P = Eps Q"
-  using ext[of P Q, OF assms] by simp
-
 lemma residue_cong:
   assumes eq: "eventually (\<lambda>z. f z = g z) (at z)" and "z = z'"
   shows   "residue f z = residue g z'"
 proof -
   from assms have eq': "eventually (\<lambda>z. g z = f z) (at z)"
     by (simp add: eq_commute)
   let ?P = "\<lambda>f c e. (\<forall>\<epsilon>>0. \<epsilon> < e \<longrightarrow>
    (f has_contour_integral of_real (2 * pi) * \<i> * c) (circlepath z \<epsilon>))"
   have "residue f z = residue g z" unfolding residue_def
   proof (rule Eps_cong)
     fix c :: complex
     have "\<exists>e>0. ?P g c e"
       if "\<exists>e>0. ?P f c e" and "eventually (\<lambda>z. f z = g z) (at z)" for f g
     proof -
       from that(1) obtain e where e: "e > 0" "?P f c e"
         by blast
       from that(2) obtain e' where e': "e' > 0" "\<And>z'. z' \<noteq> z \<Longrightarrow> dist z' z < e' \<Longrightarrow> f z' = g z'"
         unfolding eventually_at by blast
       have "?P g c (min e e')"
       proof (intro allI exI impI, goal_cases)
         case (1 \<epsilon>)
         hence "(f has_contour_integral of_real (2 * pi) * \<i> * c) (circlepath z \<epsilon>)"
           using e(2) by auto
         thus ?case
         proof (rule has_contour_integral_eq)
           fix z' assume "z' \<in> path_image (circlepath z \<epsilon>)"
           hence "dist z' z < e'" and "z' \<noteq> z"
             using 1 by (auto simp: dist_commute)
           with e'(2)[of z'] show "f z' = g z'" by simp
         qed
       qed
       moreover from e and e' have "min e e' > 0" by auto
       ultimately show ?thesis by blast
     qed
     from this[OF _ eq] and this[OF _ eq']
       show "(\<exists>e>0. ?P f c e) \<longleftrightarrow> (\<exists>e>0. ?P g c e)"
       by blast
   qed
   with assms show ?thesis by simp
 qed
 
 lemma contour_integral_circlepath_eq:
   assumes "open s" and f_holo:"f holomorphic_on (s-{z})" and "0<e1" "e1\<le>e2"
     and e2_cball:"cball z e2 \<subseteq> s"
   shows
     "f contour_integrable_on circlepath z e1"
     "f contour_integrable_on circlepath z e2"
     "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
 proof -
   define l where "l \<equiv> linepath (z+e2) (z+e1)"
   have [simp]:"valid_path l" "pathstart l=z+e2" "pathfinish l=z+e1" unfolding l_def by auto
   have "e2>0" using \<open>e1>0\<close> \<open>e1\<le>e2\<close> by auto
   have zl_img:"z\<notin>path_image l"
     proof
       assume "z \<in> path_image l"
       then have "e2 \<le> cmod (e2 - e1)"
         using segment_furthest_le[of z "z+e2" "z+e1" "z+e2",simplified] \<open>e1>0\<close> \<open>e2>0\<close> unfolding l_def
         by (auto simp add:closed_segment_commute)
       thus False using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
         apply (subst (asm) norm_of_real)
         by auto
     qed
   define g where "g \<equiv> circlepath z e2 +++ l +++ reversepath (circlepath z e1) +++ reversepath l"
   show [simp]: "f contour_integrable_on circlepath z e2" "f contour_integrable_on (circlepath z e1)"
     proof -
       show "f contour_integrable_on circlepath z e2"
         apply (intro contour_integrable_continuous_circlepath[OF
                 continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
         using \<open>e2>0\<close> e2_cball by auto
       show "f contour_integrable_on (circlepath z e1)"
         apply (intro contour_integrable_continuous_circlepath[OF
                       continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
         using \<open>e1>0\<close> \<open>e1\<le>e2\<close> e2_cball by auto
     qed
   have [simp]:"f contour_integrable_on l"
     proof -
       have "closed_segment (z + e2) (z + e1) \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
         by (intro closed_segment_subset,auto simp add:dist_norm)
       hence "closed_segment (z + e2) (z + e1) \<subseteq> s - {z}" using zl_img e2_cball unfolding l_def
         by auto
       then show "f contour_integrable_on l" unfolding l_def
         apply (intro contour_integrable_continuous_linepath[OF
                       continuous_on_subset[OF holomorphic_on_imp_continuous_on[OF f_holo]]])
         by auto
     qed
   let ?ig="\<lambda>g. contour_integral g f"
   have "(f has_contour_integral 0) g"
     proof (rule Cauchy_theorem_global[OF _ f_holo])
       show "open (s - {z})" using \<open>open s\<close> by auto
       show "valid_path g" unfolding g_def l_def by auto
       show "pathfinish g = pathstart g" unfolding g_def l_def by auto
     next
       have path_img:"path_image g \<subseteq> cball z e2"
         proof -
           have "closed_segment (z + e2) (z + e1) \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1>0\<close> \<open>e1\<le>e2\<close>
             by (intro closed_segment_subset,auto simp add:dist_norm)
           moreover have "sphere z \<bar>e1\<bar> \<subseteq> cball z e2" using \<open>e2>0\<close> \<open>e1\<le>e2\<close> \<open>e1>0\<close> by auto
           ultimately show ?thesis unfolding g_def l_def using \<open>e2>0\<close>
             by (simp add: path_image_join closed_segment_commute)
         qed
       show "path_image g \<subseteq> s - {z}"
         proof -
           have "z\<notin>path_image g" using zl_img
             unfolding g_def l_def by (auto simp add: path_image_join closed_segment_commute)
           moreover note \<open>cball z e2 \<subseteq> s\<close> and path_img
           ultimately show ?thesis by auto
         qed
       show "winding_number g w = 0" when"w \<notin> s - {z}" for w
         proof -
           have "winding_number g w = 0" when "w\<notin>s" using that e2_cball
             apply (intro winding_number_zero_outside[OF _ _ _ _ path_img])
             by (auto simp add:g_def l_def)
           moreover have "winding_number g z=0"
             proof -
               let ?Wz="\<lambda>g. winding_number g z"
               have "?Wz g = ?Wz (circlepath z e2) + ?Wz l + ?Wz (reversepath (circlepath z e1))
                   + ?Wz (reversepath l)"
                 using \<open>e2>0\<close> \<open>e1>0\<close> zl_img unfolding g_def l_def
                 by (subst winding_number_join,auto simp add:path_image_join closed_segment_commute)+
               also have "... = ?Wz (circlepath z e2) + ?Wz (reversepath (circlepath z e1))"
                 using zl_img
                 apply (subst (2) winding_number_reversepath)
                 by (auto simp add:l_def closed_segment_commute)
               also have "... = 0"
                 proof -
                   have "?Wz (circlepath z e2) = 1" using \<open>e2>0\<close>
                     by (auto intro: winding_number_circlepath_centre)
                   moreover have "?Wz (reversepath (circlepath z e1)) = -1" using \<open>e1>0\<close>
                     apply (subst winding_number_reversepath)
                     by (auto intro: winding_number_circlepath_centre)
                   ultimately show ?thesis by auto
                 qed
               finally show ?thesis .
             qed
           ultimately show ?thesis using that by auto
         qed
     qed
   then have "0 = ?ig g" using contour_integral_unique by simp
   also have "... = ?ig (circlepath z e2) + ?ig l + ?ig (reversepath (circlepath z e1))
       + ?ig (reversepath l)"
     unfolding g_def
     by (auto simp add:contour_integrable_reversepath_eq)
   also have "... = ?ig (circlepath z e2)  - ?ig (circlepath z e1)"
     by (auto simp add:contour_integral_reversepath)
   finally show "contour_integral (circlepath z e2) f = contour_integral (circlepath z e1) f"
     by simp
 qed
 
 lemma base_residue:
   assumes "open s" "z\<in>s" "r>0" and f_holo:"f holomorphic_on (s - {z})"
     and r_cball:"cball z r \<subseteq> s"
   shows "(f has_contour_integral 2 * pi * \<i> * (residue f z)) (circlepath z r)"
 proof -
   obtain e where "e>0" and e_cball:"cball z e \<subseteq> s"
     using open_contains_cball[of s] \<open>open s\<close> \<open>z\<in>s\<close> by auto
   define c where "c \<equiv> 2 * pi * \<i>"
   define i where "i \<equiv> contour_integral (circlepath z e) f / c"
   have "(f has_contour_integral c*i) (circlepath z \<epsilon>)" when "\<epsilon>>0" "\<epsilon><e" for \<epsilon>
     proof -
       have "contour_integral (circlepath z e) f = contour_integral (circlepath z \<epsilon>) f"
           "f contour_integrable_on circlepath z \<epsilon>"
           "f contour_integrable_on circlepath z e"
         using \<open>\<epsilon><e\<close>
         by (intro contour_integral_circlepath_eq[OF \<open>open s\<close> f_holo \<open>\<epsilon>>0\<close> _ e_cball],auto)+
       then show ?thesis unfolding i_def c_def
         by (auto intro:has_contour_integral_integral)
     qed
   then have "\<exists>e>0. \<forall>\<epsilon>>0. \<epsilon><e \<longrightarrow> (f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
     unfolding residue_def c_def
     apply (rule_tac someI[of _ i],intro  exI[where x=e])
     by (auto simp add:\<open>e>0\<close> c_def)
   then obtain e' where "e'>0"
       and e'_def:"\<forall>\<epsilon>>0. \<epsilon><e' \<longrightarrow> (f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
     by auto
   let ?int="\<lambda>e. contour_integral (circlepath z e) f"
   define  \<epsilon> where "\<epsilon> \<equiv> Min {r,e'} / 2"
   have "\<epsilon>>0" "\<epsilon>\<le>r" "\<epsilon><e'" using \<open>r>0\<close> \<open>e'>0\<close> unfolding \<epsilon>_def by auto
   have "(f has_contour_integral c * (residue f z)) (circlepath z \<epsilon>)"
     using e'_def[rule_format,OF \<open>\<epsilon>>0\<close> \<open>\<epsilon><e'\<close>] .
   then show ?thesis unfolding c_def
     using contour_integral_circlepath_eq[OF \<open>open s\<close> f_holo \<open>\<epsilon>>0\<close> \<open>\<epsilon>\<le>r\<close> r_cball]
     by (auto elim: has_contour_integral_eqpath[of _ _ "circlepath z \<epsilon>" "circlepath z r"])
 qed
 
 lemma residue_holo:
   assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s"
   shows "residue f z = 0"
 proof -
   define c where "c \<equiv> 2 * pi * \<i>"
   obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
     using open_contains_cball_eq by blast
   have "(f has_contour_integral c*residue f z) (circlepath z e)"
     using f_holo
     by (auto intro: base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
   moreover have "(f has_contour_integral 0) (circlepath z e)"
     using f_holo e_cball \<open>e>0\<close>
     by (auto intro: Cauchy_theorem_convex_simple[of _ "cball z e"])
   ultimately have "c*residue f z =0"
     using has_contour_integral_unique by blast
   thus ?thesis unfolding c_def  by auto
 qed
 
 lemma residue_const:"residue (\<lambda>_. c) z = 0"
   by (intro residue_holo[of "UNIV::complex set"],auto intro:holomorphic_intros)
 
 lemma residue_add:
   assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
       and g_holo:"g holomorphic_on s - {z}"
   shows "residue (\<lambda>z. f z + g z) z= residue f z + residue g z"
 proof -
   define c where "c \<equiv> 2 * pi * \<i>"
   define fg where "fg \<equiv> (\<lambda>z. f z+g z)"
   obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
     using open_contains_cball_eq by blast
   have "(fg has_contour_integral c * residue fg z) (circlepath z e)"
     unfolding fg_def using f_holo g_holo
     apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
     by (auto intro:holomorphic_intros)
   moreover have "(fg has_contour_integral c*residue f z + c* residue g z) (circlepath z e)"
     unfolding fg_def using f_holo g_holo
     by (auto intro: has_contour_integral_add base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
   ultimately have "c*(residue f z + residue g z) = c * residue fg z"
     using has_contour_integral_unique by (auto simp add:distrib_left)
   thus ?thesis unfolding fg_def
     by (auto simp add:c_def)
 qed
 
 lemma residue_lmul:
   assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
   shows "residue (\<lambda>z. c * (f z)) z= c * residue f z"
 proof (cases "c=0")
   case True
   thus ?thesis using residue_const by auto
 next
   case False
   define c' where "c' \<equiv> 2 * pi * \<i>"
   define f' where "f' \<equiv> (\<lambda>z. c * (f z))"
   obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
     using open_contains_cball_eq by blast
   have "(f' has_contour_integral c' * residue f' z) (circlepath z e)"
     unfolding f'_def using f_holo
     apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c'_def])
     by (auto intro:holomorphic_intros)
   moreover have "(f' has_contour_integral c * (c' * residue f z)) (circlepath z e)"
     unfolding f'_def using f_holo
     by (auto intro: has_contour_integral_lmul
       base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c'_def])
   ultimately have "c' * residue f' z  = c * (c' * residue f z)"
     using has_contour_integral_unique by auto
   thus ?thesis unfolding f'_def c'_def using False
     by (auto simp add:field_simps)
 qed
 
 lemma residue_rmul:
   assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
   shows "residue (\<lambda>z. (f z) * c) z= residue f z * c"
 using residue_lmul[OF assms,of c] by (auto simp add:algebra_simps)
 
 lemma residue_div:
   assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
   shows "residue (\<lambda>z. (f z) / c) z= residue f z / c "
 using residue_lmul[OF assms,of "1/c"] by (auto simp add:algebra_simps)
 
 lemma residue_neg:
   assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
   shows "residue (\<lambda>z. - (f z)) z= - residue f z"
 using residue_lmul[OF assms,of "-1"] by auto
 
 lemma residue_diff:
   assumes "open s" "z \<in> s" and f_holo: "f holomorphic_on s - {z}"
       and g_holo:"g holomorphic_on s - {z}"
   shows "residue (\<lambda>z. f z - g z) z= residue f z - residue g z"
 using residue_add[OF assms(1,2,3),of "\<lambda>z. - g z"] residue_neg[OF assms(1,2,4)]
 by (auto intro:holomorphic_intros g_holo)
 
 lemma residue_simple:
   assumes "open s" "z\<in>s" and f_holo:"f holomorphic_on s"
   shows "residue (\<lambda>w. f w / (w - z)) z = f z"
 proof -
   define c where "c \<equiv> 2 * pi * \<i>"
   define f' where "f' \<equiv> \<lambda>w. f w / (w - z)"
   obtain e where "e>0" and e_cball:"cball z e \<subseteq> s" using \<open>open s\<close> \<open>z\<in>s\<close>
     using open_contains_cball_eq by blast
   have "(f' has_contour_integral c * f z) (circlepath z e)"
     unfolding f'_def c_def using \<open>e>0\<close> f_holo e_cball
     by (auto intro!: Cauchy_integral_circlepath_simple holomorphic_intros)
   moreover have "(f' has_contour_integral c * residue f' z) (circlepath z e)"
     unfolding f'_def using f_holo
     apply (intro base_residue[OF \<open>open s\<close> \<open>z\<in>s\<close> \<open>e>0\<close> _ e_cball,folded c_def])
     by (auto intro!:holomorphic_intros)
   ultimately have "c * f z = c * residue f' z"
     using has_contour_integral_unique by blast
   thus ?thesis unfolding c_def f'_def  by auto
 qed
 
 lemma residue_simple':
   assumes s: "open s" "z \<in> s" and holo: "f holomorphic_on (s - {z})"
       and lim: "((\<lambda>w. f w * (w - z)) \<longlongrightarrow> c) (at z)"
   shows   "residue f z = c"
 proof -
   define g where "g = (\<lambda>w. if w = z then c else f w * (w - z))"
   from holo have "(\<lambda>w. f w * (w - z)) holomorphic_on (s - {z})" (is "?P")
     by (force intro: holomorphic_intros)
   also have "?P \<longleftrightarrow> g holomorphic_on (s - {z})"
     by (intro holomorphic_cong refl) (simp_all add: g_def)
   finally have *: "g holomorphic_on (s - {z})" .
 
   note lim
   also have "(\<lambda>w. f w * (w - z)) \<midarrow>z\<rightarrow> c \<longleftrightarrow> g \<midarrow>z\<rightarrow> g z"
     by (intro filterlim_cong refl) (simp_all add: g_def [abs_def] eventually_at_filter)
   finally have **: "g \<midarrow>z\<rightarrow> g z" .
 
   have g_holo: "g holomorphic_on s"
     by (rule no_isolated_singularity'[where K = "{z}"])
        (insert assms * **, simp_all add: at_within_open_NO_MATCH)
   from s and this have "residue (\<lambda>w. g w / (w - z)) z = g z"
     by (rule residue_simple)
   also have "\<forall>\<^sub>F za in at z. g za / (za - z) = f za"
     unfolding eventually_at by (auto intro!: exI[of _ 1] simp: field_simps g_def)
   hence "residue (\<lambda>w. g w / (w - z)) z = residue f z"
     by (intro residue_cong refl)
   finally show ?thesis
     by (simp add: g_def)
 qed
 
 lemma residue_holomorphic_over_power:
   assumes "open A" "z0 \<in> A" "f holomorphic_on A"
   shows   "residue (\<lambda>z. f z / (z - z0) ^ Suc n) z0 = (deriv ^^ n) f z0 / fact n"
 proof -
   let ?f = "\<lambda>z. f z / (z - z0) ^ Suc n"
   from assms(1,2) obtain r where r: "r > 0" "cball z0 r \<subseteq> A"
     by (auto simp: open_contains_cball)
   have "(?f has_contour_integral 2 * pi * \<i> * residue ?f z0) (circlepath z0 r)"
     using r assms by (intro base_residue[of A]) (auto intro!: holomorphic_intros)
   moreover have "(?f has_contour_integral 2 * pi * \<i> / fact n * (deriv ^^ n) f z0) (circlepath z0 r)"
     using assms r
     by (intro Cauchy_has_contour_integral_higher_derivative_circlepath)
        (auto intro!: holomorphic_on_subset[OF assms(3)] holomorphic_on_imp_continuous_on)
   ultimately have "2 * pi * \<i> * residue ?f z0 = 2 * pi * \<i> / fact n * (deriv ^^ n) f z0"
     by (rule has_contour_integral_unique)
   thus ?thesis by (simp add: field_simps)
 qed
 
 lemma residue_holomorphic_over_power':
   assumes "open A" "0 \<in> A" "f holomorphic_on A"
   shows   "residue (\<lambda>z. f z / z ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
   using residue_holomorphic_over_power[OF assms] by simp
 
 theorem residue_fps_expansion_over_power_at_0:
   assumes "f has_fps_expansion F"
   shows   "residue (\<lambda>z. f z / z ^ Suc n) 0 = fps_nth F n"
 proof -
   from has_fps_expansion_imp_holomorphic[OF assms] guess s . note s = this
   have "residue (\<lambda>z. f z / (z - 0) ^ Suc n) 0 = (deriv ^^ n) f 0 / fact n"
     using assms s unfolding has_fps_expansion_def
     by (intro residue_holomorphic_over_power[of s]) (auto simp: zero_ereal_def)
   also from assms have "\<dots> = fps_nth F n"
     by (subst fps_nth_fps_expansion) auto
   finally show ?thesis by simp
 qed
 
 lemma residue_pole_order:
   fixes f::"complex \<Rightarrow> complex" and z::complex
   defines "n \<equiv> nat (- zorder f z)" and "h \<equiv> zor_poly f z"
   assumes f_iso:"isolated_singularity_at f z"
     and pole:"is_pole f z"
   shows "residue f z = ((deriv ^^ (n - 1)) h z / fact (n-1))"
 proof -
   define g where "g \<equiv> \<lambda>x. if x=z then 0 else inverse (f x)"
   obtain e where [simp]:"e>0" and f_holo:"f holomorphic_on ball z e - {z}"
     using f_iso analytic_imp_holomorphic unfolding isolated_singularity_at_def by blast
   obtain r where "0 < n" "0 < r" and r_cball:"cball z r \<subseteq> ball z e" and h_holo: "h holomorphic_on cball z r"
       and h_divide:"(\<forall>w\<in>cball z r. (w\<noteq>z \<longrightarrow> f w = h w / (w - z) ^ n) \<and> h w \<noteq> 0)"
   proof -
     obtain r where r:"zorder f z < 0" "h z \<noteq> 0" "r>0" "cball z r \<subseteq> ball z e" "h holomorphic_on cball z r"
         "(\<forall>w\<in>cball z r - {z}. f w = h w / (w - z) ^ n \<and> h w \<noteq> 0)"
       using zorder_exist_pole[OF f_holo,simplified,OF \<open>is_pole f z\<close>,folded n_def h_def] by auto
     have "n>0" using \<open>zorder f z < 0\<close> unfolding n_def by simp
     moreover have "(\<forall>w\<in>cball z r. (w\<noteq>z \<longrightarrow> f w = h w / (w - z) ^ n) \<and> h w \<noteq> 0)"
       using \<open>h z\<noteq>0\<close> r(6) by blast
     ultimately show ?thesis using r(3,4,5) that by blast
   qed
   have r_nonzero:"\<And>w. w \<in> ball z r - {z} \<Longrightarrow> f w \<noteq> 0"
     using h_divide by simp
   define c where "c \<equiv> 2 * pi * \<i>"
   define der_f where "der_f \<equiv> ((deriv ^^ (n - 1)) h z / fact (n-1))"
   define h' where "h' \<equiv> \<lambda>u. h u / (u - z) ^ n"
   have "(h' has_contour_integral c / fact (n - 1) * (deriv ^^ (n - 1)) h z) (circlepath z r)"
     unfolding h'_def
     proof (rule Cauchy_has_contour_integral_higher_derivative_circlepath[of z r h z "n-1",
         folded c_def Suc_pred'[OF \<open>n>0\<close>]])
       show "continuous_on (cball z r) h" using holomorphic_on_imp_continuous_on h_holo by simp
       show "h holomorphic_on ball z r" using h_holo by auto
       show " z \<in> ball z r" using \<open>r>0\<close> by auto
     qed
   then have "(h' has_contour_integral c * der_f) (circlepath z r)" unfolding der_f_def by auto
   then have "(f has_contour_integral c * der_f) (circlepath z r)"
     proof (elim has_contour_integral_eq)
       fix x assume "x \<in> path_image (circlepath z r)"
       hence "x\<in>cball z r - {z}" using \<open>r>0\<close> by auto
       then show "h' x = f x" using h_divide unfolding h'_def by auto
     qed
   moreover have "(f has_contour_integral c * residue f z) (circlepath z r)"
     using base_residue[of \<open>ball z e\<close> z,simplified,OF \<open>r>0\<close> f_holo r_cball,folded c_def]
     unfolding c_def by simp
   ultimately have "c * der_f =  c * residue f z" using has_contour_integral_unique by blast
   hence "der_f = residue f z" unfolding c_def by auto
   thus ?thesis unfolding der_f_def by auto
 qed
 
 lemma residue_simple_pole:
   assumes "isolated_singularity_at f z0"
   assumes "is_pole f z0" "zorder f z0 = - 1"
   shows   "residue f z0 = zor_poly f z0 z0"
   using assms by (subst residue_pole_order) simp_all
 
 lemma residue_simple_pole_limit:
   assumes "isolated_singularity_at f z0"
   assumes "is_pole f z0" "zorder f z0 = - 1"
   assumes "((\<lambda>x. f (g x) * (g x - z0)) \<longlongrightarrow> c) F"
   assumes "filterlim g (at z0) F" "F \<noteq> bot"
   shows   "residue f z0 = c"
 proof -
   have "residue f z0 = zor_poly f z0 z0"
     by (rule residue_simple_pole assms)+
   also have "\<dots> = c"
     apply (rule zor_poly_pole_eqI)
     using assms by auto
   finally show ?thesis .
 qed
 
 lemma
   assumes f_holo:"f holomorphic_on s" and g_holo:"g holomorphic_on s"
           and "open s" "connected s" "z \<in> s"
   assumes g_deriv:"(g has_field_derivative g') (at z)"
   assumes "f z \<noteq> 0" "g z = 0" "g' \<noteq> 0"
   shows   porder_simple_pole_deriv: "zorder (\<lambda>w. f w / g w) z = - 1"
     and   residue_simple_pole_deriv: "residue (\<lambda>w. f w / g w) z = f z / g'"
 proof -
   have [simp]:"isolated_singularity_at f z" "isolated_singularity_at g z"
     using isolated_singularity_at_holomorphic[OF _ \<open>open s\<close> \<open>z\<in>s\<close>] f_holo g_holo
     by (meson Diff_subset holomorphic_on_subset)+
   have [simp]:"not_essential f z" "not_essential g z"
     unfolding not_essential_def using f_holo g_holo assms(3,5)
     by (meson continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on)+
   have g_nconst:"\<exists>\<^sub>F w in at z. g w \<noteq>0 "
   proof (rule ccontr)
     assume "\<not> (\<exists>\<^sub>F w in at z. g w \<noteq> 0)"
     then have "\<forall>\<^sub>F w in nhds z. g w = 0"
       unfolding eventually_at eventually_nhds frequently_at using \<open>g z = 0\<close>
       by (metis open_ball UNIV_I centre_in_ball dist_commute mem_ball)
     then have "deriv g z = deriv (\<lambda>_. 0) z"
       by (intro deriv_cong_ev) auto
     then have "deriv g z = 0" by auto
     then have "g' = 0" using g_deriv DERIV_imp_deriv by blast
     then show False using \<open>g'\<noteq>0\<close> by auto
   qed
 
   have "zorder (\<lambda>w. f w / g w) z = zorder f z - zorder g z"
   proof -
     have "\<forall>\<^sub>F w in at z. f w \<noteq>0 \<and> w\<in>s"
       apply (rule non_zero_neighbour_alt)
       using assms by auto
     with g_nconst have "\<exists>\<^sub>F w in at z. f w * g w \<noteq> 0"
       by (elim frequently_rev_mp eventually_rev_mp,auto)
     then show ?thesis using zorder_divide[of f z g] by auto
   qed
   moreover have "zorder f z=0"
     apply (rule zorder_zero_eqI[OF f_holo \<open>open s\<close> \<open>z\<in>s\<close>])
     using \<open>f z\<noteq>0\<close> by auto
   moreover have "zorder g z=1"
     apply (rule zorder_zero_eqI[OF g_holo \<open>open s\<close> \<open>z\<in>s\<close>])
     subgoal using assms(8) by auto
     subgoal using DERIV_imp_deriv assms(9) g_deriv by auto
     subgoal by simp
     done
   ultimately show "zorder (\<lambda>w. f w / g w) z = - 1" by auto
 
   show "residue (\<lambda>w. f w / g w) z = f z / g'"
   proof (rule residue_simple_pole_limit[where g=id and F="at z",simplified])
     show "zorder (\<lambda>w. f w / g w) z = - 1" by fact
     show "isolated_singularity_at (\<lambda>w. f w / g w) z"
       by (auto intro: singularity_intros)
     show "is_pole (\<lambda>w. f w / g w) z"
     proof (rule is_pole_divide)
       have "\<forall>\<^sub>F x in at z. g x \<noteq> 0"
         apply (rule non_zero_neighbour)
         using g_nconst by auto
       moreover have "g \<midarrow>z\<rightarrow> 0"
         using DERIV_isCont assms(8) continuous_at g_deriv by force
       ultimately show "filterlim g (at 0) (at z)" unfolding filterlim_at by simp
       show "isCont f z"
         using assms(3,5) continuous_on_eq_continuous_at f_holo holomorphic_on_imp_continuous_on
         by auto
       show "f z \<noteq> 0" by fact
     qed
     show "filterlim id (at z) (at z)" by (simp add: filterlim_iff)
     have "((\<lambda>w. (f w * (w - z)) / g w) \<longlongrightarrow> f z / g') (at z)"
     proof (rule lhopital_complex_simple)
       show "((\<lambda>w. f w * (w - z)) has_field_derivative f z) (at z)"
         using assms by (auto intro!: derivative_eq_intros holomorphic_derivI[OF f_holo])
       show "(g has_field_derivative g') (at z)" by fact
     qed (insert assms, auto)
     then show "((\<lambda>w. (f w / g w) * (w - z)) \<longlongrightarrow> f z / g') (at z)"
       by (simp add: field_split_simps)
   qed
 qed
 
 
 subsection \<open>Poles and residues of some well-known functions\<close>
 
 (* TODO: add more material here for other functions *)
 lemma is_pole_Gamma: "is_pole Gamma (-of_nat n)"
   unfolding is_pole_def using Gamma_poles .
 
 lemma Gamma_residue:
   "residue Gamma (-of_nat n) = (-1) ^ n / fact n"
 proof (rule residue_simple')
   show "open (- (\<int>\<^sub>\<le>\<^sub>0 - {-of_nat n}) :: complex set)"
     by (intro open_Compl closed_subset_Ints) auto
   show "Gamma holomorphic_on (- (\<int>\<^sub>\<le>\<^sub>0 - {-of_nat n}) - {- of_nat n})"
     by (rule holomorphic_Gamma) auto
   show "(\<lambda>w. Gamma w * (w - (-of_nat n))) \<midarrow>(-of_nat n)\<rightarrow> (- 1) ^ n / fact n"
     using Gamma_residues[of n] by simp
 qed auto
 
-end
\ No newline at end of file
+end
diff --git a/src/HOL/Nonstandard_Analysis/NSComplex.thy b/src/HOL/Nonstandard_Analysis/NSComplex.thy
--- a/src/HOL/Nonstandard_Analysis/NSComplex.thy
+++ b/src/HOL/Nonstandard_Analysis/NSComplex.thy
@@ -1,603 +1,603 @@
 (*  Title:      HOL/Nonstandard_Analysis/NSComplex.thy
     Author:     Jacques D. Fleuriot, University of Edinburgh
     Author:     Lawrence C Paulson
 *)
 
 section \<open>Nonstandard Complex Numbers\<close>
 
 theory NSComplex
   imports NSA
 begin
 
 type_synonym hcomplex = "complex star"
 
 abbreviation hcomplex_of_complex :: "complex \<Rightarrow> complex star"
   where "hcomplex_of_complex \<equiv> star_of"
 
 abbreviation hcmod :: "complex star \<Rightarrow> real star"
   where "hcmod \<equiv> hnorm"
 
 
 subsubsection \<open>Real and Imaginary parts\<close>
 
 definition hRe :: "hcomplex \<Rightarrow> hypreal"
   where "hRe = *f* Re"
 
 definition hIm :: "hcomplex \<Rightarrow> hypreal"
   where "hIm = *f* Im"
 
 
 subsubsection \<open>Imaginary unit\<close>
 
 definition iii :: hcomplex
   where "iii = star_of \<i>"
 
 
 subsubsection \<open>Complex conjugate\<close>
 
 definition hcnj :: "hcomplex \<Rightarrow> hcomplex"
   where "hcnj = *f* cnj"
 
 
 subsubsection \<open>Argand\<close>
 
 definition hsgn :: "hcomplex \<Rightarrow> hcomplex"
   where "hsgn = *f* sgn"
 
 definition harg :: "hcomplex \<Rightarrow> hypreal"
-  where "harg = *f* arg"
+  where "harg = *f* Arg"
 
 definition  \<comment> \<open>abbreviation for \<open>cos a + i sin a\<close>\<close>
   hcis :: "hypreal \<Rightarrow> hcomplex"
   where "hcis = *f* cis"
 
 
 subsubsection \<open>Injection from hyperreals\<close>
 
 abbreviation hcomplex_of_hypreal :: "hypreal \<Rightarrow> hcomplex"
   where "hcomplex_of_hypreal \<equiv> of_hypreal"
 
 definition  \<comment> \<open>abbreviation for \<open>r * (cos a + i sin a)\<close>\<close>
   hrcis :: "hypreal \<Rightarrow> hypreal \<Rightarrow> hcomplex"
   where "hrcis = *f2* rcis"
 
 
 subsubsection \<open>\<open>e ^ (x + iy)\<close>\<close>
 
 definition hExp :: "hcomplex \<Rightarrow> hcomplex"
   where "hExp = *f* exp"
 
 definition HComplex :: "hypreal \<Rightarrow> hypreal \<Rightarrow> hcomplex"
   where "HComplex = *f2* Complex"
 
 lemmas hcomplex_defs [transfer_unfold] =
   hRe_def hIm_def iii_def hcnj_def hsgn_def harg_def hcis_def
   hrcis_def hExp_def HComplex_def
 
 lemma Standard_hRe [simp]: "x \<in> Standard \<Longrightarrow> hRe x \<in> Standard"
   by (simp add: hcomplex_defs)
 
 lemma Standard_hIm [simp]: "x \<in> Standard \<Longrightarrow> hIm x \<in> Standard"
   by (simp add: hcomplex_defs)
 
 lemma Standard_iii [simp]: "iii \<in> Standard"
   by (simp add: hcomplex_defs)
 
 lemma Standard_hcnj [simp]: "x \<in> Standard \<Longrightarrow> hcnj x \<in> Standard"
   by (simp add: hcomplex_defs)
 
 lemma Standard_hsgn [simp]: "x \<in> Standard \<Longrightarrow> hsgn x \<in> Standard"
   by (simp add: hcomplex_defs)
 
 lemma Standard_harg [simp]: "x \<in> Standard \<Longrightarrow> harg x \<in> Standard"
   by (simp add: hcomplex_defs)
 
 lemma Standard_hcis [simp]: "r \<in> Standard \<Longrightarrow> hcis r \<in> Standard"
   by (simp add: hcomplex_defs)
 
 lemma Standard_hExp [simp]: "x \<in> Standard \<Longrightarrow> hExp x \<in> Standard"
   by (simp add: hcomplex_defs)
 
 lemma Standard_hrcis [simp]: "r \<in> Standard \<Longrightarrow> s \<in> Standard \<Longrightarrow> hrcis r s \<in> Standard"
   by (simp add: hcomplex_defs)
 
 lemma Standard_HComplex [simp]: "r \<in> Standard \<Longrightarrow> s \<in> Standard \<Longrightarrow> HComplex r s \<in> Standard"
   by (simp add: hcomplex_defs)
 
 lemma hcmod_def: "hcmod = *f* cmod"
   by (rule hnorm_def)
 
 
 subsection \<open>Properties of Nonstandard Real and Imaginary Parts\<close>
 
 lemma hcomplex_hRe_hIm_cancel_iff: "\<And>w z. w = z \<longleftrightarrow> hRe w = hRe z \<and> hIm w = hIm z"
   by transfer (rule complex_eq_iff)
 
 lemma hcomplex_equality [intro?]: "\<And>z w. hRe z = hRe w \<Longrightarrow> hIm z = hIm w \<Longrightarrow> z = w"
   by transfer (rule complex_eqI)
 
 lemma hcomplex_hRe_zero [simp]: "hRe 0 = 0"
   by transfer simp
 
 lemma hcomplex_hIm_zero [simp]: "hIm 0 = 0"
   by transfer simp
 
 lemma hcomplex_hRe_one [simp]: "hRe 1 = 1"
   by transfer simp
 
 lemma hcomplex_hIm_one [simp]: "hIm 1 = 0"
   by transfer simp
 
 
 subsection \<open>Addition for Nonstandard Complex Numbers\<close>
 
 lemma hRe_add: "\<And>x y. hRe (x + y) = hRe x + hRe y"
   by transfer simp
 
 lemma hIm_add: "\<And>x y. hIm (x + y) = hIm x + hIm y"
   by transfer simp
 
 
 subsection \<open>More Minus Laws\<close>
 
 lemma hRe_minus: "\<And>z. hRe (- z) = - hRe z"
   by transfer (rule uminus_complex.sel)
 
 lemma hIm_minus: "\<And>z. hIm (- z) = - hIm z"
   by transfer (rule uminus_complex.sel)
 
 lemma hcomplex_add_minus_eq_minus: "x + y = 0 \<Longrightarrow> x = - y"
   for x y :: hcomplex
   apply (drule minus_unique)
   apply (simp add: minus_equation_iff [of x y])
   done
 
 lemma hcomplex_i_mult_eq [simp]: "iii * iii = - 1"
   by transfer (rule i_squared)
 
 lemma hcomplex_i_mult_left [simp]: "\<And>z. iii * (iii * z) = - z"
   by transfer (rule complex_i_mult_minus)
 
 lemma hcomplex_i_not_zero [simp]: "iii \<noteq> 0"
   by transfer (rule complex_i_not_zero)
 
 
 subsection \<open>More Multiplication Laws\<close>
 
 lemma hcomplex_mult_minus_one: "- 1 * z = - z"
   for z :: hcomplex
   by simp
 
 lemma hcomplex_mult_minus_one_right: "z * - 1 = - z"
   for z :: hcomplex
   by simp
 
 lemma hcomplex_mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
   for a b c :: hcomplex
   by simp
 
 lemma hcomplex_mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
   for a b c :: hcomplex
   by simp
 
 
 subsection \<open>Subtraction and Division\<close>
 
 (* TODO: delete *)
 lemma hcomplex_diff_eq_eq [simp]: "x - y = z \<longleftrightarrow> x = z + y"
   for x y z :: hcomplex
   by (rule diff_eq_eq)
 
 
 subsection \<open>Embedding Properties for \<^term>\<open>hcomplex_of_hypreal\<close> Map\<close>
 
 lemma hRe_hcomplex_of_hypreal [simp]: "\<And>z. hRe (hcomplex_of_hypreal z) = z"
   by transfer (rule Re_complex_of_real)
 
 lemma hIm_hcomplex_of_hypreal [simp]: "\<And>z. hIm (hcomplex_of_hypreal z) = 0"
   by transfer (rule Im_complex_of_real)
 
 lemma hcomplex_of_epsilon_not_zero [simp]: "hcomplex_of_hypreal \<epsilon> \<noteq> 0"
   by (simp add: epsilon_not_zero)
 
 
 subsection \<open>\<open>HComplex\<close> theorems\<close>
 
 lemma hRe_HComplex [simp]: "\<And>x y. hRe (HComplex x y) = x"
   by transfer simp
 
 lemma hIm_HComplex [simp]: "\<And>x y. hIm (HComplex x y) = y"
   by transfer simp
 
 lemma hcomplex_surj [simp]: "\<And>z. HComplex (hRe z) (hIm z) = z"
   by transfer (rule complex_surj)
 
 lemma hcomplex_induct [case_names rect(*, induct type: hcomplex*)]:
   "(\<And>x y. P (HComplex x y)) \<Longrightarrow> P z"
   by (rule hcomplex_surj [THEN subst]) blast
 
 
 subsection \<open>Modulus (Absolute Value) of Nonstandard Complex Number\<close>
 
 lemma hcomplex_of_hypreal_abs:
   "hcomplex_of_hypreal \<bar>x\<bar> = hcomplex_of_hypreal (hcmod (hcomplex_of_hypreal x))"
   by simp
 
 lemma HComplex_inject [simp]: "\<And>x y x' y'. HComplex x y = HComplex x' y' \<longleftrightarrow> x = x' \<and> y = y'"
   by transfer (rule complex.inject)
 
 lemma HComplex_add [simp]:
   "\<And>x1 y1 x2 y2. HComplex x1 y1 + HComplex x2 y2 = HComplex (x1 + x2) (y1 + y2)"
   by transfer (rule complex_add)
 
 lemma HComplex_minus [simp]: "\<And>x y. - HComplex x y = HComplex (- x) (- y)"
   by transfer (rule complex_minus)
 
 lemma HComplex_diff [simp]:
   "\<And>x1 y1 x2 y2. HComplex x1 y1 - HComplex x2 y2 = HComplex (x1 - x2) (y1 - y2)"
   by transfer (rule complex_diff)
 
 lemma HComplex_mult [simp]:
   "\<And>x1 y1 x2 y2. HComplex x1 y1 * HComplex x2 y2 = HComplex (x1*x2 - y1*y2) (x1*y2 + y1*x2)"
   by transfer (rule complex_mult)
 
 text \<open>\<open>HComplex_inverse\<close> is proved below.\<close>
 
 lemma hcomplex_of_hypreal_eq: "\<And>r. hcomplex_of_hypreal r = HComplex r 0"
   by transfer (rule complex_of_real_def)
 
 lemma HComplex_add_hcomplex_of_hypreal [simp]:
   "\<And>x y r. HComplex x y + hcomplex_of_hypreal r = HComplex (x + r) y"
   by transfer (rule Complex_add_complex_of_real)
 
 lemma hcomplex_of_hypreal_add_HComplex [simp]:
   "\<And>r x y. hcomplex_of_hypreal r + HComplex x y = HComplex (r + x) y"
   by transfer (rule complex_of_real_add_Complex)
 
 lemma HComplex_mult_hcomplex_of_hypreal:
   "\<And>x y r. HComplex x y * hcomplex_of_hypreal r = HComplex (x * r) (y * r)"
   by transfer (rule Complex_mult_complex_of_real)
 
 lemma hcomplex_of_hypreal_mult_HComplex:
   "\<And>r x y. hcomplex_of_hypreal r * HComplex x y = HComplex (r * x) (r * y)"
   by transfer (rule complex_of_real_mult_Complex)
 
 lemma i_hcomplex_of_hypreal [simp]: "\<And>r. iii * hcomplex_of_hypreal r = HComplex 0 r"
   by transfer (rule i_complex_of_real)
 
 lemma hcomplex_of_hypreal_i [simp]: "\<And>r. hcomplex_of_hypreal r * iii = HComplex 0 r"
   by transfer (rule complex_of_real_i)
 
 
 subsection \<open>Conjugation\<close>
 
 lemma hcomplex_hcnj_cancel_iff [iff]: "\<And>x y. hcnj x = hcnj y \<longleftrightarrow> x = y"
   by transfer (rule complex_cnj_cancel_iff)
 
 lemma hcomplex_hcnj_hcnj [simp]: "\<And>z. hcnj (hcnj z) = z"
   by transfer (rule complex_cnj_cnj)
 
 lemma hcomplex_hcnj_hcomplex_of_hypreal [simp]:
   "\<And>x. hcnj (hcomplex_of_hypreal x) = hcomplex_of_hypreal x"
   by transfer (rule complex_cnj_complex_of_real)
 
 lemma hcomplex_hmod_hcnj [simp]: "\<And>z. hcmod (hcnj z) = hcmod z"
   by transfer (rule complex_mod_cnj)
 
 lemma hcomplex_hcnj_minus: "\<And>z. hcnj (- z) = - hcnj z"
   by transfer (rule complex_cnj_minus)
 
 lemma hcomplex_hcnj_inverse: "\<And>z. hcnj (inverse z) = inverse (hcnj z)"
   by transfer (rule complex_cnj_inverse)
 
 lemma hcomplex_hcnj_add: "\<And>w z. hcnj (w + z) = hcnj w + hcnj z"
   by transfer (rule complex_cnj_add)
 
 lemma hcomplex_hcnj_diff: "\<And>w z. hcnj (w - z) = hcnj w - hcnj z"
   by transfer (rule complex_cnj_diff)
 
 lemma hcomplex_hcnj_mult: "\<And>w z. hcnj (w * z) = hcnj w * hcnj z"
   by transfer (rule complex_cnj_mult)
 
 lemma hcomplex_hcnj_divide: "\<And>w z. hcnj (w / z) = hcnj w / hcnj z"
   by transfer (rule complex_cnj_divide)
 
 lemma hcnj_one [simp]: "hcnj 1 = 1"
   by transfer (rule complex_cnj_one)
 
 lemma hcomplex_hcnj_zero [simp]: "hcnj 0 = 0"
   by transfer (rule complex_cnj_zero)
 
 lemma hcomplex_hcnj_zero_iff [iff]: "\<And>z. hcnj z = 0 \<longleftrightarrow> z = 0"
   by transfer (rule complex_cnj_zero_iff)
 
 lemma hcomplex_mult_hcnj: "\<And>z. z * hcnj z = hcomplex_of_hypreal ((hRe z)\<^sup>2 + (hIm z)\<^sup>2)"
   by transfer (rule complex_mult_cnj)
 
 
 subsection \<open>More Theorems about the Function \<^term>\<open>hcmod\<close>\<close>
 
 lemma hcmod_hcomplex_of_hypreal_of_nat [simp]:
   "hcmod (hcomplex_of_hypreal (hypreal_of_nat n)) = hypreal_of_nat n"
   by simp
 
 lemma hcmod_hcomplex_of_hypreal_of_hypnat [simp]:
   "hcmod (hcomplex_of_hypreal(hypreal_of_hypnat n)) = hypreal_of_hypnat n"
   by simp
 
 lemma hcmod_mult_hcnj: "\<And>z. hcmod (z * hcnj z) = (hcmod z)\<^sup>2"
   by transfer (rule complex_mod_mult_cnj)
 
 lemma hcmod_triangle_ineq2 [simp]: "\<And>a b. hcmod (b + a) - hcmod b \<le> hcmod a"
   by transfer (rule complex_mod_triangle_ineq2)
 
 lemma hcmod_diff_ineq [simp]: "\<And>a b. hcmod a - hcmod b \<le> hcmod (a + b)"
   by transfer (rule norm_diff_ineq)
 
 
 subsection \<open>Exponentiation\<close>
 
 lemma hcomplexpow_0 [simp]: "z ^ 0 = 1"
   for z :: hcomplex
   by (rule power_0)
 
 lemma hcomplexpow_Suc [simp]: "z ^ (Suc n) = z * (z ^ n)"
   for z :: hcomplex
   by (rule power_Suc)
 
 lemma hcomplexpow_i_squared [simp]: "iii\<^sup>2 = -1"
   by transfer (rule power2_i)
 
 lemma hcomplex_of_hypreal_pow: "\<And>x. hcomplex_of_hypreal (x ^ n) = hcomplex_of_hypreal x ^ n"
   by transfer (rule of_real_power)
 
 lemma hcomplex_hcnj_pow: "\<And>z. hcnj (z ^ n) = hcnj z ^ n"
   by transfer (rule complex_cnj_power)
 
 lemma hcmod_hcomplexpow: "\<And>x. hcmod (x ^ n) = hcmod x ^ n"
   by transfer (rule norm_power)
 
 lemma hcpow_minus:
   "\<And>x n. (- x :: hcomplex) pow n = (if ( *p* even) n then (x pow n) else - (x pow n))"
   by transfer simp
 
 lemma hcpow_mult: "(r * s) pow n = (r pow n) * (s pow n)"
   for r s :: hcomplex
   by (fact hyperpow_mult)
 
 lemma hcpow_zero2 [simp]: "\<And>n. 0 pow (hSuc n) = (0::'a::semiring_1 star)"
   by transfer (rule power_0_Suc)
 
 lemma hcpow_not_zero [simp,intro]: "\<And>r n. r \<noteq> 0 \<Longrightarrow> r pow n \<noteq> (0::hcomplex)"
   by (fact hyperpow_not_zero)
 
 lemma hcpow_zero_zero: "r pow n = 0 \<Longrightarrow> r = 0"
   for r :: hcomplex
   by (blast intro: ccontr dest: hcpow_not_zero)
 
 
 subsection \<open>The Function \<^term>\<open>hsgn\<close>\<close>
 
 lemma hsgn_zero [simp]: "hsgn 0 = 0"
   by transfer (rule sgn_zero)
 
 lemma hsgn_one [simp]: "hsgn 1 = 1"
   by transfer (rule sgn_one)
 
 lemma hsgn_minus: "\<And>z. hsgn (- z) = - hsgn z"
   by transfer (rule sgn_minus)
 
 lemma hsgn_eq: "\<And>z. hsgn z = z / hcomplex_of_hypreal (hcmod z)"
   by transfer (rule sgn_eq)
 
 lemma hcmod_i: "\<And>x y. hcmod (HComplex x y) = ( *f* sqrt) (x\<^sup>2 + y\<^sup>2)"
   by transfer (rule complex_norm)
 
 lemma hcomplex_eq_cancel_iff1 [simp]:
   "hcomplex_of_hypreal xa = HComplex x y \<longleftrightarrow> xa = x \<and> y = 0"
   by (simp add: hcomplex_of_hypreal_eq)
 
 lemma hcomplex_eq_cancel_iff2 [simp]:
   "HComplex x y = hcomplex_of_hypreal xa \<longleftrightarrow> x = xa \<and> y = 0"
   by (simp add: hcomplex_of_hypreal_eq)
 
 lemma HComplex_eq_0 [simp]: "\<And>x y. HComplex x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
   by transfer (rule Complex_eq_0)
 
 lemma HComplex_eq_1 [simp]: "\<And>x y. HComplex x y = 1 \<longleftrightarrow> x = 1 \<and> y = 0"
   by transfer (rule Complex_eq_1)
 
 lemma i_eq_HComplex_0_1: "iii = HComplex 0 1"
   by transfer (simp add: complex_eq_iff)
 
 lemma HComplex_eq_i [simp]: "\<And>x y. HComplex x y = iii \<longleftrightarrow> x = 0 \<and> y = 1"
   by transfer (rule Complex_eq_i)
 
 lemma hRe_hsgn [simp]: "\<And>z. hRe (hsgn z) = hRe z / hcmod z"
   by transfer (rule Re_sgn)
 
 lemma hIm_hsgn [simp]: "\<And>z. hIm (hsgn z) = hIm z / hcmod z"
   by transfer (rule Im_sgn)
 
 lemma HComplex_inverse: "\<And>x y. inverse (HComplex x y) = HComplex (x / (x\<^sup>2 + y\<^sup>2)) (- y / (x\<^sup>2 + y\<^sup>2))"
   by transfer (rule complex_inverse)
 
 lemma hRe_mult_i_eq[simp]: "\<And>y. hRe (iii * hcomplex_of_hypreal y) = 0"
   by transfer simp
 
 lemma hIm_mult_i_eq [simp]: "\<And>y. hIm (iii * hcomplex_of_hypreal y) = y"
   by transfer simp
 
 lemma hcmod_mult_i [simp]: "\<And>y. hcmod (iii * hcomplex_of_hypreal y) = \<bar>y\<bar>"
   by transfer (simp add: norm_complex_def)
 
 lemma hcmod_mult_i2 [simp]: "\<And>y. hcmod (hcomplex_of_hypreal y * iii) = \<bar>y\<bar>"
   by transfer (simp add: norm_complex_def)
 
 
 subsubsection \<open>\<open>harg\<close>\<close>
 
 lemma cos_harg_i_mult_zero [simp]: "\<And>y. y \<noteq> 0 \<Longrightarrow> ( *f* cos) (harg (HComplex 0 y)) = 0"
   by transfer (simp add: Complex_eq)
 
 
 subsection \<open>Polar Form for Nonstandard Complex Numbers\<close>
 
 lemma complex_split_polar2: "\<forall>n. \<exists>r a. (z n) = complex_of_real r * Complex (cos a) (sin a)"
   unfolding Complex_eq by (auto intro: complex_split_polar)
 
 lemma hcomplex_split_polar:
   "\<And>z. \<exists>r a. z = hcomplex_of_hypreal r * (HComplex (( *f* cos) a) (( *f* sin) a))"
   by transfer (simp add: Complex_eq complex_split_polar)
 
 lemma hcis_eq:
   "\<And>a. hcis a = hcomplex_of_hypreal (( *f* cos) a) + iii * hcomplex_of_hypreal (( *f* sin) a)"
   by transfer (simp add: complex_eq_iff)
 
 lemma hrcis_Ex: "\<And>z. \<exists>r a. z = hrcis r a"
   by transfer (rule rcis_Ex)
 
 lemma hRe_hcomplex_polar [simp]:
   "\<And>r a. hRe (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = r * ( *f* cos) a"
   by transfer simp
 
 lemma hRe_hrcis [simp]: "\<And>r a. hRe (hrcis r a) = r * ( *f* cos) a"
   by transfer (rule Re_rcis)
 
 lemma hIm_hcomplex_polar [simp]:
   "\<And>r a. hIm (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = r * ( *f* sin) a"
   by transfer simp
 
 lemma hIm_hrcis [simp]: "\<And>r a. hIm (hrcis r a) = r * ( *f* sin) a"
   by transfer (rule Im_rcis)
 
 lemma hcmod_unit_one [simp]: "\<And>a. hcmod (HComplex (( *f* cos) a) (( *f* sin) a)) = 1"
   by transfer (simp add: cmod_unit_one)
 
 lemma hcmod_complex_polar [simp]:
   "\<And>r a. hcmod (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = \<bar>r\<bar>"
   by transfer (simp add: Complex_eq cmod_complex_polar)
 
 lemma hcmod_hrcis [simp]: "\<And>r a. hcmod(hrcis r a) = \<bar>r\<bar>"
   by transfer (rule complex_mod_rcis)
 
 text \<open>\<open>(r1 * hrcis a) * (r2 * hrcis b) = r1 * r2 * hrcis (a + b)\<close>\<close>
 
 lemma hcis_hrcis_eq: "\<And>a. hcis a = hrcis 1 a"
   by transfer (rule cis_rcis_eq)
 declare hcis_hrcis_eq [symmetric, simp]
 
 lemma hrcis_mult: "\<And>a b r1 r2. hrcis r1 a * hrcis r2 b = hrcis (r1 * r2) (a + b)"
   by transfer (rule rcis_mult)
 
 lemma hcis_mult: "\<And>a b. hcis a * hcis b = hcis (a + b)"
   by transfer (rule cis_mult)
 
 lemma hcis_zero [simp]: "hcis 0 = 1"
   by transfer (rule cis_zero)
 
 lemma hrcis_zero_mod [simp]: "\<And>a. hrcis 0 a = 0"
   by transfer (rule rcis_zero_mod)
 
 lemma hrcis_zero_arg [simp]: "\<And>r. hrcis r 0 = hcomplex_of_hypreal r"
   by transfer (rule rcis_zero_arg)
 
 lemma hcomplex_i_mult_minus [simp]: "\<And>x. iii * (iii * x) = - x"
   by transfer (rule complex_i_mult_minus)
 
 lemma hcomplex_i_mult_minus2 [simp]: "iii * iii * x = - x"
   by simp
 
 lemma hcis_hypreal_of_nat_Suc_mult:
   "\<And>a. hcis (hypreal_of_nat (Suc n) * a) = hcis a * hcis (hypreal_of_nat n * a)"
   by transfer (simp add: distrib_right cis_mult)
 
 lemma NSDeMoivre: "\<And>a. (hcis a) ^ n = hcis (hypreal_of_nat n * a)"
   by transfer (rule DeMoivre)
 
 lemma hcis_hypreal_of_hypnat_Suc_mult:
   "\<And>a n. hcis (hypreal_of_hypnat (n + 1) * a) = hcis a * hcis (hypreal_of_hypnat n * a)"
   by transfer (simp add: distrib_right cis_mult)
 
 lemma NSDeMoivre_ext: "\<And>a n. (hcis a) pow n = hcis (hypreal_of_hypnat n * a)"
   by transfer (rule DeMoivre)
 
 lemma NSDeMoivre2: "\<And>a r. (hrcis r a) ^ n = hrcis (r ^ n) (hypreal_of_nat n * a)"
   by transfer (rule DeMoivre2)
 
 lemma DeMoivre2_ext: "\<And>a r n. (hrcis r a) pow n = hrcis (r pow n) (hypreal_of_hypnat n * a)"
   by transfer (rule DeMoivre2)
 
 lemma hcis_inverse [simp]: "\<And>a. inverse (hcis a) = hcis (- a)"
   by transfer (rule cis_inverse)
 
 lemma hrcis_inverse: "\<And>a r. inverse (hrcis r a) = hrcis (inverse r) (- a)"
   by transfer (simp add: rcis_inverse inverse_eq_divide [symmetric])
 
 lemma hRe_hcis [simp]: "\<And>a. hRe (hcis a) = ( *f* cos) a"
   by transfer simp
 
 lemma hIm_hcis [simp]: "\<And>a. hIm (hcis a) = ( *f* sin) a"
   by transfer simp
 
 lemma cos_n_hRe_hcis_pow_n: "( *f* cos) (hypreal_of_nat n * a) = hRe (hcis a ^ n)"
   by (simp add: NSDeMoivre)
 
 lemma sin_n_hIm_hcis_pow_n: "( *f* sin) (hypreal_of_nat n * a) = hIm (hcis a ^ n)"
   by (simp add: NSDeMoivre)
 
 lemma cos_n_hRe_hcis_hcpow_n: "( *f* cos) (hypreal_of_hypnat n * a) = hRe (hcis a pow n)"
   by (simp add: NSDeMoivre_ext)
 
 lemma sin_n_hIm_hcis_hcpow_n: "( *f* sin) (hypreal_of_hypnat n * a) = hIm (hcis a pow n)"
   by (simp add: NSDeMoivre_ext)
 
 lemma hExp_add: "\<And>a b. hExp (a + b) = hExp a * hExp b"
   by transfer (rule exp_add)
 
 
 subsection \<open>\<^term>\<open>hcomplex_of_complex\<close>: the Injection from type \<^typ>\<open>complex\<close> to to \<^typ>\<open>hcomplex\<close>\<close>
 
 lemma hcomplex_of_complex_i: "iii = hcomplex_of_complex \<i>"
   by (rule iii_def)
 
 lemma hRe_hcomplex_of_complex: "hRe (hcomplex_of_complex z) = hypreal_of_real (Re z)"
   by transfer (rule refl)
 
 lemma hIm_hcomplex_of_complex: "hIm (hcomplex_of_complex z) = hypreal_of_real (Im z)"
   by transfer (rule refl)
 
 lemma hcmod_hcomplex_of_complex: "hcmod (hcomplex_of_complex x) = hypreal_of_real (cmod x)"
   by transfer (rule refl)
 
 
 subsection \<open>Numerals and Arithmetic\<close>
 
 lemma hcomplex_of_hypreal_eq_hcomplex_of_complex:
   "hcomplex_of_hypreal (hypreal_of_real x) = hcomplex_of_complex (complex_of_real x)"
   by transfer (rule refl)
 
 lemma hcomplex_hypreal_numeral:
   "hcomplex_of_complex (numeral w) = hcomplex_of_hypreal(numeral w)"
   by transfer (rule of_real_numeral [symmetric])
 
 lemma hcomplex_hypreal_neg_numeral:
   "hcomplex_of_complex (- numeral w) = hcomplex_of_hypreal(- numeral w)"
   by transfer (rule of_real_neg_numeral [symmetric])
 
 lemma hcomplex_numeral_hcnj [simp]: "hcnj (numeral v :: hcomplex) = numeral v"
   by transfer (rule complex_cnj_numeral)
 
 lemma hcomplex_numeral_hcmod [simp]: "hcmod (numeral v :: hcomplex) = (numeral v :: hypreal)"
   by transfer (rule norm_numeral)
 
 lemma hcomplex_neg_numeral_hcmod [simp]: "hcmod (- numeral v :: hcomplex) = (numeral v :: hypreal)"
   by transfer (rule norm_neg_numeral)
 
 lemma hcomplex_numeral_hRe [simp]: "hRe (numeral v :: hcomplex) = numeral v"
   by transfer (rule complex_Re_numeral)
 
 lemma hcomplex_numeral_hIm [simp]: "hIm (numeral v :: hcomplex) = 0"
   by transfer (rule complex_Im_numeral)
 
 end
diff --git a/src/HOL/Series.thy b/src/HOL/Series.thy
--- a/src/HOL/Series.thy
+++ b/src/HOL/Series.thy
@@ -1,1333 +1,1329 @@
 (*  Title       : Series.thy
     Author      : Jacques D. Fleuriot
     Copyright   : 1998  University of Cambridge
 
 Converted to Isar and polished by lcp
 Converted to sum and polished yet more by TNN
 Additional contributions by Jeremy Avigad
 *)
 
 section \<open>Infinite Series\<close>
 
 theory Series
 imports Limits Inequalities
 begin
 
 subsection \<open>Definition of infinite summability\<close>
 
 definition sums :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a \<Rightarrow> bool"
     (infixr "sums" 80)
   where "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> s"
 
 definition summable :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> bool"
   where "summable f \<longleftrightarrow> (\<exists>s. f sums s)"
 
 definition suminf :: "(nat \<Rightarrow> 'a::{topological_space, comm_monoid_add}) \<Rightarrow> 'a"
     (binder "\<Sum>" 10)
   where "suminf f = (THE s. f sums s)"
 
 text\<open>Variants of the definition\<close>
 lemma sums_def': "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i = 0..n. f i) \<longlonglongrightarrow> s"
   unfolding sums_def
   apply (subst filterlim_sequentially_Suc [symmetric])
   apply (simp only: lessThan_Suc_atMost atLeast0AtMost)
   done
 
 lemma sums_def_le: "f sums s \<longleftrightarrow> (\<lambda>n. \<Sum>i\<le>n. f i) \<longlonglongrightarrow> s"
   by (simp add: sums_def' atMost_atLeast0)
 
 lemma bounded_imp_summable:
   fixes a :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder,linorder_topology,linordered_comm_semiring_strict}"
   assumes 0: "\<And>n. a n \<ge> 0" and bounded: "\<And>n. (\<Sum>k\<le>n. a k) \<le> B"
   shows "summable a" 
 proof -
   have "bdd_above (range(\<lambda>n. \<Sum>k\<le>n. a k))"
     by (meson bdd_aboveI2 bounded)
   moreover have "incseq (\<lambda>n. \<Sum>k\<le>n. a k)"
     by (simp add: mono_def "0" sum_mono2)
   ultimately obtain s where "(\<lambda>n. \<Sum>k\<le>n. a k) \<longlonglongrightarrow> s"
     using LIMSEQ_incseq_SUP by blast
   then show ?thesis
     by (auto simp: sums_def_le summable_def)
 qed
 
 
 subsection \<open>Infinite summability on topological monoids\<close>
 
 lemma sums_subst[trans]: "f = g \<Longrightarrow> g sums z \<Longrightarrow> f sums z"
   by simp
 
 lemma sums_cong: "(\<And>n. f n = g n) \<Longrightarrow> f sums c \<longleftrightarrow> g sums c"
-  by (drule ext) simp
+  by presburger
 
 lemma sums_summable: "f sums l \<Longrightarrow> summable f"
   by (simp add: sums_def summable_def, blast)
 
 lemma summable_iff_convergent: "summable f \<longleftrightarrow> convergent (\<lambda>n. \<Sum>i<n. f i)"
   by (simp add: summable_def sums_def convergent_def)
 
 lemma summable_iff_convergent': "summable f \<longleftrightarrow> convergent (\<lambda>n. sum f {..n})"
-  by (simp_all only: summable_iff_convergent convergent_def
-        lessThan_Suc_atMost [symmetric] filterlim_sequentially_Suc[of "\<lambda>n. sum f {..<n}"])
+  by (simp add: convergent_def summable_def sums_def_le)
 
 lemma suminf_eq_lim: "suminf f = lim (\<lambda>n. \<Sum>i<n. f i)"
   by (simp add: suminf_def sums_def lim_def)
 
 lemma sums_zero[simp, intro]: "(\<lambda>n. 0) sums 0"
   unfolding sums_def by simp
 
 lemma summable_zero[simp, intro]: "summable (\<lambda>n. 0)"
   by (rule sums_zero [THEN sums_summable])
 
 lemma sums_group: "f sums s \<Longrightarrow> 0 < k \<Longrightarrow> (\<lambda>n. sum f {n * k ..< n * k + k}) sums s"
   apply (simp only: sums_def sum.nat_group tendsto_def eventually_sequentially)
   apply (erule all_forward imp_forward exE| assumption)+
-  apply (rule_tac x="N" in exI)
   by (metis le_square mult.commute mult.left_neutral mult_le_cancel2 mult_le_mono)
 
 lemma suminf_cong: "(\<And>n. f n = g n) \<Longrightarrow> suminf f = suminf g"
-  by (rule arg_cong[of f g], rule ext) simp
+  by presburger
 
 lemma summable_cong:
   fixes f g :: "nat \<Rightarrow> 'a::real_normed_vector"
   assumes "eventually (\<lambda>x. f x = g x) sequentially"
   shows "summable f = summable g"
 proof -
   from assms obtain N where N: "\<forall>n\<ge>N. f n = g n"
     by (auto simp: eventually_at_top_linorder)
   define C where "C = (\<Sum>k<N. f k - g k)"
   from eventually_ge_at_top[of N]
   have "eventually (\<lambda>n. sum f {..<n} = C + sum g {..<n}) sequentially"
   proof eventually_elim
     case (elim n)
     then have "{..<n} = {..<N} \<union> {N..<n}"
       by auto
     also have "sum f ... = sum f {..<N} + sum f {N..<n}"
       by (intro sum.union_disjoint) auto
     also from N have "sum f {N..<n} = sum g {N..<n}"
       by (intro sum.cong) simp_all
     also have "sum f {..<N} + sum g {N..<n} = C + (sum g {..<N} + sum g {N..<n})"
       unfolding C_def by (simp add: algebra_simps sum_subtractf)
     also have "sum g {..<N} + sum g {N..<n} = sum g ({..<N} \<union> {N..<n})"
       by (intro sum.union_disjoint [symmetric]) auto
     also from elim have "{..<N} \<union> {N..<n} = {..<n}"
       by auto
     finally show "sum f {..<n} = C + sum g {..<n}" .
   qed
   from convergent_cong[OF this] show ?thesis
     by (simp add: summable_iff_convergent convergent_add_const_iff)
 qed
 
 lemma sums_finite:
   assumes [simp]: "finite N"
     and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
   shows "f sums (\<Sum>n\<in>N. f n)"
 proof -
   have eq: "sum f {..<n + Suc (Max N)} = sum f N" for n
     by (rule sum.mono_neutral_right) (auto simp: add_increasing less_Suc_eq_le f)
   show ?thesis
     unfolding sums_def
     by (rule LIMSEQ_offset[of _ "Suc (Max N)"])
       (simp add: eq atLeast0LessThan del: add_Suc_right)
 qed
 
 corollary sums_0: "(\<And>n. f n = 0) \<Longrightarrow> (f sums 0)"
     by (metis (no_types) finite.emptyI sum.empty sums_finite)
 
 lemma summable_finite: "finite N \<Longrightarrow> (\<And>n. n \<notin> N \<Longrightarrow> f n = 0) \<Longrightarrow> summable f"
   by (rule sums_summable) (rule sums_finite)
 
 lemma sums_If_finite_set: "finite A \<Longrightarrow> (\<lambda>r. if r \<in> A then f r else 0) sums (\<Sum>r\<in>A. f r)"
   using sums_finite[of A "(\<lambda>r. if r \<in> A then f r else 0)"] by simp
 
 lemma summable_If_finite_set[simp, intro]: "finite A \<Longrightarrow> summable (\<lambda>r. if r \<in> A then f r else 0)"
   by (rule sums_summable) (rule sums_If_finite_set)
 
 lemma sums_If_finite: "finite {r. P r} \<Longrightarrow> (\<lambda>r. if P r then f r else 0) sums (\<Sum>r | P r. f r)"
   using sums_If_finite_set[of "{r. P r}"] by simp
 
 lemma summable_If_finite[simp, intro]: "finite {r. P r} \<Longrightarrow> summable (\<lambda>r. if P r then f r else 0)"
   by (rule sums_summable) (rule sums_If_finite)
 
 lemma sums_single: "(\<lambda>r. if r = i then f r else 0) sums f i"
   using sums_If_finite[of "\<lambda>r. r = i"] by simp
 
 lemma summable_single[simp, intro]: "summable (\<lambda>r. if r = i then f r else 0)"
   by (rule sums_summable) (rule sums_single)
 
 context
   fixes f :: "nat \<Rightarrow> 'a::{t2_space,comm_monoid_add}"
 begin
 
 lemma summable_sums[intro]: "summable f \<Longrightarrow> f sums (suminf f)"
   by (simp add: summable_def sums_def suminf_def)
      (metis convergent_LIMSEQ_iff convergent_def lim_def)
 
 lemma summable_LIMSEQ: "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> suminf f"
   by (rule summable_sums [unfolded sums_def])
 
 lemma summable_LIMSEQ': "summable f \<Longrightarrow> (\<lambda>n. \<Sum>i\<le>n. f i) \<longlonglongrightarrow> suminf f"
   using sums_def_le by blast
 
 lemma sums_unique: "f sums s \<Longrightarrow> s = suminf f"
   by (metis limI suminf_eq_lim sums_def)
 
 lemma sums_iff: "f sums x \<longleftrightarrow> summable f \<and> suminf f = x"
   by (metis summable_sums sums_summable sums_unique)
 
 lemma summable_sums_iff: "summable f \<longleftrightarrow> f sums suminf f"
   by (auto simp: sums_iff summable_sums)
 
 lemma sums_unique2: "f sums a \<Longrightarrow> f sums b \<Longrightarrow> a = b"
   for a b :: 'a
   by (simp add: sums_iff)
 
 lemma sums_Uniq: "\<exists>\<^sub>\<le>\<^sub>1a. f sums a"
   for a b :: 'a
   by (simp add: sums_unique2 Uniq_def)
 
 lemma suminf_finite:
   assumes N: "finite N"
     and f: "\<And>n. n \<notin> N \<Longrightarrow> f n = 0"
   shows "suminf f = (\<Sum>n\<in>N. f n)"
   using sums_finite[OF assms, THEN sums_unique] by simp
 
 end
 
 lemma suminf_zero[simp]: "suminf (\<lambda>n. 0::'a::{t2_space, comm_monoid_add}) = 0"
   by (rule sums_zero [THEN sums_unique, symmetric])
 
 
 subsection \<open>Infinite summability on ordered, topological monoids\<close>
 
 lemma sums_le: "(\<And>n. f n \<le> g n) \<Longrightarrow> f sums s \<Longrightarrow> g sums t \<Longrightarrow> s \<le> t"
   for f g :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add,linorder_topology}"
   by (rule LIMSEQ_le) (auto intro: sum_mono simp: sums_def)
 
 context
   fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add,linorder_topology}"
 begin
 
 lemma suminf_le: "(\<And>n. f n \<le> g n) \<Longrightarrow> summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f \<le> suminf g"
   using sums_le by blast
 
 lemma sum_le_suminf:
   shows "summable f \<Longrightarrow> finite I \<Longrightarrow> (\<And>n. n \<in>- I \<Longrightarrow> 0 \<le> f n) \<Longrightarrow> sum f I \<le> suminf f"
   by (rule sums_le[OF _ sums_If_finite_set summable_sums]) auto
 
 lemma suminf_nonneg: "summable f \<Longrightarrow> (\<And>n. 0 \<le> f n) \<Longrightarrow> 0 \<le> suminf f"
   using sum_le_suminf by force
 
 lemma suminf_le_const: "summable f \<Longrightarrow> (\<And>n. sum f {..<n} \<le> x) \<Longrightarrow> suminf f \<le> x"
   by (metis LIMSEQ_le_const2 summable_LIMSEQ)
 
 lemma suminf_eq_zero_iff: 
   assumes "summable f" and pos: "\<And>n. 0 \<le> f n"
   shows "suminf f = 0 \<longleftrightarrow> (\<forall>n. f n = 0)"
 proof
-  assume "suminf f = 0" 
+  assume L: "suminf f = 0" 
   then have f: "(\<lambda>n. \<Sum>i<n. f i) \<longlonglongrightarrow> 0"
     using summable_LIMSEQ[of f] assms by simp
   then have "\<And>i. (\<Sum>n\<in>{i}. f n) \<le> 0"
-  proof (rule LIMSEQ_le_const)
-    show "\<exists>N. \<forall>n\<ge>N. (\<Sum>n\<in>{i}. f n) \<le> sum f {..<n}" for i
-      using pos by (intro exI[of _ "Suc i"] allI impI sum_mono2) auto
-  qed
+    by (metis L \<open>summable f\<close> order_refl pos sum.infinite sum_le_suminf)
   with pos show "\<forall>n. f n = 0"
-    by (auto intro!: antisym)
+    by (simp add: order.antisym)
 qed (metis suminf_zero fun_eq_iff)
 
 lemma suminf_pos_iff: "summable f \<Longrightarrow> (\<And>n. 0 \<le> f n) \<Longrightarrow> 0 < suminf f \<longleftrightarrow> (\<exists>i. 0 < f i)"
   using sum_le_suminf[of "{}"] suminf_eq_zero_iff by (simp add: less_le)
 
 lemma suminf_pos2:
   assumes "summable f" "\<And>n. 0 \<le> f n" "0 < f i"
   shows "0 < suminf f"
 proof -
   have "0 < (\<Sum>n<Suc i. f n)"
     using assms by (intro sum_pos2[where i=i]) auto
   also have "\<dots> \<le> suminf f"
     using assms by (intro sum_le_suminf) auto
   finally show ?thesis .
 qed
 
 lemma suminf_pos: "summable f \<Longrightarrow> (\<And>n. 0 < f n) \<Longrightarrow> 0 < suminf f"
   by (intro suminf_pos2[where i=0]) (auto intro: less_imp_le)
 
 end
 
 context
   fixes f :: "nat \<Rightarrow> 'a::{ordered_cancel_comm_monoid_add,linorder_topology}"
 begin
 
 lemma sum_less_suminf2:
   "summable f \<Longrightarrow> (\<And>m. m\<ge>n \<Longrightarrow> 0 \<le> f m) \<Longrightarrow> n \<le> i \<Longrightarrow> 0 < f i \<Longrightarrow> sum f {..<n} < suminf f"
   using sum_le_suminf[of f "{..< Suc i}"]
     and add_strict_increasing[of "f i" "sum f {..<n}" "sum f {..<i}"]
     and sum_mono2[of "{..<i}" "{..<n}" f]
   by (auto simp: less_imp_le ac_simps)
 
 lemma sum_less_suminf: "summable f \<Longrightarrow> (\<And>m. m\<ge>n \<Longrightarrow> 0 < f m) \<Longrightarrow> sum f {..<n} < suminf f"
   using sum_less_suminf2[of n n] by (simp add: less_imp_le)
 
 end
 
 lemma summableI_nonneg_bounded:
   fixes f :: "nat \<Rightarrow> 'a::{ordered_comm_monoid_add,linorder_topology,conditionally_complete_linorder}"
   assumes pos[simp]: "\<And>n. 0 \<le> f n"
     and le: "\<And>n. (\<Sum>i<n. f i) \<le> x"
   shows "summable f"
   unfolding summable_def sums_def [abs_def]
 proof (rule exI LIMSEQ_incseq_SUP)+
   show "bdd_above (range (\<lambda>n. sum f {..<n}))"
     using le by (auto simp: bdd_above_def)
   show "incseq (\<lambda>n. sum f {..<n})"
     by (auto simp: mono_def intro!: sum_mono2)
 qed
 
 lemma summableI[intro, simp]: "summable f"
   for f :: "nat \<Rightarrow> 'a::{canonically_ordered_monoid_add,linorder_topology,complete_linorder}"
   by (intro summableI_nonneg_bounded[where x=top] zero_le top_greatest)
 
 lemma suminf_eq_SUP_real:
   assumes X: "summable X" "\<And>i. 0 \<le> X i" shows "suminf X = (SUP i. \<Sum>n<i. X n::real)"
   by (intro LIMSEQ_unique[OF summable_LIMSEQ] X LIMSEQ_incseq_SUP)
      (auto intro!: bdd_aboveI2[where M="\<Sum>i. X i"] sum_le_suminf X monoI sum_mono2)
 
 
 subsection \<open>Infinite summability on topological monoids\<close>
 
 context
   fixes f g :: "nat \<Rightarrow> 'a::{t2_space,topological_comm_monoid_add}"
 begin
 
 lemma sums_Suc:
   assumes "(\<lambda>n. f (Suc n)) sums l"
   shows "f sums (l + f 0)"
 proof  -
   have "(\<lambda>n. (\<Sum>i<n. f (Suc i)) + f 0) \<longlonglongrightarrow> l + f 0"
     using assms by (auto intro!: tendsto_add simp: sums_def)
   moreover have "(\<Sum>i<n. f (Suc i)) + f 0 = (\<Sum>i<Suc n. f i)" for n
     unfolding lessThan_Suc_eq_insert_0
     by (simp add: ac_simps sum.atLeast1_atMost_eq image_Suc_lessThan)
   ultimately show ?thesis
     by (auto simp: sums_def simp del: sum.lessThan_Suc intro: filterlim_sequentially_Suc[THEN iffD1])
 qed
 
 lemma sums_add: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n + g n) sums (a + b)"
   unfolding sums_def by (simp add: sum.distrib tendsto_add)
 
 lemma summable_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n + g n)"
   unfolding summable_def by (auto intro: sums_add)
 
 lemma suminf_add: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f + suminf g = (\<Sum>n. f n + g n)"
   by (intro sums_unique sums_add summable_sums)
 
 end
 
 context
   fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::{t2_space,topological_comm_monoid_add}"
     and I :: "'i set"
 begin
 
 lemma sums_sum: "(\<And>i. i \<in> I \<Longrightarrow> (f i) sums (x i)) \<Longrightarrow> (\<lambda>n. \<Sum>i\<in>I. f i n) sums (\<Sum>i\<in>I. x i)"
   by (induct I rule: infinite_finite_induct) (auto intro!: sums_add)
 
 lemma suminf_sum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> (\<Sum>n. \<Sum>i\<in>I. f i n) = (\<Sum>i\<in>I. \<Sum>n. f i n)"
   using sums_unique[OF sums_sum, OF summable_sums] by simp
 
 lemma summable_sum: "(\<And>i. i \<in> I \<Longrightarrow> summable (f i)) \<Longrightarrow> summable (\<lambda>n. \<Sum>i\<in>I. f i n)"
   using sums_summable[OF sums_sum[OF summable_sums]] .
 
 end
 
 lemma sums_If_finite_set':
   fixes f g :: "nat \<Rightarrow> 'a::{t2_space,topological_ab_group_add}"
   assumes "g sums S" and "finite A" and "S' = S + (\<Sum>n\<in>A. f n - g n)"
   shows   "(\<lambda>n. if n \<in> A then f n else g n) sums S'"
 proof -
   have "(\<lambda>n. g n + (if n \<in> A then f n - g n else 0)) sums (S + (\<Sum>n\<in>A. f n - g n))"
     by (intro sums_add assms sums_If_finite_set)
   also have "(\<lambda>n. g n + (if n \<in> A then f n - g n else 0)) = (\<lambda>n. if n \<in> A then f n else g n)"
     by (simp add: fun_eq_iff)
   finally show ?thesis using assms by simp
 qed
 
 subsection \<open>Infinite summability on real normed vector spaces\<close>
 
 context
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
 begin
 
 lemma sums_Suc_iff: "(\<lambda>n. f (Suc n)) sums s \<longleftrightarrow> f sums (s + f 0)"
 proof -
   have "f sums (s + f 0) \<longleftrightarrow> (\<lambda>i. \<Sum>j<Suc i. f j) \<longlonglongrightarrow> s + f 0"
     by (subst filterlim_sequentially_Suc) (simp add: sums_def)
   also have "\<dots> \<longleftrightarrow> (\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
     by (simp add: ac_simps lessThan_Suc_eq_insert_0 image_Suc_lessThan sum.atLeast1_atMost_eq)
   also have "\<dots> \<longleftrightarrow> (\<lambda>n. f (Suc n)) sums s"
   proof
     assume "(\<lambda>i. (\<Sum>j<i. f (Suc j)) + f 0) \<longlonglongrightarrow> s + f 0"
     with tendsto_add[OF this tendsto_const, of "- f 0"] show "(\<lambda>i. f (Suc i)) sums s"
       by (simp add: sums_def)
   qed (auto intro: tendsto_add simp: sums_def)
   finally show ?thesis ..
 qed
 
 lemma summable_Suc_iff: "summable (\<lambda>n. f (Suc n)) = summable f"
 proof
   assume "summable f"
   then have "f sums suminf f"
     by (rule summable_sums)
   then have "(\<lambda>n. f (Suc n)) sums (suminf f - f 0)"
     by (simp add: sums_Suc_iff)
   then show "summable (\<lambda>n. f (Suc n))"
     unfolding summable_def by blast
 qed (auto simp: sums_Suc_iff summable_def)
 
 lemma sums_Suc_imp: "f 0 = 0 \<Longrightarrow> (\<lambda>n. f (Suc n)) sums s \<Longrightarrow> (\<lambda>n. f n) sums s"
   using sums_Suc_iff by simp
 
 end
 
 context (* Separate contexts are necessary to allow general use of the results above, here. *)
   fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
 begin
 
 lemma sums_diff: "f sums a \<Longrightarrow> g sums b \<Longrightarrow> (\<lambda>n. f n - g n) sums (a - b)"
   unfolding sums_def by (simp add: sum_subtractf tendsto_diff)
 
 lemma summable_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. f n - g n)"
   unfolding summable_def by (auto intro: sums_diff)
 
 lemma suminf_diff: "summable f \<Longrightarrow> summable g \<Longrightarrow> suminf f - suminf g = (\<Sum>n. f n - g n)"
   by (intro sums_unique sums_diff summable_sums)
 
 lemma sums_minus: "f sums a \<Longrightarrow> (\<lambda>n. - f n) sums (- a)"
   unfolding sums_def by (simp add: sum_negf tendsto_minus)
 
 lemma summable_minus: "summable f \<Longrightarrow> summable (\<lambda>n. - f n)"
   unfolding summable_def by (auto intro: sums_minus)
 
 lemma suminf_minus: "summable f \<Longrightarrow> (\<Sum>n. - f n) = - (\<Sum>n. f n)"
   by (intro sums_unique [symmetric] sums_minus summable_sums)
 
 lemma sums_iff_shift: "(\<lambda>i. f (i + n)) sums s \<longleftrightarrow> f sums (s + (\<Sum>i<n. f i))"
 proof (induct n arbitrary: s)
   case 0
   then show ?case by simp
 next
   case (Suc n)
   then have "(\<lambda>i. f (Suc i + n)) sums s \<longleftrightarrow> (\<lambda>i. f (i + n)) sums (s + f n)"
     by (subst sums_Suc_iff) simp
   with Suc show ?case
     by (simp add: ac_simps)
 qed
 
 corollary sums_iff_shift': "(\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i)) \<longleftrightarrow> f sums s"
   by (simp add: sums_iff_shift)
 
 lemma sums_zero_iff_shift:
   assumes "\<And>i. i < n \<Longrightarrow> f i = 0"
   shows "(\<lambda>i. f (i+n)) sums s \<longleftrightarrow> (\<lambda>i. f i) sums s"
   by (simp add: assms sums_iff_shift)
 
 lemma summable_iff_shift [simp]: "summable (\<lambda>n. f (n + k)) \<longleftrightarrow> summable f"
   by (metis diff_add_cancel summable_def sums_iff_shift [abs_def])
 
 lemma sums_split_initial_segment: "f sums s \<Longrightarrow> (\<lambda>i. f (i + n)) sums (s - (\<Sum>i<n. f i))"
   by (simp add: sums_iff_shift)
 
 lemma summable_ignore_initial_segment: "summable f \<Longrightarrow> summable (\<lambda>n. f(n + k))"
   by (simp add: summable_iff_shift)
 
 lemma suminf_minus_initial_segment: "summable f \<Longrightarrow> (\<Sum>n. f (n + k)) = (\<Sum>n. f n) - (\<Sum>i<k. f i)"
   by (rule sums_unique[symmetric]) (auto simp: sums_iff_shift)
 
 lemma suminf_split_initial_segment: "summable f \<Longrightarrow> suminf f = (\<Sum>n. f(n + k)) + (\<Sum>i<k. f i)"
   by (auto simp add: suminf_minus_initial_segment)
 
 lemma suminf_split_head: "summable f \<Longrightarrow> (\<Sum>n. f (Suc n)) = suminf f - f 0"
   using suminf_split_initial_segment[of 1] by simp
 
 lemma suminf_exist_split:
   fixes r :: real
   assumes "0 < r" and "summable f"
   shows "\<exists>N. \<forall>n\<ge>N. norm (\<Sum>i. f (i + n)) < r"
 proof -
   from LIMSEQ_D[OF summable_LIMSEQ[OF \<open>summable f\<close>] \<open>0 < r\<close>]
   obtain N :: nat where "\<forall> n \<ge> N. norm (sum f {..<n} - suminf f) < r"
     by auto
   then show ?thesis
     by (auto simp: norm_minus_commute suminf_minus_initial_segment[OF \<open>summable f\<close>])
 qed
 
 lemma summable_LIMSEQ_zero: 
   assumes "summable f" shows "f \<longlonglongrightarrow> 0"
 proof -
   have "Cauchy (\<lambda>n. sum f {..<n})"
     using LIMSEQ_imp_Cauchy assms summable_LIMSEQ by blast
   then show ?thesis
     unfolding  Cauchy_iff LIMSEQ_iff
     by (metis add.commute add_diff_cancel_right' diff_zero le_SucI sum.lessThan_Suc)
 qed
 
 lemma summable_imp_convergent: "summable f \<Longrightarrow> convergent f"
   by (force dest!: summable_LIMSEQ_zero simp: convergent_def)
 
 lemma summable_imp_Bseq: "summable f \<Longrightarrow> Bseq f"
   by (simp add: convergent_imp_Bseq summable_imp_convergent)
 
 end
 
 lemma summable_minus_iff: "summable (\<lambda>n. - f n) \<longleftrightarrow> summable f"
   for f :: "nat \<Rightarrow> 'a::real_normed_vector"
   by (auto dest: summable_minus)  (* used two ways, hence must be outside the context above *)
 
 lemma (in bounded_linear) sums: "(\<lambda>n. X n) sums a \<Longrightarrow> (\<lambda>n. f (X n)) sums (f a)"
   unfolding sums_def by (drule tendsto) (simp only: sum)
 
 lemma (in bounded_linear) summable: "summable (\<lambda>n. X n) \<Longrightarrow> summable (\<lambda>n. f (X n))"
   unfolding summable_def by (auto intro: sums)
 
 lemma (in bounded_linear) suminf: "summable (\<lambda>n. X n) \<Longrightarrow> f (\<Sum>n. X n) = (\<Sum>n. f (X n))"
   by (intro sums_unique sums summable_sums)
 
 lemmas sums_of_real = bounded_linear.sums [OF bounded_linear_of_real]
 lemmas summable_of_real = bounded_linear.summable [OF bounded_linear_of_real]
 lemmas suminf_of_real = bounded_linear.suminf [OF bounded_linear_of_real]
 
 lemmas sums_scaleR_left = bounded_linear.sums[OF bounded_linear_scaleR_left]
 lemmas summable_scaleR_left = bounded_linear.summable[OF bounded_linear_scaleR_left]
 lemmas suminf_scaleR_left = bounded_linear.suminf[OF bounded_linear_scaleR_left]
 
 lemmas sums_scaleR_right = bounded_linear.sums[OF bounded_linear_scaleR_right]
 lemmas summable_scaleR_right = bounded_linear.summable[OF bounded_linear_scaleR_right]
 lemmas suminf_scaleR_right = bounded_linear.suminf[OF bounded_linear_scaleR_right]
 
 lemma summable_const_iff: "summable (\<lambda>_. c) \<longleftrightarrow> c = 0"
   for c :: "'a::real_normed_vector"
 proof -
   have "\<not> summable (\<lambda>_. c)" if "c \<noteq> 0"
   proof -
     from that have "filterlim (\<lambda>n. of_nat n * norm c) at_top sequentially"
       by (subst mult.commute)
         (auto intro!: filterlim_tendsto_pos_mult_at_top filterlim_real_sequentially)
     then have "\<not> convergent (\<lambda>n. norm (\<Sum>k<n. c))"
       by (intro filterlim_at_infinity_imp_not_convergent filterlim_at_top_imp_at_infinity)
         (simp_all add: sum_constant_scaleR)
     then show ?thesis
       unfolding summable_iff_convergent using convergent_norm by blast
   qed
   then show ?thesis by auto
 qed
 
 
 subsection \<open>Infinite summability on real normed algebras\<close>
 
 context
   fixes f :: "nat \<Rightarrow> 'a::real_normed_algebra"
 begin
 
 lemma sums_mult: "f sums a \<Longrightarrow> (\<lambda>n. c * f n) sums (c * a)"
   by (rule bounded_linear.sums [OF bounded_linear_mult_right])
 
 lemma summable_mult: "summable f \<Longrightarrow> summable (\<lambda>n. c * f n)"
   by (rule bounded_linear.summable [OF bounded_linear_mult_right])
 
 lemma suminf_mult: "summable f \<Longrightarrow> suminf (\<lambda>n. c * f n) = c * suminf f"
   by (rule bounded_linear.suminf [OF bounded_linear_mult_right, symmetric])
 
 lemma sums_mult2: "f sums a \<Longrightarrow> (\<lambda>n. f n * c) sums (a * c)"
   by (rule bounded_linear.sums [OF bounded_linear_mult_left])
 
 lemma summable_mult2: "summable f \<Longrightarrow> summable (\<lambda>n. f n * c)"
   by (rule bounded_linear.summable [OF bounded_linear_mult_left])
 
 lemma suminf_mult2: "summable f \<Longrightarrow> suminf f * c = (\<Sum>n. f n * c)"
   by (rule bounded_linear.suminf [OF bounded_linear_mult_left])
 
 end
 
 lemma sums_mult_iff:
   fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra,field}"
   assumes "c \<noteq> 0"
   shows "(\<lambda>n. c * f n) sums (c * d) \<longleftrightarrow> f sums d"
   using sums_mult[of f d c] sums_mult[of "\<lambda>n. c * f n" "c * d" "inverse c"]
   by (force simp: field_simps assms)
 
 lemma sums_mult2_iff:
   fixes f :: "nat \<Rightarrow> 'a::{real_normed_algebra,field}"
   assumes "c \<noteq> 0"
   shows   "(\<lambda>n. f n * c) sums (d * c) \<longleftrightarrow> f sums d"
   using sums_mult_iff[OF assms, of f d] by (simp add: mult.commute)
 
 lemma sums_of_real_iff:
   "(\<lambda>n. of_real (f n) :: 'a::real_normed_div_algebra) sums of_real c \<longleftrightarrow> f sums c"
   by (simp add: sums_def of_real_sum[symmetric] tendsto_of_real_iff del: of_real_sum)
 
 
 subsection \<open>Infinite summability on real normed fields\<close>
 
 context
   fixes c :: "'a::real_normed_field"
 begin
 
 lemma sums_divide: "f sums a \<Longrightarrow> (\<lambda>n. f n / c) sums (a / c)"
   by (rule bounded_linear.sums [OF bounded_linear_divide])
 
 lemma summable_divide: "summable f \<Longrightarrow> summable (\<lambda>n. f n / c)"
   by (rule bounded_linear.summable [OF bounded_linear_divide])
 
 lemma suminf_divide: "summable f \<Longrightarrow> suminf (\<lambda>n. f n / c) = suminf f / c"
   by (rule bounded_linear.suminf [OF bounded_linear_divide, symmetric])
 
 lemma summable_inverse_divide: "summable (inverse \<circ> f) \<Longrightarrow> summable (\<lambda>n. c / f n)"
   by (auto dest: summable_mult [of _ c] simp: field_simps)
 
 lemma sums_mult_D: "(\<lambda>n. c * f n) sums a \<Longrightarrow> c \<noteq> 0 \<Longrightarrow> f sums (a/c)"
   using sums_mult_iff by fastforce
 
 lemma summable_mult_D: "summable (\<lambda>n. c * f n) \<Longrightarrow> c \<noteq> 0 \<Longrightarrow> summable f"
   by (auto dest: summable_divide)
 
 
 text \<open>Sum of a geometric progression.\<close>
 
 lemma geometric_sums:
   assumes "norm c < 1"
   shows "(\<lambda>n. c^n) sums (1 / (1 - c))"
 proof -
   have neq_0: "c - 1 \<noteq> 0"
     using assms by auto
   then have "(\<lambda>n. c ^ n / (c - 1) - 1 / (c - 1)) \<longlonglongrightarrow> 0 / (c - 1) - 1 / (c - 1)"
     by (intro tendsto_intros assms)
   then have "(\<lambda>n. (c ^ n - 1) / (c - 1)) \<longlonglongrightarrow> 1 / (1 - c)"
     by (simp add: nonzero_minus_divide_right [OF neq_0] diff_divide_distrib)
   with neq_0 show "(\<lambda>n. c ^ n) sums (1 / (1 - c))"
     by (simp add: sums_def geometric_sum)
 qed
 
 lemma summable_geometric: "norm c < 1 \<Longrightarrow> summable (\<lambda>n. c^n)"
   by (rule geometric_sums [THEN sums_summable])
 
 lemma suminf_geometric: "norm c < 1 \<Longrightarrow> suminf (\<lambda>n. c^n) = 1 / (1 - c)"
   by (rule sums_unique[symmetric]) (rule geometric_sums)
 
 lemma summable_geometric_iff [simp]: "summable (\<lambda>n. c ^ n) \<longleftrightarrow> norm c < 1"
 proof
   assume "summable (\<lambda>n. c ^ n :: 'a :: real_normed_field)"
   then have "(\<lambda>n. norm c ^ n) \<longlonglongrightarrow> 0"
     by (simp add: norm_power [symmetric] tendsto_norm_zero_iff summable_LIMSEQ_zero)
   from order_tendstoD(2)[OF this zero_less_one] obtain n where "norm c ^ n < 1"
     by (auto simp: eventually_at_top_linorder)
   then show "norm c < 1" using one_le_power[of "norm c" n]
     by (cases "norm c \<ge> 1") (linarith, simp)
 qed (rule summable_geometric)
 
 end
 
 text \<open>Biconditional versions for constant factors\<close>
 context
   fixes c :: "'a::real_normed_field"
 begin
 
 lemma summable_cmult_iff [simp]: "summable (\<lambda>n. c * f n) \<longleftrightarrow> c=0 \<or> summable f"
 proof -
   have "\<lbrakk>summable (\<lambda>n. c * f n); c \<noteq> 0\<rbrakk> \<Longrightarrow> summable f"
     using summable_mult_D by blast
   then show ?thesis
     by (auto simp: summable_mult)
 qed
 
 lemma summable_divide_iff [simp]: "summable (\<lambda>n. f n / c) \<longleftrightarrow> c=0 \<or> summable f"
 proof -
   have "\<lbrakk>summable (\<lambda>n. f n / c); c \<noteq> 0\<rbrakk> \<Longrightarrow> summable f"
     by (auto dest: summable_divide [where c = "1/c"])
   then show ?thesis
     by (auto simp: summable_divide)
 qed
 
 end
 
 lemma power_half_series: "(\<lambda>n. (1/2::real)^Suc n) sums 1"
 proof -
   have 2: "(\<lambda>n. (1/2::real)^n) sums 2"
     using geometric_sums [of "1/2::real"] by auto
   have "(\<lambda>n. (1/2::real)^Suc n) = (\<lambda>n. (1 / 2) ^ n / 2)"
     by (simp add: mult.commute)
   then show ?thesis
     using sums_divide [OF 2, of 2] by simp
 qed
 
 
 subsection \<open>Telescoping\<close>
 
 lemma telescope_sums:
   fixes c :: "'a::real_normed_vector"
   assumes "f \<longlonglongrightarrow> c"
   shows "(\<lambda>n. f (Suc n) - f n) sums (c - f 0)"
   unfolding sums_def
 proof (subst filterlim_sequentially_Suc [symmetric])
   have "(\<lambda>n. \<Sum>k<Suc n. f (Suc k) - f k) = (\<lambda>n. f (Suc n) - f 0)"
     by (simp add: lessThan_Suc_atMost atLeast0AtMost [symmetric] sum_Suc_diff)
   also have "\<dots> \<longlonglongrightarrow> c - f 0"
     by (intro tendsto_diff LIMSEQ_Suc[OF assms] tendsto_const)
   finally show "(\<lambda>n. \<Sum>n<Suc n. f (Suc n) - f n) \<longlonglongrightarrow> c - f 0" .
 qed
 
 lemma telescope_sums':
   fixes c :: "'a::real_normed_vector"
   assumes "f \<longlonglongrightarrow> c"
   shows "(\<lambda>n. f n - f (Suc n)) sums (f 0 - c)"
   using sums_minus[OF telescope_sums[OF assms]] by (simp add: algebra_simps)
 
 lemma telescope_summable:
   fixes c :: "'a::real_normed_vector"
   assumes "f \<longlonglongrightarrow> c"
   shows "summable (\<lambda>n. f (Suc n) - f n)"
   using telescope_sums[OF assms] by (simp add: sums_iff)
 
 lemma telescope_summable':
   fixes c :: "'a::real_normed_vector"
   assumes "f \<longlonglongrightarrow> c"
   shows "summable (\<lambda>n. f n - f (Suc n))"
   using summable_minus[OF telescope_summable[OF assms]] by (simp add: algebra_simps)
 
 
 subsection \<open>Infinite summability on Banach spaces\<close>
 
 text \<open>Cauchy-type criterion for convergence of series (c.f. Harrison).\<close>
 
 lemma summable_Cauchy: "summable f \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m\<ge>N. \<forall>n. norm (sum f {m..<n}) < e)" (is "_ = ?rhs")
   for f :: "nat \<Rightarrow> 'a::banach"
 proof
   assume f: "summable f"
   show ?rhs
   proof clarify
     fix e :: real
     assume "0 < e"
     then obtain M where M: "\<And>m n. \<lbrakk>m\<ge>M; n\<ge>M\<rbrakk> \<Longrightarrow> norm (sum f {..<m} - sum f {..<n}) < e"
       using f by (force simp add: summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff)
     have "norm (sum f {m..<n}) < e" if "m \<ge> M" for m n
     proof (cases m n rule: linorder_class.le_cases)
       assume "m \<le> n"
       then show ?thesis
         by (metis (mono_tags, hide_lams) M atLeast0LessThan order_trans sum_diff_nat_ivl that zero_le)
     next
       assume "n \<le> m"
       then show ?thesis
         by (simp add: \<open>0 < e\<close>)
     qed
     then show "\<exists>N. \<forall>m\<ge>N. \<forall>n. norm (sum f {m..<n}) < e"
       by blast
   qed
 next
   assume r: ?rhs
   then show "summable f"
     unfolding summable_iff_convergent Cauchy_convergent_iff [symmetric] Cauchy_iff
   proof clarify
     fix e :: real
     assume "0 < e"
     with r obtain N where N: "\<And>m n. m \<ge> N \<Longrightarrow> norm (sum f {m..<n}) < e"
       by blast
     have "norm (sum f {..<m} - sum f {..<n}) < e" if "m\<ge>N" "n\<ge>N" for m n
     proof (cases m n rule: linorder_class.le_cases)
       assume "m \<le> n"
       then show ?thesis
-        by (metis Groups_Big.sum_diff N finite_lessThan lessThan_minus_lessThan lessThan_subset_iff norm_minus_commute \<open>m\<ge>N\<close>)
+        by (metis N finite_lessThan lessThan_minus_lessThan lessThan_subset_iff norm_minus_commute sum_diff \<open>m\<ge>N\<close>)
     next
       assume "n \<le> m"
       then show ?thesis
-        by (metis Groups_Big.sum_diff N finite_lessThan lessThan_minus_lessThan lessThan_subset_iff \<open>n\<ge>N\<close>)
+        by (metis N finite_lessThan lessThan_minus_lessThan lessThan_subset_iff sum_diff \<open>n\<ge>N\<close>)
     qed
     then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (sum f {..<m} - sum f {..<n}) < e"
       by blast
   qed
 qed
 
 lemma summable_Cauchy':
   fixes f :: "nat \<Rightarrow> 'a :: banach"
-  assumes "eventually (\<lambda>m. \<forall>n\<ge>m. norm (sum f {m..<n}) \<le> g m) sequentially"
-  assumes "filterlim g (nhds 0) sequentially"
+  assumes ev: "eventually (\<lambda>m. \<forall>n\<ge>m. norm (sum f {m..<n}) \<le> g m) sequentially"
+  assumes g0: "g \<longlonglongrightarrow> 0"
   shows "summable f"
 proof (subst summable_Cauchy, intro allI impI, goal_cases)
   case (1 e)
-  from order_tendstoD(2)[OF assms(2) this] and assms(1)
-  have "eventually (\<lambda>m. \<forall>n. norm (sum f {m..<n}) < e) at_top"
+  then have "\<forall>\<^sub>F x in sequentially. g x < e"
+    using g0 order_tendstoD(2) by blast
+  with ev have "eventually (\<lambda>m. \<forall>n. norm (sum f {m..<n}) < e) at_top"
   proof eventually_elim
     case (elim m)
     show ?case
     proof
       fix n
       from elim show "norm (sum f {m..<n}) < e"
         by (cases "n \<ge> m") auto
     qed
   qed
   thus ?case by (auto simp: eventually_at_top_linorder)
 qed
 
 context
   fixes f :: "nat \<Rightarrow> 'a::banach"
 begin
 
 text \<open>Absolute convergence imples normal convergence.\<close>
 
 lemma summable_norm_cancel: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> summable f"
   unfolding summable_Cauchy
   apply (erule all_forward imp_forward ex_forward | assumption)+
   apply (fastforce simp add: order_le_less_trans [OF norm_sum] order_le_less_trans [OF abs_ge_self])
   done
 
 lemma summable_norm: "summable (\<lambda>n. norm (f n)) \<Longrightarrow> norm (suminf f) \<le> (\<Sum>n. norm (f n))"
   by (auto intro: LIMSEQ_le tendsto_norm summable_norm_cancel summable_LIMSEQ norm_sum)
 
 text \<open>Comparison tests.\<close>
 
 lemma summable_comparison_test: 
   assumes fg: "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n" and g: "summable g"
   shows "summable f"
 proof -
   obtain N where N: "\<And>n. n\<ge>N \<Longrightarrow> norm (f n) \<le> g n" 
     using assms by blast
   show ?thesis
   proof (clarsimp simp add: summable_Cauchy)
     fix e :: real
     assume "0 < e"
     then obtain Ng where Ng: "\<And>m n. m \<ge> Ng \<Longrightarrow> norm (sum g {m..<n}) < e" 
       using g by (fastforce simp: summable_Cauchy)
     with N have "norm (sum f {m..<n}) < e" if "m\<ge>max N Ng" for m n
     proof -
       have "norm (sum f {m..<n}) \<le> sum g {m..<n}"
         using N that by (force intro: sum_norm_le)
       also have "... \<le> norm (sum g {m..<n})"
         by simp
       also have "... < e"
         using Ng that by auto
       finally show ?thesis .
     qed
     then show "\<exists>N. \<forall>m\<ge>N. \<forall>n. norm (sum f {m..<n}) < e" 
       by blast
   qed
 qed
 
 lemma summable_comparison_test_ev:
   "eventually (\<lambda>n. norm (f n) \<le> g n) sequentially \<Longrightarrow> summable g \<Longrightarrow> summable f"
   by (rule summable_comparison_test) (auto simp: eventually_at_top_linorder)
 
 text \<open>A better argument order.\<close>
 lemma summable_comparison_test': "summable g \<Longrightarrow> (\<And>n. n \<ge> N \<Longrightarrow> norm (f n) \<le> g n) \<Longrightarrow> summable f"
   by (rule summable_comparison_test) auto
 
 
 subsection \<open>The Ratio Test\<close>
 
 lemma summable_ratio_test:
   assumes "c < 1" "\<And>n. n \<ge> N \<Longrightarrow> norm (f (Suc n)) \<le> c * norm (f n)"
   shows "summable f"
 proof (cases "0 < c")
   case True
   show "summable f"
   proof (rule summable_comparison_test)
     show "\<exists>N'. \<forall>n\<ge>N'. norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
     proof (intro exI allI impI)
       fix n
       assume "N \<le> n"
       then show "norm (f n) \<le> (norm (f N) / (c ^ N)) * c ^ n"
       proof (induct rule: inc_induct)
         case base
         with True show ?case by simp
       next
         case (step m)
         have "norm (f (Suc m)) / c ^ Suc m * c ^ n \<le> norm (f m) / c ^ m * c ^ n"
           using \<open>0 < c\<close> \<open>c < 1\<close> assms(2)[OF \<open>N \<le> m\<close>] by (simp add: field_simps)
         with step show ?case by simp
       qed
     qed
     show "summable (\<lambda>n. norm (f N) / c ^ N * c ^ n)"
       using \<open>0 < c\<close> \<open>c < 1\<close> by (intro summable_mult summable_geometric) simp
   qed
 next
   case False
   have "f (Suc n) = 0" if "n \<ge> N" for n
   proof -
     from that have "norm (f (Suc n)) \<le> c * norm (f n)"
       by (rule assms(2))
     also have "\<dots> \<le> 0"
       using False by (simp add: not_less mult_nonpos_nonneg)
     finally show ?thesis
       by auto
   qed
   then show "summable f"
     by (intro sums_summable[OF sums_finite, of "{.. Suc N}"]) (auto simp: not_le Suc_less_eq2)
 qed
 
 end
 
 
 text \<open>Relations among convergence and absolute convergence for power series.\<close>
 
 lemma Abel_lemma:
   fixes a :: "nat \<Rightarrow> 'a::real_normed_vector"
   assumes r: "0 \<le> r"
     and r0: "r < r0"
     and M: "\<And>n. norm (a n) * r0^n \<le> M"
   shows "summable (\<lambda>n. norm (a n) * r^n)"
 proof (rule summable_comparison_test')
   show "summable (\<lambda>n. M * (r / r0) ^ n)"
     using assms by (auto simp add: summable_mult summable_geometric)
   show "norm (norm (a n) * r ^ n) \<le> M * (r / r0) ^ n" for n
     using r r0 M [of n] dual_order.order_iff_strict
     by (fastforce simp add: abs_mult field_simps)
 qed
 
 
 text \<open>Summability of geometric series for real algebras.\<close>
 
 lemma complete_algebra_summable_geometric:
   fixes x :: "'a::{real_normed_algebra_1,banach}"
   assumes "norm x < 1"
   shows "summable (\<lambda>n. x ^ n)"
 proof (rule summable_comparison_test)
   show "\<exists>N. \<forall>n\<ge>N. norm (x ^ n) \<le> norm x ^ n"
     by (simp add: norm_power_ineq)
   from assms show "summable (\<lambda>n. norm x ^ n)"
     by (simp add: summable_geometric)
 qed
 
 
 subsection \<open>Cauchy Product Formula\<close>
 
 text \<open>
   Proof based on Analysis WebNotes: Chapter 07, Class 41
   \<^url>\<open>http://www.math.unl.edu/~webnotes/classes/class41/prp77.htm\<close>
 \<close>
 
 lemma Cauchy_product_sums:
   fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
   assumes a: "summable (\<lambda>k. norm (a k))"
     and b: "summable (\<lambda>k. norm (b k))"
   shows "(\<lambda>k. \<Sum>i\<le>k. a i * b (k - i)) sums ((\<Sum>k. a k) * (\<Sum>k. b k))"
 proof -
   let ?S1 = "\<lambda>n::nat. {..<n} \<times> {..<n}"
   let ?S2 = "\<lambda>n::nat. {(i,j). i + j < n}"
   have S1_mono: "\<And>m n. m \<le> n \<Longrightarrow> ?S1 m \<subseteq> ?S1 n" by auto
   have S2_le_S1: "\<And>n. ?S2 n \<subseteq> ?S1 n" by auto
   have S1_le_S2: "\<And>n. ?S1 (n div 2) \<subseteq> ?S2 n" by auto
   have finite_S1: "\<And>n. finite (?S1 n)" by simp
   with S2_le_S1 have finite_S2: "\<And>n. finite (?S2 n)" by (rule finite_subset)
 
   let ?g = "\<lambda>(i,j). a i * b j"
   let ?f = "\<lambda>(i,j). norm (a i) * norm (b j)"
   have f_nonneg: "\<And>x. 0 \<le> ?f x" by auto
   then have norm_sum_f: "\<And>A. norm (sum ?f A) = sum ?f A"
     unfolding real_norm_def
     by (simp only: abs_of_nonneg sum_nonneg [rule_format])
 
   have "(\<lambda>n. (\<Sum>k<n. a k) * (\<Sum>k<n. b k)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
     by (intro tendsto_mult summable_LIMSEQ summable_norm_cancel [OF a] summable_norm_cancel [OF b])
   then have 1: "(\<lambda>n. sum ?g (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
     by (simp only: sum_product sum.Sigma [rule_format] finite_lessThan)
 
   have "(\<lambda>n. (\<Sum>k<n. norm (a k)) * (\<Sum>k<n. norm (b k))) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
     using a b by (intro tendsto_mult summable_LIMSEQ)
   then have "(\<lambda>n. sum ?f (?S1 n)) \<longlonglongrightarrow> (\<Sum>k. norm (a k)) * (\<Sum>k. norm (b k))"
     by (simp only: sum_product sum.Sigma [rule_format] finite_lessThan)
   then have "convergent (\<lambda>n. sum ?f (?S1 n))"
     by (rule convergentI)
   then have Cauchy: "Cauchy (\<lambda>n. sum ?f (?S1 n))"
     by (rule convergent_Cauchy)
   have "Zfun (\<lambda>n. sum ?f (?S1 n - ?S2 n)) sequentially"
   proof (rule ZfunI, simp only: eventually_sequentially norm_sum_f)
     fix r :: real
     assume r: "0 < r"
     from CauchyD [OF Cauchy r] obtain N
       where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (sum ?f (?S1 m) - sum ?f (?S1 n)) < r" ..
     then have "\<And>m n. N \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> norm (sum ?f (?S1 m - ?S1 n)) < r"
       by (simp only: sum_diff finite_S1 S1_mono)
     then have N: "\<And>m n. N \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> sum ?f (?S1 m - ?S1 n) < r"
       by (simp only: norm_sum_f)
     show "\<exists>N. \<forall>n\<ge>N. sum ?f (?S1 n - ?S2 n) < r"
     proof (intro exI allI impI)
       fix n
       assume "2 * N \<le> n"
       then have n: "N \<le> n div 2" by simp
       have "sum ?f (?S1 n - ?S2 n) \<le> sum ?f (?S1 n - ?S1 (n div 2))"
         by (intro sum_mono2 finite_Diff finite_S1 f_nonneg Diff_mono subset_refl S1_le_S2)
       also have "\<dots> < r"
         using n div_le_dividend by (rule N)
       finally show "sum ?f (?S1 n - ?S2 n) < r" .
     qed
   qed
   then have "Zfun (\<lambda>n. sum ?g (?S1 n - ?S2 n)) sequentially"
     apply (rule Zfun_le [rule_format])
     apply (simp only: norm_sum_f)
     apply (rule order_trans [OF norm_sum sum_mono])
     apply (auto simp add: norm_mult_ineq)
     done
   then have 2: "(\<lambda>n. sum ?g (?S1 n) - sum ?g (?S2 n)) \<longlonglongrightarrow> 0"
     unfolding tendsto_Zfun_iff diff_0_right
     by (simp only: sum_diff finite_S1 S2_le_S1)
   with 1 have "(\<lambda>n. sum ?g (?S2 n)) \<longlonglongrightarrow> (\<Sum>k. a k) * (\<Sum>k. b k)"
     by (rule Lim_transform2)
   then show ?thesis
     by (simp only: sums_def sum.triangle_reindex)
 qed
 
 lemma Cauchy_product:
   fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
   assumes "summable (\<lambda>k. norm (a k))"
     and "summable (\<lambda>k. norm (b k))"
   shows "(\<Sum>k. a k) * (\<Sum>k. b k) = (\<Sum>k. \<Sum>i\<le>k. a i * b (k - i))"
   using assms by (rule Cauchy_product_sums [THEN sums_unique])
 
 lemma summable_Cauchy_product:
   fixes a b :: "nat \<Rightarrow> 'a::{real_normed_algebra,banach}"
   assumes "summable (\<lambda>k. norm (a k))"
     and "summable (\<lambda>k. norm (b k))"
   shows "summable (\<lambda>k. \<Sum>i\<le>k. a i * b (k - i))"
   using Cauchy_product_sums[OF assms] by (simp add: sums_iff)
 
 
 subsection \<open>Series on \<^typ>\<open>real\<close>s\<close>
 
 lemma summable_norm_comparison_test:
   "\<exists>N. \<forall>n\<ge>N. norm (f n) \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. norm (f n))"
   by (rule summable_comparison_test) auto
 
 lemma summable_rabs_comparison_test: "\<exists>N. \<forall>n\<ge>N. \<bar>f n\<bar> \<le> g n \<Longrightarrow> summable g \<Longrightarrow> summable (\<lambda>n. \<bar>f n\<bar>)"
   for f :: "nat \<Rightarrow> real"
   by (rule summable_comparison_test) auto
 
 lemma summable_rabs_cancel: "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> summable f"
   for f :: "nat \<Rightarrow> real"
   by (rule summable_norm_cancel) simp
 
 lemma summable_rabs: "summable (\<lambda>n. \<bar>f n\<bar>) \<Longrightarrow> \<bar>suminf f\<bar> \<le> (\<Sum>n. \<bar>f n\<bar>)"
   for f :: "nat \<Rightarrow> real"
   by (fold real_norm_def) (rule summable_norm)
 
 lemma summable_zero_power [simp]: "summable (\<lambda>n. 0 ^ n :: 'a::{comm_ring_1,topological_space})"
 proof -
   have "(\<lambda>n. 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then 0^0 else 0)"
     by (intro ext) (simp add: zero_power)
   moreover have "summable \<dots>" by simp
   ultimately show ?thesis by simp
 qed
 
 lemma summable_zero_power' [simp]: "summable (\<lambda>n. f n * 0 ^ n :: 'a::{ring_1,topological_space})"
 proof -
   have "(\<lambda>n. f n * 0 ^ n :: 'a) = (\<lambda>n. if n = 0 then f 0 * 0^0 else 0)"
     by (intro ext) (simp add: zero_power)
   moreover have "summable \<dots>" by simp
   ultimately show ?thesis by simp
 qed
 
 lemma summable_power_series:
   fixes z :: real
   assumes le_1: "\<And>i. f i \<le> 1"
     and nonneg: "\<And>i. 0 \<le> f i"
     and z: "0 \<le> z" "z < 1"
   shows "summable (\<lambda>i. f i * z^i)"
 proof (rule summable_comparison_test[OF _ summable_geometric])
   show "norm z < 1"
     using z by (auto simp: less_imp_le)
   show "\<And>n. \<exists>N. \<forall>na\<ge>N. norm (f na * z ^ na) \<le> z ^ na"
     using z
     by (auto intro!: exI[of _ 0] mult_left_le_one_le simp: abs_mult nonneg power_abs less_imp_le le_1)
 qed
 
 lemma summable_0_powser: "summable (\<lambda>n. f n * 0 ^ n :: 'a::real_normed_div_algebra)"
 proof -
   have A: "(\<lambda>n. f n * 0 ^ n) = (\<lambda>n. if n = 0 then f n else 0)"
     by (intro ext) auto
   then show ?thesis
     by (subst A) simp_all
 qed
 
 lemma summable_powser_split_head:
   "summable (\<lambda>n. f (Suc n) * z ^ n :: 'a::real_normed_div_algebra) = summable (\<lambda>n. f n * z ^ n)"
 proof -
   have "summable (\<lambda>n. f (Suc n) * z ^ n) \<longleftrightarrow> summable (\<lambda>n. f (Suc n) * z ^ Suc n)"
     (is "?lhs \<longleftrightarrow> ?rhs")
   proof
     show ?rhs if ?lhs
       using summable_mult2[OF that, of z]
       by (simp add: power_commutes algebra_simps)
     show ?lhs if ?rhs
       using summable_mult2[OF that, of "inverse z"]
       by (cases "z \<noteq> 0", subst (asm) power_Suc2) (simp_all add: algebra_simps)
   qed
   also have "\<dots> \<longleftrightarrow> summable (\<lambda>n. f n * z ^ n)" by (rule summable_Suc_iff)
   finally show ?thesis .
 qed
 
 lemma summable_powser_ignore_initial_segment:
   fixes f :: "nat \<Rightarrow> 'a :: real_normed_div_algebra"
   shows "summable (\<lambda>n. f (n + m) * z ^ n) \<longleftrightarrow> summable (\<lambda>n. f n * z ^ n)"
 proof (induction m)
   case (Suc m)
   have "summable (\<lambda>n. f (n + Suc m) * z ^ n) = summable (\<lambda>n. f (Suc n + m) * z ^ n)"
     by simp
   also have "\<dots> = summable (\<lambda>n. f (n + m) * z ^ n)"
     by (rule summable_powser_split_head)
   also have "\<dots> = summable (\<lambda>n. f n * z ^ n)"
     by (rule Suc.IH)
   finally show ?case .
 qed simp_all
 
 lemma powser_split_head:
   fixes f :: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}"
   assumes "summable (\<lambda>n. f n * z ^ n)"
   shows "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z"
     and "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0"
     and "summable (\<lambda>n. f (Suc n) * z ^ n)"
 proof -
   from assms show "summable (\<lambda>n. f (Suc n) * z ^ n)"
     by (subst summable_powser_split_head)
   from suminf_mult2[OF this, of z]
     have "(\<Sum>n. f (Suc n) * z ^ n) * z = (\<Sum>n. f (Suc n) * z ^ Suc n)"
     by (simp add: power_commutes algebra_simps)
   also from assms have "\<dots> = suminf (\<lambda>n. f n * z ^ n) - f 0"
     by (subst suminf_split_head) simp_all
   finally show "suminf (\<lambda>n. f n * z ^ n) = f 0 + suminf (\<lambda>n. f (Suc n) * z ^ n) * z"
     by simp
   then show "suminf (\<lambda>n. f (Suc n) * z ^ n) * z = suminf (\<lambda>n. f n * z ^ n) - f 0"
     by simp
 qed
 
 lemma summable_partial_sum_bound:
   fixes f :: "nat \<Rightarrow> 'a :: banach"
     and e :: real
   assumes summable: "summable f"
     and e: "e > 0"
   obtains N where "\<And>m n. m \<ge> N \<Longrightarrow> norm (\<Sum>k=m..n. f k) < e"
 proof -
   from summable have "Cauchy (\<lambda>n. \<Sum>k<n. f k)"
     by (simp add: Cauchy_convergent_iff summable_iff_convergent)
   from CauchyD [OF this e] obtain N
     where N: "\<And>m n. m \<ge> N \<Longrightarrow> n \<ge> N \<Longrightarrow> norm ((\<Sum>k<m. f k) - (\<Sum>k<n. f k)) < e"
     by blast
   have "norm (\<Sum>k=m..n. f k) < e" if m: "m \<ge> N" for m n
   proof (cases "n \<ge> m")
     case True
     with m have "norm ((\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k)) < e"
       by (intro N) simp_all
     also from True have "(\<Sum>k<Suc n. f k) - (\<Sum>k<m. f k) = (\<Sum>k=m..n. f k)"
       by (subst sum_diff [symmetric]) (simp_all add: sum.last_plus)
     finally show ?thesis .
   next
     case False
     with e show ?thesis by simp_all
   qed
   then show ?thesis by (rule that)
 qed
 
 lemma powser_sums_if:
   "(\<lambda>n. (if n = m then (1 :: 'a::{ring_1,topological_space}) else 0) * z^n) sums z^m"
 proof -
   have "(\<lambda>n. (if n = m then 1 else 0) * z^n) = (\<lambda>n. if n = m then z^n else 0)"
     by (intro ext) auto
   then show ?thesis
     by (simp add: sums_single)
 qed
 
 lemma
   fixes f :: "nat \<Rightarrow> real"
   assumes "summable f"
     and "inj g"
     and pos: "\<And>x. 0 \<le> f x"
   shows summable_reindex: "summable (f \<circ> g)"
     and suminf_reindex_mono: "suminf (f \<circ> g) \<le> suminf f"
     and suminf_reindex: "(\<And>x. x \<notin> range g \<Longrightarrow> f x = 0) \<Longrightarrow> suminf (f \<circ> g) = suminf f"
 proof -
   from \<open>inj g\<close> have [simp]: "\<And>A. inj_on g A"
     by (rule subset_inj_on) simp
 
   have smaller: "\<forall>n. (\<Sum>i<n. (f \<circ> g) i) \<le> suminf f"
   proof
     fix n
     have "\<forall> n' \<in> (g ` {..<n}). n' < Suc (Max (g ` {..<n}))"
       by (metis Max_ge finite_imageI finite_lessThan not_le not_less_eq)
     then obtain m where n: "\<And>n'. n' < n \<Longrightarrow> g n' < m"
       by blast
 
     have "(\<Sum>i<n. f (g i)) = sum f (g ` {..<n})"
       by (simp add: sum.reindex)
     also have "\<dots> \<le> (\<Sum>i<m. f i)"
       by (rule sum_mono2) (auto simp add: pos n[rule_format])
     also have "\<dots> \<le> suminf f"
       using \<open>summable f\<close>
       by (rule sum_le_suminf) (simp_all add: pos)
     finally show "(\<Sum>i<n. (f \<circ>  g) i) \<le> suminf f"
       by simp
   qed
 
   have "incseq (\<lambda>n. \<Sum>i<n. (f \<circ> g) i)"
     by (rule incseq_SucI) (auto simp add: pos)
   then obtain  L where L: "(\<lambda> n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> L"
     using smaller by(rule incseq_convergent)
   then have "(f \<circ> g) sums L"
     by (simp add: sums_def)
   then show "summable (f \<circ> g)"
     by (auto simp add: sums_iff)
 
   then have "(\<lambda>n. \<Sum>i<n. (f \<circ> g) i) \<longlonglongrightarrow> suminf (f \<circ> g)"
     by (rule summable_LIMSEQ)
   then show le: "suminf (f \<circ> g) \<le> suminf f"
     by(rule LIMSEQ_le_const2)(blast intro: smaller[rule_format])
 
   assume f: "\<And>x. x \<notin> range g \<Longrightarrow> f x = 0"
 
   from \<open>summable f\<close> have "suminf f \<le> suminf (f \<circ> g)"
   proof (rule suminf_le_const)
     fix n
     have "\<forall> n' \<in> (g -` {..<n}). n' < Suc (Max (g -` {..<n}))"
       by(auto intro: Max_ge simp add: finite_vimageI less_Suc_eq_le)
     then obtain m where n: "\<And>n'. g n' < n \<Longrightarrow> n' < m"
       by blast
     have "(\<Sum>i<n. f i) = (\<Sum>i\<in>{..<n} \<inter> range g. f i)"
       using f by(auto intro: sum.mono_neutral_cong_right)
     also have "\<dots> = (\<Sum>i\<in>g -` {..<n}. (f \<circ> g) i)"
       by (rule sum.reindex_cong[where l=g])(auto)
     also have "\<dots> \<le> (\<Sum>i<m. (f \<circ> g) i)"
       by (rule sum_mono2)(auto simp add: pos n)
     also have "\<dots> \<le> suminf (f \<circ> g)"
       using \<open>summable (f \<circ> g)\<close> by (rule sum_le_suminf) (simp_all add: pos)
     finally show "sum f {..<n} \<le> suminf (f \<circ> g)" .
   qed
   with le show "suminf (f \<circ> g) = suminf f"
     by (rule antisym)
 qed
 
 lemma sums_mono_reindex:
   assumes subseq: "strict_mono g"
     and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
   shows "(\<lambda>n. f (g n)) sums c \<longleftrightarrow> f sums c"
   unfolding sums_def
 proof
   assume lim: "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c"
   have "(\<lambda>n. \<Sum>k<n. f (g k)) = (\<lambda>n. \<Sum>k<g n. f k)"
   proof
     fix n :: nat
     from subseq have "(\<Sum>k<n. f (g k)) = (\<Sum>k\<in>g`{..<n}. f k)"
       by (subst sum.reindex) (auto intro: strict_mono_imp_inj_on)
     also from subseq have "\<dots> = (\<Sum>k<g n. f k)"
       by (intro sum.mono_neutral_left ballI zero)
         (auto simp: strict_mono_less strict_mono_less_eq)
     finally show "(\<Sum>k<n. f (g k)) = (\<Sum>k<g n. f k)" .
   qed
   also from LIMSEQ_subseq_LIMSEQ[OF lim subseq] have "\<dots> \<longlonglongrightarrow> c"
     by (simp only: o_def)
   finally show "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c" .
 next
   assume lim: "(\<lambda>n. \<Sum>k<n. f (g k)) \<longlonglongrightarrow> c"
   define g_inv where "g_inv n = (LEAST m. g m \<ge> n)" for n
   from filterlim_subseq[OF subseq] have g_inv_ex: "\<exists>m. g m \<ge> n" for n
     by (auto simp: filterlim_at_top eventually_at_top_linorder)
   then have g_inv: "g (g_inv n) \<ge> n" for n
     unfolding g_inv_def by (rule LeastI_ex)
   have g_inv_least: "m \<ge> g_inv n" if "g m \<ge> n" for m n
     using that unfolding g_inv_def by (rule Least_le)
   have g_inv_least': "g m < n" if "m < g_inv n" for m n
     using that g_inv_least[of n m] by linarith
   have "(\<lambda>n. \<Sum>k<n. f k) = (\<lambda>n. \<Sum>k<g_inv n. f (g k))"
   proof
     fix n :: nat
     {
       fix k
       assume k: "k \<in> {..<n} - g`{..<g_inv n}"
       have "k \<notin> range g"
       proof (rule notI, elim imageE)
         fix l
         assume l: "k = g l"
         have "g l < g (g_inv n)"
           by (rule less_le_trans[OF _ g_inv]) (use k l in simp_all)
         with subseq have "l < g_inv n"
           by (simp add: strict_mono_less)
         with k l show False
           by simp
       qed
       then have "f k = 0"
         by (rule zero)
     }
     with g_inv_least' g_inv have "(\<Sum>k<n. f k) = (\<Sum>k\<in>g`{..<g_inv n}. f k)"
       by (intro sum.mono_neutral_right) auto
     also from subseq have "\<dots> = (\<Sum>k<g_inv n. f (g k))"
       using strict_mono_imp_inj_on by (subst sum.reindex) simp_all
     finally show "(\<Sum>k<n. f k) = (\<Sum>k<g_inv n. f (g k))" .
   qed
   also {
     fix K n :: nat
     assume "g K \<le> n"
     also have "n \<le> g (g_inv n)"
       by (rule g_inv)
     finally have "K \<le> g_inv n"
       using subseq by (simp add: strict_mono_less_eq)
   }
   then have "filterlim g_inv at_top sequentially"
     by (auto simp: filterlim_at_top eventually_at_top_linorder)
   with lim have "(\<lambda>n. \<Sum>k<g_inv n. f (g k)) \<longlonglongrightarrow> c"
     by (rule filterlim_compose)
   finally show "(\<lambda>n. \<Sum>k<n. f k) \<longlonglongrightarrow> c" .
 qed
 
 lemma summable_mono_reindex:
   assumes subseq: "strict_mono g"
     and zero: "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
   shows "summable (\<lambda>n. f (g n)) \<longleftrightarrow> summable f"
   using sums_mono_reindex[of g f, OF assms] by (simp add: summable_def)
 
 lemma suminf_mono_reindex:
   fixes f :: "nat \<Rightarrow> 'a::{t2_space,comm_monoid_add}"
   assumes "strict_mono g" "\<And>n. n \<notin> range g \<Longrightarrow> f n = 0"
   shows   "suminf (\<lambda>n. f (g n)) = suminf f"
 proof (cases "summable f")
   case True
   with sums_mono_reindex [of g f, OF assms]
     and summable_mono_reindex [of g f, OF assms]
   show ?thesis
     by (simp add: sums_iff)
 next
   case False
   then have "\<not>(\<exists>c. f sums c)"
     unfolding summable_def by blast
   then have "suminf f = The (\<lambda>_. False)"
     by (simp add: suminf_def)
   moreover from False have "\<not> summable (\<lambda>n. f (g n))"
     using summable_mono_reindex[of g f, OF assms] by simp
   then have "\<not>(\<exists>c. (\<lambda>n. f (g n)) sums c)"
     unfolding summable_def by blast
   then have "suminf (\<lambda>n. f (g n)) = The (\<lambda>_. False)"
     by (simp add: suminf_def)
   ultimately show ?thesis by simp
 qed
 
 lemma summable_bounded_partials:
   fixes f :: "nat \<Rightarrow> 'a :: {real_normed_vector,complete_space}"
   assumes bound: "eventually (\<lambda>x0. \<forall>a\<ge>x0. \<forall>b>a. norm (sum f {a<..b}) \<le> g a) sequentially"
   assumes g: "g \<longlonglongrightarrow> 0"
   shows   "summable f" unfolding summable_iff_convergent'
 proof (intro Cauchy_convergent CauchyI', goal_cases)
   case (1 \<epsilon>)
   with g have "eventually (\<lambda>x. \<bar>g x\<bar> < \<epsilon>) sequentially"
     by (auto simp: tendsto_iff)
   from eventually_conj[OF this bound] obtain x0 where x0:
     "\<And>x. x \<ge> x0 \<Longrightarrow> \<bar>g x\<bar> < \<epsilon>" "\<And>a b. x0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> norm (sum f {a<..b}) \<le> g a" 
     unfolding eventually_at_top_linorder by auto
   show ?case
   proof (intro exI[of _ x0] allI impI)
     fix m n assume mn: "x0 \<le> m" "m < n"
     have "dist (sum f {..m}) (sum f {..n}) = norm (sum f {..n} - sum f {..m})"
       by (simp add: dist_norm norm_minus_commute)
     also have "sum f {..n} - sum f {..m} = sum f ({..n} - {..m})"
       using mn by (intro Groups_Big.sum_diff [symmetric]) auto
     also have "{..n} - {..m} = {m<..n}" using mn by auto
     also have "norm (sum f {m<..n}) \<le> g m" using mn by (intro x0) auto
     also have "\<dots> \<le> \<bar>g m\<bar>" by simp
     also have "\<dots> < \<epsilon>" using mn by (intro x0) auto
     finally show "dist (sum f {..m}) (sum f {..n}) < \<epsilon>" .
   qed
 qed
 
 end