src/HOL/Analysis/Complex_Analysis_Basics.thy
 author paulson Mon, 04 Nov 2019 17:06:18 +0000 changeset 71029 934e0044e94b parent 71001 3e374c65f96b child 71030 b6e69c71a9f6 permissions -rw-r--r--
Moved or deleted some out of place material, also eliminating obsolete naming conventions
```
(*  Author: John Harrison, Marco Maggesi, Graziano Gentili, Gianni Ciolli, Valentina Bruno
Ported from "hol_light/Multivariate/canal.ml" by L C Paulson (2014)
*)

section \<open>Complex Analysis Basics\<close>

theory Complex_Analysis_Basics
imports Derivative "HOL-Library.Nonpos_Ints"
begin

(* TODO FIXME: A lot of the things in here have nothing to do with complex analysis *)

subsection\<^marker>\<open>tag unimportant\<close>\<open>General lemmas\<close>

lemma nonneg_Reals_cmod_eq_Re: "z \<in> \<real>\<^sub>\<ge>\<^sub>0 \<Longrightarrow> norm z = Re z"

lemma fact_cancel:
fixes c :: "'a::real_field"
shows "of_nat (Suc n) * c / (fact (Suc n)) = c / (fact n)"
using of_nat_neq_0 by force

lemma vector_derivative_cnj_within:
assumes "at x within A \<noteq> bot" and "f differentiable at x within A"
shows   "vector_derivative (\<lambda>z. cnj (f z)) (at x within A) =
cnj (vector_derivative f (at x within A))" (is "_ = cnj ?D")
proof -
let ?D = "vector_derivative f (at x within A)"
from assms have "(f has_vector_derivative ?D) (at x within A)"
by (subst (asm) vector_derivative_works)
hence "((\<lambda>x. cnj (f x)) has_vector_derivative cnj ?D) (at x within A)"
by (rule has_vector_derivative_cnj)
thus ?thesis using assms by (auto dest: vector_derivative_within)
qed

lemma vector_derivative_cnj:
assumes "f differentiable at x"
shows   "vector_derivative (\<lambda>z. cnj (f z)) (at x) = cnj (vector_derivative f (at x))"
using assms by (intro vector_derivative_cnj_within) auto

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

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

lemma uniformly_continuous_on_cmul_right [continuous_intros]:
fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. f x * c)"
using bounded_linear.uniformly_continuous_on[OF bounded_linear_mult_left] .

lemma uniformly_continuous_on_cmul_left[continuous_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_on_norm_id [continuous_intros]: "continuous_on S norm"
by (intro continuous_on_id continuous_on_norm)

lemma
shows open_halfspace_Re_lt: "open {z. Re(z) < b}"
and open_halfspace_Re_gt: "open {z. Re(z) > b}"
and closed_halfspace_Re_ge: "closed {z. Re(z) \<ge> b}"
and closed_halfspace_Re_le: "closed {z. Re(z) \<le> b}"
and closed_halfspace_Re_eq: "closed {z. Re(z) = b}"
and open_halfspace_Im_lt: "open {z. Im(z) < b}"
and open_halfspace_Im_gt: "open {z. Im(z) > b}"
and closed_halfspace_Im_ge: "closed {z. Im(z) \<ge> b}"
and closed_halfspace_Im_le: "closed {z. Im(z) \<le> b}"
and closed_halfspace_Im_eq: "closed {z. Im(z) = b}"
by (intro open_Collect_less closed_Collect_le closed_Collect_eq continuous_on_Re
continuous_on_Im continuous_on_id continuous_on_const)+

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

lemma closed_Real_halfspace_Re_le: "closed (\<real> \<inter> {w. Re w \<le> x})"
by (simp add: closed_Int closed_complex_Reals closed_halfspace_Re_le)

lemma closed_nonpos_Reals_complex [simp]: "closed (\<real>\<^sub>\<le>\<^sub>0 :: complex set)"
proof -
have "\<real>\<^sub>\<le>\<^sub>0 = \<real> \<inter> {z. Re(z) \<le> 0}"
using complex_nonpos_Reals_iff complex_is_Real_iff by auto
then show ?thesis
by (metis closed_Real_halfspace_Re_le)
qed

lemma closed_Real_halfspace_Re_ge: "closed (\<real> \<inter> {w. x \<le> Re(w)})"
using closed_halfspace_Re_ge

lemma closed_nonneg_Reals_complex [simp]: "closed (\<real>\<^sub>\<ge>\<^sub>0 :: complex set)"
proof -
have "\<real>\<^sub>\<ge>\<^sub>0 = \<real> \<inter> {z. Re(z) \<ge> 0}"
using complex_nonneg_Reals_iff complex_is_Real_iff by auto
then show ?thesis
by (metis closed_Real_halfspace_Re_ge)
qed

lemma closed_real_abs_le: "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
proof -
have "{w \<in> \<real>. \<bar>Re w\<bar> \<le> r} = (\<real> \<inter> {w. Re w \<le> r}) \<inter> (\<real> \<inter> {w. Re w \<ge> -r})"
by auto
then show "closed {w \<in> \<real>. \<bar>Re w\<bar> \<le> r}"
by (simp add: closed_Int closed_Real_halfspace_Re_ge closed_Real_halfspace_Re_le)
qed

lemma real_lim:
fixes l::complex
assumes "(f \<longlongrightarrow> l) F" and "\<not> trivial_limit F" and "eventually P F" and "\<And>a. P a \<Longrightarrow> f a \<in> \<real>"
shows  "l \<in> \<real>"
proof (rule Lim_in_closed_set[OF closed_complex_Reals _ assms(2,1)])
show "eventually (\<lambda>x. f x \<in> \<real>) F"
using assms(3, 4) by (auto intro: eventually_mono)
qed

lemma real_lim_sequentially:
fixes l::complex
shows "(f \<longlongrightarrow> 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 sum_in_Reals)

lemma Lim_null_comparison_Re:
assumes "eventually (\<lambda>x. norm(f x) \<le> Re(g x)) F" "(g \<longlongrightarrow> 0) F" shows "(f \<longlongrightarrow> 0) F"
by (rule Lim_null_comparison[OF assms(1)] tendsto_eq_intros assms(2))+ simp

lemma closed_segment_same_Re:
assumes "Re a = Re b"
shows   "closed_segment a b = {z. Re z = Re a \<and> Im z \<in> closed_segment (Im a) (Im b)}"
proof safe
fix z assume "z \<in> closed_segment a b"
then obtain u where u: "u \<in> {0..1}" "z = a + of_real u * (b - a)"
by (auto simp: closed_segment_def scaleR_conv_of_real algebra_simps)
from assms show "Re z = Re a" by (auto simp: u)
from u(1) show "Im z \<in> closed_segment (Im a) (Im b)"
by (intro closed_segmentI[of u]) (auto simp: u algebra_simps)
next
fix z assume [simp]: "Re z = Re a" and "Im z \<in> closed_segment (Im a) (Im b)"
then obtain u where u: "u \<in> {0..1}" "Im z = Im a + of_real u * (Im b - Im a)"
by (auto simp: closed_segment_def scaleR_conv_of_real algebra_simps)
from u(1) show "z \<in> closed_segment a b" using assms
by (intro closed_segmentI[of u]) (auto simp: u algebra_simps scaleR_conv_of_real complex_eq_iff)
qed

lemma closed_segment_same_Im:
assumes "Im a = Im b"
shows   "closed_segment a b = {z. Im z = Im a \<and> Re z \<in> closed_segment (Re a) (Re b)}"
proof safe
fix z assume "z \<in> closed_segment a b"
then obtain u where u: "u \<in> {0..1}" "z = a + of_real u * (b - a)"
by (auto simp: closed_segment_def scaleR_conv_of_real algebra_simps)
from assms show "Im z = Im a" by (auto simp: u)
from u(1) show "Re z \<in> closed_segment (Re a) (Re b)"
by (intro closed_segmentI[of u]) (auto simp: u algebra_simps)
next
fix z assume [simp]: "Im z = Im a" and "Re z \<in> closed_segment (Re a) (Re b)"
then obtain u where u: "u \<in> {0..1}" "Re z = Re a + of_real u * (Re b - Re a)"
by (auto simp: closed_segment_def scaleR_conv_of_real algebra_simps)
from u(1) show "z \<in> closed_segment a b" using assms
by (intro closed_segmentI[of u]) (auto simp: u algebra_simps scaleR_conv_of_real complex_eq_iff)
qed

subsection\<open>Holomorphic functions\<close>

definition\<^marker>\<open>tag important\<close> holomorphic_on :: "[complex \<Rightarrow> complex, complex set] \<Rightarrow> bool"
(infixl "(holomorphic'_on)" 50)
where "f holomorphic_on s \<equiv> \<forall>x\<in>s. f field_differentiable (at x within s)"

named_theorems\<^marker>\<open>tag important\<close> holomorphic_intros "structural introduction rules for holomorphic_on"

lemma holomorphic_onI [intro?]: "(\<And>x. x \<in> s \<Longrightarrow> f field_differentiable (at x within s)) \<Longrightarrow> f holomorphic_on s"

lemma holomorphic_onD [dest?]: "\<lbrakk>f holomorphic_on s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x within s)"

lemma holomorphic_on_imp_differentiable_on:
"f holomorphic_on s \<Longrightarrow> f differentiable_on s"
unfolding holomorphic_on_def differentiable_on_def

lemma holomorphic_on_imp_differentiable_at:
"\<lbrakk>f holomorphic_on s; open s; x \<in> s\<rbrakk> \<Longrightarrow> f field_differentiable (at x)"
using at_within_open holomorphic_on_def by fastforce

lemma holomorphic_on_empty [holomorphic_intros]: "f holomorphic_on {}"

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 field_differentiable_def has_field_derivative_def at_within_open [of _ s])

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

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

lemma holomorphic_transform: "\<lbrakk>f holomorphic_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g holomorphic_on s"
by (metis field_differentiable_transform_within linordered_field_no_ub holomorphic_on_def)

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

lemma holomorphic_on_linear [simp, holomorphic_intros]: "((*) c) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_linear)

lemma holomorphic_on_const [simp, holomorphic_intros]: "(\<lambda>z. c) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_const)

lemma holomorphic_on_ident [simp, holomorphic_intros]: "(\<lambda>x. x) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_ident)

lemma holomorphic_on_id [simp, holomorphic_intros]: "id holomorphic_on s"
unfolding id_def by (rule holomorphic_on_ident)

lemma holomorphic_on_compose:
"f holomorphic_on s \<Longrightarrow> g holomorphic_on (f ` s) \<Longrightarrow> (g o f) holomorphic_on s"
using field_differentiable_compose_within[of f _ s g]
by (auto simp: holomorphic_on_def)

lemma holomorphic_on_compose_gen:
"f holomorphic_on s \<Longrightarrow> g holomorphic_on t \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> (g o f) holomorphic_on s"
by (metis holomorphic_on_compose holomorphic_on_subset)

lemma holomorphic_on_balls_imp_entire:
assumes "\<not>bdd_above A" "\<And>r. r \<in> A \<Longrightarrow> f holomorphic_on ball c r"
shows   "f holomorphic_on B"
proof (rule holomorphic_on_subset)
show "f holomorphic_on UNIV" unfolding holomorphic_on_def
proof
fix z :: complex
from \<open>\<not>bdd_above A\<close> obtain r where r: "r \<in> A" "r > norm (z - c)"
by (meson bdd_aboveI not_le)
with assms(2) have "f holomorphic_on ball c r" by blast
moreover from r have "z \<in> ball c r" by (auto simp: dist_norm norm_minus_commute)
ultimately show "f field_differentiable at z"
by (auto simp: holomorphic_on_def at_within_open[of _ "ball c r"])
qed
qed auto

lemma holomorphic_on_balls_imp_entire':
assumes "\<And>r. r > 0 \<Longrightarrow> f holomorphic_on ball c r"
shows   "f holomorphic_on B"
proof (rule holomorphic_on_balls_imp_entire)
{
fix M :: real
have "\<exists>x. x > max M 0" by (intro gt_ex)
hence "\<exists>x>0. x > M" by auto
}
thus "\<not>bdd_above {(0::real)<..}" unfolding bdd_above_def
by (auto simp: not_le)
qed (insert assms, auto)

lemma holomorphic_on_minus [holomorphic_intros]: "f holomorphic_on s \<Longrightarrow> (\<lambda>z. -(f z)) holomorphic_on s"
by (metis field_differentiable_minus holomorphic_on_def)

"\<lbrakk>f holomorphic_on s; g holomorphic_on s\<rbrakk> \<Longrightarrow> (\<lambda>z. f z + g z) holomorphic_on s"

lemma holomorphic_on_diff [holomorphic_intros]:
"\<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 field_differentiable_diff)

lemma holomorphic_on_mult [holomorphic_intros]:
"\<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 field_differentiable_mult)

lemma holomorphic_on_inverse [holomorphic_intros]:
"\<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 field_differentiable_inverse)

lemma holomorphic_on_divide [holomorphic_intros]:
"\<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 field_differentiable_divide)

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

lemma holomorphic_on_sum [holomorphic_intros]:
"(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) holomorphic_on s"
unfolding holomorphic_on_def by (metis field_differentiable_sum)

lemma holomorphic_on_prod [holomorphic_intros]:
"(\<And>i. i \<in> I \<Longrightarrow> (f i) holomorphic_on s) \<Longrightarrow> (\<lambda>x. prod (\<lambda>i. f i x) I) holomorphic_on s"
by (induction I rule: infinite_finite_induct) (auto intro: holomorphic_intros)

lemma holomorphic_pochhammer [holomorphic_intros]:
"f holomorphic_on A \<Longrightarrow> (\<lambda>s. pochhammer (f s) n) holomorphic_on A"
by (induction n) (auto intro!: holomorphic_intros simp: pochhammer_Suc)

lemma holomorphic_on_scaleR [holomorphic_intros]:
"f holomorphic_on A \<Longrightarrow> (\<lambda>x. c *\<^sub>R f x) holomorphic_on A"
by (auto simp: scaleR_conv_of_real intro!: holomorphic_intros)

lemma holomorphic_on_Un [holomorphic_intros]:
assumes "f holomorphic_on A" "f holomorphic_on B" "open A" "open B"
shows   "f holomorphic_on (A \<union> B)"
using assms by (auto simp: holomorphic_on_def  at_within_open[of _ A]
at_within_open[of _ B]  at_within_open[of _ "A \<union> B"] open_Un)

lemma holomorphic_on_If_Un [holomorphic_intros]:
assumes "f holomorphic_on A" "g holomorphic_on B" "open A" "open B"
assumes "\<And>z. z \<in> A \<Longrightarrow> z \<in> B \<Longrightarrow> f z = g z"
shows   "(\<lambda>z. if z \<in> A then f z else g z) holomorphic_on (A \<union> B)" (is "?h holomorphic_on _")
proof (intro holomorphic_on_Un)
note \<open>f holomorphic_on A\<close>
also have "f holomorphic_on A \<longleftrightarrow> ?h holomorphic_on A"
by (intro holomorphic_cong) auto
finally show \<dots> .
next
note \<open>g holomorphic_on B\<close>
also have "g holomorphic_on B \<longleftrightarrow> ?h holomorphic_on B"
using assms by (intro holomorphic_cong) auto
finally show \<dots> .
qed (insert assms, auto)

lemma DERIV_deriv_iff_field_differentiable:
"DERIV f x :> deriv f x \<longleftrightarrow> f field_differentiable at x"
unfolding field_differentiable_def by (metis DERIV_imp_deriv)

lemma holomorphic_derivI:
"\<lbrakk>f holomorphic_on S; open S; x \<in> S\<rbrakk>
\<Longrightarrow> (f has_field_derivative deriv f x) (at x within T)"
by (metis DERIV_deriv_iff_field_differentiable at_within_open  holomorphic_on_def has_field_derivative_at_within)

lemma complex_derivative_chain:
"f field_differentiable at x \<Longrightarrow> g field_differentiable at (f x)
\<Longrightarrow> deriv (g o f) x = deriv g (f x) * deriv f x"
by (metis DERIV_deriv_iff_field_differentiable DERIV_chain DERIV_imp_deriv)

lemma deriv_linear [simp]: "deriv (\<lambda>w. c * w) = (\<lambda>z. c)"
by (metis DERIV_imp_deriv DERIV_cmult_Id)

lemma deriv_ident [simp]: "deriv (\<lambda>w. w) = (\<lambda>z. 1)"
by (metis DERIV_imp_deriv DERIV_ident)

lemma deriv_id [simp]: "deriv id = (\<lambda>z. 1)"

lemma deriv_const [simp]: "deriv (\<lambda>w. c) = (\<lambda>z. 0)"
by (metis DERIV_imp_deriv DERIV_const)

"\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
\<Longrightarrow> deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
unfolding DERIV_deriv_iff_field_differentiable[symmetric]
by (auto intro!: DERIV_imp_deriv derivative_intros)

lemma deriv_diff [simp]:
"\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
\<Longrightarrow> deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
unfolding DERIV_deriv_iff_field_differentiable[symmetric]
by (auto intro!: DERIV_imp_deriv derivative_intros)

lemma deriv_mult [simp]:
"\<lbrakk>f field_differentiable at z; g field_differentiable at z\<rbrakk>
\<Longrightarrow> deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
unfolding DERIV_deriv_iff_field_differentiable[symmetric]
by (auto intro!: DERIV_imp_deriv derivative_eq_intros)

lemma deriv_cmult:
"f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. c * f w) z = c * deriv f z"
by simp

lemma deriv_cmult_right:
"f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w * c) z = deriv f z * c"
by simp

lemma deriv_inverse [simp]:
"\<lbrakk>f field_differentiable at z; f z \<noteq> 0\<rbrakk>
\<Longrightarrow> deriv (\<lambda>w. inverse (f w)) z = - deriv f z / f z ^ 2"
unfolding DERIV_deriv_iff_field_differentiable[symmetric]
by (safe intro!: DERIV_imp_deriv derivative_eq_intros) (auto simp: field_split_simps power2_eq_square)

lemma deriv_divide [simp]:
"\<lbrakk>f field_differentiable at z; g field_differentiable at z; g z \<noteq> 0\<rbrakk>
\<Longrightarrow> deriv (\<lambda>w. f w / g w) z = (deriv f z * g z - f z * deriv g z) / g z ^ 2"

lemma deriv_cdivide_right:
"f field_differentiable at z \<Longrightarrow> deriv (\<lambda>w. f w / c) z = deriv f z / c"

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> deriv f z = deriv g z"
unfolding holomorphic_on_def
by (rule DERIV_imp_deriv)
(metis DERIV_deriv_iff_field_differentiable has_field_derivative_transform_within_open at_within_open)

lemma deriv_compose_linear:
"f field_differentiable at (c * z) \<Longrightarrow> deriv (\<lambda>w. f (c * w)) z = c * deriv f (c * z)"
apply (rule DERIV_imp_deriv)
unfolding DERIV_deriv_iff_field_differentiable [symmetric]
by (metis (full_types) DERIV_chain2 DERIV_cmult_Id mult.commute)

lemma nonzero_deriv_nonconstant:
assumes df: "DERIV f \<xi> :> df" and S: "open S" "\<xi> \<in> S" and "df \<noteq> 0"
shows "\<not> f constant_on S"
unfolding constant_on_def
by (metis \<open>df \<noteq> 0\<close> has_field_derivative_transform_within_open [OF df S] DERIV_const DERIV_unique)

lemma holomorphic_nonconstant:
assumes holf: "f holomorphic_on S" and "open S" "\<xi> \<in> S" "deriv f \<xi> \<noteq> 0"
shows "\<not> f constant_on S"
by (rule nonzero_deriv_nonconstant [of f "deriv f \<xi>" \<xi> S])
(use assms in \<open>auto simp: holomorphic_derivI\<close>)

subsection\<^marker>\<open>tag unimportant\<close>\<open>Caratheodory characterization\<close>

lemma field_differentiable_caratheodory_at:
"f field_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]

lemma field_differentiable_caratheodory_within:
"f field_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]

subsection\<open>Analyticity on a set\<close>

definition\<^marker>\<open>tag important\<close> 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)"

named_theorems\<^marker>\<open>tag important\<close> analytic_intros "introduction rules for proving analyticity"

lemma analytic_imp_holomorphic: "f analytic_on S \<Longrightarrow> f holomorphic_on S"
by (simp add: at_within_open [OF _ open_ball] analytic_on_def holomorphic_on_def)
(metis centre_in_ball field_differentiable_at_within)

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 field_differentiable (at x)"
apply (auto simp: analytic_on_def holomorphic_on_def)
by (metis open_ball centre_in_ball field_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>\<T>) \<longleftrightarrow> (\<forall>T \<in> \<T>. f analytic_on T)"
by (auto simp: analytic_on_def)

lemma analytic_on_UN: "f analytic_on (\<Union>i\<in>I. S i) \<longleftrightarrow> (\<forall>i\<in>I. f analytic_on (S i))"
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 (rule exI [where x="\<Union>{U. open U \<and> f analytic_on U}"], auto)
apply (metis 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 [analytic_intros,simp]: "((*) c) analytic_on S"

lemma analytic_on_const [analytic_intros,simp]: "(\<lambda>z. c) analytic_on S"
by (metis analytic_on_def holomorphic_on_const zero_less_one)

lemma analytic_on_ident [analytic_intros,simp]: "(\<lambda>x. x) analytic_on S"

lemma analytic_on_id [analytic_intros]: "id analytic_on S"
unfolding id_def by (rule analytic_on_ident)

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 field_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 [analytic_intros]:
"f analytic_on S \<Longrightarrow> (\<lambda>z. -(f z)) analytic_on S"
by (metis analytic_on_holomorphic holomorphic_on_minus)

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 (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 [analytic_intros]:
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 [analytic_intros]:
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 [analytic_intros]:
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 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 [analytic_intros]:
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 [analytic_intros]:
"f analytic_on S \<Longrightarrow> (\<lambda>z. (f z) ^ n) analytic_on S"
by (induct n) (auto simp: analytic_on_mult)

lemma analytic_on_sum [analytic_intros]:
"(\<And>i. i \<in> I \<Longrightarrow> (f i) analytic_on S) \<Longrightarrow> (\<lambda>x. sum (\<lambda>i. f i x) I) analytic_on S"
by (induct I rule: infinite_finite_induct) (auto simp: analytic_on_const analytic_on_add)

lemma deriv_left_inverse:
assumes "f holomorphic_on S" and "g holomorphic_on T"
and "open S" and "open T"
and "f ` S \<subseteq> T"
and [simp]: "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z"
and "w \<in> S"
shows "deriv f w * deriv g (f w) = 1"
proof -
have "deriv f w * deriv g (f w) = deriv g (f w) * deriv f w"
also have "... = deriv (g o f) w"
using assms
by (metis analytic_on_imp_differentiable_at analytic_on_open complex_derivative_chain image_subset_iff)
also have "... = deriv id w"
proof (rule complex_derivative_transform_within_open [where s=S])
show "g \<circ> f holomorphic_on S"
by (rule assms holomorphic_on_compose_gen holomorphic_intros)+
qed (use assms in auto)
also have "... = 1"
by simp
finally show ?thesis .
qed

subsection\<^marker>\<open>tag unimportant\<close>\<open>Analyticity at a point\<close>

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: holomorphic_on_subset)
done
next
assume ?rhs
then show ?lhs
qed

subsection\<^marker>\<open>tag unimportant\<close>\<open>Combining theorems for derivative with ``analytic at'' hypotheses\<close>

lemma
assumes "f analytic_on {z}" "g analytic_on {z}"
shows complex_derivative_add_at: "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
and complex_derivative_diff_at: "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
and complex_derivative_mult_at: "deriv (\<lambda>w. f w * g w) z =
f z * deriv g z + deriv 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 "deriv (\<lambda>w. f w + g w) z = deriv f z + deriv g z"
using s
apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
done
show "deriv (\<lambda>w. f w - g w) z = deriv f z - deriv g z"
apply (rule DERIV_imp_deriv [OF DERIV_diff])
using s
apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
done
show "deriv (\<lambda>w. f w * g w) z = f z * deriv g z + deriv f z * g z"
apply (rule DERIV_imp_deriv [OF DERIV_mult'])
using s
apply (auto simp: holomorphic_on_open field_differentiable_def DERIV_deriv_iff_field_differentiable)
done
qed

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

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

subsection\<^marker>\<open>tag unimportant\<close>\<open>Complex differentiation of sequences and series\<close>

(* TODO: Could probably be simplified using Uniform_Limit *)
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) \<longlongrightarrow> l) sequentially"
shows "\<exists>g. \<forall>x \<in> S. ((\<lambda>n. f n x) \<longlongrightarrow> 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) \<longlongrightarrow> 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 "(\<lambda>n. f n x) \<longlonglongrightarrow> l"
by (rule tf)
next show "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. cmod (f' n x * h - g' x * h) \<le> e * cmod h"
unfolding eventually_sequentially by (blast intro: **)
qed (metis has_field_derivative_def df)
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 sum_distrib_left)
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 "\<And>e. e>0 \<Longrightarrow> \<forall>\<^sub>F n in sequentially. \<forall>x\<in>S. \<forall>h. cmod ((\<Sum>i<n. h * f' i x) - g' x * h) \<le> e * cmod h"
unfolding eventually_sequentially by (blast intro: **)
qed
qed

lemma field_differentiable_series:
fixes f :: "nat \<Rightarrow> 'a::{real_normed_field,banach} \<Rightarrow> 'a"
assumes "convex S" "open S"
assumes "\<And>n x. x \<in> S \<Longrightarrow> (f n has_field_derivative f' n x) (at x)"
assumes "uniformly_convergent_on S (\<lambda>n x. \<Sum>i<n. f' i x)"
assumes "x0 \<in> S" "summable (\<lambda>n. f n x0)" and x: "x \<in> S"
shows  "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)"
proof -
from assms(4) obtain g' where A: "uniform_limit S (\<lambda>n x. \<Sum>i<n. f' i x) g' sequentially"
unfolding uniformly_convergent_on_def by blast
from x and \<open>open S\<close> have S: "at x within S = at x" by (rule at_within_open)
have "\<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)"
by (intro has_field_derivative_series[of S f f' g' x0] assms A has_field_derivative_at_within)
then obtain g where g: "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>n. f n x) sums g x"
"\<And>x. x \<in> S \<Longrightarrow> (g has_field_derivative g' x) (at x within S)" by blast
from g(2)[OF x] have g': "(g has_derivative (*) (g' x)) (at x)"
have "((\<lambda>x. \<Sum>n. f n x) has_derivative (*) (g' x)) (at x)"
by (rule has_derivative_transform_within_open[OF g' \<open>open S\<close> x])
(insert g, auto simp: sums_iff)
thus "(\<lambda>x. \<Sum>n. f n x) field_differentiable (at x)" unfolding differentiable_def
by (auto simp: summable_def field_differentiable_def has_field_derivative_def)
qed

subsection\<^marker>\<open>tag unimportant\<close>\<open>Bound theorem\<close>

lemma field_differentiable_bound:
fixes S :: "'a::real_normed_field 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])
apply (erule df [unfolded has_field_derivative_def])
apply (rule onorm_le, simp_all add: norm_mult mult_right_mono assms)
done

subsection\<^marker>\<open>tag unimportant\<close>\<open>Inverse function theorem for complex derivatives\<close>

lemma has_field_derivative_inverse_basic:
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

subsection\<^marker>\<open>tag unimportant\<close> \<open>Taylor on Complex Numbers\<close>

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

lemma field_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> norm (f (Suc n) x) \<le> B"
and w: "w \<in> S"
and z: "z \<in> S"
shows "norm(f 0 z - (\<Sum>i\<le>n. f i w * (z-w) ^ i / (fact i)))
\<le> B * norm(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)) / (fact i) +
f (Suc i) u * (z-u)^i / (fact i)) =
f (Suc n) u * (z-u) ^ n / (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)) / (fact i) +
f (Suc i) u * (z-u) ^ i / (fact i)) =
f (Suc n) u * (z-u) ^ n / (fact n) +
f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n) / (fact (Suc n)) -
f (Suc n) u * ((1 + of_nat n) * (z-u) ^ n) / (fact (Suc n))"
using Suc by simp
also have "... = f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n))"
proof -
have "(fact(Suc n)) *
(f(Suc n) u *(z-u) ^ n / (fact n) +
f(Suc(Suc n)) u *((z-u) *(z-u) ^ n) / (fact(Suc n)) -
f(Suc n) u *((1 + of_nat n) *(z-u) ^ n) / (fact(Suc n))) =
((fact(Suc n)) *(f(Suc n) u *(z-u) ^ n)) / (fact n) +
((fact(Suc n)) *(f(Suc(Suc n)) u *((z-u) *(z-u) ^ n)) / (fact(Suc n))) -
((fact(Suc n)) *(f(Suc n) u *(of_nat(Suc n) *(z-u) ^ n))) / (fact(Suc n))"
by (simp add: algebra_simps del: fact_Suc)
also have "... = ((fact (Suc n)) * (f (Suc n) u * (z-u) ^ n)) / (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 ac_simps) simp
also have "... = f (Suc (Suc n)) u * ((z-u) * (z-u) ^ n)"
finally show ?thesis
by (simp add: mult_left_cancel [where c = "(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 / (fact i)))
has_field_derivative f (Suc n) u * (z-u) ^ n / (fact n))
(at u within S)"
apply (intro derivative_eq_intros)
apply (blast intro: assms \<open>u \<in> S\<close>)
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 "norm (f (Suc n) u) * norm (z - u) ^ n \<le> norm (f (Suc n) u) * norm (u - z) ^ n"
by (metis norm_minus_commute order_refl)
also have "... \<le> norm (f (Suc n) u) * norm (z - w) ^ n"
by (metis mult_left_mono norm_ge_zero power_mono segment_bound [OF u])
also have "... \<le> B * norm (z - w) ^ n"
by (metis norm_ge_zero zero_le_power mult_right_mono  B [OF us])
finally have "norm (f (Suc n) u) * norm (z - u) ^ n \<le> B * norm (z - w) ^ n" .
} note cmod_bound = this
have "(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)) = (\<Sum>i\<le>n. (f i z / (fact i)) * 0 ^ i)"
by simp
also have "\<dots> = f 0 z / (fact 0)"
by (subst sum_zero_power) simp
finally have "norm (f 0 z - (\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)))
\<le> norm ((\<Sum>i\<le>n. f i w * (z - w) ^ i / (fact i)) -
(\<Sum>i\<le>n. f i z * (z - z) ^ i / (fact i)))"
also have "... \<le> B * norm (z - w) ^ n / (fact n) * norm (w - z)"
apply (rule field_differentiable_bound
[where f' = "\<lambda>w. f (Suc n) w * (z - w)^n / (fact n)"
and S = "closed_segment w z", OF convex_closed_segment])
apply (auto simp: ends_in_segment DERIV_subset [OF sum_deriv wzs]
norm_divide norm_mult norm_power divide_le_cancel cmod_bound)
done
also have "...  \<le> B * norm (z - w) ^ Suc n / (fact n)"
finally show ?thesis .
qed

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 / (fact i)))
\<le> B * cmod(z - w)^(Suc n) / fact n"
using assms by (rule field_Taylor)

text\<open>Something more like the traditional MVT for real components\<close>

lemma complex_mvt_line:
assumes "\<And>u. u \<in> closed_segment w z \<Longrightarrow> (f has_field_derivative f'(u)) (at u)"
shows "\<exists>u. u \<in> closed_segment w z \<and> Re(f z) - Re(f w) = Re(f'(u) * (z - w))"
proof -
have twz: "\<And>t. (1 - t) *\<^sub>R w + t *\<^sub>R z = w + t *\<^sub>R (z - w)"
note assms[unfolded has_field_derivative_def, derivative_intros]
show ?thesis
apply (cut_tac mvt_simple
[of 0 1 "Re o f o (\<lambda>t. (1 - t) *\<^sub>R w +  t *\<^sub>R z)"
"\<lambda>u. Re o (\<lambda>h. f'((1 - u) *\<^sub>R w + u *\<^sub>R z) * h) o (\<lambda>t. t *\<^sub>R (z - w))"])
apply auto
apply (rule_tac x="(1 - x) *\<^sub>R w + x *\<^sub>R z" in exI)
apply (auto simp: closed_segment_def twz) []
apply (intro derivative_eq_intros has_derivative_at_withinI, simp_all)
apply (force simp: twz closed_segment_def)
done
qed

lemma complex_Taylor_mvt:
assumes "\<And>i x. \<lbrakk>x \<in> closed_segment w z; i \<le> n\<rbrakk> \<Longrightarrow> ((f i) has_field_derivative f (Suc i) x) (at x)"
shows "\<exists>u. u \<in> closed_segment w z \<and>
Re (f 0 z) =
Re ((\<Sum>i = 0..n. f i w * (z - w) ^ i / (fact i)) +
(f (Suc n) u * (z-u)^n / (fact n)) * (z - w))"
proof -
{ fix u
assume u: "u \<in> closed_segment w z"
have "(\<Sum>i = 0..n.
(f (Suc i) u * (z-u) ^ i - of_nat i * (f i u * (z-u) ^ (i - Suc 0))) /
(fact i)) =
f (Suc 0) u -
(f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
(fact (Suc n)) +
(\<Sum>i = 0..n.
(f (Suc (Suc i)) u * ((z-u) ^ Suc i) - of_nat (Suc i) * (f (Suc i) u * (z-u) ^ i)) /
(fact (Suc i)))"
by (subst sum_Suc_reindex) simp
also have "... = f (Suc 0) u -
(f (Suc (Suc n)) u * ((z-u) ^ Suc n) - (of_nat (Suc n)) * (z-u) ^ n * f (Suc n) u) /
(fact (Suc n)) +
(\<Sum>i = 0..n.
f (Suc (Suc i)) u * ((z-u) ^ Suc i) / (fact (Suc i))  -
f (Suc i) u * (z-u) ^ i / (fact i))"
by (simp only: diff_divide_distrib fact_cancel ac_simps)
also have "... = f (Suc 0) u -
(f (Suc (Suc n)) u * (z-u) ^ Suc n - of_nat (Suc n) * (z-u) ^ n * f (Suc n) u) /
(fact (Suc n)) +
f (Suc (Suc n)) u * (z-u) ^ Suc n / (fact (Suc n)) - f (Suc 0) u"
by (subst sum_Suc_diff) auto
also have "... = f (Suc n) u * (z-u) ^ n / (fact n)"
by (simp only: algebra_simps diff_divide_distrib fact_cancel)
finally have "(\<Sum>i = 0..n. (f (Suc i) u * (z - u) ^ i
- of_nat i * (f i u * (z-u) ^ (i - Suc 0))) / (fact i)) =
f (Suc n) u * (z - u) ^ n / (fact n)" .
then have "((\<lambda>u. \<Sum>i = 0..n. f i u * (z - u) ^ i / (fact i)) has_field_derivative
f (Suc n) u * (z - u) ^ n / (fact n))  (at u)"
apply (intro derivative_eq_intros)+
apply (force intro: u assms)
apply (rule refl)+
apply (auto simp: ac_simps)
done
}
then show ?thesis
apply (cut_tac complex_mvt_line [of w z "\<lambda>u. \<Sum>i = 0..n. f i u * (z-u) ^ i / (fact i)"
"\<lambda>u. (f (Suc n) u * (z-u)^n / (fact n))"])
apply (auto simp add: intro: open_closed_segment)
done
qed

subsection\<^marker>\<open>tag unimportant\<close> \<open>Polynomal function extremal theorem, from HOL Light\<close>

lemma polyfun_extremal_lemma: (*COMPLEX_POLYFUN_EXTREMAL_LEMMA in HOL Light*)
fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
assumes "0 < e"
shows "\<exists>M. \<forall>z. M \<le> norm(z) \<longrightarrow> norm (\<Sum>i\<le>n. c(i) * z^i) \<le> e * norm(z) ^ (Suc n)"
proof (induct n)
case 0 with assms
show ?case
apply (rule_tac x="norm (c 0) / e" in exI)
apply (auto simp: field_simps)
done
next
case (Suc n)
obtain M where M: "\<And>z. M \<le> norm z \<Longrightarrow> norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
using Suc assms by blast
show ?case
proof (rule exI [where x= "max M (1 + norm(c(Suc n)) / e)"], clarsimp simp del: power_Suc)
fix z::'a
assume z1: "M \<le> norm z" and "1 + norm (c (Suc n)) / e \<le> norm z"
then have z2: "e + norm (c (Suc n)) \<le> e * norm z"
using assms by (simp add: field_simps)
have "norm (\<Sum>i\<le>n. c i * z^i) \<le> e * norm z ^ Suc n"
using M [OF z1] by simp
then have "norm (\<Sum>i\<le>n. c i * z^i) + norm (c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)"
by simp
then have "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc n + norm (c (Suc n) * z ^ Suc n)"
by (blast intro: norm_triangle_le elim: )
also have "... \<le> (e + norm (c (Suc n))) * norm z ^ Suc n"
by (simp add: norm_power norm_mult algebra_simps)
also have "... \<le> (e * norm z) * norm z ^ Suc n"
by (metis z2 mult.commute mult_left_mono norm_ge_zero norm_power)
finally show "norm ((\<Sum>i\<le>n. c i * z^i) + c (Suc n) * z ^ Suc n) \<le> e * norm z ^ Suc (Suc n)"
by simp
qed
qed

lemma polyfun_extremal: (*COMPLEX_POLYFUN_EXTREMAL in HOL Light*)
fixes c :: "nat \<Rightarrow> 'a::real_normed_div_algebra"
assumes k: "c k \<noteq> 0" "1\<le>k" and kn: "k\<le>n"
shows "eventually (\<lambda>z. norm (\<Sum>i\<le>n. c(i) * z^i) \<ge> B) at_infinity"
using kn
proof (induction n)
case 0
then show ?case
using k  by simp
next
case (Suc m)
let ?even = ?case
show ?even
proof (cases "c (Suc m) = 0")
case True
then show ?even using Suc k
by auto (metis antisym_conv less_eq_Suc_le not_le)
next
case False
then obtain M where M:
"\<And>z. M \<le> norm z \<Longrightarrow> norm (\<Sum>i\<le>m. c i * z^i) \<le> norm (c (Suc m)) / 2 * norm z ^ Suc m"
using polyfun_extremal_lemma [of "norm(c (Suc m)) / 2" c m] Suc
by auto
have "\<exists>b. \<forall>z. b \<le> norm z \<longrightarrow> B \<le> norm (\<Sum>i\<le>Suc m. c i * z^i)"
proof (rule exI [where x="max M (max 1 (\<bar>B\<bar> / (norm(c (Suc m)) / 2)))"], clarsimp simp del: power_Suc)
fix z::'a
assume z1: "M \<le> norm z" "1 \<le> norm z"
and "\<bar>B\<bar> * 2 / norm (c (Suc m)) \<le> norm z"
then have z2: "\<bar>B\<bar> \<le> norm (c (Suc m)) * norm z / 2"
using False by (simp add: field_simps)
have nz: "norm z \<le> norm z ^ Suc m"
by (metis \<open>1 \<le> norm z\<close> One_nat_def less_eq_Suc_le power_increasing power_one_right zero_less_Suc)
have *: "\<And>y x. norm (c (Suc m)) * norm z / 2 \<le> norm y - norm x \<Longrightarrow> B \<le> norm (x + y)"
by (metis abs_le_iff add.commute norm_diff_ineq order_trans z2)
have "norm z * norm (c (Suc m)) + 2 * norm (\<Sum>i\<le>m. c i * z^i)
\<le> norm (c (Suc m)) * norm z + norm (c (Suc m)) * norm z ^ Suc m"
using M [of z] Suc z1  by auto
also have "... \<le> 2 * (norm (c (Suc m)) * norm z ^ Suc m)"
using nz by (simp add: mult_mono del: power_Suc)
finally show "B \<le> norm ((\<Sum>i\<le>m. c i * z^i) + c (Suc m) * z ^ Suc m)"
using Suc.IH
apply (auto simp: eventually_at_infinity)
apply (rule *)
apply (simp add: field_simps norm_mult norm_power)
done
qed
then show ?even