src/HOL/Multivariate_Analysis/Complex_Analysis_Basics.thy

New complex analysis material

(*  Author: John Harrison, Marco Maggesi, Graziano Gentili, Gianni Ciolli, Valentina Bruno
    Ported from "hol_light/Multivariate/canal.ml" by L C Paulson (2014)
*)

header {* Complex Analysis Basics *}

theory Complex_Analysis_Basics
imports  "~~/src/HOL/Multivariate_Analysis/Cartesian_Euclidean_Space"

begin

subsection {*Complex number lemmas *}

lemma abs_sqrt_wlog:
  fixes x::"'a::linordered_idom"
  assumes "!!x::'a. x\<ge>0 \<Longrightarrow> P x (x\<^sup>2)" shows "P \<bar>x\<bar> (x\<^sup>2)"
by (metis abs_ge_zero assms power2_abs)

lemma complex_abs_le_norm: "abs(Re z) + abs(Im z) \<le> sqrt(2) * norm z"
proof (cases z)
  case (Complex x y)
  show ?thesis
    apply (rule power2_le_imp_le)
    apply (auto simp: real_sqrt_mult [symmetric] Complex)
    apply (rule abs_sqrt_wlog [where x=x])
    apply (rule abs_sqrt_wlog [where x=y])
    apply (simp add: power2_sum add_commute sum_squares_bound)
    done
qed

lemma continuous_Re: "continuous_on UNIV Re"
  by (metis (poly_guards_query) bounded_linear.continuous_on bounded_linear_Re 
            continuous_on_cong continuous_on_id)

lemma continuous_Im: "continuous_on UNIV Im"
  by (metis (poly_guards_query) bounded_linear.continuous_on bounded_linear_Im 
            continuous_on_cong continuous_on_id)

lemma open_halfspace_Re_lt: "open {z. Re(z) < b}"
proof -
  have "{z. Re(z) < b} = Re -`{..<b}"
    by blast
  then show ?thesis
    by (auto simp: continuous_Re continuous_imp_open_vimage [of UNIV])
qed

lemma open_halfspace_Re_gt: "open {z. Re(z) > b}"
proof -
  have "{z. Re(z) > b} = Re -`{b<..}"
    by blast
  then show ?thesis
    by (auto simp: continuous_Re continuous_imp_open_vimage [of UNIV])
qed

lemma closed_halfspace_Re_ge: "closed {z. Re(z) \<ge> b}"
proof -
  have "{z. Re(z) \<ge> b} = - {z. Re(z) < b}"
    by auto
  then show ?thesis
    by (simp add: closed_def open_halfspace_Re_lt)
qed

lemma closed_halfspace_Re_le: "closed {z. Re(z) \<le> b}"
proof -
  have "{z. Re(z) \<le> b} = - {z. Re(z) > b}"
    by auto
  then show ?thesis
    by (simp add: closed_def open_halfspace_Re_gt)
qed

lemma closed_halfspace_Re_eq: "closed {z. Re(z) = b}"
proof -
  have "{z. Re(z) = b} = {z. Re(z) \<le> b} \<inter> {z. Re(z) \<ge> b}"
    by auto
  then show ?thesis
    by (auto simp: closed_Int closed_halfspace_Re_le closed_halfspace_Re_ge)
qed

lemma open_halfspace_Im_lt: "open {z. Im(z) < b}"
proof -
  have "{z. Im(z) < b} = Im -`{..<b}"
    by blast
  then show ?thesis
    by (auto simp: continuous_Im continuous_imp_open_vimage [of UNIV])
qed

lemma open_halfspace_Im_gt: "open {z. Im(z) > b}"
proof -
  have "{z. Im(z) > b} = Im -`{b<..}"
    by blast
  then show ?thesis
    by (auto simp: continuous_Im continuous_imp_open_vimage [of UNIV])
qed

lemma closed_halfspace_Im_ge: "closed {z. Im(z) \<ge> b}"
proof -
  have "{z. Im(z) \<ge> b} = - {z. Im(z) < b}"
    by auto
  then show ?thesis
    by (simp add: closed_def open_halfspace_Im_lt)
qed

lemma closed_halfspace_Im_le: "closed {z. Im(z) \<le> b}"
proof -
  have "{z. Im(z) \<le> b} = - {z. Im(z) > b}"
    by auto
  then show ?thesis
    by (simp add: closed_def open_halfspace_Im_gt)
qed

lemma closed_halfspace_Im_eq: "closed {z. Im(z) = b}"
proof -
  have "{z. Im(z) = b} = {z. Im(z) \<le> b} \<inter> {z. Im(z) \<ge> b}"
    by auto
  then show ?thesis
    by (auto simp: closed_Int closed_halfspace_Im_le closed_halfspace_Im_ge)
qed

lemma complex_is_Real_iff: "z \<in> \<real> \<longleftrightarrow> Im z = 0"
  by (metis Complex_in_Reals Im_complex_of_real Reals_cases complex_surj)

lemma closed_complex_Reals: "closed (Reals :: complex set)"
proof -
  have "-(Reals :: complex set) = {z. Im(z) < 0} \<union> {z. 0 < Im(z)}"
    by (auto simp: complex_is_Real_iff)
  then show ?thesis
    by (metis closed_def open_Un open_halfspace_Im_gt open_halfspace_Im_lt)
qed


lemma linear_times:
  fixes c::"'a::{real_algebra,real_vector}" shows "linear (\<lambda>x. c * x)"
  by (auto simp: linearI distrib_left)

lemma bilinear_times:
  fixes c::"'a::{real_algebra,real_vector}" shows "bilinear (\<lambda>x y::'a. x*y)"
  unfolding bilinear_def
  by (auto simp: distrib_left distrib_right intro!: linearI)

lemma linear_cnj: "linear cnj"
  by (auto simp: linearI cnj_def)

lemma tendsto_mult_left:
  fixes c::"'a::real_normed_algebra" 
  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. c * (f x)) ---> c * l) F"
by (rule tendsto_mult [OF tendsto_const])

lemma tendsto_mult_right:
  fixes c::"'a::real_normed_algebra" 
  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. (f x) * c) ---> l * c) F"
by (rule tendsto_mult [OF _ tendsto_const])

lemma tendsto_Re_upper:
  assumes "~ (trivial_limit F)" 
          "(f ---> l) F" 
          "eventually (\<lambda>x. Re(f x) \<le> b) F"
    shows  "Re(l) \<le> b"
  by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Re)

lemma tendsto_Re_lower:
  assumes "~ (trivial_limit F)" 
          "(f ---> l) F" 
          "eventually (\<lambda>x. b \<le> Re(f x)) F"
    shows  "b \<le> Re(l)"
  by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Re)

lemma tendsto_Im_upper:
  assumes "~ (trivial_limit F)" 
          "(f ---> l) F" 
          "eventually (\<lambda>x. Im(f x) \<le> b) F"
    shows  "Im(l) \<le> b"
  by (metis assms tendsto_le [OF _ tendsto_const]  tendsto_Im)

lemma tendsto_Im_lower:
  assumes "~ (trivial_limit F)" 
          "(f ---> l) F" 
          "eventually (\<lambda>x. b \<le> Im(f x)) F"
    shows  "b \<le> Im(l)"
  by (metis assms tendsto_le [OF _ _ tendsto_const]  tendsto_Im)

subsection{*General lemmas*}

lemma continuous_mult_left:
  fixes c::"'a::real_normed_algebra" 
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. c * f x)"
by (rule continuous_mult [OF continuous_const])

lemma continuous_mult_right:
  fixes c::"'a::real_normed_algebra" 
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. f x * c)"
by (rule continuous_mult [OF _ continuous_const])

lemma continuous_on_mult_left:
  fixes c::"'a::real_normed_algebra" 
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. c * f x)"
by (rule continuous_on_mult [OF continuous_on_const])

lemma continuous_on_mult_right:
  fixes c::"'a::real_normed_algebra" 
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. f x * c)"
by (rule continuous_on_mult [OF _ continuous_on_const])

lemma uniformly_continuous_on_cmul_right [continuous_on_intros]:
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
  assumes "uniformly_continuous_on s f"
  shows "uniformly_continuous_on s (\<lambda>x. f x * c)"
proof (cases "c=0")
  case True then show ?thesis
    by (simp add: uniformly_continuous_on_const)
next
  case False show ?thesis
    apply (rule bounded_linear.uniformly_continuous_on)
    apply (metis bounded_linear_ident)
    using assms
    apply (auto simp: uniformly_continuous_on_def dist_norm)
    apply (drule_tac x = "e / norm c" in spec, auto)
    apply (metis divide_pos_pos zero_less_norm_iff False)
    apply (rule_tac x=d in exI, auto)
    apply (drule_tac x = x in bspec, assumption)
    apply (drule_tac x = "x'" in bspec)
    apply (auto simp: False less_divide_eq)
    by (metis dual_order.strict_trans2 left_diff_distrib norm_mult_ineq)
qed

lemma uniformly_continuous_on_cmul_left[continuous_on_intros]:
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
  assumes "uniformly_continuous_on s f"
    shows "uniformly_continuous_on s (\<lambda>x. c * f x)"
by (metis assms bounded_linear.uniformly_continuous_on bounded_linear_mult_right)

lemma continuous_within_norm_id [continuous_intros]: "continuous (at x within S) norm"
  by (rule continuous_norm [OF continuous_ident])

lemma continuous_on_norm_id [continuous_intros]: "continuous_on S norm"
  by (metis continuous_on_eq continuous_on_id continuous_on_norm)


subsection{*DERIV stuff*}

(*move some premises to a sensible order. Use more \<And> symbols.*)

lemma DERIV_continuous_on: "\<lbrakk>\<And>x. x \<in> s \<Longrightarrow> DERIV f x :> D\<rbrakk> \<Longrightarrow> continuous_on s f"
  by (metis DERIV_continuous continuous_at_imp_continuous_on)

lemma DERIV_subset: 
  "(f has_field_derivative f') (at x within s) \<Longrightarrow> t \<subseteq> s 
   \<Longrightarrow> (f has_field_derivative f') (at x within t)"
  by (simp add: has_field_derivative_def has_derivative_within_subset)

lemma lambda_zero: "(\<lambda>h::'a::mult_zero. 0) = op * 0"
  by auto

lemma lambda_one: "(\<lambda>x::'a::monoid_mult. x) = op * 1"
  by auto

lemma has_derivative_zero_constant:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  assumes "convex s"
      and d0: "\<And>x. x\<in>s \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within s)"
    shows "\<exists>c. \<forall>x\<in>s. f x = c"
proof (cases "s={}")
  case False
  then obtain x where "x \<in> s"
    by auto
  have d0': "\<forall>x\<in>s. (f has_derivative (\<lambda>h. 0)) (at x within s)"
    by (metis d0)
  have "\<And>y. y \<in> s \<Longrightarrow> f x = f y"
  proof -
    case goal1
    then show ?case
      using differentiable_bound[OF assms(1) d0', of 0 x y] and `x \<in> s`
      unfolding onorm_const
      by auto
  qed
  then show ?thesis 
    by metis
next
  case True
  then show ?thesis by auto
qed

lemma has_derivative_zero_unique:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
  assumes "convex s"
      and "\<And>x. x\<in>s \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within s)"
      and "a \<in> s"
      and "x \<in> s"
    shows "f x = f a"
  using assms
  by (metis has_derivative_zero_constant)

lemma has_derivative_zero_connected_constant:
  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
  assumes "connected s"
      and "open s"
      and "finite k"
      and "continuous_on s f"
      and "\<forall>x\<in>(s - k). (f has_derivative (\<lambda>h. 0)) (at x within s)"
    obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c"
proof (cases "s = {}")
  case True
  then show ?thesis
by (metis empty_iff that)
next
  case False
  then obtain c where "c \<in> s"
    by (metis equals0I)
  then show ?thesis
    by (metis has_derivative_zero_unique_strong_connected assms that)
qed

lemma DERIV_zero_connected_constant:
  fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
  assumes "connected s"
      and "open s"
      and "finite k"
      and "continuous_on s f"
      and "\<forall>x\<in>(s - k). DERIV f x :> 0"
    obtains c where "\<And>x. x \<in> s \<Longrightarrow> f(x) = c"
using has_derivative_zero_connected_constant [OF assms(1-4)] assms
by (metis DERIV_const Derivative.has_derivative_const Diff_iff at_within_open 
          frechet_derivative_at has_field_derivative_def)

lemma DERIV_zero_constant:
  fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
  shows    "\<lbrakk>convex s;
             \<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)\<rbrakk> 
             \<Longrightarrow> \<exists>c. \<forall>x \<in> s. f(x) = c"
  unfolding has_field_derivative_def
  by (auto simp: lambda_zero intro: has_derivative_zero_constant)

lemma DERIV_zero_unique:
  fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
  assumes "convex s"
      and d0: "\<And>x. x\<in>s \<Longrightarrow> (f has_field_derivative 0) (at x within s)"
      and "a \<in> s"
      and "x \<in> s"
    shows "f x = f a"
apply (rule has_derivative_zero_unique [where f=f, OF assms(1) _ assms(3,4)])
by (metis d0 has_field_derivative_imp_has_derivative lambda_zero)

lemma DERIV_zero_connected_unique:
  fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
  assumes "connected s"
      and "open s"
      and d0: "\<And>x. x\<in>s \<Longrightarrow> DERIV f x :> 0"
      and "a \<in> s"
      and "x \<in> s"
    shows "f x = f a" 
    apply (rule Integration.has_derivative_zero_unique_strong_connected [of s "{}" f])
    using assms
    apply auto
    apply (metis DERIV_continuous_on)
  by (metis at_within_open has_field_derivative_def lambda_zero)

lemma DERIV_transform_within:
  assumes "(f has_field_derivative f') (at a within s)"
      and "0 < d" "a \<in> s"
      and "\<And>x. x\<in>s \<Longrightarrow> dist x a < d \<Longrightarrow> f x = g x"
    shows "(g has_field_derivative f') (at a within s)"
  using assms unfolding has_field_derivative_def
  by (blast intro: Derivative.has_derivative_transform_within)

lemma DERIV_transform_within_open:
  assumes "DERIV f a :> f'"
      and "open s" "a \<in> s"
      and "\<And>x. x\<in>s \<Longrightarrow> f x = g x"
    shows "DERIV g a :> f'"
  using assms unfolding has_field_derivative_def
by (metis has_derivative_transform_within_open)

lemma DERIV_transform_at:
  assumes "DERIV f a :> f'"
      and "0 < d"
      and "\<And>x. dist x a < d \<Longrightarrow> f x = g x"
    shows "DERIV g a :> f'"
  by (blast intro: assms DERIV_transform_within)


subsection{*Holomorphic functions*}

lemma has_derivative_ident[has_derivative_intros, simp]: 
     "FDERIV complex_of_real x :> complex_of_real"
  by (simp add: has_derivative_def tendsto_const bounded_linear_of_real)

lemma has_real_derivative:
  fixes f :: "real\<Rightarrow>real" 
  assumes "(f has_derivative f') F"
    obtains c where "(f has_derivative (\<lambda>x. x * c)) F"
proof -
  obtain c where "f' = (\<lambda>x. x * c)"
    by (metis assms derivative_linear real_bounded_linear)
  then show ?thesis
    by (metis assms that)
qed

lemma has_real_derivative_iff:
  fixes f :: "real\<Rightarrow>real" 
  shows "(\<exists>f'. (f has_derivative (\<lambda>x. x * f')) F) = (\<exists>D. (f has_derivative D) F)"
by (auto elim: has_real_derivative)

definition complex_differentiable :: "[complex \<Rightarrow> complex, complex filter] \<Rightarrow> bool"
           (infixr "(complex'_differentiable)" 50)  
  where "f complex_differentiable F \<equiv> \<exists>f'. (f has_field_derivative f') F"

definition DD :: "['a \<Rightarrow> 'a::real_normed_field, 'a] \<Rightarrow> 'a" --{*for real, complex?*}
  where "DD f x \<equiv> THE f'. (f has_derivative (\<lambda>x. x * f')) (at x)"

definition holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
           (infixl "(holomorphic'_on)" 50)
  where "f holomorphic_on s \<equiv> \<forall>x \<in> s. \<exists>f'. (f has_field_derivative f') (at x within s)"
  
lemma holomorphic_on_empty: "f holomorphic_on {}"
  by (simp add: holomorphic_on_def)

lemma holomorphic_on_differentiable:
     "f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. f complex_differentiable (at x within s))"
unfolding holomorphic_on_def complex_differentiable_def has_field_derivative_def
by (metis mult_commute_abs)

lemma holomorphic_on_open:
    "open s \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>f'. DERIV f x :> f')"
  by (auto simp: holomorphic_on_def has_field_derivative_def at_within_open [of _ s])

lemma complex_differentiable_imp_continuous_at: 
    "f complex_differentiable (at x) \<Longrightarrow> continuous (at x) f"
  by (metis DERIV_continuous complex_differentiable_def)

lemma holomorphic_on_imp_continuous_on: 
    "f holomorphic_on s \<Longrightarrow> continuous_on s f"
  by (metis DERIV_continuous continuous_on_eq_continuous_within holomorphic_on_def) 

lemma has_derivative_within_open:
  "a \<in> s \<Longrightarrow> open s \<Longrightarrow> (f has_field_derivative f') (at a within s) \<longleftrightarrow> DERIV f a :> f'"
  by (simp add: has_field_derivative_def) (metis has_derivative_within_open)

lemma holomorphic_on_subset:
    "f holomorphic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f holomorphic_on t"
  unfolding holomorphic_on_def
  by (metis DERIV_subset subsetD)

lemma complex_differentiable_within_subset:
    "\<lbrakk>f complex_differentiable (at x within s); t \<subseteq> s\<rbrakk>
     \<Longrightarrow> f complex_differentiable (at x within t)"
  unfolding complex_differentiable_def
  by (metis DERIV_subset)

lemma complex_differentiable_at_within:
    "\<lbrakk>f complex_differentiable (at x)\<rbrakk>
     \<Longrightarrow> f complex_differentiable (at x within s)"
  unfolding complex_differentiable_def
  by (metis DERIV_subset top_greatest)


lemma has_derivative_mult_right:
  fixes c:: "'a :: real_normed_algebra"
  shows "((op * c) has_derivative (op * c)) F"
by (rule has_derivative_mult_right [OF has_derivative_id])

lemma complex_differentiable_linear:
     "(op * c) complex_differentiable F"
proof -
  have "\<And>u::complex. (\<lambda>x. x * u) = op * u"
    by (rule ext) (simp add: mult_ac)
  then show ?thesis
    unfolding complex_differentiable_def has_field_derivative_def
    by (force intro: has_derivative_mult_right)
qed

lemma complex_differentiable_const:
  "(\<lambda>z. c) complex_differentiable F"
  unfolding complex_differentiable_def has_field_derivative_def
  apply (rule exI [where x=0])
  by (metis Derivative.has_derivative_const lambda_zero) 

lemma complex_differentiable_id:
  "(\<lambda>z. z) complex_differentiable F"
  unfolding complex_differentiable_def has_field_derivative_def
  apply (rule exI [where x=1])
  apply (simp add: lambda_one [symmetric])
  done

(*DERIV_minus*)
lemma field_differentiable_minus:
  assumes "(f has_field_derivative f') F" 
    shows "((\<lambda>z. - (f z)) has_field_derivative -f') F"
  apply (rule has_derivative_imp_has_field_derivative[OF has_derivative_minus])
  using assms 
  by (auto simp: has_field_derivative_def field_simps mult_commute_abs)

(*DERIV_add*)
lemma field_differentiable_add:
  assumes "(f has_field_derivative f') F" "(g has_field_derivative g') F"
    shows "((\<lambda>z. f z + g z) has_field_derivative f' + g') F"
  apply (rule has_derivative_imp_has_field_derivative[OF has_derivative_add])
  using assms 
  by (auto simp: has_field_derivative_def field_simps mult_commute_abs)

(*DERIV_diff*)
lemma field_differentiable_diff:
  assumes "(f has_field_derivative f') F" "(g has_field_derivative g') F"
    shows "((\<lambda>z. f z - g z) has_field_derivative f' - g') F"
by (simp only: assms diff_conv_add_uminus field_differentiable_add field_differentiable_minus)

lemma complex_differentiable_minus:
    "f complex_differentiable F \<Longrightarrow> (\<lambda>z. -(f z)) complex_differentiable F"
  using assms unfolding complex_differentiable_def
  by (metis field_differentiable_minus)

lemma complex_differentiable_add:
  assumes "f complex_differentiable F" "g complex_differentiable F"
    shows "(\<lambda>z. f z + g z) complex_differentiable F"
  using assms unfolding complex_differentiable_def
  by (metis field_differentiable_add)

lemma complex_differentiable_diff:
  assumes "f complex_differentiable F" "g complex_differentiable F"
    shows "(\<lambda>z. f z - g z) complex_differentiable F"
  using assms unfolding complex_differentiable_def
  by (metis field_differentiable_diff)

lemma complex_differentiable_inverse:
  assumes "f complex_differentiable (at a within s)" "f a \<noteq> 0"
  shows "(\<lambda>z. inverse (f z)) complex_differentiable (at a within s)"
  using assms unfolding complex_differentiable_def
  by (metis DERIV_inverse_fun)

lemma complex_differentiable_mult:
  assumes "f complex_differentiable (at a within s)" 
          "g complex_differentiable (at a within s)"
    shows "(\<lambda>z. f z * g z) complex_differentiable (at a within s)"
  using assms unfolding complex_differentiable_def
  by (metis DERIV_mult [of f _ a s g])
  
lemma complex_differentiable_divide:
  assumes "f complex_differentiable (at a within s)" 
          "g complex_differentiable (at a within s)"
          "g a \<noteq> 0"
    shows "(\<lambda>z. f z / g z) complex_differentiable (at a within s)"
  using assms unfolding complex_differentiable_def
  by (metis DERIV_divide [of f _ a s g])

lemma complex_differentiable_power:
  assumes "f complex_differentiable (at a within s)" 
    shows "(\<lambda>z. f z ^ n) complex_differentiable (at a within s)"
  using assms unfolding complex_differentiable_def
  by (metis DERIV_power)

lemma complex_differentiable_transform_within:
  "0 < d \<Longrightarrow>
        x \<in> s \<Longrightarrow>
        (\<And>x'. x' \<in> s \<Longrightarrow> dist x' x < d \<Longrightarrow> f x' = g x') \<Longrightarrow>
        f complex_differentiable (at x within s)
        \<Longrightarrow> g complex_differentiable (at x within s)"
  unfolding complex_differentiable_def has_field_derivative_def
  by (blast intro: has_derivative_transform_within)

lemma complex_differentiable_compose_within:
  assumes "f complex_differentiable (at a within s)" 
          "g complex_differentiable (at (f a) within f`s)"
    shows "(g o f) complex_differentiable (at a within s)"
  using assms unfolding complex_differentiable_def
  by (metis DERIV_image_chain)

lemma complex_differentiable_within_open:
     "\<lbrakk>a \<in> s; open s\<rbrakk> \<Longrightarrow> f complex_differentiable at a within s \<longleftrightarrow> 
                          f complex_differentiable at a"
  unfolding complex_differentiable_def
  by (metis at_within_open)

lemma holomorphic_transform:
     "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s"
  apply (auto simp: holomorphic_on_def has_field_derivative_def)
  by (metis complex_differentiable_def complex_differentiable_transform_within has_field_derivative_def linordered_field_no_ub)

lemma holomorphic_eq:
     "(\<And>x. x \<in> s \<Longrightarrow> f x = g x) \<Longrightarrow> f holomorphic_on s \<longleftrightarrow> g holomorphic_on s"
  by (metis holomorphic_transform)

subsection{*Holomorphic*}

lemma holomorphic_on_linear:
     "(op * c) holomorphic_on s"
  unfolding holomorphic_on_def  by (metis DERIV_cmult_Id)

lemma holomorphic_on_const:
     "(\<lambda>z. c) holomorphic_on s"
  unfolding holomorphic_on_def  
  by (metis DERIV_const)

lemma holomorphic_on_id:
     "id holomorphic_on s"
  unfolding holomorphic_on_def id_def  
  by (metis DERIV_ident)

lemma holomorphic_on_compose:
  "f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s)
           \<Longrightarrow> (g o f) holomorphic_on s"
  unfolding holomorphic_on_def
  by (metis DERIV_image_chain imageI)

lemma holomorphic_on_compose_gen:
  "\<lbrakk>f holomorphic_on s; g holomorphic_on t; f ` s \<subseteq> t\<rbrakk> \<Longrightarrow> (g o f) holomorphic_on s"
  unfolding holomorphic_on_def
  by (metis DERIV_image_chain DERIV_subset image_subset_iff)

lemma holomorphic_on_minus:
  "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
  unfolding holomorphic_on_def
by (metis DERIV_minus)

lemma holomorphic_on_add:
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"
  unfolding holomorphic_on_def
  by (metis DERIV_add)

lemma holomorphic_on_diff:
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z - g z) holomorphic_on s"
  unfolding holomorphic_on_def
  by (metis DERIV_diff)

lemma holomorphic_on_mult:
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z * g z) holomorphic_on s"
  unfolding holomorphic_on_def
  by auto (metis DERIV_mult)

lemma holomorphic_on_inverse:
  "\<lbrakk>f holomorphic_on s; \<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. inverse (f z)) holomorphic_on s"
  unfolding holomorphic_on_def
  by (metis DERIV_inverse')

lemma holomorphic_on_divide:
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s; \<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>z. f z / g z) holomorphic_on s"
  unfolding holomorphic_on_def
  by (metis (full_types) DERIV_divide)

lemma holomorphic_on_power:
  "f holomorphic_on s \<Longrightarrow> (\<lambda>z. (f z)^n) holomorphic_on s"
  unfolding holomorphic_on_def
  by (metis DERIV_power)

lemma holomorphic_on_setsum:
  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s)
           \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) holomorphic_on s"
  unfolding holomorphic_on_def
  apply (induct I rule: finite_induct) 
  apply (force intro: DERIV_const DERIV_add)+
  done

lemma DERIV_imp_DD: "DERIV f x :> f' \<Longrightarrow> DD f x = f'"
    apply (simp add: DD_def has_field_derivative_def mult_commute_abs)
    apply (rule the_equality, assumption)
    apply (metis DERIV_unique has_field_derivative_def)
    done

lemma DD_iff_derivative_differentiable:
  fixes f :: "real\<Rightarrow>real" 
  shows   "DERIV f x :> DD f x \<longleftrightarrow> f differentiable at x"
unfolding DD_def differentiable_def 
by (metis (full_types) DD_def DERIV_imp_DD has_field_derivative_def has_real_derivative_iff 
          mult_commute_abs)

lemma DD_iff_derivative_complex_differentiable:
  fixes f :: "complex\<Rightarrow>complex" 
  shows "DERIV f x :> DD f x \<longleftrightarrow> f complex_differentiable at x"
unfolding DD_def complex_differentiable_def
by (metis DD_def DERIV_imp_DD)

lemma complex_differentiable_compose:
  "f complex_differentiable at z \<Longrightarrow> g complex_differentiable at (f z)
          \<Longrightarrow> (g o f) complex_differentiable at z"
by (metis complex_differentiable_at_within complex_differentiable_compose_within)

lemma complex_derivative_chain:
  fixes z::complex
  shows
  "f complex_differentiable at z \<Longrightarrow> g complex_differentiable at (f z)
          \<Longrightarrow> DD (g o f) z = DD g (f z) * DD f z"
by (metis DD_iff_derivative_complex_differentiable DERIV_chain DERIV_imp_DD)

lemma comp_derivative_chain:
  fixes z::real
  shows "\<lbrakk>f differentiable at z; g differentiable at (f z)\<rbrakk> 
         \<Longrightarrow> DD (g o f) z = DD g (f z) * DD f z"
by (metis DD_iff_derivative_differentiable DERIV_chain DERIV_imp_DD)

lemma complex_derivative_linear: "DD (\<lambda>w. c * w) = (\<lambda>z. c)"
by (metis DERIV_imp_DD DERIV_cmult_Id)

lemma complex_derivative_ident: "DD (\<lambda>w. w) = (\<lambda>z. 1)"
by (metis DERIV_imp_DD DERIV_ident)

lemma complex_derivative_const: "DD (\<lambda>w. c) = (\<lambda>z. 0)"
by (metis DERIV_imp_DD DERIV_const)

lemma complex_derivative_add:
  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>  
   \<Longrightarrow> DD (\<lambda>w. f w + g w) z = DD f z + DD g z"
  unfolding complex_differentiable_def
  by (rule DERIV_imp_DD) (metis (poly_guards_query) DERIV_add DERIV_imp_DD)  

lemma complex_derivative_diff:
  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>  
   \<Longrightarrow> DD (\<lambda>w. f w - g w) z = DD f z - DD g z"
  unfolding complex_differentiable_def
  by (rule DERIV_imp_DD) (metis (poly_guards_query) DERIV_diff DERIV_imp_DD)

lemma complex_derivative_mult:
  "\<lbrakk>f complex_differentiable at z; g complex_differentiable at z\<rbrakk>  
   \<Longrightarrow> DD (\<lambda>w. f w * g w) z = f z * DD g z + DD f z * g z"
  unfolding complex_differentiable_def
  by (rule DERIV_imp_DD) (metis DERIV_imp_DD DERIV_mult')

lemma complex_derivative_cmult:
  "f complex_differentiable at z \<Longrightarrow> DD (\<lambda>w. c * f w) z = c * DD f z"
  unfolding complex_differentiable_def
  by (metis DERIV_cmult DERIV_imp_DD)

lemma complex_derivative_cmult_right:
  "f complex_differentiable at z \<Longrightarrow> DD (\<lambda>w. f w * c) z = DD f z * c"
  unfolding complex_differentiable_def
  by (metis DERIV_cmult_right DERIV_imp_DD)

lemma complex_derivative_transform_within_open:
  "\<lbrakk>f holomorphic_on s; g holomorphic_on s; open s; z \<in> s; \<And>w. w \<in> s \<Longrightarrow> f w = g w\<rbrakk> 
   \<Longrightarrow> DD f z = DD g z"
  unfolding holomorphic_on_def
  by (rule DERIV_imp_DD) (metis has_derivative_within_open DERIV_imp_DD DERIV_transform_within_open)

lemma complex_derivative_compose_linear:
    "f complex_differentiable at (c * z) \<Longrightarrow> DD (\<lambda>w. f (c * w)) z = c * DD f (c * z)"
apply (rule DERIV_imp_DD)
apply (simp add: DD_iff_derivative_complex_differentiable [symmetric])
apply (metis DERIV_chain' DERIV_cmult_Id comm_semiring_1_class.normalizing_semiring_rules(7))  
done

subsection{*Caratheodory characterization.*}

(*REPLACE the original version. BUT IN WHICH FILE??*)
thm Deriv.CARAT_DERIV

lemma complex_differentiable_caratheodory_at:
  "f complex_differentiable (at z) \<longleftrightarrow>
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z) g)"
  using CARAT_DERIV [of f]
  by (simp add: complex_differentiable_def has_field_derivative_def)

lemma complex_differentiable_caratheodory_within:
  "f complex_differentiable (at z within s) \<longleftrightarrow>
         (\<exists>g. (\<forall>w. f(w) - f(z) = g(w) * (w - z)) \<and> continuous (at z within s) g)"
  using DERIV_caratheodory_within [of f]
  by (simp add: complex_differentiable_def has_field_derivative_def)

subsection{*analyticity on a set*}

definition analytic_on (infixl "(analytic'_on)" 50)  
  where
   "f analytic_on s \<equiv> \<forall>x \<in> s. \<exists>e. 0 < e \<and> f holomorphic_on (ball x e)"

lemma analytic_imp_holomorphic:
     "f analytic_on s \<Longrightarrow> f holomorphic_on s"
  unfolding analytic_on_def holomorphic_on_def
  apply (simp add: has_derivative_within_open [OF _ open_ball])
  apply (metis DERIV_subset dist_self mem_ball top_greatest)
  done

lemma analytic_on_open:
     "open s \<Longrightarrow> f analytic_on s \<longleftrightarrow> f holomorphic_on s"
apply (auto simp: analytic_imp_holomorphic)
apply (auto simp: analytic_on_def holomorphic_on_def)
by (metis holomorphic_on_def holomorphic_on_subset open_contains_ball)

lemma analytic_on_imp_differentiable_at:
  "f analytic_on s \<Longrightarrow> x \<in> s \<Longrightarrow> f complex_differentiable (at x)"
 apply (auto simp: analytic_on_def holomorphic_on_differentiable)
by (metis Topology_Euclidean_Space.open_ball centre_in_ball complex_differentiable_within_open)

lemma analytic_on_subset:
     "f analytic_on s \<Longrightarrow> t \<subseteq> s \<Longrightarrow> f analytic_on t"
  by (auto simp: analytic_on_def)

lemma analytic_on_Un:
    "f analytic_on (s \<union> t) \<longleftrightarrow> f analytic_on s \<and> f analytic_on t"
  by (auto simp: analytic_on_def)

lemma analytic_on_Union:
  "f analytic_on (\<Union> s) \<longleftrightarrow> (\<forall>t \<in> s. f analytic_on t)"
  by (auto simp: analytic_on_def)
  
lemma analytic_on_holomorphic:
  "f analytic_on s \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f holomorphic_on t)"
  (is "?lhs = ?rhs")
proof -
  have "?lhs \<longleftrightarrow> (\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t)"
  proof safe
    assume "f analytic_on s"
    then show "\<exists>t. open t \<and> s \<subseteq> t \<and> f analytic_on t"
      apply (simp add: analytic_on_def)
      apply (rule exI [where x="\<Union>{u. open u \<and> f analytic_on u}"], auto)
      apply (metis Topology_Euclidean_Space.open_ball analytic_on_open centre_in_ball)
      by (metis analytic_on_def)
  next
    fix t
    assume "open t" "s \<subseteq> t" "f analytic_on t" 
    then show "f analytic_on s"
        by (metis analytic_on_subset)
  qed
  also have "... \<longleftrightarrow> ?rhs"
    by (auto simp: analytic_on_open)
  finally show ?thesis .
qed

lemma analytic_on_linear: "(op * c) analytic_on s"
  apply (simp add: analytic_on_holomorphic holomorphic_on_linear)
  by (metis open_UNIV top_greatest)

lemma analytic_on_const: "(\<lambda>z. c) analytic_on s"
  unfolding analytic_on_def
  by (metis holomorphic_on_const zero_less_one)

lemma analytic_on_id: "id analytic_on s"
  unfolding analytic_on_def
  apply (simp add: holomorphic_on_id)
  by (metis gt_ex)

lemma analytic_on_compose:
  assumes f: "f analytic_on s"
      and g: "g analytic_on (f ` s)"
    shows "(g o f) analytic_on s"
unfolding analytic_on_def
proof (intro ballI)
  fix x
  assume x: "x \<in> s"
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball x e" using f
    by (metis analytic_on_def)
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball (f x) e'" using g
    by (metis analytic_on_def g image_eqI x) 
  have "isCont f x"
    by (metis analytic_on_imp_differentiable_at complex_differentiable_imp_continuous_at f x)
  with e' obtain d where d: "0 < d" and fd: "f ` ball x d \<subseteq> ball (f x) e'"
     by (auto simp: continuous_at_ball)
  have "g \<circ> f holomorphic_on ball x (min d e)" 
    apply (rule holomorphic_on_compose)
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
    by (metis fd gh holomorphic_on_subset image_mono min.cobounded1 subset_ball)
  then show "\<exists>e>0. g \<circ> f holomorphic_on ball x e"
    by (metis d e min_less_iff_conj) 
qed

lemma analytic_on_compose_gen:
  "f analytic_on s \<Longrightarrow> g analytic_on t \<Longrightarrow> (\<And>z. z \<in> s \<Longrightarrow> f z \<in> t)
             \<Longrightarrow> g o f analytic_on s"
by (metis analytic_on_compose analytic_on_subset image_subset_iff)

lemma analytic_on_neg:
  "f analytic_on s \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on s"
by (metis analytic_on_holomorphic holomorphic_on_minus)

lemma analytic_on_add:
  assumes f: "f analytic_on s"
      and g: "g analytic_on s"
    shows "(\<lambda>z. f z + g z) analytic_on s"
unfolding analytic_on_def
proof (intro ballI)
  fix z
  assume z: "z \<in> s"
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
    by (metis analytic_on_def)
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
    by (metis analytic_on_def g z) 
  have "(\<lambda>z. f z + g z) holomorphic_on ball z (min e e')" 
    apply (rule holomorphic_on_add) 
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
  then show "\<exists>e>0. (\<lambda>z. f z + g z) holomorphic_on ball z e"
    by (metis e e' min_less_iff_conj)
qed

lemma analytic_on_diff:
  assumes f: "f analytic_on s"
      and g: "g analytic_on s"
    shows "(\<lambda>z. f z - g z) analytic_on s"
unfolding analytic_on_def
proof (intro ballI)
  fix z
  assume z: "z \<in> s"
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
    by (metis analytic_on_def)
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
    by (metis analytic_on_def g z) 
  have "(\<lambda>z. f z - g z) holomorphic_on ball z (min e e')" 
    apply (rule holomorphic_on_diff) 
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
  then show "\<exists>e>0. (\<lambda>z. f z - g z) holomorphic_on ball z e"
    by (metis e e' min_less_iff_conj)
qed

lemma analytic_on_mult:
  assumes f: "f analytic_on s"
      and g: "g analytic_on s"
    shows "(\<lambda>z. f z * g z) analytic_on s"
unfolding analytic_on_def
proof (intro ballI)
  fix z
  assume z: "z \<in> s"
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
    by (metis analytic_on_def)
  obtain e' where e': "0 < e'" and gh: "g holomorphic_on ball z e'" using g
    by (metis analytic_on_def g z) 
  have "(\<lambda>z. f z * g z) holomorphic_on ball z (min e e')" 
    apply (rule holomorphic_on_mult) 
    apply (metis fh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
    by (metis gh holomorphic_on_subset min.bounded_iff order_refl subset_ball)
  then show "\<exists>e>0. (\<lambda>z. f z * g z) holomorphic_on ball z e"
    by (metis e e' min_less_iff_conj)
qed

lemma analytic_on_inverse:
  assumes f: "f analytic_on s"
      and nz: "(\<And>z. z \<in> s \<Longrightarrow> f z \<noteq> 0)"
    shows "(\<lambda>z. inverse (f z)) analytic_on s"
unfolding analytic_on_def
proof (intro ballI)
  fix z
  assume z: "z \<in> s"
  then obtain e where e: "0 < e" and fh: "f holomorphic_on ball z e" using f
    by (metis analytic_on_def)
  have "continuous_on (ball z e) f"
    by (metis fh holomorphic_on_imp_continuous_on)
  then obtain e' where e': "0 < e'" and nz': "\<And>y. dist z y < e' \<Longrightarrow> f y \<noteq> 0" 
    by (metis Topology_Euclidean_Space.open_ball centre_in_ball continuous_on_open_avoid e z nz)  
  have "(\<lambda>z. inverse (f z)) holomorphic_on ball z (min e e')" 
    apply (rule holomorphic_on_inverse)
    apply (metis fh holomorphic_on_subset min.cobounded2 min.commute subset_ball)
    by (metis nz' mem_ball min_less_iff_conj) 
  then show "\<exists>e>0. (\<lambda>z. inverse (f z)) holomorphic_on ball z e"
    by (metis e e' min_less_iff_conj)
qed


lemma analytic_on_divide:
  assumes f: "f analytic_on s"
      and g: "g analytic_on s"
      and nz: "(\<And>z. z \<in> s \<Longrightarrow> g z \<noteq> 0)"
    shows "(\<lambda>z. f z / g z) analytic_on s"
unfolding divide_inverse
by (metis analytic_on_inverse analytic_on_mult f g nz)

lemma analytic_on_power:
  "f analytic_on s \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on s"
by (induct n) (auto simp: analytic_on_const analytic_on_mult)

lemma analytic_on_setsum:
  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on s)
           \<Longrightarrow> (\<lambda>x. setsum (\<lambda>i. f i x) I) analytic_on s"
  by (induct I rule: finite_induct) (auto simp: analytic_on_const analytic_on_add)

subsection{*analyticity at a point.*}

lemma analytic_at_ball:
  "f analytic_on {z} \<longleftrightarrow> (\<exists>e. 0<e \<and> f holomorphic_on ball z e)"
by (metis analytic_on_def singleton_iff)

lemma analytic_at:
    "f analytic_on {z} \<longleftrightarrow> (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s)"
by (metis analytic_on_holomorphic empty_subsetI insert_subset)

lemma analytic_on_analytic_at:
    "f analytic_on s \<longleftrightarrow> (\<forall>z \<in> s. f analytic_on {z})"
by (metis analytic_at_ball analytic_on_def)

lemma analytic_at_two:
  "f analytic_on {z} \<and> g analytic_on {z} \<longleftrightarrow>
   (\<exists>s. open s \<and> z \<in> s \<and> f holomorphic_on s \<and> g holomorphic_on s)"
  (is "?lhs = ?rhs")
proof 
  assume ?lhs
  then obtain s t 
    where st: "open s" "z \<in> s" "f holomorphic_on s"
              "open t" "z \<in> t" "g holomorphic_on t"
    by (auto simp: analytic_at)
  show ?rhs
    apply (rule_tac x="s \<inter> t" in exI)
    using st
    apply (auto simp: Diff_subset holomorphic_on_subset)
    done
next
  assume ?rhs 
  then show ?lhs
    by (force simp add: analytic_at)
qed

subsection{*Combining theorems for derivative with ``analytic at'' hypotheses*}

lemma 
  assumes "f analytic_on {z}" "g analytic_on {z}"
  shows complex_derivative_add_at: "DD (\<lambda>w. f w + g w) z = DD f z + DD g z"
    and complex_derivative_diff_at: "DD (\<lambda>w. f w - g w) z = DD f z - DD g z"
    and complex_derivative_mult_at: "DD (\<lambda>w. f w * g w) z =
           f z * DD g z + DD f z * g z"
proof -
  obtain s where s: "open s" "z \<in> s" "f holomorphic_on s" "g holomorphic_on s"
    using assms by (metis analytic_at_two)
  show "DD (\<lambda>w. f w + g w) z = DD f z + DD g z"
    apply (rule DERIV_imp_DD [OF DERIV_add])
    using s
    apply (auto simp: holomorphic_on_open complex_differentiable_def DD_iff_derivative_complex_differentiable)
    done
  show "DD (\<lambda>w. f w - g w) z = DD f z - DD g z"
    apply (rule DERIV_imp_DD [OF DERIV_diff])
    using s
    apply (auto simp: holomorphic_on_open complex_differentiable_def DD_iff_derivative_complex_differentiable)
    done
  show "DD (\<lambda>w. f w * g w) z = f z * DD g z + DD f z * g z"
    apply (rule DERIV_imp_DD [OF DERIV_mult'])
    using s
    apply (auto simp: holomorphic_on_open complex_differentiable_def DD_iff_derivative_complex_differentiable)
    done
qed

lemma complex_derivative_cmult_at:
  "f analytic_on {z} \<Longrightarrow>  DD (\<lambda>w. c * f w) z = c * DD f z"
by (auto simp: complex_derivative_mult_at complex_derivative_const analytic_on_const)

lemma complex_derivative_cmult_right_at:
  "f analytic_on {z} \<Longrightarrow>  DD (\<lambda>w. f w * c) z = DD f z * c"
by (auto simp: complex_derivative_mult_at complex_derivative_const analytic_on_const)

text{*A composition lemma for functions of mixed type*}
lemma has_vector_derivative_real_complex:
  fixes f :: "complex \<Rightarrow> complex"
  assumes "DERIV f (of_real a) :> f'"
  shows "((\<lambda>x. f (of_real x)) has_vector_derivative f') (at a)"
  using diff_chain_at [OF has_derivative_ident, of f "op * f'" a] assms
  unfolding has_field_derivative_def has_vector_derivative_def o_def
  by (auto simp: mult_ac scaleR_conv_of_real)

subsection{*Complex differentiation of sequences and series*}

lemma has_complex_derivative_sequence:
  fixes s :: "complex set"
  assumes cvs: "convex s"
      and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
      and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s \<longrightarrow> norm (f' n x - g' x) \<le> e"
      and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) ---> l) sequentially"
    shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) ---> g x) sequentially \<and> 
                       (g has_field_derivative (g' x)) (at x within s)"
proof -
  from assms obtain x l where x: "x \<in> s" and tf: "((\<lambda>n. f n x) ---> l) sequentially"
    by blast
  { fix e::real assume e: "e > 0"
    then obtain N where N: "\<forall>n\<ge>N. \<forall>x. x \<in> s \<longrightarrow> cmod (f' n x - g' x) \<le> e"
      by (metis conv)    
    have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
    proof (rule exI [of _ N], clarify)
      fix n y h
      assume "N \<le> n" "y \<in> s"
      then have "cmod (f' n y - g' y) \<le> e"
        by (metis N)
      then have "cmod h * cmod (f' n y - g' y) \<le> cmod h * e"
        by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
      then show "cmod (f' n y * h - g' y * h) \<le> e * cmod h"
        by (simp add: norm_mult [symmetric] field_simps)
    qed
  } note ** = this
  show ?thesis
  unfolding has_field_derivative_def
  proof (rule has_derivative_sequence [OF cvs _ _ x])
    show "\<forall>n. \<forall>x\<in>s. (f n has_derivative (op * (f' n x))) (at x within s)"
      by (metis has_field_derivative_def df)
  next show "(\<lambda>n. f n x) ----> l"
    by (rule tf)
  next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
    by (blast intro: **)
  qed
qed


lemma has_complex_derivative_series:
  fixes s :: "complex set"
  assumes cvs: "convex s"
      and df:  "\<And>n x. x \<in> s \<Longrightarrow> (f n has_field_derivative f' n x) (at x within s)"
      and conv: "\<And>e. 0 < e \<Longrightarrow> \<exists>N. \<forall>n x. n \<ge> N \<longrightarrow> x \<in> s 
                \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
      and "\<exists>x l. x \<in> s \<and> ((\<lambda>n. f n x) sums l)"
    shows "\<exists>g. \<forall>x \<in> s. ((\<lambda>n. f n x) sums g x) \<and> ((g has_field_derivative g' x) (at x within s))"
proof -
  from assms obtain x l where x: "x \<in> s" and sf: "((\<lambda>n. f n x) sums l)"
    by blast
  { fix e::real assume e: "e > 0"
    then obtain N where N: "\<forall>n x. n \<ge> N \<longrightarrow> x \<in> s 
            \<longrightarrow> cmod ((\<Sum>i<n. f' i x) - g' x) \<le> e"
      by (metis conv)    
    have "\<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
    proof (rule exI [of _ N], clarify)
      fix n y h
      assume "N \<le> n" "y \<in> s"
      then have "cmod ((\<Sum>i<n. f' i y) - g' y) \<le> e"
        by (metis N)
      then have "cmod h * cmod ((\<Sum>i<n. f' i y) - g' y) \<le> cmod h * e"
        by (auto simp: antisym_conv2 mult_le_cancel_left norm_triangle_ineq2)
      then show "cmod ((\<Sum>i<n. h * f' i y) - g' y * h) \<le> e * cmod h"
        by (simp add: norm_mult [symmetric] field_simps setsum_right_distrib)
    qed
  } note ** = this
  show ?thesis
  unfolding has_field_derivative_def
  proof (rule has_derivative_series [OF cvs _ _ x])
    fix n x
    assume "x \<in> s"
    then show "((f n) has_derivative (\<lambda>z. z * f' n x)) (at x within s)"
      by (metis df has_field_derivative_def mult_commute_abs)
  next show " ((\<lambda>n. f n x) sums l)"
    by (rule sf)
  next show "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x\<in>s. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
    by (blast intro: **)
  qed
qed

subsection{*Bound theorem*}

lemma complex_differentiable_bound:
  fixes s :: "complex set"
  assumes cvs: "convex s"
      and df:  "\<And>z. z \<in> s \<Longrightarrow> (f has_field_derivative f' z) (at z within s)"
      and dn:  "\<And>z. z \<in> s \<Longrightarrow> norm (f' z) \<le> B"
      and "x \<in> s"  "y \<in> s"
    shows "norm(f x - f y) \<le> B * norm(x - y)"
  apply (rule differentiable_bound [OF cvs])
  using assms
  apply (auto simp: has_field_derivative_def Ball_def onorm_def)
  apply (rule cSUP_least)
  apply (auto simp: norm_mult)
  apply (metis norm_one)
  done

subsection{*Inverse function theorem for complex derivatives.*}

lemma has_complex_derivative_inverse_basic:
  fixes f :: "complex \<Rightarrow> complex"
  shows "DERIV f (g y) :> f' \<Longrightarrow>
        f' \<noteq> 0 \<Longrightarrow>
        continuous (at y) g \<Longrightarrow>
        open t \<Longrightarrow>
        y \<in> t \<Longrightarrow>
        (\<And>z. z \<in> t \<Longrightarrow> f (g z) = z)
        \<Longrightarrow> DERIV g y :> inverse (f')"
  unfolding has_field_derivative_def
  apply (rule has_derivative_inverse_basic)
  apply (auto simp:  bounded_linear_mult_right)
  done

(*Used only once, in Multivariate/cauchy.ml. *)
lemma has_complex_derivative_inverse_strong:
  fixes f :: "complex \<Rightarrow> complex"
  shows "DERIV f x :> f' \<Longrightarrow>
         f' \<noteq> 0 \<Longrightarrow>
         open s \<Longrightarrow>
         x \<in> s \<Longrightarrow>
         continuous_on s f \<Longrightarrow>
         (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
         \<Longrightarrow> DERIV g (f x) :> inverse (f')"
  unfolding has_field_derivative_def
  apply (rule has_derivative_inverse_strong [of s x f g ])
  using assms 
  by auto

lemma has_complex_derivative_inverse_strong_x:
  fixes f :: "complex \<Rightarrow> complex"
  shows  "DERIV f (g y) :> f' \<Longrightarrow>
          f' \<noteq> 0 \<Longrightarrow>
          open s \<Longrightarrow>
          continuous_on s f \<Longrightarrow>
          g y \<in> s \<Longrightarrow> f(g y) = y \<Longrightarrow>
          (\<And>z. z \<in> s \<Longrightarrow> g (f z) = z)
          \<Longrightarrow> DERIV g y :> inverse (f')"
  unfolding has_field_derivative_def
  apply (rule has_derivative_inverse_strong_x [of s g y f])
  using assms 
  by auto

subsection{*Further useful properties of complex conjugation*}

lemma lim_cnj: "((\<lambda>x. cnj(f x)) ---> cnj l) F \<longleftrightarrow> (f ---> l) F"
  by (simp add: tendsto_iff dist_complex_def Complex.complex_cnj_diff [symmetric])

lemma sums_cnj: "((\<lambda>x. cnj(f x)) sums cnj l) \<longleftrightarrow> (f sums l)"
  by (simp add: sums_def lim_cnj cnj_setsum [symmetric])

lemma continuous_within_cnj: "continuous (at z within s) cnj"
by (metis bounded_linear_cnj linear_continuous_within)

lemma continuous_on_cnj: "continuous_on s cnj"
by (metis bounded_linear_cnj linear_continuous_on)

subsection{*Some limit theorems about real part of real series etc.*}

lemma real_lim:
  fixes l::complex
  assumes "(f ---> l) F" and " ~(trivial_limit F)" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
  shows  "l \<in> \<real>"
proof -
  have 1: "((\<lambda>i. Im (f i)) ---> Im l) F"
    by (metis assms(1) tendsto_Im) 
  then have  "((\<lambda>i. Im (f i)) ---> 0) F"
    by (smt2 Lim_eventually assms(3) assms(4) complex_is_Real_iff eventually_elim2)
  then show ?thesis using 1
    by (metis 1 assms(2) complex_is_Real_iff tendsto_unique) 
qed

lemma real_lim_sequentially:
  fixes l::complex
  shows "(f ---> l) sequentially \<Longrightarrow> (\<exists>N. \<forall>n\<ge>N. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
by (rule real_lim [where F=sequentially]) (auto simp: eventually_sequentially)

lemma real_series: 
  fixes l::complex
  shows "f sums l \<Longrightarrow> (\<And>n. f n \<in> \<real>) \<Longrightarrow> l \<in> \<real>"
unfolding sums_def
by (metis real_lim_sequentially setsum_in_Reals)


lemma Lim_null_comparison_Re:
   "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F \<Longrightarrow>  (g ---> 0) F \<Longrightarrow> (f ---> 0) F"
  by (metis Lim_null_comparison complex_Re_zero tendsto_Re)


lemma norm_setsum_bound:
  fixes n::nat
  shows" \<lbrakk>\<And>n. N \<le> n \<Longrightarrow> norm (f n) \<le> g n; N \<le> m\<rbrakk>
       \<Longrightarrow> norm (setsum f {m..<n}) \<le> setsum g {m..<n}"
apply (induct n, auto)
by (metis dual_order.trans le_cases le_neq_implies_less norm_triangle_mono)

(*Replaces the one in Series*)
lemma summable_comparison_test:
  fixes f:: "nat \<Rightarrow> 'a::banach" 
  shows "summable g \<Longrightarrow> (\<And>n. n \<ge> N \<Longrightarrow> norm(f n) \<le> g n) \<Longrightarrow> summable f"
apply (auto simp: Series.summable_Cauchy)  
apply (drule_tac x = e in spec, auto)
apply (rule_tac x="max N Na" in exI, auto)
by (smt2 norm_setsum_bound)

(*MOVE TO Finite_Cartesian_Product*)
lemma sums_vec_nth :
  assumes "f sums a"
  shows "(\<lambda>x. f x $ i) sums a $ i"
using assms unfolding sums_def
by (auto dest: tendsto_vec_nth [where i=i])

lemma summable_vec_nth :
  assumes "summable f"
  shows "summable (\<lambda>x. f x $ i)"
using assms unfolding summable_def
by (blast intro: sums_vec_nth)

(*REPLACE THE ONES IN COMPLEX.THY*)
lemma Re_setsum: "Re(setsum f s) = setsum (%x. Re(f x)) s"
apply (cases "finite s")
  by (induct s rule: finite_induct) auto

lemma Im_setsum: "Im(setsum f s) = setsum (%x. Im(f x)) s"
apply (cases "finite s")
  by (induct s rule: finite_induct) auto

lemma of_real_setsum: "of_real (setsum f s) = setsum (%x. of_real(f x)) s"
apply (cases "finite s")
  by (induct s rule: finite_induct) auto

lemma of_real_setprod: "of_real (setprod f s) = setprod (%x. of_real(f x)) s"
apply (cases "finite s")
  by (induct s rule: finite_induct) auto

lemma sums_Re:
  assumes "f sums a"
  shows "(\<lambda>x. Re (f x)) sums Re a"
using assms 
by (auto simp: sums_def Re_setsum [symmetric] isCont_tendsto_compose [of a Re])

lemma sums_Im:
  assumes "f sums a"
  shows "(\<lambda>x. Im (f x)) sums Im a"
using assms 
by (auto simp: sums_def Im_setsum [symmetric] isCont_tendsto_compose [of a Im])

lemma summable_Re:
  assumes "summable f"
  shows "summable (\<lambda>x. Re (f x))"
using assms unfolding summable_def
by (blast intro: sums_Re)

lemma summable_Im:
  assumes "summable f"
  shows "summable (\<lambda>x. Im (f x))"
using assms unfolding summable_def
by (blast intro: sums_Im)

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

lemma summable_complex_of_real [simp]:
  "summable (\<lambda>n. complex_of_real (f n)) = summable f"
apply (auto simp: Series.summable_Cauchy)  
apply (drule_tac x = e in spec, auto)
apply (rule_tac x=N in exI)
apply (auto simp: of_real_setsum [symmetric])
done


(* ------------------------------------------------------------------------- *)
(* A kind of complex Taylor theorem.                                         *)
(* ------------------------------------------------------------------------- *)


lemma setsum_Suc_reindex:
  fixes f :: "nat \<Rightarrow> 'a::ab_group_add"
    shows  "setsum f {0..n} = f 0 - f (Suc n) + setsum (\<lambda>i. f (Suc i)) {0..n}"
by (induct n) auto

(*REPLACE THE REAL VERSIONS*)
lemma mult_left_cancel:
  fixes c:: "'a::ring_no_zero_divisors"
  shows "c \<noteq> 0 \<Longrightarrow> (c*a=c*b) = (a=b)"
by simp 

lemma mult_right_cancel:
  fixes c:: "'a::ring_no_zero_divisors"
  shows "c \<noteq> 0 \<Longrightarrow> (a*c=b*c) = (a=b)"
by simp 

lemma complex_taylor:
  assumes s: "convex s" 
      and f: "\<And>i x. x \<in> s \<Longrightarrow> i \<le> n \<Longrightarrow> (f i has_field_derivative f (Suc i) x) (at x within s)"
      and B: "\<And>x. x \<in> s \<Longrightarrow> cmod (f (Suc n) x) \<le> B"
      and w: "w \<in> s"
      and z: "z \<in> s"
    shows "cmod(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / of_nat (fact i)))
          \<le> B * cmod(z - w)^(Suc n) / fact n"
proof -
  have wzs: "closed_segment w z \<subseteq> s" using assms
    by (metis convex_contains_segment)
  { fix u
    assume "u \<in> closed_segment w z"
    then have "u \<in> s"
      by (metis wzs subsetD)
    have "(\<Sum>i\<le>n. f i u * (- of_nat i * (z-u)^(i - 1)) / of_nat (fact i) +
                      f (Suc i) u * (z-u)^i / of_nat (fact i)) = 
              f (Suc n) u * (z-u) ^ n / of_nat (fact n)"
    proof (induction n)
      case 0 show ?case by simp
    next
      case (Suc n)
      have "(\<Sum>i\<le>Suc n. f i u * (- of_nat i * (z-u) ^ (i - 1)) / of_nat (fact i) +
                             f (Suc i) u * (z-u) ^ i / of_nat (fact i)) =  
           f (Suc n) u * (z-u) ^ n / of_nat (fact n) +
           f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / of_nat (fact (Suc n)) -
           f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / of_nat (fact (Suc n))"
        using Suc by simp
      also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / of_nat (fact (Suc n))"
      proof -
        have "of_nat(fact(Suc n)) *
             (f(Suc n) u *(z-u) ^ n / of_nat(fact n) +
               f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / of_nat(fact(Suc n)) -
               f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / of_nat(fact(Suc n))) =
            (of_nat(fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / of_nat(fact n) +
            (of_nat(fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / of_nat(fact(Suc n))) -
            (of_nat(fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / of_nat(fact(Suc n))"
          by (simp add: algebra_simps del: fact_Suc)
        also have "... =
                   (of_nat (fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / of_nat (fact n) +
                   (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
                   (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
          by (simp del: fact_Suc)
        also have "... = 
                   (of_nat (Suc n) * (f (Suc n) u * (z-u) ^ n)) +
                   (f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)) -
                   (f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n))"
          by (simp only: fact_Suc of_nat_mult mult_ac) simp
        also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
          by (simp add: algebra_simps)
        finally show ?thesis
        by (simp add: mult_left_cancel [where c = "of_nat (fact (Suc n))", THEN iffD1] del: fact_Suc)
      qed
      finally show ?case .
    qed
    then have "((\<lambda>v. (\<Sum>i\<le>n. f i v * (z - v)^i / of_nat (fact i))) 
                has_field_derivative f (Suc n) u * (z-u) ^ n / of_nat (fact n))
               (at u within s)"
      apply (intro DERIV_intros DERIV_power[THEN DERIV_cong])
      apply (blast intro: assms `u \<in> s`)
      apply (rule refl)+
      apply (auto simp: field_simps)
      done
  } note sum_deriv = this
  { fix u
    assume u: "u \<in> closed_segment w z"
    then have us: "u \<in> s"
      by (metis wzs subsetD)
    have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> cmod (f (Suc n) u) * cmod (u - z) ^ n"
      by (metis norm_minus_commute order_refl)
    also have "... \<le> cmod (f (Suc n) u) * cmod (z - w) ^ n"
      by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
    also have "... \<le> B * cmod (z - w) ^ n"
      by (metis norm_ge_zero zero_le_power mult_right_mono  B [OF us])
    finally have "cmod (f (Suc n) u) * cmod (z - u) ^ n \<le> B * cmod (z - w) ^ n" .
  } note cmod_bound = this
  have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / of_nat (fact i)) = (\<Sum>i\<le>n. (f i z / of_nat (fact i)) * 0 ^ i)"
    by simp
  also have "\<dots> = f 0 z / of_nat (fact 0)"
    by (subst setsum_zero_power) simp
  finally have "cmod (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / of_nat (fact i))) 
            \<le> cmod ((\<Sum>i\<le>n. f i w * (z - w) ^ i / of_nat (fact i)) -
                    (\<Sum>i\<le>n. f i z * (z - z) ^ i / of_nat (fact i)))"
    by (simp add: norm_minus_commute)
  also have "... \<le> B * cmod (z - w) ^ n / real_of_nat (fact n) * cmod (w - z)"
    apply (rule complex_differentiable_bound 
      [where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / of_nat(fact n)"
         and s = "closed_segment w z", OF convex_segment])
    apply (auto simp: ends_in_segment real_of_nat_def DERIV_subset [OF sum_deriv wzs]
                  norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
    done
  also have "...  \<le> B * cmod (z - w) ^ Suc n / real (fact n)"
    by (simp add: algebra_simps norm_minus_commute real_of_nat_def)
  finally show ?thesis .
qed

end